I recently presented a class on math education during the STEM seminar at the Latter-day Saint Home Educators conference. The transcript, handout, and PowerPoint are linked in the side bar, but I found so many great quotes (that I wanted to use but ran out of time) that I feel like putting a few of them up on my blog.
From the Mathematical Association of America website:
Japanese cognitive psychologist, Giyoo Hatano, gave the following five "characterizations" of long-term knowledge acquisition, with which, he felt, most cognitive psychologists would agree. And with which, most mathematics education researchers would agree.
Knowledge is acquired by construction, not by transmission alone. Compelling evidence for this is provided by the work on procedural bugs and misconceptions -- it is highly unlikely that students acquire them from direct teaching. For example, young children often make systematic subtraction errors, the most common of which is always subtracting the smaller digit from the larger, regardless of position, and many preservice elementary teachers believe "division (always) makes smaller." Surely, no one taught them this.
Knowledge acquisition involves restructuring -- not only does the amount of a person's knowledge gradually increase, it gets reorganized. Children do not think like miniature or incomplete adults. For example, in attributing unknown properties to animate objects, Hatano found young children rely on similarity-based inference, whereas older children and adults use category-based inference. He finds studies of conceptual change, both in the history of science and in cognitive development, especially relevant because fundamental conceptual change is perhaps the most radical kind of (mental) restructuring. [Cf. Carey, "Conceptual differences between children and adults," Mind and Language 3, 1988; Kuhn, The Structure of Scientific Revolutions, University of Chicago Press, 1970.]
The process of knowledge acquisition is constrained both internally, by what one already knows, and externally, by cultural artifacts such as shared language and notation. This explains, in part, why different individuals acquire similar, but not identical, knowledge.
Knowledge is domain specific. This serves cognitive economy -- in problem solving, one need only access relevant knowledge. However, what is acquired in one domain can be transferred to another (e.g., through analogy) or generalized to a variety of domains (e.g., by abstracting structural commonalties).
Knowledge acquisition is "situated," i.e., it reflects how it was originally acquired and has been used -- it consists not only of abstract rules, laws, and formulas, but also of personal experiences. Becoming an expert, say in mathematics or physics, may be a process of "desituating" one's knowledge to make it less context-bound, less tied to surface features.
From the Stella's Stunners website:
There are several reasons for students' resistance to problem solving. One is that some hard work may well be involved — how much is unknown. Students these days are typically overbooked, overscheduled, and caught up in the spell cast by e-mails, texting, Facebook, video games, and all of the other engrossing ways of spending time. Who wants to sit and stare at a problem, waiting for an idea to hit? Another inhibitor is that we simply do not like to be in situations where we feel frustrated and incompetent. And related to this discomfort, for some students, is the fact that hard thinking evokes other problems with considerably more emotional weight: "Why didn't Dad come home last night?" "What if Mom loses her job?" "What if I'm pregnant?" It can be difficult in such circumstances to entice students to engage in problem solving, thinking in ways not previously experienced, for an unknown length of time, and with no certainty of success.
But education is all about training minds to be imaginative, savvy, persistent, and resourceful — exactly the characteristics our public leaders say our students lack. Trudging through mathematics textbooks, year after year, is not in itself going to help students become the skilled mathematicians, scientists, technicians, or even literate citizens our global economy requires. It is the opportunity to grapple with and solve non-routine problems, problems that are not necessarily clearly defined, that provides students, our prospective adults, with the intellectual robustness that our country needs in its citizens.
From "The Teaching of Arithmetic I: The Story of an Experiment, by L.P. Benezet
I went into a certain eighth-grade room one day and was accompanied by a
stenographer who took down, verbatim, the answers given me by the children. I was trying to get the
children to tell me, in their own words, that if you have two fractions with the same numerator, the one
with the smaller denominator is the larger. I quote typical answers.
• "The smaller number in fractions is always the largest."
• "If you had one thing and cut it into pieces the smaller piece will be the bigger. I mean the one
you could cut the least pieces in would be the bigger pieces."
• "The denominator that is smallest is the largest."
The average layman will think that this must have been a group of half-wits, but I can assure you that it
is typical of the attempts of fourteen-year-old children from any part of the country to put their ideas into
English. The trouble was not with the children or with the teacher; it was with the curriculum. If the
course of study required that the children master long division before leaving the fourth grade and
fractions before finishing the fifth, then the teacher had to spend hours and hours on this work to the
neglect of giving children practice in speaking the English language.
...I feel that it is all nonsense to take eight years to get children through the ordinary
arithmetic assignment of the elementary schools. What possible need has a ten-year-old
child for a knowledge of long division? The whole subject of arithmetic could be postponed
until the seventh year of school, and it could be mastered in two years' study by any normal
child.
And from an article in USAToday:
Math as rules starts early. Kids learn in elementary school that you can "add a zero to multiply by ten." And it's true, 237 x 10 = 2370. Never mind why. But then when kids learn decimals, the rule fails: 2.37 x 10 is not 2.370. One approach is to simply add another rule. But that's not the best way.
Common Core saves us from plug-and-chug. In fact, math is based on a collection of ideas that do make sense. The rules come from the ideas. Common Core asks students to learn math this way, with both computational fluency and understanding of the ideas.
Learning math this way leads to deeper understanding, obviates the need for endless rule-memorizing and provides the intellectual flexibility to apply math in new situations, ones for which the rules need to be adapted. (It's also a lot more fun.) Combining computational fluency with understanding makes for problem solvers who can genuinely use their math. This is what businesses want and what is necessary to use math in a quantitative discipline.
Here is what good math learning produces: Students who can compute correctly and wisely, choosing the best way to do a given computation; students who can explain what they are doing when they solve a problem or use math to analyze a situation; and students who have the flexibility and understanding to find the best approach to a new problem.
I thoroughly agree with all of the above--including the attempt to revamp math education in the U.S.--but with one caveat. It's impossible to TEACH math in a way that gets a GROUP of children to understand--to deeply rather than superficially understand. Each person must build their own structure of understanding within their own brain--it cannot be mass produced. Sorry educators, you have good intentions, and the OUTCOME you're after is the right one. But it's not within your power to construct something in someone else's brain.
Individual tutoring was for centuries the education of rich men's children, and the results were and continue to be superior to teaching in groups. The human brain is the most complex structure in the universe, so finding even two children who are at exactly the same point in their understanding of math concepts, so that you can instruct them together, is fruitless. Even teaching one child is very hit and miss--you still must find a way to know exactly where he's at so that you can help him move his understanding of the topic along.
I strongly believe that when teaching DOES yield deep learning, it is a serendipitous combination of things that were happening in the child's real life (life outside of school is real)--problems he was faced with, things he was thinking about, etc.--together with just the right topic being presented at the right time with the right spin by the right person. Anything else leads to the "memorize, test, forget" cycle. Since all those "rights" rarely occur, most of the math understanding we move forward in our lives with is something we created in our down-time: playing games, building things, pondering on why numbers do this or that, etc.
As Lev Vygotsky said, “Practical experience … shows that direct teaching of concepts is impossible and fruitless. A teacher who tries to do this usually accomplishes nothing but empty verbalism, a parrot-like repetition of words by the child, simulating a knowledge of the corresponding concepts but actually covering up a vacuum.”
Teachers do a great deal of good in the practical day-to-day running of the world, but if schools were abolished, children would still learn. As Hugh Nibley said, "My only job as a teacher is to save my students time." In other words, a teacher is someone to point you in the right direction when you're on a quest to learn something. The key: it has to be the LEARNER'S quest, not the teacher's, or nothing of importance will happen in the learner's mind.
