Best education book I've read this month

In the too-short-to-be-a-book-review department: the best education book I've read this month is The End of Ignorance, by John Mighton. It's about math, but it's also about brains and learning and schools and human potential.   Definitely recommended.

Mama Squirrel's Reading List: Math and more

In my library pile:

Kepler's Conjecture: How some of the greatest minds in history helped solve one of the oldest math problems in the world, by George G. Szpiro

Painless Algebra, by Lynette Long, Ph.D. (a Barron's guide I'm looking at maybe for Dollygirl in the fall)

The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom, by James W. Stigler and James Hiebert

Mathematics: The Science of Patterns, by Keith Devlin.  This looks like a very cool book: it's not a how-to, it's more of a why.

Dollygirl's School Plans: Term 3, Week 8

"We are aware that our own discursive talk is usually a waste of time and a strain on the scholars' attention, so we (of the P.N.E.U.) confine ourselves to affording two things,––knowledge, and a keen sympathy in the interest roused by that knowledge. It is our part to see that every child knows and can tell, whether by way of oral narrative or written essay."  ~~ Charlotte Mason, Philosophy of Education

Monday:
(Daily opening time: hymns, poetry, current events)
Old Testament: Book of Numbers
Saxon Algebra 1/2
History of England: Who was Henry III?
French, using Complete French Smart 7 workbook
Apologia General Science:  how do "calories" relate to the ideas about food consumption and combustion we have been discussing?
Shakespeare's King John
Grammar: a review passage drawn from literature (but definitely easier than the last one we did)
Ourselves Book II: how reading philosophy helps instruct the conscience

"Perhaps the gravest defect in school curricula is that they fail to give a comprehensive, intelligent and interesting introduction to history."  ~~ Philosophy of Education

Tuesday:
New Testament: Gospel of Mark
Saxon Algebra 1/2
Money Matters for Teens
French
Plutarch's Life of Cicero (we are doing two Plutarch lessons this week)
Map quizzes with Seterra Onlinne
The White Mountains
Handwriting with Fix It...Write
History: work on Century Chart

"...the desire for knowledge for its own sake, on the other hand, finds satisfaction in knowledge itself."  ~~ Philosophy of Education

Wednesday:
Old Testament: Book of Numbers
Money Matters for Teens
French
History of Music for Young People: lesson on Mendelssohn we missed last week
Apologia General Science: about metabolic rates in humans and animals
Ivanhoe
A little more grammar practice
University of Waterloo Gauss mathematics competition

"These young students have the powers of perfect recollection and just application because they have read with attention and concentration and have in every case reproduced what they have read in narration, or, the gist of some portion of it, in writing." ~~ Philosophy of Education 

Thursday:
New Testament: Gospel of Mark
The Spring of the Year: chapter on phoebes that was missed last week
French
History of England: about Henry III and parliaments
Picture Talk: Jan Vermeer
Key to Geometry
The White Mountains
Handwriting

"Children recognise with incipient weariness the doctored tale as soon as it is begun to be told, but the human story with its evil and its good never flags in interest." ~~ Philosophy of Education

Friday: 
Basic Bible Studies by Francis Schaeffer
Balance Benders Level 3
Apologia General Science: finishing off Module 12, about life and energy
Plutarch's Life of Cicero
Transcription (copywork)
Ivanhoe
Key to Geometry
French History: reading more from De Joinville's Memoirs of the Crusades
Notes in the Book of Centuries

Math, how it's going, and Heather's helpful math post

Heather at To Sow a Seed has a good homeschool math post this week.  That might not sound like enough to get you to click over there, but trust me, she has some important things to say both about math and about homeschooling.

Right now Dollygirl is doing a mixture of Saxon Algebra 1/2 (2nd edition), Key to Geometry, and Balance Benders; also Money Matters by Larry Burkett, but that is more consumer education and citizenship than math.

I know how you're supposed to do Saxon.  That's not how we're doing it.  That would be exactly how to make Dollygirl never want to do math again.  The closest comparison I can make is that I'm teaching it much like I did Miquon, minus the Cuisenaire rods.  There are topics that Dollygirl is already very good at; there are new things she needs practice with.  During a math lesson (which doesn't mean One Saxon Lesson), I try to go over something new or to expand on a concept we've been working on (right now it's rate problems and unit multipliers); we might do some sample problems together on that, or do a few other questions on more familiar topics, such as finding the lowest common multiple or changing improper fractions to mixed numbers.  She might do those orally (if they're that sort of question), might do them on scratch paper (the same idea as working at the blackboard).  I watch while she's working those out and offer a little direction if she needs it; we check the Solutions Manual and if everything lines up, we move on; if not, we go back to the point where she got off track.

Then I usually assign either a few word problems, or, depending on what's in the Saxon lesson, a certain number of the shorter-type questions such as "solve for x."  So it might take us a few days to get through one Saxon lesson, or we might stop partway through and move on, or we might do a whole problem set and then skip the next, or we might even go back to a very early lesson for some arithmetic review.

(I always took Ruth Beechick seriously when she said "teach the child, not the book.")

How I became a math teacher?

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.

Math Archives #1: Can they do enough math to know they're being cheated?

First posted April 2012; but this is a post-within-a-post, and part of it is from 2007.

I had planned to repost this 2007 post today (both the part about our own homeschool and the comparison with the third grade math class at the end of the post), and then someone sent me a link to a recent Macleans' Magazine article on the sorry situation in Canadian math teaching.  It reminded me even more of the educational Blerwm (see the old post) that continues to spew, particularly in the elementary schools.  If this situation doesn't make you furious for our children--that is, the children of this generation, even if we homeschoolers have taught our own offspring better--I don't know what would. And it's not just that they grow up cheated on math:  the same applies to standards in reading, writing, and other skills that, until recently, were considered within the normal scope of a child's education.

And what makes me even angrier for these children is that we non-experts, the home-teaching parents who may or may not have college-level math courses or education credentials (many homeschoolers do have advanced degrees), seem to be doing better than the current average at math education, almost without trying.  Some of it's the curriculum homeschoolers use--certain popular programs are known to be a level or two over traditional North American math goals, so kids using them would seem a bit ahead anyway. But even if we take math slow and simple, we have this crazy advantage over the current hands-tied school situation: most of us parents, especially those of us over a certain age, were taught with traditional math methods, and that's what we pass on to our kids.  Here's how you multiply fractions, here's how you divide them.  None of this messing with paper strips.  

It doesn't matter why we do it, though, so much as whether or not it works.  Can our kids add, subtract, multiply, divide?  Can they make change?  Can they figure out a percentage?  Do they just have a good sense of how numbers work?  Apparently the kids taught with the any-way-that-works-for-you method can't, and don't. When they get to high school, where math is still taught using more traditional methods, a lot of them flounder.

Are you laughing in disbelief at this point?  I'm more ready to spit.  Crayons has been suggesting that she might like to go to public school for grade six, just to try it out like Ponytails did.  Sorry: with this amount of un-teaching going on in Canadian schools, that would be my last choice for her for next year.  

Here's the relevant part from our 2007 post:

But on the other hand, there was an article today in the local paper about math teaching in public schools, that tipped things back towards thinking again that we must be doing all right.

"Recently [the grade 3 teacher] taught the children to count by fives, using Popsicle sticks. She had them sit in a circle and line up four Popsicle sticks in a row, with a coloured one laid diagonally across each pile.

"Then she asked how many Popsicle sticks there were. One student crawled into the middle of the circle and counted up the piles: "Five, 10, 15, 20, 25, 30, 35, 40 45 . . ." he said and paused at the final two sticks. "Forty-seven" he called.

"The class applauded him. 'Good job!' she praised, and then sent the children to sit down with worksheets where they again had to add the "bundles" of lines arranged five to a pile.

"Instead of having the children write down the correct totals, though, she had them choose the right answer from some numbers printed on the bottom of the sheet. They were to cut out the right number and glue it in the proper spot.

"The children were enjoying cutting and feeling the texture of the glue stick under their fingernails.

"'Children at this age are very visual and very kinesthetic,' she said. They learn by seeing and often need to move around while learning, even if it's just working with glue."

OK, I know it's still September, and maybe that was a review lesson--but cutting and pasting answers in grade 3? And Crayons (grade 1) has been doing that same kind of counting-by-fives-plus-whatever's-left. Without crawling on the floor, I might add. Or needing to get glue stick under her fingernails.

Countdown to School: Interview with The Apprentice

What are you studying at university?

I am in a research-based science program, with a specialization in math. This will effectively give me a double degree in science and math. This program has a strong focus on conducting research, reading and analyzing papers, presentation and scientific writing, and pedagogy (teaching).

Do you think of yourself as a "math person?"  Do you see the world mathematically?

I don't actually see myself as that much of a math person. Although I was one of the stronger math students at my high school, I've met an awful lot of people at university who are more talented in math than I am. Not that I don't have good math skills, but when you talk to these people about a problem, or see their proofs, you realize that it's more than just algorithms, it's more like learning a language. You can memorize vocabulary lists and learn grammar, but constructing speech itself is more of an art. I know how to order a beer in French, but I certainly can't write poetry.

What got you interested in pursuing post-secondary mathematics?

My short answer is that I wanted to do physics but I'm better at math, so I decided to go at it from the maths side. The first two years of my program, regardless of your specialization, have mandatory full-year math courses. The program tried hard to show how this math could be applied to other branches of science, but to many of my classmates it didn't even feel like part of science, just an annoying course to get through. As I've taken more and more math courses, I've found them linking together and applying to other disciplines. I've always found math useful and relevant. I feel like my degree is really just in research science, but when I was able to pick a focus I wanted to have a tool that would be useful in any of my scientific aspirations.

Would you like to be a math teacher?  Why or why not?

I would absolutely love to be a math teacher. I spent a lot of time in high school acting as a math tutor at the school's math help centre, and I also peer tutored a math class. After physics lessons I often found myself surrounded by students wanting me to run over the problems (I loved doing that but it meant I never got my own homework done...). At this point I feel like I may end up in something similar to teaching or perhaps administration. Ontario teaching jobs are very hard to come by no matter how good you are. While I'd prefer to teach on the undergraduate level anyway, this generally requires a PhD. I'm not sure at this point that I would want to pursue that much school. However, I am keeping learning and pedagogy in my sights, this year I am taking a child development course and hope to focus an individual project on the effectiveness of various mathematics teaching methods.

What were some of your early math experiences?  Do you think they were important to your later interest in math?

The earliest math experience I can remember is learning addition and subtraction on a homemade scale constructed from string and yogurt cups. ("This piece weighs five. The piece on the other side weighs two. What do we need to add to make them balance?") Later achievements involved hopping back and forth on a number line, and learning fractions through cooking. I think these experiences were really important, especially watching my parents use things like fractions in everyday life. Learning things like the fact that you can fill the half-teaspoon measure halfway full to get a quarter-teaspoon without dirtying two spoons (or with my dad, if you pour out half a bottle of engine oil, that's 500 mL) have always reminded me how useful math is in real life. Learning that numbers and electricity could be connected in small electronics led to an interest in physics.

Was there a time you did not feel "good at math," or that you disliked it?

There are always times that I don't feel good at math. The problem is that until you get really really specialized, everything has to stay connected. It may not always be clear that eventually two concepts will merge together, so if something seems kind of pointless at the time and you just learn the bare minimum, it can come back to haunt you. I'm sure I disliked it at various times in elementary school. What I've found in tutoring others is if they're really not getting it, you need to take a step back and try a completely different direction or explanation. A lot of students I found just needed a reason for what they were doing to understand (e.g. trig gives you ratios between angles and sides of a triangle, so if you don't know one of them, you can find it). This really frustrated me at one point when I was tutoring students in the "lower" academic stream. Those students might have been there because English wasn't their first language, because they had a learning disability, or sometimes even because they just didn't like school. But none of them were stupid. I had a supervising teacher actually stop me mid-explanation and tell me that I couldn't give a student the logic behind a concept, I should just teach them the algorithm for doing it, because they weren't capable of understanding. No wonder they didn't think they were good at math.

What parts (types) of mathematics do you enjoy studying most, and which ones not so much?

I really like working with mathematical computer programs and understanding how they work. I did a research project on this last year looking at molecular electronics using several programs. Really the part of math I don't like is certain testable things that you only have to know for a course, but later you would just look up or use the computer to solve. I understand that you need to know the process, and I'm interested in knowing the process, but I tend to make computational errors by hand which can slow me down or give the wrong result.

If you were teaching your twelve-year-old self, what would you do do make math a good subject?

If I was teaching my younger self, I'd have lots of practice and worksheets because practice makes things much easier. But I'd also have experiments where you can collect real numerical data and apply the concepts to that, so you can see how they're used and they become relevant.

Any other thoughts?

One of the best math resources I've come across is Khan Academy.  I'd highly recommend it for high school level math (it has other grades too including some university). The conceptual teaching is excellent and there are lots of practice questions too. You can earn points for watching the videos and doing questions, which I loved when I was a kid and still enjoy.  (We didn't use Khan Academy; I just liked playing games and getting points.)

Thanks, Apprentice!

Countdown to School: Middle School Math, Part Two (or, Getting there is half the fun; come share it with me.*)

Part One is here.

First, a bit of Charlotte Mason catechism.

Why was Charlotte Mason disappointed in the results of her early teaching experiences?

"The children, no doubt, 'got on'––a little; but each one of them had the makings in her of a noble character, of a fine mind, and where was the lever to lift each of these little worlds?"

Why didn't Charlotte Mason think much of just plugging away at long arithmetic problems (among other things)?

"If education is to secure the step-by-step progress of the individual and the race, it must mean something over and above the daily plodding at small tasks which goes by the name."

What should be done during lessons?

"This is the sort of thing that the children should go through, more or less, in every lesson––a tracing of effect from cause, or of cause from effect; a comparing of things to find out wherein they are alike, and wherein they differ; a conclusion as to causes or consequences from certain premises."

How is teaching math like French?

"Supposing, for instance, that by good teaching you secure the child's attention to the verb avoir, he will remember it; that is to say, some infinitely slight growth of brain tissue will record and retain that one French verb. But one verb is nothing; you want the child to learn French, and for this you must not only fix his attention upon each new lesson, but each must be so linked into the last that it is impossible for him to recall one without the other following in its train. The physical effect of such a method appears to be that each new growth of the brain tissue is, so to speak, laid upon the last; that is, to put it figuratively, a certain tract of the brain may be conceived of as being overlaid with French."

What was that again?

"Every Lesson must recall the Last.––Let every lesson gain the child's entire attention, and let each new lesson be so interlaced with the last that the one must recall the other; that again, recalls the one before it, and so on to the beginning."

Second, those who want to teach Charlotte Mason-style math should read "You don't need a composition program," a post at Higher Up and Further In.  The subject is different, the principles are much the same.  Not expecting more than is reasonable for a child's age, either in attention span or final output (hour-long grade one math, anyone?).  Trusting the process through many weeks or months or years of faithful narration, copywork, and dictation from worthy books, and without undue interference (also known as classroom clutter or busywork).

Now ineffective and even harmful curriculums do exist in math, as well as in English and other subjects. We've run into a couple of them ourselves.  The Deputy Headmistress recently mentioned a book that her teenager asked her to read aloud because she was having difficulty, but even reading it aloud could not make a good book out of a poor one, and they agreed to drop it.  There is an important, but obviously difficult, distinction between needing to persevere with something, even when it's difficult or doesn't seem immediately rewarding, and deciding to modify it or even let it go.

And the testimonies of those who have "taught math Charlotte Mason-style," especially in the upper years when we all seem to turn to the same few textbook publishers, are fewer and farther between than those whose children have become excellent readers and writers.  That is a fact.  However, lack of recognition doesn't mean it doesn't exist.  What of the way of the will and the way of the reason; the three instruments of education (environment, habit, living ideas)?  What of students' ability to visualize, as they do in their spelling, their picture study, their narrations?  What about simply having more doors opened?  What about "not how much does he know, but how much does he care?"

Finally, in a world that is already too utilitarian, too unpoetical, too competitive, analytical and computer-driven, why and how should a "human curriculum" promote mathematics?  And what kind of mathematics?  Where do our middle-schoolers, somewhere in the transition between elementary mathematics and senior-high algebra and geometry, fit into that vision?

Those are the questions.  Stay tuned for an attempt at some answers.

Part Three is here.

In which the chambered nautilus flunks the test (a math lesson for Crayons)

Fifth grader Crayons/Dollygirl has been learning a bit of math history from John Tiner's Exploring the World of Mathematics.  In this week's lesson I had just intended to finish up the chapter on Number Patterns, but the Fibonacci business got away from me a bit.  But that's a good thing.

Here's the lesson as I plan to present it tomorrow, making use of online resources (including one with an unexpected surprise).

1.  Review what we have learned so far about Fibonacci numbers:  that they run in the sequence 1, 1, 2, 3, 5, 8, 13 and so on, with each pair of numbers adding up to the number following right after.

2.  Construct squares following the sequence from graph paper--that is, two with sides 1 unit long, one with sides 2 units long, and so on.  Colour them and cut them out.

3.  Arrange them as if you were packing them in a box, starting with the smallest ones in the centre.  See diagram in the book if you're not sure. Then watch this online animation.  At the end of the animation, watch the drawing of the spiral.  Can you see how that works?  This is called a Fibonacci spiral.

4.  Places in nature where Fibonacci spirals occur:  in spiral galaxies, in your inner ear, in pine cones, in cauliflower.  (I'm thinking maybe also in fiddleheads? I'm not sure about those.)  But not, according to this blog post and its accompanying slide show, in that classic example (cited in Tiner's book), the chambered nautilus.  So much for that. (Maybe it works for some people?)

5.  Another use of Fibonacci spirals:  in art, what is referred to as the Golden Mean or Golden Ratio.  Apparently our eyes just like to follow things that move around in those proportions. Here's a neat blog post showing photographic examples.  (According to the post, the photos were not deliberately planned to match up with the spiral: they just do because they're good photos.  Or they're good photos because they just do.)

6.  So a fun followup might be to find other examples of paintings or photographs that follow these proportions.  Or to deliberately create a drawing--or maybe just a colour pattern--that follows it, and see how that works.  Do you like the way it turned out--why or why not?