The question mark in the subject line is deliberate. I am a very un-mathy person. I like Scrabble, cryptograms, word-based logic problems, crosswords; I shy away from number puzzles. Not that I can't do math or handle numbers, at least in the everyday world; it's just that if I were sign up for a course in something that interested me, it probably wouldn't be math. My idea of probably the dullest job in the world is accounting.
But teaching math--that does interest me. Ever since I started looking at math curricula and homeschooling The Apprentice (and the later Squirrelings), close to two decades ago, I've been fascinated by the history of how math has been taught, especially over the last century, especially at the elementary levels. I was there for a lot of it, good, bad, and ugly. What might be abstractions for some are clear memories for me. I liked this, I learned from that; or not. This teacher knew how to get math ideas across; that one made us fall asleep.
And I like teaching elementary math (and basic algebra and geometry) at home, seeing the girls learn new ideas and gain confidence in their numeracy. Even when they struggle or complain that math is hard or boring...that's a challenge. I like being able to work at our own pace, and to use whatever's handy for illustrations. I like feeling free to just say "here's the rule, here's how you do it" when that makes more sense than endless demonstrations and discovery learning.
I liked using our stash of rods and hundred charts and games and software. I liked helping people who didn't get Miquon Math. (I still think it's a brilliant primary curriculum.) I liked reading about people like John Holt and John Mighton who believed that all children could learn math if it was carefully taught.
So does that make me a math teacher? Well, I do teach math, and as I said, I have been teaching math to at least one child each year, sometimes two, for almost twenty years. (Sometimes Mr. Fixit has been the math teacher too.) While I'm not a mathematician, did not major in math, do not even have math credits beyond Grade 13 (and I struggled for that one), I seem to have steered the Squirrelings towards acceptable levels of numeracy. Other homeschooling parents, many without specialization in math, have done the same.
How? I can't speak for all the other families out there. For some it might be nothing more than buying a solid textbook or workbook series and doing whatever comes next. For myself, I just decided that the process of teaching elementary math was not that much more mysterious than the teaching of any other subject. If I could teach reading, writing, history, there was no particular reason I couldn't also handle elementary arithmetic and middle-school math topics. And since I was very aware of the booby traps and swamps in my own math adventures, I was determined to avoid as many of them as possible, including the infamous Fifth Grade Slough of Despond (girls often fall into it around the time they meet up with Giant Long Division). If public school gave me only a mediocre appreciation of math, I could do at least somewhat better with my own girls.
Now here's the big point.
In Ontario, scores on standardized math tests are dropping. Why? Some blame the teachers. Some blame the curriculum. Some blame society. Or the weather.
One proposed solution is to have elementary math taught only by math specialists. Because even the classroom teachers don't seem to be able to teach math well using the new approaches. Does that imply that there's a) something wrong with the students, b) something wrong with the teachers, or c) something wrong with the curriculum? Votes?
It reminds me of a situation where an office bought a huge, expensive, complicated copier that required advanced training just to make ordinary copies. Yes, if you were properly trained on it, you could use it to copy, sort and bind entire encyclopedias, but most of the usual copying chores were much more mundane. It would have made more sense to buy a simpler machine, and send the occasional complicated jobs to a print shop. It didn't make sense to blame the office staff, either, just because they didn't want to be full-time slaves of the Copying Beast. And it wouldn't have made sense to put blame on the clients--because, in the end, they didn't care how big or expensive the copier was--they just wanted their letters and documents.
And what we really need is for schools to teach math (and other subjects), in a way that the teachers can handle, in a way that delivers what the children require, in ways that help them to grow and learn and include numbers and measurement and shapes and mathematical relationships in their lives. Because they aren't impressed by how big the machine is, either, if it's not working for them.
For them, all you administrators out there. Take it from this question-marked math teacher.
Linked from Math Teachers at Play Carnival #71.
Showing posts with label hundred chart. Show all posts
Showing posts with label hundred chart. Show all posts
Friday, February 07, 2014
Tuesday, February 24, 2009
A Family Journey Through Miquon Math
In 1996 we started using Miquon Math with our kindergarten-aged Apprentice. A couple of years later, now with an Internet connection and making all kinds of online discoveries, I found the Miquon-Key e-mail list [now defunct].Unfortunately my first stint on the list came to a quick close as the Apprentice rebelled against Miquon (too much, too long, wasn't working anymore), and we moved on. But when Ponytails got to first grade, I resubscribed; and at the same time, I found new ways of working with the curriculum and wanted to share them with the list.
With permission of the list owner, I've dug out some of those posts--mostly for my own benefit, as Crayons is doing the same math now that Ponytails was a few years ago. I've edited them to use our blog nicknames and to take out some repetitive details. I've also added links to things I posted here after we started the blog.
The sequence of the six Miquon Math workbooks is Orange, Red, Blue, Green, Yellow and Purple. The average speed on them for most homeschoolers is about two workbooks a year.
The teacher's manuals for the program are the Lab Sheet Annotations, Notes to Teachers, and the First Grade Diary. There is no "Lesson 1", "Lesson 2" and so on; each topic is introduced generally in the Annotations, and then notes are given for the worksheets. Often there are activities suggested to go along with the sheets, such as chalkboard games. Many Miquon users just have their children work right through the books, especially if they are independent learners; but we had more success mixing up the topics and not depending too heavily on the worksheets.
August 1999: Introducing Myself
This is our fourth year using Miquon and I never imagined there was a loop just for us. I've been reading through the posts and I already see I'm in good company. Our Apprentice just started the yellow book today. She loves the rods, but doing operations with them doesn't make sense to her. She loves the books, likes adding and multiplying (HATES subtracting), but would rather use a hundred chart, break-apart cubes, two rulers one above the other (anybody tried that? it's a "magic adding machine"), turning a question into a word problem, just about anything except putting two rods together and figuring out how long they are.
Answering a question about using two rulers as an adding machine:
I can't take credit for this idea--I got it from a library book but I can't remember which one! You need two rulers, metric works best because there are more numbers. I use two identical transparent ones. This idea doesn't work well with very young children, but school age children find it fun.
To add 3+12, set the rulers against one another so that the 3 on the upper one is above the 0 on the lower one. Find the 12 on the lower ruler, and whatever number is above the 12 is the answer! To subtract 10-7, place the 10 on the upper ruler over the 7 on the lower one. Find the 0 on the lower ruler, and whatever number is above the 0 is the difference.
November 1999
We started the Yellow Book this fall and for some reason we just can't get into it; my daughter groans "oh no, Miquon" when I get the book out. We are playing a lot of math games, doing questions with a muffin tin and macaroni (if you put 10 pieces in each compartment, you can do addition and subtraction up to 120) and with coloured cubes and pieces of coloured paper for the units, tens and hundreds (if I give you 5 cubes for the blue tens "column" and 4 for the red units "column", what number do you have? and if I give you 3 more for the blue and 7 for the red?--regrouping is necessary here)....
I never saw anyone as sure yesterday that 48-30 equalled 2 (on D-21). She took away the three tens and then insisted that you had to do something with two (I think she was thinking 10-8). I had to get the muffin tin out and prove the answer was 18, and I still don't think she believed me.
December 1999
I read a very interesting article by Ruth Beechick (I think it was in her Question and Answer Book). To summarize...She describes a class where the children solved problems with some kind of flat counters. The teacher would say "make three piles of two" or "eight piles of ten" and so on. The kids got comfortable with the idea of not counting every single one. As I remember it, she eventually suggested to some of them that they could write out the solutions so that they wouldn't have to count out all those piles every time; she showed them how they could take two piles of two, three piles of two, four piles of two etc. and write the answers going across the page; and then they caught on and started writing out the threes, fours etc. The idea spread around the classroom and soon many of the kids were writing out their own tables for reference.
Later she suggested that if they learned some of those facts they wouldn't have to refer to the tables all the time, and again the idea caught on; the kids saw the usefulness of carrying the facts around in their heads instead of on their papers. I don't know how well they all learned their times tables, but I'm sure that was a happier classroom than the one in "Hans Christian Andersen!" (If you never saw the movie, the schoolchildren are droning "two and two are four, four and four are eight...")
December 1999 again
In spite of my complaints [about the Yellow book], we have found one way to make some of the addition and subtraction a little clearer (for example, page E-53, which is addition and subtraction with a sum or difference of 100, 200, 1000, etc.) We get out our pile of pennies, dimes and Canadian dollar coins, and do the problems as if they were money.
We have a couple of favourite scenarios we use with this. One is "you had this much money in your piggy bank this morning. Your sister came along and "borrowed" a few coins. When you came back later, you found you only had so much money left. How much exactly did she take?"
The opposite is, "you had this much money; grandpa came for a visit and slipped a few coins in; when you counted all your money you found you had this much; so how much money did he put in?"
One problem was x + 32 = 100. My dd didn't know how to do this mentally, so I said, "Pretend I'm 30 years old. In how many years will I be 100?" "70," she said. "Now it's two years later; I'm 32 years old. Now how many years will it be until I'm 100?" "About 72" she said. "Really? But two years ago you told me it would only be 70 years. How can it be more now?" She thought about this, and then decided 68 made more sense.
[There is a long gap here because The Apprentice switched to another math program, and I unsubscribed from the Miquon-Key list until our second daughter, Ponytails, was in the first grade and starting Miquon.]
August 2003
I'm starting Miquon again with renewed enthusiasm for the program, and I'm trying to use it more creatively. One difference from our first time through is that Ponytails is a bit older than The Apprentice was when she started the workbooks.
Another difference this time around is that Ponytails has had lots of experience playing with rods over the last couple of years. She's not quite comfortable yet thinking of them in terms of their actual length; she thinks of them as a blue rod or a green rod, not as a four or a five; but I think actually that's good; it means she can think about them for themselves without worrying too much yet what they represent.
September 2003
I inadvertently found a way to heighten Ponytails' interest in learning the lengths of Cuisenaire rods. This morning I held up a yellow rod and asked her how many white rods long it would be. She said four. I said "Okay--prove it." She took the bait and said, "Okay then--I WILL." She lined up the white rods and admitted that she needed five. When I held up a light green and asked how many whites that would take, she said, "Three--and I'll PROVE it to you." And she did. We continued like that, and she kept on PROVING stuff to me--and the math got done.
September 2003
I found a homemade substitute for the wooden rod tubes described in the First Grade Diary (wooden tubes just long enough to hold one 10 cm orange rod, or the equivalent in shorter rods). (They are described in the entry for Sept. 30.)
Cut a piece of index card to a 10 cm length and wrap it fairly tightly around an orange rod, trying to crease the edges as you go (to keep the rectangular shape). Secure each end well with tape, and there you go. I made four in under 10 minutes. They're obviously not made to last forever, but at that price they could be replaced easily when needed.
The purpose of the tubes, if you don't have the Diary, is to play hidden-rod sorts of games. The simplest version would be to insert two rods that add up to ten, show the child one end of the tube, and ask what colour should appear at the other end. A more complicated version would be to use three (or more) rods, show both ends and ask what is hidden in the middle.
September 2003: Centimeter Cubes
We use these for a lot of things (besides giving the toddler something to play with [update: I don't remember what our toddler was doing playing with something so small--I must have been right beside her at the time because obviously cm cubes aren't an appropriate toy for children small enough to eat them]); I used them a lot when my oldest was learning subtraction, because you can't break a piece off a Cuisenaire rod! We also used them later on to demonstrate multiplication/division--twenty cubes snapped together, then broken into four groups of five and so on.
Young children can use them to do patterns with--two reds, a blue, two reds, a blue. You can draw a pattern with markers and have them copy it.
You can build things with them besides just straight trains. Build a 3-D model for the child to copy, something made out of several cubes stuck together. (Easy to do with Duplo or Lego too.)
You can use them to visualize word problems; for instance, you can use a piece of green construction paper to represent a field, and use some white cubes to represent sheep in the field--for counting, adding, subtracting. Or a blue piece with different coloured "fish".
We also use them to play a simple guessing game at about the K-1 level. You snap a few together (5 or 6), hold it behind your back and break a couple off; hold out the remaining piece and have the child guess how many are still behind your back. (You can do the break-off game with Duplo or Lego blocks too. Anything that snaps. Or even with small groups of non-joining objects, such as crayons or coins.)
And I suppose you could use them for actually measuring things.
October 2003: After A Month of Grade One
....I have to say, I am more impressed than ever with Miquon and I'm glad I chose it for my first grader (I might have been discouraged after my older one's so-so experience).
The amazing part of this that we are doing a maximum of 20 minutes of math a day, doing a lot of rod games and the sorts of short activities suggested in First Grade Diary, and Ponytails is NEVERTHELESS picking up not only addition concepts but has started subtraction as well and ENJOYS it.
What a great program this is. I keep wondering why I see all these entire unused sets of Miquon at used curriculum sales; it makes me wonder if people just give up too soon.
October 2003
On the difficulty of problems such as x - 3 = 2:
Ponytails and I played around with some problems like this today using coloured popsicle sticks and a small empty box which had originally contained frozen chicken wings. I put six popsicle sticks in the box without showing her how many, and said "I was all by myself for dinner and I was hungry, so I took out four chicken wings to cook. (I removed four sticks.) The next time I looked in the box (I showed her), there were two left. How many were there before I ate all those chicken wings?" She looked at the two in the box and the four on the table and said "six." We repeated this with some different sets of numbers, and she had no trouble with that.
Then we went to our blackboard and I drew an empty square and told her that was for "something." I wrote the problem "Something" - 4 = 2, and reminded her about the "chicken wings." I also showed her again with the appropriate Cuisenaire rods. She caught on to that quickly. I did one more simple one with her, and then she wanted to write one. She wrote "Something" - 5 = 10. I asked her, "Do you know how to do that one?" She wasn't sure. I took a yellow and an orange rod and said, "here's our 5, and here's our 10." She still wasn't sure how much that would be, so I had her figure out how long they would be together in white rods, and she got it.
October 2003: Odd and Even
I notice in the First Grade Diary that the topic of odd and even is suddenly introduced at about this point in the school year, although there's no comment there about how it was first presented (there are some suggestions in the Annotations, though), and odd-and-even worksheets aren't included until the Red Book.
I asked Ponytails if she knew what odd and even numbers were, and she said, "One is odd. Two is even. Three is odd. Four is even, and like that." I asked her if she knew how you know if something is an odd number, and she gave me a pretty hilarious explanation that I can't even try to reproduce here. (Some of our adult explanations must sound just like that to her.)
So obviously she does have some idea that some numbers are odd or even, but doesn't really understand what that means.....One explanation given in the Miquon materials was that you can build an even number with red rods, but an odd number you can't. I also saw an arrangement of cubes (similar to an activity in Family Math) where you take one cube, two cubes, three cubes etc. and show how you can arrange the even ones in pairs but not the odd ones.
January 2004
I've noticed that some topics are covered VERY briefly in the Miquon worksheets, for instance, skip counting. In the Red Book, page F-15 is a dot to dot based on counting by 3's; F-18 is counting by 4's; F-21 is counting by 5's and 6's. Obviously a one-page puzzle is not going to be enough for a child who doesn't yet know how to count by 3's or whatever. Another topic, similarly, is time; there are a few sheets in the back of the orange book, but if the child doesn't really get telling time to the hour or half hour the first time around, there's no point in going on to the quarter hours right away.
However, this isn't a complaint! I'm just pointing out why I think it doesn't work as well just to go through the books in a rigid sequence. The curriculum obviously expects that the children are going to be learning skip counting, for instance; and if you look in the Annotations and First Grade Diary, there are extra suggestions for activities for the different topics. If I see pages on a topic I think Ponytails has already mastered (like the simple counting dot-to-dots at the beginning of the Orange Book), I use those sheets to see for sure that she knows what she's doing and has no difficulty.
But I guess what I want to say--to make it short--is that as we move into the Red Book, I'm using the topics in the book as a guide to what we should be covering--more addition/subtraction/multiplication, work in fractions, skip counting, doubles, finding tens, inequalities, and eventually division concepts. But I know there's no way she's going to learn those things, like counting by threes, by doing one worksheet. The sheets are there for her to practice on or to nail down something I think she's almost there on; but the bulk of our time will be spent on finding out how numbers work, and the ways we can move them around and do things with them.
February 2004: A Measurement Story
We're going to spend the next two weeks doing the T pages of the orange book, which are about measurement. When I looked at the Labsheet Annotations for this section, there were some classroom experiences described that sounded a bit difficult to replicate with just one student, so I took the description and rewrote it as a story for my daughter. (She always likes to hear what "the Miquon teacher" did with her class according to the First Grade Diary.)
February 2004
We are doing some of the C (adding) pages in the Red book, and the focus is grouping for tens. I've noticed that it helps if we start off with a no-book rod activity and then move on to the worksheet. That way too, if Ponytails isn't following the concrete activity, I know it's not time yet to move on to the written part.
Here are a couple of quick ones we've done or are doing with those pages. For C-12, which is 10 + 2, 10 + 4, and so on, I made two piles of rods; one pile was just orange rods, the other was all the other ones. We took turns pulling just one rod out of the mixed pile, but as many oranges in the other hand as we wanted, and "guessing" what number the other person had made. (Guessing is the wrong word, figuring out is probably better.) So I took two orange rods and a red one, and my dd said "10, 20, 22. You have 22." She took four orange rods and a blue one, and I said "49." I tried to make mostly numbers in the teens and twenties to make sure she understood those, but she kept taking as many orange rods as she could to try to make it really hard. I think we got up to 129.
Anyway, after that, the worksheet was no problem.
We're going to do C-15 today, which brings in the concept of grouping for 10s. I think I'll take a group of four or five rods (similar to the groupings on the page) and show her how to add them quickly by building pairs that make ten. I think this follows very well from the C-12 activity--instead of pulling an orange rod, you're pulling two rods that make ten.
March 2004
Cat and Mouse Game (comments)
Last Friday I planned to have Ponytails do the Red Book, page C-25, the dice game where the cat tries to beat the mouse to the mouse hole. The way it turned out, The Apprentice (seventh grader) had some spare time that morning so I asked her to play the game with her sister instead. I stuck around to watch.
What I quickly realized is that for any child who regularly plays board games, this game is so simple that it quickly becomes boring. (The corresponding subtraction game, D-9, has the same problem.) After one quick round, I suggested that maybe they could spice up the game a bit by designating one or two of the numbers as "go back that many spaces." They decided to make three of the numbers "go forward" and three "go back."
The inevitable happened: with three forward and three back, they ended up going back as much as they went forward! I didn't want to interfere, so I let them play until they'd both been sent back to start several times; then I suggested again that maybe just one of the numbers should be "go back." They decided on 6, and played that way until one of them won.
Just wanted to share about this--that pages like this that may seem too simple can still be made interesting if you add an extra twist!
March 2004
We are doing 20 minutes of math a day but it's not all from the books; we spend about half the time doing activities with the rods or other manipulatives, practicing skip counting and counting backwards, measuring things and so on. We got through the Orange Book, most of it except for some bits like clock arithmetic, in the first half of the school year, and are now mostly doing the Red Book. At this rate, she will be done it by June, no problem.
Today Ponytails begged me to teach her "the dividing thing, you know, there's a line with a dot on each side." So I did. :-)
May 2004
Ponytails is almost through the Red book, but we didn't do every page in the books, and we didn't do them in order. What we did do, especially with the Orange book, was use the First Grade Diary as a model. Not exactly every day as described (that would be impossible), but as kind of a plan for what typical math lessons would look like--often incorporating two or three seemingly unrelated activities, and having only one of those (at the most) be a worksheet. We did pages from the Red book (such as the Odd/Even section) before finishing the Orange book. We've worked on concepts like telling time and geometry all through the year, not just for the few pages on that that are included in the Orange book.
I've seen a strong thread of "tens awareness" building through the first two books, even though "place value" as a topic doesn't come up until a later book. If you keep that in mind, Miquon seems to make more sense. You do many activities based around the orange rod--finding combinations of rods that make ten (this covers both adding and subtracting), and learning to group for tens when given a string of numbers to add.
There's also a lot around the concept of equations. You start with very simple equations, like 3 + 3 = 6; later you get into equations where you have to figure something out on each side, like 3 + 3 = 2 + something. (There's a Miquon game called "How much wood" or "Lumberyard" which introduces this concept.) Then inequalities, and greater-and-less-than.
So what I'm saying is that if you do a lot of hands-on activities--using the blackboard, a hundred chart, a number line, the rods, small objects for oral word problems--involving tens and equations, you'll help your child get in touch with some of these basic concepts. Then what you'll find is that the worksheets are kind of like the follow-up activities to the concepts that he's already learning, and you'll find it easier to pick which sheets he should be doing next.
May 2004: Numberlines
The Lab Sheet Annotations is not the best laid-out teacher’s manual....I’ve added “improvements” to my copy, including colour-coded notes at the top of the pages to show which of them relate to which workbook. It does leave a lot up to the teacher to decide. I originally had the impression that Miquon students visited the math lab and were just turned loose with balances and blocks. I think that was partly true (according to the manuals), but there was also a sense of progression and growth in understanding that could only come through a certain amount of planning and guidance by the teacher.
I responded to questions by a list member about Section N.
Thank you for your questions about this, because it led to a really interesting afternoon reading carefully through that section, and I got some good ideas for number line activities. You mention a problem with Louis’ answer in the Grasshopper game on page 227, and you’re right–it’s a typo where he says the grasshopper goes back 3 jumps and it should say units instead. This game description is almost identical to one in the First Grade Diary, but a few of the numbers were changed, and maybe the word got changed accidentally in the process. (There’s also a typo at the bottom of page 229, top of 230; the section heading and a whole paragraph are repeated.)
About the Number Line topic in general...I hated number line exercises in grade school, and I’ve read that some math educators don’t believe that children understand them. Miquon takes the opposite approach, making them concrete and integral to the program; work on the number lines relates very closely to the work in adding and subtracting, and later, multiplication and fractions. It’s just too bad that they’re hidden in this little section and that the N pages are the ones you’re probably going to skip–what are you supposed to do with a bunch of blank number lines?
The number lines and other activities in this section are interesting because they don’t depend on the usual plus-minus operation signs. For that reason, I think they could be useful and non-threatening for kids who get scared by minus signs. The activities described in the Annotations go from very simple (the basic description of a number line) to quite complex; they should be done throughout the year and the program (there are “N” sheets used in three different workbooks), getting more complex as you go on.
Think of all the different kinds of number lines there are to use. “Real” number lines (on paper or chalkboard) can vary by beginning at points other than 0 or 1, can be marked with fractional units, can be marked in multiples of 2, 3, 5 etc. But there are other number lines that children may find easier to understand: thermometers, rulers, measuring tapes, board games (the kind where you roll the dice and move along a numbered track–there are a couple of simple ones in the workbooks), even life-size number lines drawn with chalk on the driveway. A hundred chart (we have a poster-size one on the wall) is great for a lot of number-line-type activities (Ruth Beechick’s booklet on math gives ideas for using one). We’ve also used concrete things like a tower of blocks to illustrate concepts of increasing and decreasing–going up and down our “skyscraper” in an imaginary elevator, stopping at different floors and sometimes pretending that the elevator goes to several levels of basement (negative numbers).
My take on this topic? I think it’s like many of the other sections, like addition: you start simple, then add little twists in (if the children don’t ask about them themselves), and keep adding on to the possibilities. But slowly, a little at a time, as they’re ready. Talking about functions in primary-level math seems way beyond what most of us expect (we’d like to stick safely with our basic adding and subtracting!). But the interesting part of it...for me...is that playing with numbers in this way is going to incorporate a great amount of adding, subtracting and more, and in a painless way. I also like its practicality as children learn to measure with rulers and apply what they’re learning in one area to the other. And I like its possibilities for bigger questions: where do you end up when you are on the third floor and the elevator goes down five floors? What happens if you want to put two jumping rules together? If we had a number line that went on out the door, where would one million be?
May 2004 again
A list member wrote, "Regarding the Functions section, I'm not even sure I would know how to approach or explain this. Any suggestion would be so helpful."
Sheets N2-N4 can be used any way you want; there are suggestions on page 231; you might also look at the problems on page 225.
I like the game presentation of functions here. This is one place where I think the Annotations does do a good job taking you through a series of exercises about number lines and functions, especially at the Green and Yellow levels. It's suggested that you stick to the terminology of the number line games, using words such as: "try my rule," "make up a rule," "2 goes to 4," "start," "beginning number," "land," "landing number," "undoing rules" (inverses), "jail rules," "standstill points," "anywhere goes to anywhere plus four" or "box goes to box plus four." Functions, as far as this book goes, are just more and more complex rules for number line games.
At the Orange level, number lines stay simple, for instance with the basic Gus and Happy game. At the higher levels, the rules of the game become more involved; students experiment with combining more than one rule and with a simple form of graphing the functions (something we spent a lot of time on in high school Functions and Relations).
At one point it's suggested that you try using a number line labelled with letters of the alphabet instead of numbers. You could play that game out loud, too, without written letters: if the rule of the game is to tell me the letter that is my letter plus two more, what is the letter if my letter is B? (That's how substitution ciphers work...also known as secret codes.) This even makes me think of the notes on a piano: if I play C and we make up a rule that says play the note that is two notes lower, you play A. Of course this would only work for adding/subtracting functions, not multiplying/dividing ones.
I wouldn't even worry about the more complex "rules of the game" until your student is very comfortable doing the more basic number line activities. If you notice, there are relatively few worksheets in this section for a topic that can become quite complex; my feeling is (based on the notes) that they originally did a lot of chalkboard work with it. To be honest, I can't even remember going through these upper-level sheets with my oldest daughter, but I know she "did" the Green book. Shows you how much we missed the first time through...
June 2004
We did K-7 this morning, which is one of two pages dealing with factors of 24 (before the topic of factors is gone into as such). I decided to keep the book closed this time until we had done some concrete work with the rods.
I had Ponytails build a train of 24 (two oranges and a purple) and then asked her to find all the one-colour trains she could of that length. I had intended to wait until she had found them all before doing any recording, but as soon as she lined up six purples, she said "Six fours! Can we write that down?" So we went on like that until she had found three pairs of multiplication facts equalling 24. (After she found twelve twos, I asked her to imagine she had some "twelve rods" and asked how many of those would fit into 24.)
At this point, doing the multiplication part of K-7 would be optional since you have really just done it anyway, but I had Ponytails fill in the blanks anyway. There are division facts (the inverses of the multiplication facts) going down the right hand side of the page, and those are something we're still just getting into, but she didn't have a lot of trouble with them if I phrased it as "how many 4's in 24?"
Oh--and something funny only Miquon users would appreciate. The other day we were in the car and pulled up to a red light beside a sport vehicle. Ponytails pointed: "Look, Mom! Four fours!" I looked, and she was right: on the side, the truck said 4x4.
November 2004: Blue Book
Ponytails is in her second year now with Miquon and....I find we're using the rods a fair amount, especially for two-digit addition and subtraction,and also a hundred chart. (We made a big one--big is good!-- that has cardboard number disks attached with sticky Velcro.) We use the hundred chart almost every day for oral adding and subtracting problems, and she's getting very good at moving the right number of spaces DOWN for tens and ACROSS for units. (This is great place value stuff.) (Oh, a note about doing hundreds with the rods: we just cut some hundred-sized squares of orange paper.)
We've been playing around with assigning different values to the rods than the usual white = one . This was helpful, for instance, on the "diagnostic page" E-46 which had questions like 60 + 40 + 70 + 30 . I showed Ponytails that if each white rod = 10, she could use the rods easily to add those numbers together (in addition to grouping for tens).
We did E-44 (arranging three given numbers into number sentences) using rods and "symbol cards" (half-pieces of index cards marked with signs for plus, minus, equals etc.). (Haven't done E-45 yet where they are to choose their own three numbers.) After she arranged the rods and cards to make true number sentences, she copied the sentences onto the worksheet.
For E-48 and E-49 I wrote all the questions (10+19, 23-13 etc.) on half-index cards and did them orally using the hundred chart (she did not write the answers). I wrote the addition questions on one side in one colour, and the subtraction ones on the reverse in another colour). This way (since it wasn't done on a worksheet) we can use a few of the same questions as review on another day. Yesterday I picked out three of the addition questions, had her do them with the hundred chart, and then asked her to do the reverse sides as well.
We've also been spending some time on doubles, using pages F-23 and F-25. We started by working down the "1" column on F-25, doing doubles up to 256 (my dd figured that one out with rods). Because she seemed really interested in this, I re-read her a story we have in a Childcraft math book about why there is no mathematical possibility of vampires (because if you double the number of vampires every week, everybody in the world would be a vampire in less than a year :-)); it's just a variation of the story about the servant who asks for one grain of rice, then two, then four etc. until he owns the whole kingdom. I also showed her the doubling function on a calculator (2 x = = = = etc.)
All this is to say that if you have children who are active/easily distracted and who are "global"...I think the word is global...enough to enjoy challenging questions and big numbers even though they may sometimes get stuck on easy ones...I think Miquon continues to be a good choice even past the first grade stage, if you can keep the activities concrete and work within their attention span.
April 2005: Number Lines
We are in the middle of the green book, having just completed the section on multi-digit addition and now working on number line games.
We also found a game today in Family Math that is good for place value and addition. If you have the book, it's called Dollar Digit and it's on page 112. You need a pile of dimes and a pile of pennies; one die; and two sheets of paper each marked in two columns (one for dimes, one for pennies). You number each paper 1-7 down the left hand side. For each turn, someone rolls the die but all the players use that same number. You can either take that number of pennies or that number of dimes (but not mixed), and place them beside the number 1 on your sheet. Next turn, someone else rolls but all players again use the number from that roll. Any time you get 10 pennies, you have to trade them for a dime. After 7 rolls, you add up what you have, and the closest one to a dollar wins. (You can make a rule that you can't go over, if you want.)
May 2005
Ponytails is transitioning into the Yellow book, although we haven't finished all the end-of-the-book topics in the Green yet.
Anyway, although Miquon's transition into subtraction is slow, I am really impressed with the reasoning behind it. I told Ponytails that this kind of math--that is, learning to make life easier for yourself by changing questions like 48-13 into 50-15) is "Smart Math." Using-your-brain math, rather than having the teacher tell you what to do and you just doing it. I think she pretty much understood that.
May 2005: Functions
A list member asked about Page N-7: "Why is it written 'box arrow box' then the rule to add or subtract a number. Why not just one box?"
What you're working with here are functions. That's why they're set up as they are...the book is giving you a kind of junior version of math that we did in high school.
The Annotations is really helpful with this section. If you present it as they do, it will probably make more sense. Imagine a series of number line games, where each person involved makes up a different rule for how you're allowed to move. (Kind of like how the different chess pieces all have their own rules for moving. Or each person having their own dance step.) Billy's rule says that you start on "box", and you end up on "box plus three." Whatever "box" is, you add three to "box" to make your move. (In this case, you're jumping by threes then.) You can also say "anywhere" or "starting point" instead of "box" if that makes more sense.
So if you start with 4 and follow "Billy's rule," you end up at ? (7). And so on.
Then "you try Billy's rule." Same thing. Each time you're just adding three.
Then make up some examples of your own.
If you follow the Annotation suggestions, you'll see some variations that you can use while you're playing around with number lines. For instance, what if somebody has a rule that "box" goes to "box plus three", and someone else's rule says "box" goes to "box times two?" If they have a race, who gets further along, or is there a point where one overtakes the other?
Or you could even play "guess my rule." Show a series of moves and see if your child can guess what rule you were following.
May 2005 again
A list member mentioned that they always skipped the Factor House Game in the O pages.
LOL, and I thought we were the only ones . We just did those pages, and I couldn't see the point in cutting out the little houses (other than the scissors practice).
But the sections after that have been great--square numbers and area are so much fun with rods! We built pyramids out of rod squares as suggested in the Annotations, but we built them off-center (rather than each new layer right in the middle, we lined up the edges on one corner of the bottom layer, if that makes sense), so that you could see the different colours of rods and how many extra were added. The next day, I printed out some cm graph paper and we did the same thing only with paper and crayons. We made up a story that a famly started out with a small house (a red, 2 cm square) and then needed to enlarge their house, so they added a red rod on each side and then filled in the gap with a white rod, making it as large as a green, 3 cm square. Then they had more children, so they needed to enlarge again, and so on. We went through several squares this way, writing out the formula each time. (i.e. 3 squared = 2 squared + 2 + 2 + 1; 4 squared = 3 squared + 3 + 3 + 1.) There's no way all this would have made any sense to Ponytails (and to me) without the rods.
December 2005
A list member asked for help with worksheet H-47.
There are different ways you can illustrate these problem besides with the number line. You could turn each question into a word problem, like this: "If you have two thirds of a pizza, and I have two thirds of a pizza, how many thirds of a pizza do we have? Is there another way to say that? Or, is that more or less than a whole pizza? How much more? It's okay to say we have four thirds of a pizza, and it's also okay to say we have one and a third pizzas."
When you look at each question, try saying it like this "Two two-thirds." "Four two-thirds." "How many two-thirds are there in six-thirds? What IS six-thirds, let's figure that one out first. (Use whatever manipulatives you want.) Oh, six-thirds is the same as two whole things. So how many two-thirds are there in two whole things? (Use manipulatives to figure it out, if you need to.) Do you want to think it through with pizzas again? Well, if you have two-thirds of a pizza, and I have two-thirds, how much do we have left? Another two-thirds. So there are three two-thirds pieces in two whole pizzas."
The last one: "Some number of thirds is the same as two wholes. How many thirds can we fit into two wholes?" (Use manipulatives.)
The manipulatives could include Cuisenaire rods, a paper "Hershey bar" (divided into sections), paper or plastic "pizzas", or plastic fraction pieces.
It's a good idea to vary the shapes if you're doing this kind of exercise on more than one day; you can get into a rut of always thinking about fractions in terms of round things like cakes and pizzas, so it's good to remind kids that fractions come in other shapes like squares and rectangles too. And even, if they're ready to think about this, in groups of things rather than in pieces of a whole. For that, you can use cereal or beans or cubes (white rods work fine) or whatever small objects you want to divide into groups...these kind of problems are definitely more of a mental challenge, but you can turn them into word problems as well. Stories about classes of children, or teams, or a bag of treats that we have to divide up, all work well.
The rods are good fraction manipulatives in themselves, and that's one thing you miss out on if your kids think of them only as representing the numbers one to ten. It can actually be a lot of fun playing around with these ideas...my third grader liked this part of math. For instance, you take a brown rod. "This is two." (You may get protests.) "Well, today it is a two. Maybe it's a two-dollar bill (or coin if you're Canadian). Okay, let's make it be money. If this is a two-dollar bill, show me a one-dollar bill." (Child shows you a purple.) "Show me half a dollar. Show me a quarter dollar." (Short sidetrack into why quarters are called quarters.) "Okay, let's try something else. (Show a light green rod.) This is one. Show me two. Show me three."
The fun, for Ponytails anyway, was when I asked her to do some for me. I think she has caught on to the idea that what we're doing is about relationships. Rods are about relationships. Fractions are about relationships. Fractions are also about dividing and multiplying.
March 30, 2006
The power of the rods is that they can represent so many different concepts. Any one of the rods can represent "one"; then all the others can be identified in relation to that rod. The most usual set of names we give them is to say that the white rod is one, and then red is two and so on; but that is just the beginning of what you can do with them. If your child works with them from the beginning and understands their relationships, then they should have no trouble in using them for more difficult concepts like fractions. (If the orange rod is two, then the yellow rod is one.)
I'm reading a book called The Mathematical Mystery Tour, written for adults--kind of a fictionalized history of mathematics. It got mixed reviews on Amazon as far as its scope and attention to details goes (and whether it was boring or not). The point, though, is that in the first chapter the characters are talking about Pythagoras and early Greek mathematics, which was based only on integers. The mathematician in the chapter points out that the ancient Greeks substituted their geometrical knowledge for the algebra they lacked. In other words, they could draw a diagram of a problem (in the dirt, usually) and figure out how to express the answer as a ratio (since they had only integers to work with). The diagrams in the book look exactly like something you might build with Cuisenaire rods!
When I read that chapter, I realized again what a powerful set of ideas we are giving our children with these manipulatives. Primary-aged children, like the Greeks, do not have any knowledge of algebra; we do not ask them yet to "solve for x." But my third grader is able to *see* the answers, using rods, to problems that she would otherwise not be able to solve.
April 2006: Cookin' with Math
May 2006: Math Stuff
January 2007: Trains Game
A list member asked about number names for rods.
For that reason, I might even suggest that you don't worry about whether the "2" rod is 2 at this point, but stick to the colours and the relationships between the colours. Two reds make a purple, two purples make a brown. How many ways can you make a train as long as an orange rod? You can play trading sorts of games (as if the rods were money)--what will you trade me for a yellow rod? You could even develop that into a sort of playing store--for instance, if a doll is priced at a yellow rod, she could pay for it with five white rods, two reds and a white, etc.
There's a game with variations that's given in the primary-level Idea Book for Cuisenaire Rods, and we've played it many times to help build awareness of the colours without getting hung up on the numbers.
The basic game is that you dump about 40-50 rods on the table or the floor, and then take turns trying to find a rod that the other person can't "match." You hold up one of the rods (any rod except a white one), and the other person has to find two rods that make a train as long as your rod. (Not one, or three--just two.) If they can, then you set those three rods aside and the other person gets a turn to hold up a rod and stump you. You continue the game until somebody holds up a rod that can't be matched with the rods that are still on the table; then that person wins the game.
For instance, you hold up an orange rod; the other person can then match it with two yellows, or a purple and a dark green, or a light green and a black, or whatever. But if it's near the end of the game and you hold up a light green (3 cm) rod, and all the red (2 cm) rods have been used up, then there's no way they can match that, and you win.
April 2007: Yay for Cuisenaire Rods
June 2007: The Second First Year of Miquon Math
August 2007: Math with Lore
September 2007: More Math Lore
December 2007: On Using Balances
Lore Rasmussen did use balance scales to demonstrate equations, and somewhere in the teachers' books, can't remember where now, there are suggestions for simple homemade balances that work more-or-less well enough to get the concept across. (For example, margarine tubs suspended from a coat hanger suspended from a broomstick.)
When our oldest was small, we made a balance with yogurt cups, so now whenever we come across an equation that needs to be simplified on both sides, we say "this one is a yogurt-cup problem."
May 2008: Names of Blocks
Another question about memorizing the colours and number names of Cuisenaire rods.
Yes and no--some of the fascination of the rods is in the fact that they can represent anything. If the white is 1,000, the orange is 10,000; if the orange is 1000, the white is 100; if the orange is 100, the white is 10 and so on.
It is convenient for them to learn, though, that for most of the basic arithmetic they'll be doing, the white does represent 1 and the orange 10. One way I taught the relationships between them was to have the child measure all the rods with a white rod. How many white rods fit into a blue, an orange...Also, the stair-patterns and pyramids children often make with the rods are good for showing those relationships--they will remember that first you take a white, then a red, then a light green.
One other way to help them remember might be to connect particular rods with the ages of children in the family. Make up imaginary names for the rods, if people-names don't confuse them in addition to colour-names and number-names. The light green rod is John, he's three; the yellow rod is Joe, he's five; and you're the dark green, six. Jackie is the orange rod, ten--she's twice as big as Joe. Stand them up as if they were little people and have some fun with them. (If John stands on Joe's shoulders, are they as big as Jackie?)
July 2008: Crayons' Grade Two: Math (planning process and detailed Blue Book notes)
With permission of the list owner, I've dug out some of those posts--mostly for my own benefit, as Crayons is doing the same math now that Ponytails was a few years ago. I've edited them to use our blog nicknames and to take out some repetitive details. I've also added links to things I posted here after we started the blog.
The sequence of the six Miquon Math workbooks is Orange, Red, Blue, Green, Yellow and Purple. The average speed on them for most homeschoolers is about two workbooks a year.
The teacher's manuals for the program are the Lab Sheet Annotations, Notes to Teachers, and the First Grade Diary. There is no "Lesson 1", "Lesson 2" and so on; each topic is introduced generally in the Annotations, and then notes are given for the worksheets. Often there are activities suggested to go along with the sheets, such as chalkboard games. Many Miquon users just have their children work right through the books, especially if they are independent learners; but we had more success mixing up the topics and not depending too heavily on the worksheets.
August 1999: Introducing Myself
This is our fourth year using Miquon and I never imagined there was a loop just for us. I've been reading through the posts and I already see I'm in good company. Our Apprentice just started the yellow book today. She loves the rods, but doing operations with them doesn't make sense to her. She loves the books, likes adding and multiplying (HATES subtracting), but would rather use a hundred chart, break-apart cubes, two rulers one above the other (anybody tried that? it's a "magic adding machine"), turning a question into a word problem, just about anything except putting two rods together and figuring out how long they are.
Answering a question about using two rulers as an adding machine:
I can't take credit for this idea--I got it from a library book but I can't remember which one! You need two rulers, metric works best because there are more numbers. I use two identical transparent ones. This idea doesn't work well with very young children, but school age children find it fun.
To add 3+12, set the rulers against one another so that the 3 on the upper one is above the 0 on the lower one. Find the 12 on the lower ruler, and whatever number is above the 12 is the answer! To subtract 10-7, place the 10 on the upper ruler over the 7 on the lower one. Find the 0 on the lower ruler, and whatever number is above the 0 is the difference.
November 1999
We started the Yellow Book this fall and for some reason we just can't get into it; my daughter groans "oh no, Miquon" when I get the book out. We are playing a lot of math games, doing questions with a muffin tin and macaroni (if you put 10 pieces in each compartment, you can do addition and subtraction up to 120) and with coloured cubes and pieces of coloured paper for the units, tens and hundreds (if I give you 5 cubes for the blue tens "column" and 4 for the red units "column", what number do you have? and if I give you 3 more for the blue and 7 for the red?--regrouping is necessary here)....
I never saw anyone as sure yesterday that 48-30 equalled 2 (on D-21). She took away the three tens and then insisted that you had to do something with two (I think she was thinking 10-8). I had to get the muffin tin out and prove the answer was 18, and I still don't think she believed me.
December 1999
I read a very interesting article by Ruth Beechick (I think it was in her Question and Answer Book). To summarize...She describes a class where the children solved problems with some kind of flat counters. The teacher would say "make three piles of two" or "eight piles of ten" and so on. The kids got comfortable with the idea of not counting every single one. As I remember it, she eventually suggested to some of them that they could write out the solutions so that they wouldn't have to count out all those piles every time; she showed them how they could take two piles of two, three piles of two, four piles of two etc. and write the answers going across the page; and then they caught on and started writing out the threes, fours etc. The idea spread around the classroom and soon many of the kids were writing out their own tables for reference.
Later she suggested that if they learned some of those facts they wouldn't have to refer to the tables all the time, and again the idea caught on; the kids saw the usefulness of carrying the facts around in their heads instead of on their papers. I don't know how well they all learned their times tables, but I'm sure that was a happier classroom than the one in "Hans Christian Andersen!" (If you never saw the movie, the schoolchildren are droning "two and two are four, four and four are eight...")
December 1999 again
In spite of my complaints [about the Yellow book], we have found one way to make some of the addition and subtraction a little clearer (for example, page E-53, which is addition and subtraction with a sum or difference of 100, 200, 1000, etc.) We get out our pile of pennies, dimes and Canadian dollar coins, and do the problems as if they were money.
We have a couple of favourite scenarios we use with this. One is "you had this much money in your piggy bank this morning. Your sister came along and "borrowed" a few coins. When you came back later, you found you only had so much money left. How much exactly did she take?"
The opposite is, "you had this much money; grandpa came for a visit and slipped a few coins in; when you counted all your money you found you had this much; so how much money did he put in?"
One problem was x + 32 = 100. My dd didn't know how to do this mentally, so I said, "Pretend I'm 30 years old. In how many years will I be 100?" "70," she said. "Now it's two years later; I'm 32 years old. Now how many years will it be until I'm 100?" "About 72" she said. "Really? But two years ago you told me it would only be 70 years. How can it be more now?" She thought about this, and then decided 68 made more sense.
[There is a long gap here because The Apprentice switched to another math program, and I unsubscribed from the Miquon-Key list until our second daughter, Ponytails, was in the first grade and starting Miquon.]
August 2003
I'm starting Miquon again with renewed enthusiasm for the program, and I'm trying to use it more creatively. One difference from our first time through is that Ponytails is a bit older than The Apprentice was when she started the workbooks.
Another difference this time around is that Ponytails has had lots of experience playing with rods over the last couple of years. She's not quite comfortable yet thinking of them in terms of their actual length; she thinks of them as a blue rod or a green rod, not as a four or a five; but I think actually that's good; it means she can think about them for themselves without worrying too much yet what they represent.
September 2003
I inadvertently found a way to heighten Ponytails' interest in learning the lengths of Cuisenaire rods. This morning I held up a yellow rod and asked her how many white rods long it would be. She said four. I said "Okay--prove it." She took the bait and said, "Okay then--I WILL." She lined up the white rods and admitted that she needed five. When I held up a light green and asked how many whites that would take, she said, "Three--and I'll PROVE it to you." And she did. We continued like that, and she kept on PROVING stuff to me--and the math got done.
September 2003
I found a homemade substitute for the wooden rod tubes described in the First Grade Diary (wooden tubes just long enough to hold one 10 cm orange rod, or the equivalent in shorter rods). (They are described in the entry for Sept. 30.)
Cut a piece of index card to a 10 cm length and wrap it fairly tightly around an orange rod, trying to crease the edges as you go (to keep the rectangular shape). Secure each end well with tape, and there you go. I made four in under 10 minutes. They're obviously not made to last forever, but at that price they could be replaced easily when needed.
The purpose of the tubes, if you don't have the Diary, is to play hidden-rod sorts of games. The simplest version would be to insert two rods that add up to ten, show the child one end of the tube, and ask what colour should appear at the other end. A more complicated version would be to use three (or more) rods, show both ends and ask what is hidden in the middle.
September 2003: Centimeter Cubes
We use these for a lot of things (besides giving the toddler something to play with [update: I don't remember what our toddler was doing playing with something so small--I must have been right beside her at the time because obviously cm cubes aren't an appropriate toy for children small enough to eat them]); I used them a lot when my oldest was learning subtraction, because you can't break a piece off a Cuisenaire rod! We also used them later on to demonstrate multiplication/division--twenty cubes snapped together, then broken into four groups of five and so on.
Young children can use them to do patterns with--two reds, a blue, two reds, a blue. You can draw a pattern with markers and have them copy it.
You can build things with them besides just straight trains. Build a 3-D model for the child to copy, something made out of several cubes stuck together. (Easy to do with Duplo or Lego too.)
You can use them to visualize word problems; for instance, you can use a piece of green construction paper to represent a field, and use some white cubes to represent sheep in the field--for counting, adding, subtracting. Or a blue piece with different coloured "fish".
We also use them to play a simple guessing game at about the K-1 level. You snap a few together (5 or 6), hold it behind your back and break a couple off; hold out the remaining piece and have the child guess how many are still behind your back. (You can do the break-off game with Duplo or Lego blocks too. Anything that snaps. Or even with small groups of non-joining objects, such as crayons or coins.)
And I suppose you could use them for actually measuring things
October 2003: After A Month of Grade One
....I have to say, I am more impressed than ever with Miquon and I'm glad I chose it for my first grader (I might have been discouraged after my older one's so-so experience).
The amazing part of this that we are doing a maximum of 20 minutes of math a day, doing a lot of rod games and the sorts of short activities suggested in First Grade Diary, and Ponytails is NEVERTHELESS picking up not only addition concepts but has started subtraction as well and ENJOYS it.
What a great program this is. I keep wondering why I see all these entire unused sets of Miquon at used curriculum sales; it makes me wonder if people just give up too soon.
October 2003
On the difficulty of problems such as x - 3 = 2:
Ponytails and I played around with some problems like this today using coloured popsicle sticks and a small empty box which had originally contained frozen chicken wings. I put six popsicle sticks in the box without showing her how many, and said "I was all by myself for dinner and I was hungry, so I took out four chicken wings to cook. (I removed four sticks.) The next time I looked in the box (I showed her), there were two left. How many were there before I ate all those chicken wings?" She looked at the two in the box and the four on the table and said "six." We repeated this with some different sets of numbers, and she had no trouble with that.
Then we went to our blackboard and I drew an empty square and told her that was for "something." I wrote the problem "Something" - 4 = 2, and reminded her about the "chicken wings." I also showed her again with the appropriate Cuisenaire rods. She caught on to that quickly. I did one more simple one with her, and then she wanted to write one. She wrote "Something" - 5 = 10. I asked her, "Do you know how to do that one?" She wasn't sure. I took a yellow and an orange rod and said, "here's our 5, and here's our 10." She still wasn't sure how much that would be, so I had her figure out how long they would be together in white rods, and she got it.
October 2003: Odd and Even
I notice in the First Grade Diary that the topic of odd and even is suddenly introduced at about this point in the school year, although there's no comment there about how it was first presented (there are some suggestions in the Annotations, though), and odd-and-even worksheets aren't included until the Red Book.
I asked Ponytails if she knew what odd and even numbers were, and she said, "One is odd. Two is even. Three is odd. Four is even, and like that." I asked her if she knew how you know if something is an odd number, and she gave me a pretty hilarious explanation that I can't even try to reproduce here. (Some of our adult explanations must sound just like that to her.)
So obviously she does have some idea that some numbers are odd or even, but doesn't really understand what that means.....One explanation given in the Miquon materials was that you can build an even number with red rods, but an odd number you can't. I also saw an arrangement of cubes (similar to an activity in Family Math) where you take one cube, two cubes, three cubes etc. and show how you can arrange the even ones in pairs but not the odd ones.
January 2004
I've noticed that some topics are covered VERY briefly in the Miquon worksheets, for instance, skip counting. In the Red Book, page F-15 is a dot to dot based on counting by 3's; F-18 is counting by 4's; F-21 is counting by 5's and 6's. Obviously a one-page puzzle is not going to be enough for a child who doesn't yet know how to count by 3's or whatever. Another topic, similarly, is time; there are a few sheets in the back of the orange book, but if the child doesn't really get telling time to the hour or half hour the first time around, there's no point in going on to the quarter hours right away.
However, this isn't a complaint! I'm just pointing out why I think it doesn't work as well just to go through the books in a rigid sequence. The curriculum obviously expects that the children are going to be learning skip counting, for instance; and if you look in the Annotations and First Grade Diary, there are extra suggestions for activities for the different topics. If I see pages on a topic I think Ponytails has already mastered (like the simple counting dot-to-dots at the beginning of the Orange Book), I use those sheets to see for sure that she knows what she's doing and has no difficulty.
But I guess what I want to say--to make it short--is that as we move into the Red Book, I'm using the topics in the book as a guide to what we should be covering--more addition/subtraction/multiplication, work in fractions, skip counting, doubles, finding tens, inequalities, and eventually division concepts. But I know there's no way she's going to learn those things, like counting by threes, by doing one worksheet. The sheets are there for her to practice on or to nail down something I think she's almost there on; but the bulk of our time will be spent on finding out how numbers work, and the ways we can move them around and do things with them.
February 2004: A Measurement Story
We're going to spend the next two weeks doing the T pages of the orange book, which are about measurement. When I looked at the Labsheet Annotations for this section, there were some classroom experiences described that sounded a bit difficult to replicate with just one student, so I took the description and rewrote it as a story for my daughter. (She always likes to hear what "the Miquon teacher" did with her class according to the First Grade Diary.)
February 2004
We are doing some of the C (adding) pages in the Red book, and the focus is grouping for tens. I've noticed that it helps if we start off with a no-book rod activity and then move on to the worksheet. That way too, if Ponytails isn't following the concrete activity, I know it's not time yet to move on to the written part.
Here are a couple of quick ones we've done or are doing with those pages. For C-12, which is 10 + 2, 10 + 4, and so on, I made two piles of rods; one pile was just orange rods, the other was all the other ones. We took turns pulling just one rod out of the mixed pile, but as many oranges in the other hand as we wanted, and "guessing" what number the other person had made. (Guessing is the wrong word, figuring out is probably better.) So I took two orange rods and a red one, and my dd said "10, 20, 22. You have 22." She took four orange rods and a blue one, and I said "49." I tried to make mostly numbers in the teens and twenties to make sure she understood those, but she kept taking as many orange rods as she could to try to make it really hard. I think we got up to 129.
Anyway, after that, the worksheet was no problem.
We're going to do C-15 today, which brings in the concept of grouping for 10s. I think I'll take a group of four or five rods (similar to the groupings on the page) and show her how to add them quickly by building pairs that make ten. I think this follows very well from the C-12 activity--instead of pulling an orange rod, you're pulling two rods that make ten.
March 2004
Cat and Mouse Game (comments)
Last Friday I planned to have Ponytails do the Red Book, page C-25, the dice game where the cat tries to beat the mouse to the mouse hole. The way it turned out, The Apprentice (seventh grader) had some spare time that morning so I asked her to play the game with her sister instead. I stuck around to watch.
What I quickly realized is that for any child who regularly plays board games, this game is so simple that it quickly becomes boring. (The corresponding subtraction game, D-9, has the same problem.) After one quick round, I suggested that maybe they could spice up the game a bit by designating one or two of the numbers as "go back that many spaces." They decided to make three of the numbers "go forward" and three "go back."
The inevitable happened: with three forward and three back, they ended up going back as much as they went forward! I didn't want to interfere, so I let them play until they'd both been sent back to start several times; then I suggested again that maybe just one of the numbers should be "go back." They decided on 6, and played that way until one of them won.
Just wanted to share about this--that pages like this that may seem too simple can still be made interesting if you add an extra twist!
March 2004
We are doing 20 minutes of math a day but it's not all from the books; we spend about half the time doing activities with the rods or other manipulatives, practicing skip counting and counting backwards, measuring things and so on. We got through the Orange Book, most of it except for some bits like clock arithmetic, in the first half of the school year, and are now mostly doing the Red Book. At this rate, she will be done it by June, no problem.
Today Ponytails begged me to teach her "the dividing thing, you know, there's a line with a dot on each side." So I did. :-)
May 2004
Ponytails is almost through the Red book, but we didn't do every page in the books, and we didn't do them in order. What we did do, especially with the Orange book, was use the First Grade Diary as a model. Not exactly every day as described (that would be impossible), but as kind of a plan for what typical math lessons would look like--often incorporating two or three seemingly unrelated activities, and having only one of those (at the most) be a worksheet. We did pages from the Red book (such as the Odd/Even section) before finishing the Orange book. We've worked on concepts like telling time and geometry all through the year, not just for the few pages on that that are included in the Orange book.
I've seen a strong thread of "tens awareness" building through the first two books, even though "place value" as a topic doesn't come up until a later book. If you keep that in mind, Miquon seems to make more sense. You do many activities based around the orange rod--finding combinations of rods that make ten (this covers both adding and subtracting), and learning to group for tens when given a string of numbers to add.
There's also a lot around the concept of equations. You start with very simple equations, like 3 + 3 = 6; later you get into equations where you have to figure something out on each side, like 3 + 3 = 2 + something. (There's a Miquon game called "How much wood" or "Lumberyard" which introduces this concept.) Then inequalities, and greater-and-less-than.
So what I'm saying is that if you do a lot of hands-on activities--using the blackboard, a hundred chart, a number line, the rods, small objects for oral word problems--involving tens and equations, you'll help your child get in touch with some of these basic concepts. Then what you'll find is that the worksheets are kind of like the follow-up activities to the concepts that he's already learning, and you'll find it easier to pick which sheets he should be doing next.
May 2004: Numberlines
The Lab Sheet Annotations is not the best laid-out teacher’s manual....I’ve added “improvements” to my copy, including colour-coded notes at the top of the pages to show which of them relate to which workbook. It does leave a lot up to the teacher to decide. I originally had the impression that Miquon students visited the math lab and were just turned loose with balances and blocks. I think that was partly true (according to the manuals), but there was also a sense of progression and growth in understanding that could only come through a certain amount of planning and guidance by the teacher.
I responded to questions by a list member about Section N.
Thank you for your questions about this, because it led to a really interesting afternoon reading carefully through that section, and I got some good ideas for number line activities. You mention a problem with Louis’ answer in the Grasshopper game on page 227, and you’re right–it’s a typo where he says the grasshopper goes back 3 jumps and it should say units instead. This game description is almost identical to one in the First Grade Diary, but a few of the numbers were changed, and maybe the word got changed accidentally in the process. (There’s also a typo at the bottom of page 229, top of 230; the section heading and a whole paragraph are repeated.)
About the Number Line topic in general...I hated number line exercises in grade school, and I’ve read that some math educators don’t believe that children understand them. Miquon takes the opposite approach, making them concrete and integral to the program; work on the number lines relates very closely to the work in adding and subtracting, and later, multiplication and fractions. It’s just too bad that they’re hidden in this little section and that the N pages are the ones you’re probably going to skip–what are you supposed to do with a bunch of blank number lines?
The number lines and other activities in this section are interesting because they don’t depend on the usual plus-minus operation signs. For that reason, I think they could be useful and non-threatening for kids who get scared by minus signs. The activities described in the Annotations go from very simple (the basic description of a number line) to quite complex; they should be done throughout the year and the program (there are “N” sheets used in three different workbooks), getting more complex as you go on.
Think of all the different kinds of number lines there are to use. “Real” number lines (on paper or chalkboard) can vary by beginning at points other than 0 or 1, can be marked with fractional units, can be marked in multiples of 2, 3, 5 etc. But there are other number lines that children may find easier to understand: thermometers, rulers, measuring tapes, board games (the kind where you roll the dice and move along a numbered track–there are a couple of simple ones in the workbooks), even life-size number lines drawn with chalk on the driveway. A hundred chart (we have a poster-size one on the wall) is great for a lot of number-line-type activities (Ruth Beechick’s booklet on math gives ideas for using one). We’ve also used concrete things like a tower of blocks to illustrate concepts of increasing and decreasing–going up and down our “skyscraper” in an imaginary elevator, stopping at different floors and sometimes pretending that the elevator goes to several levels of basement (negative numbers).
My take on this topic? I think it’s like many of the other sections, like addition: you start simple, then add little twists in (if the children don’t ask about them themselves), and keep adding on to the possibilities. But slowly, a little at a time, as they’re ready. Talking about functions in primary-level math seems way beyond what most of us expect (we’d like to stick safely with our basic adding and subtracting!). But the interesting part of it...for me...is that playing with numbers in this way is going to incorporate a great amount of adding, subtracting and more, and in a painless way. I also like its practicality as children learn to measure with rulers and apply what they’re learning in one area to the other. And I like its possibilities for bigger questions: where do you end up when you are on the third floor and the elevator goes down five floors? What happens if you want to put two jumping rules together? If we had a number line that went on out the door, where would one million be?
May 2004 again
A list member wrote, "Regarding the Functions section, I'm not even sure I would know how to approach or explain this. Any suggestion would be so helpful."
Sheets N2-N4 can be used any way you want; there are suggestions on page 231; you might also look at the problems on page 225.
I like the game presentation of functions here. This is one place where I think the Annotations does do a good job taking you through a series of exercises about number lines and functions, especially at the Green and Yellow levels. It's suggested that you stick to the terminology of the number line games, using words such as: "try my rule," "make up a rule," "2 goes to 4," "start," "beginning number," "land," "landing number," "undoing rules" (inverses), "jail rules," "standstill points," "anywhere goes to anywhere plus four" or "box goes to box plus four." Functions, as far as this book goes, are just more and more complex rules for number line games.
At the Orange level, number lines stay simple, for instance with the basic Gus and Happy game. At the higher levels, the rules of the game become more involved; students experiment with combining more than one rule and with a simple form of graphing the functions (something we spent a lot of time on in high school Functions and Relations).
At one point it's suggested that you try using a number line labelled with letters of the alphabet instead of numbers. You could play that game out loud, too, without written letters: if the rule of the game is to tell me the letter that is my letter plus two more, what is the letter if my letter is B? (That's how substitution ciphers work...also known as secret codes.) This even makes me think of the notes on a piano: if I play C and we make up a rule that says play the note that is two notes lower, you play A. Of course this would only work for adding/subtracting functions, not multiplying/dividing ones.
I wouldn't even worry about the more complex "rules of the game" until your student is very comfortable doing the more basic number line activities. If you notice, there are relatively few worksheets in this section for a topic that can become quite complex; my feeling is (based on the notes) that they originally did a lot of chalkboard work with it. To be honest, I can't even remember going through these upper-level sheets with my oldest daughter, but I know she "did" the Green book. Shows you how much we missed the first time through...
June 2004
We did K-7 this morning, which is one of two pages dealing with factors of 24 (before the topic of factors is gone into as such). I decided to keep the book closed this time until we had done some concrete work with the rods.
I had Ponytails build a train of 24 (two oranges and a purple) and then asked her to find all the one-colour trains she could of that length. I had intended to wait until she had found them all before doing any recording, but as soon as she lined up six purples, she said "Six fours! Can we write that down?" So we went on like that until she had found three pairs of multiplication facts equalling 24. (After she found twelve twos, I asked her to imagine she had some "twelve rods" and asked how many of those would fit into 24.)
At this point, doing the multiplication part of K-7 would be optional since you have really just done it anyway, but I had Ponytails fill in the blanks anyway. There are division facts (the inverses of the multiplication facts) going down the right hand side of the page, and those are something we're still just getting into, but she didn't have a lot of trouble with them if I phrased it as "how many 4's in 24?"
Oh--and something funny only Miquon users would appreciate. The other day we were in the car and pulled up to a red light beside a sport vehicle. Ponytails pointed: "Look, Mom! Four fours!" I looked, and she was right: on the side, the truck said 4x4.
November 2004: Blue Book
Ponytails is in her second year now with Miquon and....I find we're using the rods a fair amount, especially for two-digit addition and subtraction,and also a hundred chart. (We made a big one--big is good!-- that has cardboard number disks attached with sticky Velcro.) We use the hundred chart almost every day for oral adding and subtracting problems, and she's getting very good at moving the right number of spaces DOWN for tens and ACROSS for units. (This is great place value stuff.) (Oh, a note about doing hundreds with the rods: we just cut some hundred-sized squares of orange paper.)
We've been playing around with assigning different values to the rods than the usual white = one . This was helpful, for instance, on the "diagnostic page" E-46 which had questions like 60 + 40 + 70 + 30 . I showed Ponytails that if each white rod = 10, she could use the rods easily to add those numbers together (in addition to grouping for tens).
We did E-44 (arranging three given numbers into number sentences) using rods and "symbol cards" (half-pieces of index cards marked with signs for plus, minus, equals etc.). (Haven't done E-45 yet where they are to choose their own three numbers.) After she arranged the rods and cards to make true number sentences, she copied the sentences onto the worksheet.
For E-48 and E-49 I wrote all the questions (10+19, 23-13 etc.) on half-index cards and did them orally using the hundred chart (she did not write the answers). I wrote the addition questions on one side in one colour, and the subtraction ones on the reverse in another colour). This way (since it wasn't done on a worksheet) we can use a few of the same questions as review on another day. Yesterday I picked out three of the addition questions, had her do them with the hundred chart, and then asked her to do the reverse sides as well.
We've also been spending some time on doubles, using pages F-23 and F-25. We started by working down the "1" column on F-25, doing doubles up to 256 (my dd figured that one out with rods). Because she seemed really interested in this, I re-read her a story we have in a Childcraft math book about why there is no mathematical possibility of vampires (because if you double the number of vampires every week, everybody in the world would be a vampire in less than a year :-)); it's just a variation of the story about the servant who asks for one grain of rice, then two, then four etc. until he owns the whole kingdom. I also showed her the doubling function on a calculator (2 x = = = = etc.)
All this is to say that if you have children who are active/easily distracted and who are "global"...I think the word is global...enough to enjoy challenging questions and big numbers even though they may sometimes get stuck on easy ones...I think Miquon continues to be a good choice even past the first grade stage, if you can keep the activities concrete and work within their attention span.
April 2005: Number Lines
We are in the middle of the green book, having just completed the section on multi-digit addition and now working on number line games.
We also found a game today in Family Math that is good for place value and addition. If you have the book, it's called Dollar Digit and it's on page 112. You need a pile of dimes and a pile of pennies; one die; and two sheets of paper each marked in two columns (one for dimes, one for pennies). You number each paper 1-7 down the left hand side. For each turn, someone rolls the die but all the players use that same number. You can either take that number of pennies or that number of dimes (but not mixed), and place them beside the number 1 on your sheet. Next turn, someone else rolls but all players again use the number from that roll. Any time you get 10 pennies, you have to trade them for a dime. After 7 rolls, you add up what you have, and the closest one to a dollar wins. (You can make a rule that you can't go over, if you want.)
May 2005
Ponytails is transitioning into the Yellow book, although we haven't finished all the end-of-the-book topics in the Green yet.
Anyway, although Miquon's transition into subtraction is slow, I am really impressed with the reasoning behind it. I told Ponytails that this kind of math--that is, learning to make life easier for yourself by changing questions like 48-13 into 50-15) is "Smart Math." Using-your-brain math, rather than having the teacher tell you what to do and you just doing it. I think she pretty much understood that.
May 2005: Functions
A list member asked about Page N-7: "Why is it written 'box arrow box' then the rule to add or subtract a number. Why not just one box?"
What you're working with here are functions. That's why they're set up as they are...the book is giving you a kind of junior version of math that we did in high school.
The Annotations is really helpful with this section. If you present it as they do, it will probably make more sense. Imagine a series of number line games, where each person involved makes up a different rule for how you're allowed to move. (Kind of like how the different chess pieces all have their own rules for moving. Or each person having their own dance step.) Billy's rule says that you start on "box", and you end up on "box plus three." Whatever "box" is, you add three to "box" to make your move. (In this case, you're jumping by threes then.) You can also say "anywhere" or "starting point" instead of "box" if that makes more sense.
So if you start with 4 and follow "Billy's rule," you end up at ? (7). And so on.
Then "you try Billy's rule." Same thing. Each time you're just adding three.
Then make up some examples of your own.
If you follow the Annotation suggestions, you'll see some variations that you can use while you're playing around with number lines. For instance, what if somebody has a rule that "box" goes to "box plus three", and someone else's rule says "box" goes to "box times two?" If they have a race, who gets further along, or is there a point where one overtakes the other?
Or you could even play "guess my rule." Show a series of moves and see if your child can guess what rule you were following.
May 2005 again
A list member mentioned that they always skipped the Factor House Game in the O pages.
LOL, and I thought we were the only ones . We just did those pages, and I couldn't see the point in cutting out the little houses (other than the scissors practice).
But the sections after that have been great--square numbers and area are so much fun with rods! We built pyramids out of rod squares as suggested in the Annotations, but we built them off-center (rather than each new layer right in the middle, we lined up the edges on one corner of the bottom layer, if that makes sense), so that you could see the different colours of rods and how many extra were added. The next day, I printed out some cm graph paper and we did the same thing only with paper and crayons. We made up a story that a famly started out with a small house (a red, 2 cm square) and then needed to enlarge their house, so they added a red rod on each side and then filled in the gap with a white rod, making it as large as a green, 3 cm square. Then they had more children, so they needed to enlarge again, and so on. We went through several squares this way, writing out the formula each time. (i.e. 3 squared = 2 squared + 2 + 2 + 1; 4 squared = 3 squared + 3 + 3 + 1.) There's no way all this would have made any sense to Ponytails (and to me) without the rods.
December 2005
A list member asked for help with worksheet H-47.
There are different ways you can illustrate these problem besides with the number line. You could turn each question into a word problem, like this: "If you have two thirds of a pizza, and I have two thirds of a pizza, how many thirds of a pizza do we have? Is there another way to say that? Or, is that more or less than a whole pizza? How much more? It's okay to say we have four thirds of a pizza, and it's also okay to say we have one and a third pizzas."
When you look at each question, try saying it like this "Two two-thirds." "Four two-thirds." "How many two-thirds are there in six-thirds? What IS six-thirds, let's figure that one out first. (Use whatever manipulatives you want.) Oh, six-thirds is the same as two whole things. So how many two-thirds are there in two whole things? (Use manipulatives to figure it out, if you need to.) Do you want to think it through with pizzas again? Well, if you have two-thirds of a pizza, and I have two-thirds, how much do we have left? Another two-thirds. So there are three two-thirds pieces in two whole pizzas."
The last one: "Some number of thirds is the same as two wholes. How many thirds can we fit into two wholes?" (Use manipulatives.)
The manipulatives could include Cuisenaire rods, a paper "Hershey bar" (divided into sections), paper or plastic "pizzas", or plastic fraction pieces.
It's a good idea to vary the shapes if you're doing this kind of exercise on more than one day; you can get into a rut of always thinking about fractions in terms of round things like cakes and pizzas, so it's good to remind kids that fractions come in other shapes like squares and rectangles too. And even, if they're ready to think about this, in groups of things rather than in pieces of a whole. For that, you can use cereal or beans or cubes (white rods work fine) or whatever small objects you want to divide into groups...these kind of problems are definitely more of a mental challenge, but you can turn them into word problems as well. Stories about classes of children, or teams, or a bag of treats that we have to divide up, all work well.
The rods are good fraction manipulatives in themselves, and that's one thing you miss out on if your kids think of them only as representing the numbers one to ten. It can actually be a lot of fun playing around with these ideas...my third grader liked this part of math. For instance, you take a brown rod. "This is two." (You may get protests.) "Well, today it is a two. Maybe it's a two-dollar bill (or coin if you're Canadian). Okay, let's make it be money. If this is a two-dollar bill, show me a one-dollar bill." (Child shows you a purple.) "Show me half a dollar. Show me a quarter dollar." (Short sidetrack into why quarters are called quarters.) "Okay, let's try something else. (Show a light green rod.) This is one. Show me two. Show me three."
The fun, for Ponytails anyway, was when I asked her to do some for me. I think she has caught on to the idea that what we're doing is about relationships. Rods are about relationships. Fractions are about relationships. Fractions are also about dividing and multiplying.
March 30, 2006
The power of the rods is that they can represent so many different concepts. Any one of the rods can represent "one"; then all the others can be identified in relation to that rod. The most usual set of names we give them is to say that the white rod is one, and then red is two and so on; but that is just the beginning of what you can do with them. If your child works with them from the beginning and understands their relationships, then they should have no trouble in using them for more difficult concepts like fractions. (If the orange rod is two, then the yellow rod is one.)
I'm reading a book called The Mathematical Mystery Tour, written for adults--kind of a fictionalized history of mathematics. It got mixed reviews on Amazon as far as its scope and attention to details goes (and whether it was boring or not). The point, though, is that in the first chapter the characters are talking about Pythagoras and early Greek mathematics, which was based only on integers. The mathematician in the chapter points out that the ancient Greeks substituted their geometrical knowledge for the algebra they lacked. In other words, they could draw a diagram of a problem (in the dirt, usually) and figure out how to express the answer as a ratio (since they had only integers to work with). The diagrams in the book look exactly like something you might build with Cuisenaire rods!
When I read that chapter, I realized again what a powerful set of ideas we are giving our children with these manipulatives. Primary-aged children, like the Greeks, do not have any knowledge of algebra; we do not ask them yet to "solve for x." But my third grader is able to *see* the answers, using rods, to problems that she would otherwise not be able to solve.
April 2006: Cookin' with Math
May 2006: Math Stuff
January 2007: Trains Game
A list member asked about number names for rods.
For that reason, I might even suggest that you don't worry about whether the "2" rod is 2 at this point, but stick to the colours and the relationships between the colours. Two reds make a purple, two purples make a brown. How many ways can you make a train as long as an orange rod? You can play trading sorts of games (as if the rods were money)--what will you trade me for a yellow rod? You could even develop that into a sort of playing store--for instance, if a doll is priced at a yellow rod, she could pay for it with five white rods, two reds and a white, etc.
There's a game with variations that's given in the primary-level Idea Book for Cuisenaire Rods, and we've played it many times to help build awareness of the colours without getting hung up on the numbers.
The basic game is that you dump about 40-50 rods on the table or the floor, and then take turns trying to find a rod that the other person can't "match." You hold up one of the rods (any rod except a white one), and the other person has to find two rods that make a train as long as your rod. (Not one, or three--just two.) If they can, then you set those three rods aside and the other person gets a turn to hold up a rod and stump you. You continue the game until somebody holds up a rod that can't be matched with the rods that are still on the table; then that person wins the game.
For instance, you hold up an orange rod; the other person can then match it with two yellows, or a purple and a dark green, or a light green and a black, or whatever. But if it's near the end of the game and you hold up a light green (3 cm) rod, and all the red (2 cm) rods have been used up, then there's no way they can match that, and you win.
April 2007: Yay for Cuisenaire Rods
June 2007: The Second First Year of Miquon Math
August 2007: Math with Lore
September 2007: More Math Lore
December 2007: On Using Balances
Lore Rasmussen did use balance scales to demonstrate equations, and somewhere in the teachers' books, can't remember where now, there are suggestions for simple homemade balances that work more-or-less well enough to get the concept across. (For example, margarine tubs suspended from a coat hanger suspended from a broomstick.)
When our oldest was small, we made a balance with yogurt cups, so now whenever we come across an equation that needs to be simplified on both sides, we say "this one is a yogurt-cup problem."
May 2008: Names of Blocks
Another question about memorizing the colours and number names of Cuisenaire rods.
Yes and no--some of the fascination of the rods is in the fact that they can represent anything. If the white is 1,000, the orange is 10,000; if the orange is 1000, the white is 100; if the orange is 100, the white is 10 and so on.
It is convenient for them to learn, though, that for most of the basic arithmetic they'll be doing, the white does represent 1 and the orange 10. One way I taught the relationships between them was to have the child measure all the rods with a white rod. How many white rods fit into a blue, an orange...Also, the stair-patterns and pyramids children often make with the rods are good for showing those relationships--they will remember that first you take a white, then a red, then a light green.
One other way to help them remember might be to connect particular rods with the ages of children in the family. Make up imaginary names for the rods, if people-names don't confuse them in addition to colour-names and number-names. The light green rod is John, he's three; the yellow rod is Joe, he's five; and you're the dark green, six. Jackie is the orange rod, ten--she's twice as big as Joe. Stand them up as if they were little people and have some fun with them. (If John stands on Joe's shoulders, are they as big as Jackie?)
July 2008: Crayons' Grade Two: Math (planning process and detailed Blue Book notes)
Thursday, July 17, 2008
Crayons' Grade Two: Math
Previous math posts:
Math with Lore (2007)
The Second First Year of Miquon Math (2007)
The Primary Math Cupboard (2007)
Math Stuff (2006)
Cookin' with Math (why they're not Cuisinart Rods) (2006)
Lots of Math Posts
How do you plan a year of Miquon Math when the Lab Sheet Annotations (teacher's manual) is so vague about what you do when?
This is how I planned second-year Miquon both several years ago for Ponytails and then again recently for Crayons. I figured on her getting through the Blue and Green books this year (third and fourth books of six), so I looked at the whole scope and sequence (page 9 in the Annotations) and divided up the topics for those two books among the thirty-six weeks in the school year. When you look at the workbooks, some topics get a lot more worksheet space and/or more emphasis than others--so I give those more school weeks.
No fancy spreadsheet programs here--just a sheet of lined paper. Week 1: Odd and Even. (We could spend more time on that but I know Crayons is pretty solid on Odd/Even.) Week 2: Addition. Week 3: Addition. Week 4: Addition. Week 5: Subtraction. And so on. I make sure that the oddball topics at the ends of the books don't get too squished in at the end of the year (sometimes I redistribute those throughout the year), and I try to make room both for review and for preview.
Preview?
In the years I've used Miquon, I've noticed that, if you're doing the worksheets pretty much in sequence, you can come up quickly against a sheet that would be much more valuable if your student had a few no-worksheet opportunities to practice that topic before trying it on paper. One quick example: at one point there are some skip-counting dot-to-dot pages. Now obviously those aren't going to be enough for anybody to learn skip counting, and I don't think they were meant to be. It makes more sense to tuck "counting by fours" into several previous lessons, and then--aha! Today you get to count by fours on this puzzle!
And that's why I like to plan Miquon Math ahead for the year, instead of just opening the book. It also helps give a bit more variety to each week's lessons. We can preview a bit on geometric shapes and skip counting, work on the week's addition or subtraction, and review what we did from a couple of weeks ago--repeat a game or activity, or do a worksheet that was skipped over.
I'm not looking at using a lot of supplements for math this year, outside of our normal cache of manipulatives. We'll probably work quite a bit with a hundred chart--I find that's very helpful for learning subtraction and also for "Smart Math." "Smart Math" is using your head about arithmetic and not getting caught up in dumb mistakes kids make when they've been misled or over-taught by some of our teacherish ways to do things. The classic one is being given 100 - 99 on paper and trying to cancel out the zeros because that's what you've been shown how to do when you subtract. "Smart Math" says "100 - 99? I don't care what it looks like, you can't fool me, the answer's 1."
When it comes to putting more detail into the year's math plans--knowing what card games and so on I'm going to use for addition or subtraction--sometimes I plan a lot ahead of time, sometimes it's a night-before flip through the Annotations. This year I got lucky: I found somebody's plans for the first few weeks of the Blue Book. Mine. I forgot I had sent these to the Miquon-Key Yahoo list back when Ponytails was at this level--and there they were in the archives, saving me most of the planning work for this term.
And here they are, with a few edits. The original post only covered weeks 1-6, so I've added somewhat briefer notes for weeks 7-12. I hope maybe this will help somebody get their year started.
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I put together some rough plans for my second-grader's first six weeks of math this coming school year, starting with the Blue Book. I know they are somewhat sketchy, but I thought seeing them might help someone else who's at around the same place. They're slanted toward the things I know my daughter still needs to work on, rather
than trying to include every concept that possibly be covered using those Miquon pages. To me, that's a great thing about this program--it is very flexible and you can spend more or less time preparing for, doing, and reviewing a given activity, depending on how fast and how well they "get it" (or not) the first time through. We may not get to everything every week, but I find having the extra suggestions in place helps me plan a variety of activities as well as prepare for upcoming lessons. For instance, E46 includes one problem where it's necessary to read amounts of money such as $3.25; I will make sure she knows how to do that before asking her to do the problem. (In some cases though...I think Mrs. Rasmussen [Lore Rasmussen, author of Miquon Math] might say this too...it's probably okay just to let THEM ask YOU when they need to know. Mom, what's this mean with the funny S and the line through it? Okay, here's how you read money. Right?)
I find I'm drawing a lot on Ruth Beechick's little booklet and her hundred-chart suggestions in planning how I'm going to teach some of these concepts. I think the Lab Sheet Annotations tends to use a number line more on the actual worksheets, but they do suggest using a hundred-chart as well. We have a large poster-size one, and
another one we made with cardboard number disks attached with sticky-back Velcro. You can also find small reproducible ones on many math websites.
Anyway, here are my notes. FGD means the First Grade Diary. The other page references are to the Lab Sheet Annotations. I've avoided including games and so on that we might include outside of the Miquon materials, other than games with cards and dice; I'll probably pencil some of those in after I take a look through what we've got on hand here. The word problems are made up as we go along.
Miquon Blue Book, Weeks 1-6
Week 1
Odd and Even sheets
Play games on p. 38: Make 10, Odd or Even
Review sequence of numbers: take some cards, put them in order from
smallest to largest; play War
Telling time–review with flash cards; see game in FGD p. 171
Which would you rather have? FGD p. 171
FGD p. 186, Making true statements (introduce signs for not greater
than, not less than)
Review skip counting
Guess what number I'm thinking of (FGD p. 200)
Word problems
Week 2
Complete Odd and Even sheets
Chalkboard work related to C26, as suggested in the LSA: adding
strings of single digits, recombining them to make adding easier;
this is also fun to do with piles of Cheerios or raisins, maybe with
pennies
See explanation for C28, sequence given (practice some of these
ideas before doing the sheet next week):
1. Find "other names" for numbers such as 5 (make patterns with rods)
2. Oral questions such as "in the problem 15 + 8 = what, what do I
need to add to 15 to make the next 10?" This is not an easy concept;
try doing with money, with a number line. An idea: build 15 with
rods; then add eight white rods, and see how many are left after you
take enough to turn the 15 into 20.
3. Oral practice suggested on p. 49: If I want to add 7 + 5, what
name for 5 would help me most? (Expand to 27 + 5, 77 + 5 if they
don't see the point of doing this to add something "easy" like 7+5.)
Review writing 2-digit numbers, breaking them down into 10s and units
(what's a unit?). Use popsicle sticks, with some bundled into groups
of 10. Use dimes and pennies.
Play the games on p. 38.
Word problems
Week 3
Do addition sheets C26-C-30
C26: Adding strings of single digits
C27: Using doubles to make equivalent trains
C28: is tricky: finding missing addends for 10, then using them to
add things like 7 + 4
C30: tens and units
Practice grid problems such as C32 but with smaller numbers. Maybe
roll dice or draw cards to choose tne numbers.
Oral math questions similar to those listed for Week 4
Word problems?–time, money, measurement
Week 4
Practice writing 3-digit numbers
Practice writing money $3.25
Oral subtraction problems such as 205-200, 380-300, 380-80
Oral addition and subtraction "bridging" questions such as 46+7, 46-7
(Ruth Beechick calls it "bridging" when you are adding or subtracting
something and have to jump to the next line of the 100-chart to find
the answer.)
Oral addition, strings of single digit numbers
Practice adding 10s, 20s and 30s to things (100-chart)
Oral subtraction such as 52-51, 99-99, 30-20, 31-21 (100-chart,
number line)
Oral practice related to making change, see D16–when jumping 10s and
units, first jump the 10s, then the units. Practice with money.
Do addition sheets C31-34
C31 is a grid game, a bit tricky to figure out at first. (You are
making an addition chart.)
C32 and 33, adding numbers in grids
C34, another addition table
Week 5
Do subtraction sheets D13-16
Grid problems on D13
D14–simple subtracting
D16–practice subtracting any number from 100. This is DIFFICULT
unless you jump the 10s first, then the units (see p. 72)–like making
change
Word problems
Introduce 10 more/10 less activity (p. 98) (jumping by 10s)
Practice writing 3-digit numbers, especially ones with 0's in them
Try game based on E44&45, p. 99
Use playing cards–choose three, make up as many sentences as you can
about the three numbers chosen; can use greater than/less than,
equals/does not equal signs
Practice adding multiples of 10–example, 40+70, 60+40
Count by 20s
Practice adding & subtracting 9's instead of 10s; also 11's
Add any activities previewing week 7
Week 6
Do Addition/Subtraction pages E42-49
Adding 10s, 100s and their inverses (counting forwards and backwards
by 10s, starting at any number, up to any 3-digit number)
E46 is a "diagnostic page" including 2-digit addition and subtraction
and a money problem
Do word problems
Add any activities previewing weeks 7 and 8 (Looking ahead: geometric shapes, parts of a dollar)
Week 7
Multiplication--see the preliminary activities on page 117. Doubling, halving. Worksheets F24-F30. Looking ahead: geometric shapes.
Week 8
Multiplication (activity, The Pattern Book, that creates two booklets of math tables)
--Practice with velcro hundreds chart
Follow instructions for the booklets on page 120
Looking ahead: parts of a dollar
Week 9
Multiplication: continue booklets; follow instructions on page 122.
Worksheets F 41 and 42-- filling out a complete multiplication table; what are prime numbers?
Begin G sheets if there is time, since there are a lot for next week.
Also work on telling time.
Week 10
Addition, subtraction and multiplication together
Worksheets G13-G20 (this is a lot; they may not all get completed)
Distributive law--suggest rod examples. (This is fairly difficult.)
Practice telling time.
Looking ahead: geometric shapes, review basic division ideas from last year.
Week 11
Fractions--review what halves, thirds and quarters are. If this is going well, extend to 5ths through 8ths (sheet H25). Look at H-27 through H-29 along with concrete examples.
Looking ahead: parts of a dollar, review division.
Week 12
Fractions
H 30, parts of a dollar
H 31, measuring cups
H 32-H-42--varied problems, too many for one week--do at least up to H 36, and save the rest for review or for next term.
Related Posts:
Crayons' Grade Two: Social Studies
Math with Lore (2007)
The Second First Year of Miquon Math (2007)
The Primary Math Cupboard (2007)
Math Stuff (2006)
Cookin' with Math (why they're not Cuisinart Rods) (2006)
Lots of Math Posts
How do you plan a year of Miquon Math when the Lab Sheet Annotations (teacher's manual) is so vague about what you do when?
This is how I planned second-year Miquon both several years ago for Ponytails and then again recently for Crayons. I figured on her getting through the Blue and Green books this year (third and fourth books of six), so I looked at the whole scope and sequence (page 9 in the Annotations) and divided up the topics for those two books among the thirty-six weeks in the school year. When you look at the workbooks, some topics get a lot more worksheet space and/or more emphasis than others--so I give those more school weeks.
No fancy spreadsheet programs here--just a sheet of lined paper. Week 1: Odd and Even. (We could spend more time on that but I know Crayons is pretty solid on Odd/Even.) Week 2: Addition. Week 3: Addition. Week 4: Addition. Week 5: Subtraction. And so on. I make sure that the oddball topics at the ends of the books don't get too squished in at the end of the year (sometimes I redistribute those throughout the year), and I try to make room both for review and for preview.
Preview?
In the years I've used Miquon, I've noticed that, if you're doing the worksheets pretty much in sequence, you can come up quickly against a sheet that would be much more valuable if your student had a few no-worksheet opportunities to practice that topic before trying it on paper. One quick example: at one point there are some skip-counting dot-to-dot pages. Now obviously those aren't going to be enough for anybody to learn skip counting, and I don't think they were meant to be. It makes more sense to tuck "counting by fours" into several previous lessons, and then--aha! Today you get to count by fours on this puzzle!
And that's why I like to plan Miquon Math ahead for the year, instead of just opening the book. It also helps give a bit more variety to each week's lessons. We can preview a bit on geometric shapes and skip counting, work on the week's addition or subtraction, and review what we did from a couple of weeks ago--repeat a game or activity, or do a worksheet that was skipped over.
I'm not looking at using a lot of supplements for math this year, outside of our normal cache of manipulatives. We'll probably work quite a bit with a hundred chart--I find that's very helpful for learning subtraction and also for "Smart Math." "Smart Math" is using your head about arithmetic and not getting caught up in dumb mistakes kids make when they've been misled or over-taught by some of our teacherish ways to do things. The classic one is being given 100 - 99 on paper and trying to cancel out the zeros because that's what you've been shown how to do when you subtract. "Smart Math" says "100 - 99? I don't care what it looks like, you can't fool me, the answer's 1."
When it comes to putting more detail into the year's math plans--knowing what card games and so on I'm going to use for addition or subtraction--sometimes I plan a lot ahead of time, sometimes it's a night-before flip through the Annotations. This year I got lucky: I found somebody's plans for the first few weeks of the Blue Book. Mine. I forgot I had sent these to the Miquon-Key Yahoo list back when Ponytails was at this level--and there they were in the archives, saving me most of the planning work for this term.
And here they are, with a few edits. The original post only covered weeks 1-6, so I've added somewhat briefer notes for weeks 7-12. I hope maybe this will help somebody get their year started.
--------------------------------------------------------------
I put together some rough plans for my second-grader's first six weeks of math this coming school year, starting with the Blue Book. I know they are somewhat sketchy, but I thought seeing them might help someone else who's at around the same place. They're slanted toward the things I know my daughter still needs to work on, rather
than trying to include every concept that possibly be covered using those Miquon pages. To me, that's a great thing about this program--it is very flexible and you can spend more or less time preparing for, doing, and reviewing a given activity, depending on how fast and how well they "get it" (or not) the first time through. We may not get to everything every week, but I find having the extra suggestions in place helps me plan a variety of activities as well as prepare for upcoming lessons. For instance, E46 includes one problem where it's necessary to read amounts of money such as $3.25; I will make sure she knows how to do that before asking her to do the problem. (In some cases though...I think Mrs. Rasmussen [Lore Rasmussen, author of Miquon Math] might say this too...it's probably okay just to let THEM ask YOU when they need to know. Mom, what's this mean with the funny S and the line through it? Okay, here's how you read money. Right?)
I find I'm drawing a lot on Ruth Beechick's little booklet and her hundred-chart suggestions in planning how I'm going to teach some of these concepts. I think the Lab Sheet Annotations tends to use a number line more on the actual worksheets, but they do suggest using a hundred-chart as well. We have a large poster-size one, and
another one we made with cardboard number disks attached with sticky-back Velcro. You can also find small reproducible ones on many math websites.
Anyway, here are my notes. FGD means the First Grade Diary. The other page references are to the Lab Sheet Annotations. I've avoided including games and so on that we might include outside of the Miquon materials, other than games with cards and dice; I'll probably pencil some of those in after I take a look through what we've got on hand here. The word problems are made up as we go along.
Miquon Blue Book, Weeks 1-6
Week 1
Odd and Even sheets
Play games on p. 38: Make 10, Odd or Even
Review sequence of numbers: take some cards, put them in order from
smallest to largest; play War
Telling time–review with flash cards; see game in FGD p. 171
Which would you rather have? FGD p. 171
FGD p. 186, Making true statements (introduce signs for not greater
than, not less than)
Review skip counting
Guess what number I'm thinking of (FGD p. 200)
Word problems
Week 2
Complete Odd and Even sheets
Chalkboard work related to C26, as suggested in the LSA: adding
strings of single digits, recombining them to make adding easier;
this is also fun to do with piles of Cheerios or raisins, maybe with
pennies
See explanation for C28, sequence given (practice some of these
ideas before doing the sheet next week):
1. Find "other names" for numbers such as 5 (make patterns with rods)
2. Oral questions such as "in the problem 15 + 8 = what, what do I
need to add to 15 to make the next 10?" This is not an easy concept;
try doing with money, with a number line. An idea: build 15 with
rods; then add eight white rods, and see how many are left after you
take enough to turn the 15 into 20.
3. Oral practice suggested on p. 49: If I want to add 7 + 5, what
name for 5 would help me most? (Expand to 27 + 5, 77 + 5 if they
don't see the point of doing this to add something "easy" like 7+5.)
Review writing 2-digit numbers, breaking them down into 10s and units
(what's a unit?). Use popsicle sticks, with some bundled into groups
of 10. Use dimes and pennies.
Play the games on p. 38.
Word problems
Week 3
Do addition sheets C26-C-30
C26: Adding strings of single digits
C27: Using doubles to make equivalent trains
C28: is tricky: finding missing addends for 10, then using them to
add things like 7 + 4
C30: tens and units
Practice grid problems such as C32 but with smaller numbers. Maybe
roll dice or draw cards to choose tne numbers.
Oral math questions similar to those listed for Week 4
Word problems?–time, money, measurement
Week 4
Practice writing 3-digit numbers
Practice writing money $3.25
Oral subtraction problems such as 205-200, 380-300, 380-80
Oral addition and subtraction "bridging" questions such as 46+7, 46-7
(Ruth Beechick calls it "bridging" when you are adding or subtracting
something and have to jump to the next line of the 100-chart to find
the answer.)
Oral addition, strings of single digit numbers
Practice adding 10s, 20s and 30s to things (100-chart)
Oral subtraction such as 52-51, 99-99, 30-20, 31-21 (100-chart,
number line)
Oral practice related to making change, see D16–when jumping 10s and
units, first jump the 10s, then the units. Practice with money.
Do addition sheets C31-34
C31 is a grid game, a bit tricky to figure out at first. (You are
making an addition chart.)
C32 and 33, adding numbers in grids
C34, another addition table
Week 5
Do subtraction sheets D13-16
Grid problems on D13
D14–simple subtracting
D16–practice subtracting any number from 100. This is DIFFICULT
unless you jump the 10s first, then the units (see p. 72)–like making
change
Word problems
Introduce 10 more/10 less activity (p. 98) (jumping by 10s)
Practice writing 3-digit numbers, especially ones with 0's in them
Try game based on E44&45, p. 99
Use playing cards–choose three, make up as many sentences as you can
about the three numbers chosen; can use greater than/less than,
equals/does not equal signs
Practice adding multiples of 10–example, 40+70, 60+40
Count by 20s
Practice adding & subtracting 9's instead of 10s; also 11's
Add any activities previewing week 7
Week 6
Do Addition/Subtraction pages E42-49
Adding 10s, 100s and their inverses (counting forwards and backwards
by 10s, starting at any number, up to any 3-digit number)
E46 is a "diagnostic page" including 2-digit addition and subtraction
and a money problem
Do word problems
Add any activities previewing weeks 7 and 8 (Looking ahead: geometric shapes, parts of a dollar)
Week 7
Multiplication--see the preliminary activities on page 117. Doubling, halving. Worksheets F24-F30. Looking ahead: geometric shapes.
Week 8
Multiplication (activity, The Pattern Book, that creates two booklets of math tables)
--Practice with velcro hundreds chart
Follow instructions for the booklets on page 120
Looking ahead: parts of a dollar
Week 9
Multiplication: continue booklets; follow instructions on page 122.
Worksheets F 41 and 42-- filling out a complete multiplication table; what are prime numbers?
Begin G sheets if there is time, since there are a lot for next week.
Also work on telling time.
Week 10
Addition, subtraction and multiplication together
Worksheets G13-G20 (this is a lot; they may not all get completed)
Distributive law--suggest rod examples. (This is fairly difficult.)
Practice telling time.
Looking ahead: geometric shapes, review basic division ideas from last year.
Week 11
Fractions--review what halves, thirds and quarters are. If this is going well, extend to 5ths through 8ths (sheet H25). Look at H-27 through H-29 along with concrete examples.
Looking ahead: parts of a dollar, review division.
Week 12
Fractions
H 30, parts of a dollar
H 31, measuring cups
H 32-H-42--varied problems, too many for one week--do at least up to H 36, and save the rest for review or for next term.
Related Posts:
Crayons' Grade Two: Social Studies
Tuesday, July 15, 2008
Crayons' Grade Two Outline (updated)
School Year 2008-2009
Grade Two
Teacher’s Resources
How is My Second Grader Doing in School?—Jacobson & Raymer (language and math activities)
Teaching Children, by Diane Lopez
3-R’s booklets, by Ruth Beechick
Bible:
Term 1: 1 Samuel 1-20, Matthew 1-14, selected Psalms (also planning to buy Judy Rogers’ CD of Psalms set to music, Never Be Shaken)
Term 2: 1 Sam 21-31, 2 Sam 1-12, Matthew 15-28
Term 3: 2 Sam 13-24, 1 Kings 1 & 2; Mr. Pipes and the British Hymn Makers
(Other Bible memory work is listed under Memory Work)
Each term: lessons (1x/wk) from People of the Bible: Life and Customs.
2nd and 3rd terms: Luther’s Small Catechism, section on the Lord’s Prayer
Singing
Hymns, both our own favourites and new ones from Mr. Pipes
Folk songs: sources include Canada Is For Kids series of three CDs (borrowed from library); our own books of children’s songs and folk songs; songs from Festivals, Family and Food
(Planning to purchase colouring books to accompany the CDs)
Language Arts:
Specific grade 2 skills, taught as needed, using our own books and supplements (word puzzles, a couple of Gifted and Talented workbooks, magnetic words, Scrabble letters, children's dictionary)
Skills include:
Oral communication, including narration, telephone/manners (using Uncommon Courtesy for Kids)
Listening skills (demonstrated by oral or other responses)
Capitalization, some punctuation, plurals, complete sentences, contractions, prefixes/suffixes, alphabetizing to the second letter (using Gifted and Talented workbooks, Word Puzzles Gr 2/3 workbook, other activities)
Copywork, simple dictation (spelling words with specific patterns as well as calendar words and holiday words) (We found some basic Gr 1 and 2 word lists in Kathryn Stout’s Natural Speller)
Printing practice, using Canadian Handwriting workbooks
Memory work (see list for each term)
Reading silently and out loud, and being read to (see booklists)
Writing, mostly informal, e.g. short letters
Following written directions--cooking, crafts
Library skills (short unit at the end of the year)
Math
Begin Miquon Math Blue level (see their scope and sequence)
Typical grade 2 skills including number awareness, skip counting, understanding of place value, addition & subtraction, fraction concepts, money, time, measurement, problem solving, greater than/less than, multiplication & division concepts
Games, rod activities, hundreds chart, real-life math situations, commercial & homemade board games
Math Munchers CD-Rom (good for geometric shapes)
History and Geography
Eh? to Zed--A Canadian ABeCeDarium
Choose one letter each week and find out more about the Canadian words on that page
Term 1: An Island Story chapters 22-32, Child's History of the World chp 45, 47-51 (original edition)
David Thompson activity book (and online supplements)--covers his life and explorations of the NorthWest; we learn something about the fur trade, mapmaking, and the Rocky Mountains. (I am still looking for some appropriate biographical material to go with this.)
Term 2: AIS 33-50, CHOW 52-54; Stories for Canada's Birthday by Audrey McKim; Kids' Book of the Far North (and library books about the Arctic) [Note: I had planned to buy this book and work right through it, but on second look I decided to borrow it from the library and use it only as a resource to get us started on a study of the Arctic; we will use more books from the library as well as the Internet and our own books.]
Term 3: AIS 51-61; CHOW 55-58; Stories for Canada's Birthday; Bagley's Marco Polo (To Far Cathay)
If we have time, we will also read the D’Aulaire biographies of Abraham Lincoln and Christopher Columbus; possibly also Diane Stanley’s Joan of Arc.
HOLIDAYS
A Pioneer Thanksgiving
A Pioneer Christmas (both by Barbara Greenwood; read during the appropriate seasons)
Festivals, Family and Food, by Diana Carey and Judy Large
Brother Sun, Sister Moon (biography of St. Francis)—read during Lent
Christmas books
Science and Nature
Handbook of Nature Study (including the HNS blog), and Natural Science Through the Seasons (Partridge)—as teacher resources only.
Books to read together include:
Through the Year (Frasier et al), a simply-written science reader that is referenced in Partridge's book
Among the Forest People (Pierson)
Among the Night People (Pierson)
Nightprowlers
Linnea’s Almanac
Linnea’s Windowsill Garden (if available)
Exploring Nature Around the Year: Winter
LITERATURE
A Wonder Book and Tanglewood Tales, by Nathaniel Hawthorne (Greek myths); Pilgrim's Progress; poems from Come Hither, an anthology edited by Walter de la Mare
Term 1-- Understood Betsy; selected poems by James Whitcomb Riley; St. George and the Dragon; Hiawatha’s Childhood (picture book with stanzas from Longfellow’s poem)
Term 2-- Wind in the Willows
Term 3-- Robin Hood
TEATIME READING
This is something new we will be trying; planned readings include The Old Nurse’s Stocking Basket and Italian Peepshow, both by Eleanor Farjeon; The Door in the Wall, by Marguerite De Angeli; and poems of Walter de la Mare and Christina Rossetti.
Extra Reading (Bedtime stories, independent reading)
Heidi
Andersen’s fairy tales
Five Children and It
Farmer Boy, The Long Winter (Laura Ingalls Wilder)
Series books: Miss Bianca, Paddington, Oz books, All-of-a-Kind Family
The Story of Dr. Dolittle
holiday books
Along Came a Dog
Mr. Popper’s Penguins
Abel’s Island
The Year at Maple Hill Farm
The Tough Winter (Robert Lawson; sequel to Rabbit Hill)
The Courage of Sarah Noble
The Buffalo and the Bell (Myra Scovel; story about India)
Owls in the Family
and other books from the library
ART AND MUSIC
Composers: Mark O’Connor, Igor Stravinsky; Franz Liszt; Antonin Dvořák (some of our own records and CD’s; some borrowed from the library)
Artists: Paul Kane, Cornelius Krieghoff, William Kurelek (the “3 K’s” of Canadian art); short unit on pop art; Caspar David Friedrich; six weeks on Giotto, and six weeks on Vincent Van Gogh (one of Crayons’ favourites)
(We will use picture books and other material on these composers and artists, e.g. Stravinsky by Mike Venezia; Katie and the Sunflowers; The Yellow House (about Van Gogh); The Glorious Impossible (about Giotto); A Boy Named Giotto; Kurelek’s books Prairie Boy’s Summer, Prairie Boy’s Winter, Lumberjack, and A Northern Nativity)
Drawing and painting activities
Musical instruments--maybe start some keyboard lessons
LIFE SKILLS
Crafts--Jumbo Book of Crafts; possible sewing club with some friends; make Christmas decorations; cooking, helping at home
Phys-ed type activities
FRENCH
Aux Yeux des Enfants (short scripts based mainly on family life and seasonal topics; we use this with homemade felt-board cutouts)
MEMORY WORK
(Bible work from Teaching Children (Lopez))
TERM 1: Ps. 23, Matt. 2:1-12, poems,
TERM 2: Ps. 117, Matt. 6: 9-13, poems
TERM 3: Ps. 121, Matt. 28:1-10, poems
Related Posts:
Crayons' Grade Two: Bible
Crayons' Grade Two: Social Studies
Crayons' Grade Two: Math
Crayons' Grade Two: Language Arts
As Little As Possible
Grade Two: The Very Last First Time?
Grade Two
Teacher’s Resources
How is My Second Grader Doing in School?—Jacobson & Raymer (language and math activities)
Teaching Children, by Diane Lopez
3-R’s booklets, by Ruth Beechick
Bible:
Term 1: 1 Samuel 1-20, Matthew 1-14, selected Psalms (also planning to buy Judy Rogers’ CD of Psalms set to music, Never Be Shaken)
Term 2: 1 Sam 21-31, 2 Sam 1-12, Matthew 15-28
Term 3: 2 Sam 13-24, 1 Kings 1 & 2; Mr. Pipes and the British Hymn Makers
(Other Bible memory work is listed under Memory Work)
Each term: lessons (1x/wk) from People of the Bible: Life and Customs.
2nd and 3rd terms: Luther’s Small Catechism, section on the Lord’s Prayer
Singing
Hymns, both our own favourites and new ones from Mr. Pipes
Folk songs: sources include Canada Is For Kids series of three CDs (borrowed from library); our own books of children’s songs and folk songs; songs from Festivals, Family and Food
(Planning to purchase colouring books to accompany the CDs)
Language Arts:
Specific grade 2 skills, taught as needed, using our own books and supplements (word puzzles, a couple of Gifted and Talented workbooks, magnetic words, Scrabble letters, children's dictionary)
Skills include:
Oral communication, including narration, telephone/manners (using Uncommon Courtesy for Kids)
Listening skills (demonstrated by oral or other responses)
Capitalization, some punctuation, plurals, complete sentences, contractions, prefixes/suffixes, alphabetizing to the second letter (using Gifted and Talented workbooks, Word Puzzles Gr 2/3 workbook, other activities)
Copywork, simple dictation (spelling words with specific patterns as well as calendar words and holiday words) (We found some basic Gr 1 and 2 word lists in Kathryn Stout’s Natural Speller)
Printing practice, using Canadian Handwriting workbooks
Memory work (see list for each term)
Reading silently and out loud, and being read to (see booklists)
Writing, mostly informal, e.g. short letters
Following written directions--cooking, crafts
Library skills (short unit at the end of the year)
Math
Begin Miquon Math Blue level (see their scope and sequence)
Typical grade 2 skills including number awareness, skip counting, understanding of place value, addition & subtraction, fraction concepts, money, time, measurement, problem solving, greater than/less than, multiplication & division concepts
Games, rod activities, hundreds chart, real-life math situations, commercial & homemade board games
Math Munchers CD-Rom (good for geometric shapes)
History and Geography
Eh? to Zed--A Canadian ABeCeDarium
Choose one letter each week and find out more about the Canadian words on that page
Term 1: An Island Story chapters 22-32, Child's History of the World chp 45, 47-51 (original edition)
David Thompson activity book (and online supplements)--covers his life and explorations of the NorthWest; we learn something about the fur trade, mapmaking, and the Rocky Mountains. (I am still looking for some appropriate biographical material to go with this.)
Term 2: AIS 33-50, CHOW 52-54; Stories for Canada's Birthday by Audrey McKim; Kids' Book of the Far North (and library books about the Arctic) [Note: I had planned to buy this book and work right through it, but on second look I decided to borrow it from the library and use it only as a resource to get us started on a study of the Arctic; we will use more books from the library as well as the Internet and our own books.]
Term 3: AIS 51-61; CHOW 55-58; Stories for Canada's Birthday; Bagley's Marco Polo (To Far Cathay)
If we have time, we will also read the D’Aulaire biographies of Abraham Lincoln and Christopher Columbus; possibly also Diane Stanley’s Joan of Arc.
HOLIDAYS
A Pioneer Thanksgiving
A Pioneer Christmas (both by Barbara Greenwood; read during the appropriate seasons)
Festivals, Family and Food, by Diana Carey and Judy Large
Brother Sun, Sister Moon (biography of St. Francis)—read during Lent
Christmas books
Science and Nature
Handbook of Nature Study (including the HNS blog), and Natural Science Through the Seasons (Partridge)—as teacher resources only.
Books to read together include:
Through the Year (Frasier et al), a simply-written science reader that is referenced in Partridge's book
Among the Forest People (Pierson)
Among the Night People (Pierson)
Nightprowlers
Linnea’s Almanac
Linnea’s Windowsill Garden (if available)
Exploring Nature Around the Year: Winter
LITERATURE
A Wonder Book and Tanglewood Tales, by Nathaniel Hawthorne (Greek myths); Pilgrim's Progress; poems from Come Hither, an anthology edited by Walter de la Mare
Term 1-- Understood Betsy; selected poems by James Whitcomb Riley; St. George and the Dragon; Hiawatha’s Childhood (picture book with stanzas from Longfellow’s poem)
Term 2-- Wind in the Willows
Term 3-- Robin Hood
TEATIME READING
This is something new we will be trying; planned readings include The Old Nurse’s Stocking Basket and Italian Peepshow, both by Eleanor Farjeon; The Door in the Wall, by Marguerite De Angeli; and poems of Walter de la Mare and Christina Rossetti.
Extra Reading (Bedtime stories, independent reading)
Heidi
Andersen’s fairy tales
Five Children and It
Farmer Boy, The Long Winter (Laura Ingalls Wilder)
Series books: Miss Bianca, Paddington, Oz books, All-of-a-Kind Family
The Story of Dr. Dolittle
holiday books
Along Came a Dog
Mr. Popper’s Penguins
Abel’s Island
The Year at Maple Hill Farm
The Tough Winter (Robert Lawson; sequel to Rabbit Hill)
The Courage of Sarah Noble
The Buffalo and the Bell (Myra Scovel; story about India)
Owls in the Family
and other books from the library
ART AND MUSIC
Composers: Mark O’Connor, Igor Stravinsky; Franz Liszt; Antonin Dvořák (some of our own records and CD’s; some borrowed from the library)
Artists: Paul Kane, Cornelius Krieghoff, William Kurelek (the “3 K’s” of Canadian art); short unit on pop art; Caspar David Friedrich; six weeks on Giotto, and six weeks on Vincent Van Gogh (one of Crayons’ favourites)
(We will use picture books and other material on these composers and artists, e.g. Stravinsky by Mike Venezia; Katie and the Sunflowers; The Yellow House (about Van Gogh); The Glorious Impossible (about Giotto); A Boy Named Giotto; Kurelek’s books Prairie Boy’s Summer, Prairie Boy’s Winter, Lumberjack, and A Northern Nativity)
Drawing and painting activities
Musical instruments--maybe start some keyboard lessons
LIFE SKILLS
Crafts--Jumbo Book of Crafts; possible sewing club with some friends; make Christmas decorations; cooking, helping at home
Phys-ed type activities
FRENCH
Aux Yeux des Enfants (short scripts based mainly on family life and seasonal topics; we use this with homemade felt-board cutouts)
MEMORY WORK
(Bible work from Teaching Children (Lopez))
TERM 1: Ps. 23, Matt. 2:1-12, poems,
TERM 2: Ps. 117, Matt. 6: 9-13, poems
TERM 3: Ps. 121, Matt. 28:1-10, poems
Related Posts:
Crayons' Grade Two: Bible
Crayons' Grade Two: Social Studies
Crayons' Grade Two: Math
Crayons' Grade Two: Language Arts
As Little As Possible
Grade Two: The Very Last First Time?
Thursday, June 19, 2008
September homeschool plans for Crayons
(Tentative plans!) [July 15, 2008: Updated version]
Crayons will be in Grade 2 this fall, and will be following a modified version of Ambleside Online's Year 2. Because she's been following along with many of Ponytails' readings over the last couple of years, I have had to substitute some books. She's also a very avid independent reader, so she may be able to handle some of these books on her own (although she likes to be read to as well).
Bible:
Term 1: 1 Samuel, Matthew, selected Psalms & Proverbs; Judy Rogers CDs
Term 2: 2nd Samuel, continue Matthew, read from Acts
Term 3: Life of Solomon; continue Book of Acts
Language Arts:
Specific grade 2 skills from Gentle Language (an outline by Karen Glass), Teaching Children (Diane Lopez) and Ruth Beechick's 3-R's booklets, taught as needed, using our own books and supplements (word puzzles, a couple of Gifted and Talented workbooks, magnetic words, Scrabble letters, children's dictionary)
Skills include:
Oral communication, including narration, telephone/manners
Listening skills (demonstrated by oral or other responses)
Capitalization, some punctuation, plurals, complete sentences, contractions, prefixes/suffixes, alphabetizing to the second letter
Copywork, simple dictation (spelling words with specific patterns as well as calendar words and holiday words)
Printing practice, using Canadian Handwriting workbooks
Memory work (see list for each term)
Reading silently and out loud, and being read to (see booklists)
Writing, mostly informal, e.g. short letters
Following written directions--cooking, crafts
Math
Begin Miquon Math Blue level (see their scope and sequence)
Typical grade 2 skills including number awareness, skip counting, understanding of place value, addition & subtraction, fraction concepts, money, time, measurement, problem solving, greater than/less than, multiplication & division concepts
Games, rod activities, hundreds chart, real-life math situations, commercial & homemade board games
History and Geography
Eh? to Zed--A Canadian ABeCeDarium
Choose one letter each week and find out more about the Canadian words on that page
Term 1: An Island Story chapters 22-32, Child's History of the World chp 45, 47-51
David Thompson activity book (and online supplements)--covers his life and explorations of the NorthWest; we learn something about the fur trade, mapmaking, and the Rocky Mountains.
Term 2: AIS 33-50, CHOW 52-54, Stories for Canada's Birthday, Kids' Book of the Far North (and library books about the Arctic)
Term 3: AIS 51-61; CHOW 55-58; Stories for Canada's Birthday; Bagley's Marco Polo (To Far Cathay)
HOLIDAYS
A Pioneer Thanksgiving
A Pioneer Christmas (both by Barbara Greenwood; read during the appropriate seasons)
"Journey to a First Canadian Christmas" (story from Stories for Canada's Birthday)
Festivals, Family and Food, by Diana Carey and Judy Large (good way to learn about British holidays like Whitsun and Candlemas)
Biography
Term 1--undecided, may skip
Term 2--Brother Sun, Sister Moon (St. Francis of Assisi)
Term 3--Mr. Pipes & the British Hymn Makers (I like this for Year 2 because it ties in with Pilgrim's Progress)
Science and Nature
Topics from Handbook of Nature Study (including the HNS blog), Natural Science Through the Seasons (Partridge), and Through the Year (Frasier et al), a simply-written science reader that is referenced in Partridge's book
Possible books by term:
Term 1: Flower Fairies of the Autumn; Among the Night People (Pierson)
Term 2: Continue the Pierson series of nature books; add Linnea's Almanac
Term 3: Pagoo; Linnea's Windowsill Garden
LITERATURE
Shakespeare stories, Pilgrim's Progress
Term 1--Poems of Walter de la Mare; Understood Betsy; extra reading (see AO lists)
Term 2--Poems of Eugene Field and James Whitcomb Riley; Wind in the Willows
Term 3--Poems of Christina Rossetti; Robin Hood
ART AND MUSIC
Artist and composer: More or less follow the Ambleside Online rotation. Possibly study Andy Warhol in Term 1 since a local museum will be hosting a Warhol exhibit starting in January. Possibly do Giotto in Term 2 (we did him a few years ago, but Crayons doesn't remember)
Drawing and painting activities
Singing--folk songs, hymns, Canadian songs
Musical instruments--maybe start some keyboard lessons
LIFE SKILLS
Crafts--Jumbo Book of Crafts; possible sewing club with some friends; make Christmas decorations; cooking, helping at home; add knitting frame/corking and cat's cradles in third term
Phys-ed type activities
FRENCH
Aux Yeux des Enfants, which we usually do in Grade 1 but didn't get to this past year
MEMORY WORK
(Bible work was taken from Teaching Children (Lopez))
TERM 1: Ps. 23, Matt. 2:1-12, poems, geography songs, names of Bible books
TERM 2: Ps. 117, Matt. 6: 9-13, Lutheran catechism, etc.
TERM 3: Ps. 121, Matt. 28:1-10, catechism, etc.
(I should note here that our plans for Ponytails are more tentative at this point, so I won't be posting them for awhile.)
Crayons will be in Grade 2 this fall, and will be following a modified version of Ambleside Online's Year 2. Because she's been following along with many of Ponytails' readings over the last couple of years, I have had to substitute some books. She's also a very avid independent reader, so she may be able to handle some of these books on her own (although she likes to be read to as well).
Bible:
Term 1: 1 Samuel, Matthew, selected Psalms & Proverbs; Judy Rogers CDs
Term 2: 2nd Samuel, continue Matthew, read from Acts
Term 3: Life of Solomon; continue Book of Acts
Language Arts:
Specific grade 2 skills from Gentle Language (an outline by Karen Glass), Teaching Children (Diane Lopez) and Ruth Beechick's 3-R's booklets, taught as needed, using our own books and supplements (word puzzles, a couple of Gifted and Talented workbooks, magnetic words, Scrabble letters, children's dictionary)
Skills include:
Oral communication, including narration, telephone/manners
Listening skills (demonstrated by oral or other responses)
Capitalization, some punctuation, plurals, complete sentences, contractions, prefixes/suffixes, alphabetizing to the second letter
Copywork, simple dictation (spelling words with specific patterns as well as calendar words and holiday words)
Printing practice, using Canadian Handwriting workbooks
Memory work (see list for each term)
Reading silently and out loud, and being read to (see booklists)
Writing, mostly informal, e.g. short letters
Following written directions--cooking, crafts
Math
Begin Miquon Math Blue level (see their scope and sequence)
Typical grade 2 skills including number awareness, skip counting, understanding of place value, addition & subtraction, fraction concepts, money, time, measurement, problem solving, greater than/less than, multiplication & division concepts
Games, rod activities, hundreds chart, real-life math situations, commercial & homemade board games
History and Geography
Eh? to Zed--A Canadian ABeCeDarium
Choose one letter each week and find out more about the Canadian words on that page
Term 1: An Island Story chapters 22-32, Child's History of the World chp 45, 47-51
David Thompson activity book (and online supplements)--covers his life and explorations of the NorthWest; we learn something about the fur trade, mapmaking, and the Rocky Mountains.
Term 2: AIS 33-50, CHOW 52-54, Stories for Canada's Birthday, Kids' Book of the Far North (and library books about the Arctic)
Term 3: AIS 51-61; CHOW 55-58; Stories for Canada's Birthday; Bagley's Marco Polo (To Far Cathay)
HOLIDAYS
A Pioneer Thanksgiving
A Pioneer Christmas (both by Barbara Greenwood; read during the appropriate seasons)
"Journey to a First Canadian Christmas" (story from Stories for Canada's Birthday)
Festivals, Family and Food, by Diana Carey and Judy Large (good way to learn about British holidays like Whitsun and Candlemas)
Biography
Term 1--undecided, may skip
Term 2--Brother Sun, Sister Moon (St. Francis of Assisi)
Term 3--Mr. Pipes & the British Hymn Makers (I like this for Year 2 because it ties in with Pilgrim's Progress)
Science and Nature
Topics from Handbook of Nature Study (including the HNS blog), Natural Science Through the Seasons (Partridge), and Through the Year (Frasier et al), a simply-written science reader that is referenced in Partridge's book
Possible books by term:
Term 1: Flower Fairies of the Autumn; Among the Night People (Pierson)
Term 2: Continue the Pierson series of nature books; add Linnea's Almanac
Term 3: Pagoo; Linnea's Windowsill Garden
LITERATURE
Shakespeare stories, Pilgrim's Progress
Term 1--Poems of Walter de la Mare; Understood Betsy; extra reading (see AO lists)
Term 2--Poems of Eugene Field and James Whitcomb Riley; Wind in the Willows
Term 3--Poems of Christina Rossetti; Robin Hood
ART AND MUSIC
Artist and composer: More or less follow the Ambleside Online rotation. Possibly study Andy Warhol in Term 1 since a local museum will be hosting a Warhol exhibit starting in January. Possibly do Giotto in Term 2 (we did him a few years ago, but Crayons doesn't remember)
Drawing and painting activities
Singing--folk songs, hymns, Canadian songs
Musical instruments--maybe start some keyboard lessons
LIFE SKILLS
Crafts--Jumbo Book of Crafts; possible sewing club with some friends; make Christmas decorations; cooking, helping at home; add knitting frame/corking and cat's cradles in third term
Phys-ed type activities
FRENCH
Aux Yeux des Enfants, which we usually do in Grade 1 but didn't get to this past year
MEMORY WORK
(Bible work was taken from Teaching Children (Lopez))
TERM 1: Ps. 23, Matt. 2:1-12, poems, geography songs, names of Bible books
TERM 2: Ps. 117, Matt. 6: 9-13, Lutheran catechism, etc.
TERM 3: Ps. 121, Matt. 28:1-10, catechism, etc.
(I should note here that our plans for Ponytails are more tentative at this point, so I won't be posting them for awhile.)
Monday, April 30, 2007
Of plastic counting bears and cubes
Mr. Person at TextSavvy posted some interesting findings (Hands-On, Brains-Off) debunking a current sacred cow in elementary mathematics teaching: hands-on activities and manipulatives. He includes a quote from Education Week:
Charlotte Mason wasn't always in favour of commercial manipulatives and models, either. (You can see a set of math manipulatives (by Adolf Sonnenschein) that she described here.) [2012 update: that link has changed, but there is a similar photo here.] She didn't want pre-made models getting in the way of students doing their own thinking. She didn't want them getting too dependent on rods. (She also didn't like "drawing in chequers.")
However, she did make use of both beans and dominoes as teaching aids. (Dominoes were used as a sort of addition flashcard: young students were to learn all the combinations in the set.) Longtime CM user Lynn Hocraffer wrote an interesting article, "Seashell Math," about using a bag of shells to help her son who couldn't "see" what the numbers in arithmetic were about. She says, "We did use all the sections and activities [of the math program] with my son, but I was dense and didn't do the manipulatives until the end. Dumb me! I knew my son was a kinesthetic learner, but because he recited so well I took a while to realize he didn't know what he was saying! He could "Reason", that is he could follow in order, but it had no meaning, no application."
I've talked to a lot of homeschoolers who have gone one direction or another in choosing math materials, sometimes after trying several approaches. I know people who have had enough with the "math toys" and are now back to using the Victorian-era Ray's Arithmetic. I know other homeschoolers who never really felt they "got" math themselves until their kids started using Math-U-See with its colourful blocks. The division might be right there, between those teaching parents who are already comfortable with math and who do better without all the "blocks and whistles," and those who can communicate the same concepts without gimmicks. I believe that there are also children who learn just fine without having to see or handle manipulatives; and there are others who need that visual or kinesthetic boost to make sense of it. But is there a place for those of us who aren't math majors but still feel like we have a pretty good grasp of what needs to be taught and just prefer to teach it with some kind of manipulatives? (And for how long?--are manipulatives to be encouraged in the primary grades but not in the upper years?--or are they to be eschewed all the way along?)
I remember being in the first grade (in a rows-of-desks classroom) and not being able to figure out why 1 - 0 = 1. I was supposed to be one of the smart kids in the class; I went to the second grade room to do reading every morning. But those arithmetic drill sheets with their 0's really threw me. Nobody ever gave me a clear illustration or explanation of why 1 - 0 didn't equal 0.
The rest of my elementary math education (at a push-the-desks-into-groups school) was so forgettable that I've pretty much forgotten what we did do. I remember our spelling series perfectly (waste of time--I already knew how to spell), but I don't even think we used math textbooks. This was during the experimental '70's, in what was supposed to be the most up-to-date local school (the one with all the learning centres and extended classrooms). But we were still being taught by teachers who had learned more traditionally themselves; so what I do remember is a rather schizophrenic mishmash of times-table drills, problems on the blackboard, reams of purple "ditto" pages, and occasional forays into manipulatives. "Here are the attribute blocks. Follow the directions on the Learning Cards." We looked forward to those manipulative occasions not because we were learning anything but because--obviously--they were a chance to play in class.
And this--I think--is where Mr Person and I are getting onto common ground. According to the article he quotes, teachers like manipulatives for various reasons, one of which is that giving kids something to "play with" may keep them out of trouble longer. Educational suppliers like manipulatives that are required to use a particular curriculum--they really like them! Kindergarten suppliers had that one figured out over a hundred years ago. I don't think that homeschoolers are quite as bombarded by silliness as classroom teachers are--most of us couldn't afford all that stuff even if we wanted it. However, we too can be bewitched by all the neat stuff on the conference tables. It's colourful and it's fun. But the big question is, always--does doing or using whatever it is help you learn the subject better? (I go back to that question constantly; and it's expertly expounded in Mary Pride's book Schoolproof.)
In our own homeschool, with our children, using rods the way we use them, the answer is yes: I've posted about that here, here and here. I've written elsewhere about our oldest, who never quite saw the sense in Cuisenaire rods. She was more of a "just show me how to do it" math learner; however, I used them with her during the first years of school anyway, and had her complete the book Spatial Problem Solving with Cuisenaire Rods later on. Our middle one ("Ponytails"), on the other hand, relates to relationships, and the point of Cuisenaire rods is relationships. When I asked Ponytails whether she thought that the rods had actually helped her learn math better, she enthusiastically agreed and started talking about how the "one rod" could be a "ten" and vice versa. Of course she did use a variety of manipulatives and methods throughout the primary years, including a hundred chart and an abacus--so I suppose that was one way we avoided having her depend too much on one particular aid.
Now that she is in the fourth grade and done with Miquon Math, I notice that we hardly use the rods in day-to-day work; obviously they're not helpful in the nitty-gritty work of learning to multiply multi-digit numbers on paper! And although some of this heavier-on-the-pencil math isn't easy for her, I've never noticed my rod-user having any particular problem transferring her knowledge onto paper, either.
Is there a difference, then, between what we're doing and what Mr. Person is debunking, and what Charlotte Mason didn't like? What our teachers were doing when they occasionally hauled out the rods or the attribute blocks, or brought in the new-fangled Betamax to show us a video with a catchy song about finding area? I think there is, because in my homeschool classroom manipulatives aren't just busywork; I have no reason to want to make my lesson take longer than it has to! (The weather's good and the kids want to go outside...) If the rods or other manipulatives can illustrate a point or a relationship clearly (and more conveniently than having to count out multiple piles of beans), then we will use them. And although our preschoolers (Crayons especially) made an art form out of Cuisenaire floor constructions, that's not what we do during math time.
While I can see exactly what Mr. Person is getting at (and why one of his commenters referred to "Happy Meals" and "McMath"), I'm still going to hold on to our rods.
"Researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.And why would I, the Cuisenaire Rod enthusiast, find anything to agree with in this?
"Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods."
Charlotte Mason wasn't always in favour of commercial manipulatives and models, either. (You can see a set of math manipulatives (by Adolf Sonnenschein) that she described here.) [2012 update: that link has changed, but there is a similar photo here.] She didn't want pre-made models getting in the way of students doing their own thinking. She didn't want them getting too dependent on rods. (She also didn't like "drawing in chequers.")
However, she did make use of both beans and dominoes as teaching aids. (Dominoes were used as a sort of addition flashcard: young students were to learn all the combinations in the set.) Longtime CM user Lynn Hocraffer wrote an interesting article, "Seashell Math," about using a bag of shells to help her son who couldn't "see" what the numbers in arithmetic were about. She says, "We did use all the sections and activities [of the math program] with my son, but I was dense and didn't do the manipulatives until the end. Dumb me! I knew my son was a kinesthetic learner, but because he recited so well I took a while to realize he didn't know what he was saying! He could "Reason", that is he could follow in order, but it had no meaning, no application."
I've talked to a lot of homeschoolers who have gone one direction or another in choosing math materials, sometimes after trying several approaches. I know people who have had enough with the "math toys" and are now back to using the Victorian-era Ray's Arithmetic. I know other homeschoolers who never really felt they "got" math themselves until their kids started using Math-U-See with its colourful blocks. The division might be right there, between those teaching parents who are already comfortable with math and who do better without all the "blocks and whistles," and those who can communicate the same concepts without gimmicks. I believe that there are also children who learn just fine without having to see or handle manipulatives; and there are others who need that visual or kinesthetic boost to make sense of it. But is there a place for those of us who aren't math majors but still feel like we have a pretty good grasp of what needs to be taught and just prefer to teach it with some kind of manipulatives? (And for how long?--are manipulatives to be encouraged in the primary grades but not in the upper years?--or are they to be eschewed all the way along?)
I remember being in the first grade (in a rows-of-desks classroom) and not being able to figure out why 1 - 0 = 1. I was supposed to be one of the smart kids in the class; I went to the second grade room to do reading every morning. But those arithmetic drill sheets with their 0's really threw me. Nobody ever gave me a clear illustration or explanation of why 1 - 0 didn't equal 0.
The rest of my elementary math education (at a push-the-desks-into-groups school) was so forgettable that I've pretty much forgotten what we did do. I remember our spelling series perfectly (waste of time--I already knew how to spell), but I don't even think we used math textbooks. This was during the experimental '70's, in what was supposed to be the most up-to-date local school (the one with all the learning centres and extended classrooms). But we were still being taught by teachers who had learned more traditionally themselves; so what I do remember is a rather schizophrenic mishmash of times-table drills, problems on the blackboard, reams of purple "ditto" pages, and occasional forays into manipulatives. "Here are the attribute blocks. Follow the directions on the Learning Cards." We looked forward to those manipulative occasions not because we were learning anything but because--obviously--they were a chance to play in class.
And this--I think--is where Mr Person and I are getting onto common ground. According to the article he quotes, teachers like manipulatives for various reasons, one of which is that giving kids something to "play with" may keep them out of trouble longer. Educational suppliers like manipulatives that are required to use a particular curriculum--they really like them! Kindergarten suppliers had that one figured out over a hundred years ago. I don't think that homeschoolers are quite as bombarded by silliness as classroom teachers are--most of us couldn't afford all that stuff even if we wanted it. However, we too can be bewitched by all the neat stuff on the conference tables. It's colourful and it's fun. But the big question is, always--does doing or using whatever it is help you learn the subject better? (I go back to that question constantly; and it's expertly expounded in Mary Pride's book Schoolproof.)
In our own homeschool, with our children, using rods the way we use them, the answer is yes: I've posted about that here, here and here. I've written elsewhere about our oldest, who never quite saw the sense in Cuisenaire rods. She was more of a "just show me how to do it" math learner; however, I used them with her during the first years of school anyway, and had her complete the book Spatial Problem Solving with Cuisenaire Rods later on. Our middle one ("Ponytails"), on the other hand, relates to relationships, and the point of Cuisenaire rods is relationships. When I asked Ponytails whether she thought that the rods had actually helped her learn math better, she enthusiastically agreed and started talking about how the "one rod" could be a "ten" and vice versa. Of course she did use a variety of manipulatives and methods throughout the primary years, including a hundred chart and an abacus--so I suppose that was one way we avoided having her depend too much on one particular aid.
Now that she is in the fourth grade and done with Miquon Math, I notice that we hardly use the rods in day-to-day work; obviously they're not helpful in the nitty-gritty work of learning to multiply multi-digit numbers on paper! And although some of this heavier-on-the-pencil math isn't easy for her, I've never noticed my rod-user having any particular problem transferring her knowledge onto paper, either.
Is there a difference, then, between what we're doing and what Mr. Person is debunking, and what Charlotte Mason didn't like? What our teachers were doing when they occasionally hauled out the rods or the attribute blocks, or brought in the new-fangled Betamax to show us a video with a catchy song about finding area? I think there is, because in my homeschool classroom manipulatives aren't just busywork; I have no reason to want to make my lesson take longer than it has to! (The weather's good and the kids want to go outside...) If the rods or other manipulatives can illustrate a point or a relationship clearly (and more conveniently than having to count out multiple piles of beans), then we will use them. And although our preschoolers (Crayons especially) made an art form out of Cuisenaire floor constructions, that's not what we do during math time.
While I can see exactly what Mr. Person is getting at (and why one of his commenters referred to "Happy Meals" and "McMath"), I'm still going to hold on to our rods.
Friday, March 16, 2007
Math and More (Part Three)
(Part One, Part Two, Part Four, Part Five)
Since I was talking about Ruth Beechick’s booklet An Easy Start in Arithmetic, I should have mentioned as well that its last page is “Ways to Use the Hundred Chart,” and there is a small hundred chart on the back cover (in addition to the poster-size one that’s included with the booklets). Hundred charts are a great way to enrich a math program without doing worksheets; in fact, we did a whole year of kindergarten math for Ponytails without any written work at all. There’s something about those ten rows of ten that not only helps with counting, adding and subtracting, but reinforces place value ideas as well, especially when kids learn the trick of adding or subtracting tens (go up or down a row) . When they’ve learned that, then adding fourteen becomes “add ten” (go down a row) and then “four more” (four spaces to the right), without any fuss about “carrying the one.” And the nice thing about hundred charts is that they can be useful in any size: tiny (like the one on the booklet), bigger (a handwritten chart or a page-sized printout from the Web—Donna Young’s site has printable hundred charts), or really big. We have a poster-size hundred chart from the teacher’s store and also one that we made on poster board with detachable (Velcro) number disks.
Bible Lessons
For many people, Bible is as much a part of a basic curriculum as reading and math. I could assume that most people have a Bible around and that they could make up a simple program of reading Bible stories to their children, as parents have done for hundreds of years. However, since I was trying to stick to the books in the shopping bag, I was limited to a paperback copy of “The Great News: The New Testament, New International Version.” This actually works out fine, although for a third-grader it might have been nice to have slightly larger print. The bonus about this edition of the New Testament is that it comes with a 71-lesson “Reading Plan to Get You Started” in the front, which works out to just about two lessons per week (maybe reading on one day and reviewing or notebooking on the next). This goes beyond the simplest stories; it starts with “Who is Jesus?” and goes on to “What is Christianity All About,” “What is Real Faith,” “How Does God Want His People to Treat Others,” and “What Stories Did Jesus Tell?” I like this approach because it’s a bit of a change from just reading straight through one of the Gospels, and it lends itself to keeping a notebook as well. I might even use this myself next year with Ponytails.
History, Geography and/or Social Studies
The lack of history resources in the bag might make this the weakest part of the curriculum, although that would be pretty easy to fix using online or library books. However—sticking to the bag and the teaching resources—I decided to try something different. Teaching Children includes both a “regular” Social Studies outline and an “alternate” one for grades three through six, written by Susan Schaeffer Macaulay. In the alternate outline, third graders spend a lot of the year exploring what’s around them; not in a dumbed-down “Mr. Neighbourhood Policeman” approach, but in an inter-disciplinary program that combines local and comparative geography, local history, natural history, and what you might call cultural anthropology and sociology. In other words, they find out where they live, who lives there, and what’s around them. John Holt once suggested something like this too; someone asked him how he would tutor a boy who lived in an unusual natural environment, and he pointed out that it would be foolish not to take full advantage of every bit of local exploring, beach-digging and museum-lurking that they could squeeze in—that this would benefit the boy in ways that his book lessons never could.
It’s also suggested that the third graders “adopt” a missionary family or project and learn more about the country involved, correspond with the family and so on.
Of course this doesn’t come all ready-packaged for you, and in some ways it may sound like the vaguest part of the curriculum—go out together, do field trips, find out the names of the roads and the trees and the early settlers, and keep a scrapbook. Some of us might think we could cover our entire local area in one or two trips; others of us would immediately worry about how little we ourselves know about local birds, pond life, railroads and so on! As I said, you might not like this idea at all and then you’d have to fill in with other history and geography books. However, I do think it is suitable for children of about third-grade level who aren’t too interested yet in names-and-dates history (plus it would make a great ongoing activity for Friday afternoons). Can I tell you a funny story about this kind of exploring approach to social studies? A long time ago (I think The Apprentice was in the second grade), I read the book Who Killed Canadian History? by Jack Granatstein. Since the book dealt mostly with the weaknesses in secondary- and post-secondary-level education, I e-mailed the author to ask if he had any recommendations for teaching the elementary grades. Mr. Granatstein graciously wrote back and said he didn’t know a lot about teaching second graders but that his recommendation would be “Forts. Visit forts.” So there you have it.
As for the missionary/other countries side of the curriculum: I thought of three possibilities for this. One came out of the Child of China book that is included for literature; I found a unit study of Ancient China, by Judy Wilcox, written for homeschoolers, that’s meant to take twelve weeks and which sounds like a great addition to the last part of the year’s curriculum. Of course you have to buy that first (grin). Or you could buy the e-text A Child’s Geography, Volume Two: Exploring the Holy Land, which covers several Middle Eastern countries.
Or you could just go with whatever resources you know best: missionaries or organizations you know yourself or that your church supports.
How does that sound?
In the next post I'll talk about science and finish off with some extras.
Since I was talking about Ruth Beechick’s booklet An Easy Start in Arithmetic, I should have mentioned as well that its last page is “Ways to Use the Hundred Chart,” and there is a small hundred chart on the back cover (in addition to the poster-size one that’s included with the booklets). Hundred charts are a great way to enrich a math program without doing worksheets; in fact, we did a whole year of kindergarten math for Ponytails without any written work at all. There’s something about those ten rows of ten that not only helps with counting, adding and subtracting, but reinforces place value ideas as well, especially when kids learn the trick of adding or subtracting tens (go up or down a row) . When they’ve learned that, then adding fourteen becomes “add ten” (go down a row) and then “four more” (four spaces to the right), without any fuss about “carrying the one.” And the nice thing about hundred charts is that they can be useful in any size: tiny (like the one on the booklet), bigger (a handwritten chart or a page-sized printout from the Web—Donna Young’s site has printable hundred charts), or really big. We have a poster-size hundred chart from the teacher’s store and also one that we made on poster board with detachable (Velcro) number disks.
Bible Lessons
For many people, Bible is as much a part of a basic curriculum as reading and math. I could assume that most people have a Bible around and that they could make up a simple program of reading Bible stories to their children, as parents have done for hundreds of years. However, since I was trying to stick to the books in the shopping bag, I was limited to a paperback copy of “The Great News: The New Testament, New International Version.” This actually works out fine, although for a third-grader it might have been nice to have slightly larger print. The bonus about this edition of the New Testament is that it comes with a 71-lesson “Reading Plan to Get You Started” in the front, which works out to just about two lessons per week (maybe reading on one day and reviewing or notebooking on the next). This goes beyond the simplest stories; it starts with “Who is Jesus?” and goes on to “What is Christianity All About,” “What is Real Faith,” “How Does God Want His People to Treat Others,” and “What Stories Did Jesus Tell?” I like this approach because it’s a bit of a change from just reading straight through one of the Gospels, and it lends itself to keeping a notebook as well. I might even use this myself next year with Ponytails.
History, Geography and/or Social Studies
The lack of history resources in the bag might make this the weakest part of the curriculum, although that would be pretty easy to fix using online or library books. However—sticking to the bag and the teaching resources—I decided to try something different. Teaching Children includes both a “regular” Social Studies outline and an “alternate” one for grades three through six, written by Susan Schaeffer Macaulay. In the alternate outline, third graders spend a lot of the year exploring what’s around them; not in a dumbed-down “Mr. Neighbourhood Policeman” approach, but in an inter-disciplinary program that combines local and comparative geography, local history, natural history, and what you might call cultural anthropology and sociology. In other words, they find out where they live, who lives there, and what’s around them. John Holt once suggested something like this too; someone asked him how he would tutor a boy who lived in an unusual natural environment, and he pointed out that it would be foolish not to take full advantage of every bit of local exploring, beach-digging and museum-lurking that they could squeeze in—that this would benefit the boy in ways that his book lessons never could.
It’s also suggested that the third graders “adopt” a missionary family or project and learn more about the country involved, correspond with the family and so on.
Of course this doesn’t come all ready-packaged for you, and in some ways it may sound like the vaguest part of the curriculum—go out together, do field trips, find out the names of the roads and the trees and the early settlers, and keep a scrapbook. Some of us might think we could cover our entire local area in one or two trips; others of us would immediately worry about how little we ourselves know about local birds, pond life, railroads and so on! As I said, you might not like this idea at all and then you’d have to fill in with other history and geography books. However, I do think it is suitable for children of about third-grade level who aren’t too interested yet in names-and-dates history (plus it would make a great ongoing activity for Friday afternoons). Can I tell you a funny story about this kind of exploring approach to social studies? A long time ago (I think The Apprentice was in the second grade), I read the book Who Killed Canadian History? by Jack Granatstein. Since the book dealt mostly with the weaknesses in secondary- and post-secondary-level education, I e-mailed the author to ask if he had any recommendations for teaching the elementary grades. Mr. Granatstein graciously wrote back and said he didn’t know a lot about teaching second graders but that his recommendation would be “Forts. Visit forts.” So there you have it.
As for the missionary/other countries side of the curriculum: I thought of three possibilities for this. One came out of the Child of China book that is included for literature; I found a unit study of Ancient China, by Judy Wilcox, written for homeschoolers, that’s meant to take twelve weeks and which sounds like a great addition to the last part of the year’s curriculum. Of course you have to buy that first (grin). Or you could buy the e-text A Child’s Geography, Volume Two: Exploring the Holy Land, which covers several Middle Eastern countries.
Or you could just go with whatever resources you know best: missionaries or organizations you know yourself or that your church supports.
How does that sound?
In the next post I'll talk about science and finish off with some extras.
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