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.