When just playing around is not enough (John Mighton and Charlotte Mason)

On the subject of teaching mathematics, Charlotte Mason had a few negative things to say about manipulatives. Not that she didn't use real-life objects and counters: her books mention money, beans, and dominoes, just as a start.  But she was definitely against complicated, expensive math apparatus, and, using the  textbooks available a hundred years ago, we can assume that she also did not teach with what's now called a "discovery" approach. Children were taught to understand what they were doing, as much as possible, but they were also given the information and strategies they needed rather than being expected to invent their own methods.

This is where John Mighton, JUMP Math, and recent educational research come in.

Some of the conflict we have over how to teach math comes from confusion over words, especially words like "discovery" and "rote." The problem is that, like that swinging gate that Charlotte Mason mentioned, people on either end assume that there's no middle point. Either you teach like a colonial-era schoolmaster, chanting multiplication tables for hours, or you grab onto the latest discovery-based fads and hope that the concepts will eventually click.

This is where we need to ask ourselves persistently if what we're doing...um, adds up. Mighton uses the analogy of playing chess. Some amateurs have played many, many games of chess; they can play a long time, but they don't get much better just by playing. I'm thinking also of, maybe, someone who can play "Lady of Spain" on the acccordion (or the organ), but that's all.

The opposite of this approach, according to Mighton and the research he cites, is to teach specific small sequences of moves, whether in chess, or skating, or math. If you're trying to get good at golf, you go out and work on your swing. If you're learning to type, you type jkl; jkl; jkl; as many times as it takes; then you add a couple of new keys, and so on. This is not punitive; this is not what we criticize as "rote learning"; these are the ABC's of anything. If you give students keyboards and tell them to bang away, they will not figure out for themselves that they could get faster and more skillful by typing jkl;.

So why do we assume that a tableful of carefully-designed manipulative blocks...are you listening, Herr Froebel?...will, in themselves, teach children what they need to know, no matter how long they play with them? As Mighton points out, and I have seen this confirmed myself, there are high school students who have been through this whole program of discovery-based learning and who still cannot add fractions. But when they're taken through some "guided instruction," they can and do learn what they missed earlier on. (But what a shame to have wasted all that time.)

Guided instruction is not rote learning. It doesn't mean no drawings, no manipulatives. It doesn't mean blindly following an instructional sequence like a robot. It does mean that we teach students a little bit at a time until they can do each step perfectly. Who does that remind you of?

Oh yes--Charlotte Mason.

Perfect Accomplishment.––I can only offer a few hints on the teaching of writing, though much might be said. First, let the child accomplish something perfectly in every lesson––a stroke, a pothook, a letter...
Steps in Teaching.––Let the stroke be learned first; then the pothook; then the letters of which the pothook is an element––n, m, v, w, r, h, p, y; then o, and letters of which the curve is an element a, c, g, e, x, s, q; then looped and irregular letters––b, l, f, t, etc. One letter should be perfectly formed in a day, and the next day the same elemental forms repeated in another letter, until they become familiar. By-and-by copies, three or four of the letters they have learned grouped into a word––'man,' 'aunt'; the lesson to be the production of the written word once without a single fault in any letter.  ~~ Charlotte Mason, Home Education

Linked from The Carnival of Homeschooling: Ages and Stages Edition. 

JUMP Math Scores Again

We posted before about John Mighton and his JUMP Math program, back in 2007.

It's not a program we're currently using, although I was very impressed with the downloadable Fractions unit that we tried out at that time. It seemed like a great way to teach kids whose math anxiety makes them shut down before they've even really looked at the problems. They're sure that they won't be able to do whatever it is, but then you show them that, on the first page of a lesson, all they have to do is identify which subtraction problems would need regrouping. Oh--okay. I can do that. Then you build up, step by step. Ponytails also used some JUMP materials last year (when she was at public school) and liked them.

The Globe and Mail ran a good article this past weekend about JUMP. The contrast between a public school teacher's idea of a good problem and the JUMP approach is almost scary:

"[The] curriculum co-ordinator....says research shows that the best way to help kids understand a concept is to come up with a rich, conceptual problem that everyone in the class can help solve.

"Last year, for example, she visited the class of a primary-school teacher who had noticed that all the kids were wearing odd socks. The teacher came up with the concept of a sock factory, and the kids all brought in socks. Each child was given a different number of socks and their task, as a group, was to find a strategy that would combine them."

Teachers get rewarded for coming up with that stuff. It's like that teacher's magazine example Mary Pride used in Schoolproof about suggesting that the teacher find a great big pair of shoes (colourful if possible) and having the children measure feet to see who might fit them, write stories about the shoes, and so on. As Mary pointed out, what the children learned from that experience was probably not worth the trouble of finding the big shoes.

But you know what? I don't even understand that task, as it's described there. Much less what it has to do with primary-school math. Or why all the kids were wearing odd socks--is that a fashion thing, or is that neighbourhood so impoverished that we had better start paying as much attention to underclothed schoolchildren as we do to their math learning?

Contrast that with one of John Mighton's classes:

"Every hand in the class shoots up. The number 121,252 is not divisible by nine, one student tells him, and the remainder will be four.

"'You are brilliant,' he tells them. 'You are all brilliant.'"

Mama Squirrel's take: I would rather be brilliant in John Mighton's class than fool around with strategies of socks.

Better math, better learning (John Mighton and JUMP Math)

Do you know what neuroplasticity is?

What do you think about recent discoveries about the way our epigenetic system helps us process information?

University of Toronto professor Dr. John Mighton slung some of this around last night to a roomful of teachers and interested others, in between demonstrating why we invert and multiply when dividing fractions, and giving some hints about teaching the nine times table. (Funny, I just reminded one of my own Squirrelings about that yesterday, the fact that the digits have to add up to nine. Teachers don't know this?) He also shared the (to me) appalling information that, according to his research of provincial math curricula across Canada, there is not one province that specifically, in its curriculum guidelines, says that children must be taught to solve questions like "what is 2/3 of 9."

What he had to say went well beyond pushing his math program or his books, although he was there specifically to promote his book The End of Ignorance. As I listened I kept mentally hearing quotes from Charlotte Mason overlapping with some of the scientific findings Dr. Mighton was describing along with his own experiences tutoring children. (You can read the appendix to his book here.)

The most recent discoveries about our brains show that they can develop new abilities, rewire themselves and learn material beyond what was previously expected. It's not so much that you're born a math genius or not. The evidence points to the fact that most kids can learn anything.

So why don't we all learn way more than we do?

Dr. Mighton mentioned a Scientific American article, "The Expert Mind," that points out that you can learn the rules of chess, and play chess as an amateur for the rest of your life, not ever getting any better than a beginner. (Yep, he's got my chess-playing style nailed.) On the other hand, players who are taught small groups of powerful moves and so on become very good, very fast. (Wouldn't the same thing also apply to those who are taught to bang out songs on the piano but never go beyond that?)

He also described being inspired by a volume of letters by Sylvia Plath, in which she explains how she learned to write: by imitating great writers. He pointed out that Plath developed one of the most original voices in 20th century American poetry, so it obviously was no handicap to begin with imitation.

The problem in schools today is that kids are often expected to do without some simple training in basics that they need if they're to seriously develop their abilities--specifically in math, but in the other areas as well. The issue of whole language vs. phonics is one example; spending too much time on discovery-based math learning (including overuse of manipulatives) without teaching the needed basic skills is another. (Remember Mr. Person's blog post, Hands-On, Brains-Off?) Just because kids can work with models or manipulatives doesn't mean they can generalize enough to answer questions that are given in another context; and conversely, just because they haven't been able to learn something by playing with pizza pieces or whatever doesn't mean they can't learn those concepts if they're presented with a more "guided discovery" approach. (Some people apparently read The Myth of Ability and get the idea that John Mighton completely eschews manipulatives; this isn't so, he does use chocolate bars and other concrete examples when it helps to demonstrate a point.)

Add to this the fact that our working memories aren't always that great; you might "discover" something during a lesson, but forget it later. And this is not limited to children; Dr. Mighton mentioned (I think it was in The Myth of Ability as well) that he was once impressed by some mathematical discovery, and then realized that he himself was the one who had published the article some time before.

We need to pay more attention to the ways that kids learn and behave in groups; this might not apply so much to homeschooling (and might be a reason we're homeschooling), but it's still important to understand. Actually most of us know it instinctively already: when you were in school, didn't you have a pretty good idea who the "smart ones" and the "dumb ones" were? And if you thought you were one of the "dumb ones," the odds are that you started to limit your own ability to learn because you thought you couldn't.

And in our culture...as most of us also know...it's socially acceptable to laugh, wince and say "I just never could do math."

So we need to change that.

We need to find ways to increase students' confidence in themselves--no matter what their background, no matter how they've been labelled. [UPDATE: sorry if that sounded a bit too much like the I'm-so-special-boost-my-self-esteem thing. I'm talking only about helping students understand that they do genuinely have the ability to learn.] We need to avoid making the faulty assumption that certain parts of the population are born with less ability to learn than others. (Charlotte Mason's methods were used with children of all classes and backgrounds, blowing the Victorian idea of only-wealthy-children-can-learn to pieces.)

We need, according to Dr. Mighton, to have more confidence in teachers' reports of success with these methods. He talked specifically about the fact that his JUMP Math program has been accepted more in Western Canada than in Ontario, in spite of the fact that classroom teachers who have tried it have been more than satisfied with the results they've seen. Bureaucracy rules and change is slow.

We need to use "guided discovery" methods, especially in cases where manipulatives did not work well; where it would really make more sense to just teach what needs to be taught rather than expecting students to keep reinventing the wheel. We need to teach subjects such as mathematics in short, progressive steps, always "raising the bar" (a favourite Mighton phrase) just a little--or even just making the next step seem a little harder; never rushing ahead or adding in a lot of extra clutter. Dr. Mighton talked about one special-needs student who understood 1/4 + 1/4, and then 1/7 + 1/7, and then 1/36 + 1/36 and so on; but got agitated when asked to add 1/4 + 1/4 + 1/4. However, after being given more opportunities to add things like 1/400 + 1/400 and 1/855 + 1/855 (my examples), he suddenly asked to go back and try 1/4 + 1/4 + 1/4 again; and this time he was successful.

"Here we may, I think, trace the solitary source of weakness in a surpassingly excellent manual. It is quite true that the fundamental truths of the science of number all rest on the evidence of sense but, having used eyes and fingers upon ten balls or twenty balls, upon ten nuts, or leaves, or sheep, or what not, the child has formed the association of a given number with objects, and is able to conceive of the association of various other numbers with objects. In fact, he begins to think in numbers and not in objects, that is, he begins mathematics. Therefore I incline to think that an elaborate system of staves, cubes, etc., instead of tens, hundreds, thousands, errs by embarrassing the child's mind with too much teaching, and by making the illustration occupy a more prominent place than the thing illustrated."--Charlotte Mason, Home Education

Is this just a return to rote learning? No. Even Charlotte Mason had her students practice times tables, and insisted that learning in subjects such as mathematics and grammar must be continuous--that each bit must build on the next. As Charlotte Mason said--look at the evidence. I have personally seen our 10-year-old's success with Mighton's JUMP Math fractions unit this fall, at a time when she particularly needed to rebuild confidence in her math ability.

If kids stumble through school not being able to read, to spell, to do basic math--then is it their fault for being stupid, or our fault for not teaching them properly, or for making them think they are incapable? (I don't think Dr. Mighton blames classroom teachers but rather the system in general.) If we have astonishing success with children who were thought unable to learn--how did that happen? (Some educators could learn a lot from homeschoolers.) If we discover, or rediscover, some method that works well for a wide range of students--can we put down our prejudices and simply use what works? Don't we want all children to be good readers and writers, to go beyond the minimum, to enjoy all learning including mathematics? Shouldn't we be doing whatever it takes to reach those goals?

You can decide.

Win-win math teaching

"We underestimate children by assuming that they will only enjoy learning concepts that have obvious physical models or applications. While I would encourage a teacher to serve pieces of pie or pizza to their class to illustrate a point about fractions, this is not the only way to get kids interested in math. Children will happily play a game with numbers or mathematical symbols, even if it has no obvious connections to the everyday world, as long as the game presents a series of interesting challenges, has clear rules and outcomes, and if the person playing the game has a good chance of winning. Children are born to solve puzzles: in my experience, they are completely happy at school if they are allowed to exercise their minds and to show off to a caring adult. What children hate most is failure. They generally find mathematical rules and operations boring only because those things are often poorly taught, without passion, in a manner that produces very few winners." --John Mighton, JUMP Math: Teacher's Manual for the Fractions Unit (Second Edition), copyright 2005, JUMP Math.

Charlotte Mason might have liked Mighton

It's already the weekend and I'm still working on some of the things that stood out from last weekend's papers. In the midst of thinking about Big Words, I put off writing about another Big Idea, this one from the Globe and Mail's Books section last Saturday. Here's the review (by Keith Oatley) of John Mighton's book The End of Ignorance: Multiplying Our Human Potential.
Check out these quotes:

"As Mighton observes, there are differences in people's abilities. But do we want a school system that first accentuates such differences, and then takes them to define who people are?"

"When [Mighton] was younger, he worked as math tutor for primary-school children. He was struck by how even children labelled hopeless could learn math by means of simple procedures. It turns out that, approached in this way, math was enjoyable for children, among the easier things to learn rather than among the more difficult."

"He is brilliant at breaking down math problems into parts that children can do easily, which can then form more complex wholes....as they complete each step and repeat it with small variations, unforeseen abilities emerge, sources of pride and confidence for the children."--all quotes by Keith Oatley, Globe and Mail review

I don't know anything about the JUMP program, but that sounds very CM-friendly to me.