There is a continuing chorus of complaints about how math is taught from those who seek to reform math education. The chief complaint is the lack of transfer of knowledge. That is, students cannot seem to take their prior knowledge and apply it to problems that rely on the same knowledge but are in new or novel settings.
The reformers then talk about how we need to build students’ “depth of knowledge” to get to the holy grail of “deeper understanding”.
I’ve taken a sample of a PowerPoint which is similar to many others that have been making the rounds over the years. In it, the problem is presented as follows:
Students appear to demonstrate “deep, authentic command of mathematical concepts” when given commonly used problems.
However with more challenging problems, the same students seem
to no longer demonstrate that command.
First, we must have a clear understanding about why these problems are different from one another.
Next, we need to practice using these problems so that we understand how students may react to them.
Last, we need a source that can provide us with a variety of free problems.
The underlying message is that we haven’t been doing these things and students are getting “superficial knowledge”. The supposed proof are charts of problems that correspond to varying depths of knowledge. What’s misleading about these charts is that we look at them with many years of experience under our belts thinking “Yeah, kids should be able to do these.”
What seems to be neglected in all this, is the distinction between novice and expert and that problems seen for the first time (i.e., “new” problems) are naturally going to be harder to solve. What is needed to get students over the hump are 1) worked examples, and 2) scaffolded problems that increase in variety and difficulty.
A look back at math books from previous eras shows that in fact, we have been doing these things. Below is an excerpt from a fifth grade arithmetic textbook from 1937 called “Modern School Arithmetic” by the reformers of that era: John Clark, Arthur Otis and Caroline Hatton.
Interesting to note that the traditional modes of education seem to address the very concerns that the current slew of reformers claim has been missing. Something to keep in mind the next time you attend an NCTM conference or their equivalents.
traditional math 
Teachers appear to demonstrate “deep, authentic command of mathematical concepts” when given commonly used problems.
However with more challenging problems, the same teachers seem to no longer demonstrate that command.
I put a calculus problem up on the board recently that I knew was solvable, because I knew it had enough information. It took me twenty minutes to figure out how to solve it. That’s a problem I wrote, in a topic I have taught a dozen times.
I’ve seen Maths teachers stumped over circle geometry problems that 15 year-olds are expected answer.
Yet we expect students to be able to instantly be able to do what we cannot.
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Reblogged this on Nonpartisan Education Group.
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In helping my son prepare for the SAT and AMS tests a number of years ago, we looked at many sample test problems. We realized that transference between problems can fall apart very quickly. Testers love to find those problems. The failure is that many think this its a good way to test mathematical understanding. We found that the best way to prepare for these tests was to do as many of the sample problems as possible. What you see on the test won’t be the same, but the transference distance won’t be as great. It’s not better transferable understanding, but massive preparation.
There are all sorts of subtle variations to work problems, logarithm problems and every other type of problem you can think of. General concepts and understandings are just the start. Doing many, many problem sets and developing the confidence that you can eventually figure it out is required. There are no conceptual understandings that will replace that. Group work and engagement are no substitutes for the long term individual skill of properly completing homework P-sets. How many K-8 math teachers see homework as only a way to try? That’s a road to test failure.
When I taught college math and CS, the programming tests counted very little. That’s common in computer science education. The grade was all about individually creating a working program – the homework. that’s the way K-12 math should be. Homework P-sets are like a weekly take home test that’s graded. Everyone can get a ‘A’. That’s going to be my new theme for math – grade homework. Tests are only for keeping everyone honest.
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