I went to graduate school to read the research on cognition and instruction, the applied psychology of the classroom. Researchers have done experiments and have compared classroom methods across cultures. Some of the methods/techniques they have investigated (and found evidence for) are specific, like the spacing effect on memory. Some of the methods/techniques they have investigated are more general, such as how teachers' questioning strategies affects student learning. (See

*Why Don't Children Like School?*for a readable summary of some key research.) Overall, there is a lot of different evidence to support these ideas:

- students retain information and concepts best when these are used intermittently, spread out over a semester
- students learn concepts best when they are asked medium-to-difficult questions about why something works (or are given tough application questions), rather than asked low-level information questions -- too easy, and the concept does not stick; too hard, and the students are lost
- students learn concepts best when they are shown specific metacognitive strategies to help them evaluate their own thinking
- communication matters: the act of trying to explain an concept clearly helps students solidify their understanding of it

I am lucky I started teaching in debate, where there is no set curriculum, and I was left to my own devices to figure out what my students could use to become better debaters and more critical thinkers. I realized quickly that, although I wanted my students to understand deontological and consequentialist thinking, reading original texts was not helpful. Better to explain the ideas simply, and then give them scenarios to evaluate from both perspectives. The same with logic: formal logic was not helpful, but it was useful to show them more basic models (Toulmin's model, Venn diagrams, Ishikawa diagrams, etc.) and have them spend the time looking at an argument and clearly articulating the assumptions and analyzing its flaws. The practical applications -- what they could actually do in analyzing examples of actual, real-world arguments -- was far more important than how far we got into pure, theoretical logic. I still have a desire to write a book with a couple simple models and lots of arguments from different fields (law, politics, economics, etc.) to analyze. It would be a double benefit: the students would get exposure to key social science theories, say, the Keynes/Hayek debate/rap battle about fiat money, as well as getting lots of practice diagraming arguments.

A similar situation occurred when I first started teaching public speaking (the second course I taught on my own): it was the Wild West, and we played around with what concepts to teach and far to go into each one, and we did what seemed best for our students. When I realized that putting together a persuasive speech was extremely difficult for my students -- that we would need to spend weeks on just recognizing the difference between a logical argument and an emotional appeal -- I just said to heck with it and dropped the persuasive speech. Do I think everyone ought to learn the difference between an argument and an emotional appeal? Of course. But I also think everyone ought to learn about self-selection/survivor bias, utilitarianism, Venn diagrams, Taylor series, statistical inferences from sampling distributions, etc., etc. Public speaking can not accommodate it all. At a certain point, one has to evaluate what the students can actually do, and if one is presenting a topic for its own sake alone, then it ought to be cut. So, I added another speech, a demonstration speech, which proved challenging (but reasonably so) for them.

I am lucky that a math course was not my first or second teaching assignment, because I knew by the time I got around to teaching math to trust my gut about whether a concept was "a bridge too far" for students. The edifice of the math curriculum is imposing, because there appears to be such an absolute, inviolable need to cover every concept so students are prepared for the next course. The new teacher can be overwhelmed: "I have to cover completing the square, or else I've set them up for failure next year!" But this inviolability is mostly just an appearance: there is substantial overlap from course to course, and topics get repeated; it is far more important than students understand what they are doing than they cover every topic poorly.

Despite the fact that the four ideas I outlined above are clearly supported by research evidence, they have not really won over mathematics textbook publishers, school districts, parents, and teachers. Books continue to cover concepts in logically grouped topic units, and once the unit test is over, the concept is barely referred to or used again. Asking kids tough questions that require some problem-solving or deep thinking makes parents uncomfortable: "My child says he doesn't get it. Aren't you showing them how to do it?" (Yes, and then we ask them to do it on their own for real, in hard but achievably challenging problems, which discomfits some students!) And teaching metacognitive strategies and emphasizing clear communication are derided as fuzzy and not worth the time.

Perhaps a big part of the divide is because reformers and traditionalists have gotten hung up on the original presentation of the material: lecture or discovery. While I do like to have students experience some discovery, I always follow-up to clarify and help them clearly formulate the key ideas. If an idea does not lend itself to discovery, then I am willing to lecture -- but I immediately ask my students to put the idea into use. It seems like various, mixed instructional techniques can be appropriate. The specific presentation and instruction techniques are less important than structuring the curriculum to provide ample, challenging practice to students, plenty of feedback, and the repetition and integration of topics. Besides, discovery is only one stage in the process of mathematics; there are many meaningful things the teacher can ask them to do at other stages. The discussion reformers and traditionalist ought to have is about the curriculum and assessment, not the instruction, because then they will be able to find common ground; who disagrees with mixed review? Who disagrees with continual feedback? (The May/June 2012

*Washington Monthly*ran a fascinating focus on educational testing; here is one article about using computers to provide automated feedback to students, which seems like a good idea if it is done under the supervision of a good teacher. It is also why the Virginia Tech math emporium could be a neat model as well.)

The issue now is that the cognitive research has not yet created a rich vocabulary that details the minutiae of curriculum choices teachers face every day. The research is still quite broad-stroke. It is like buying a home organization book that gives good but broad advice, such as, "Throw out anything you are not using!," versus a book that has specific, concrete advice on how to store spices, sports equipment, toiletries, and that comes with pictures of successful examples. In the next post, I will discuss some of the key issues that I find come up with math instruction and try to develop some vocabulary around it.

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