Knowledge Transfer

A quick note, stemming from discussions at the new job about depth of knowledge (on the part of elementary/middle school students and their teachers) and related questions. One of those related questions is knowledge transfer: the ability to take knowledge from one context and apply it in a related context. The first hurdle there, of course, is recognizing that the contexts are related; the second is knowing what to keep and what to change, as the contexts change.

A complicated link of half-remembered references brought me to a paper (PDF) by one Michelle Perry describing an experiment done with 4th and 5th graders. The children in the experiment were selected based on knowing basic arithmetic (addition and multiplication under 20) but not being able to correctly fill in a blank in a problem like

\[ 4 + 6 + 3 = \_\_ + 3 \]

which tests the conception of the \(=\) sign as a statement of equivalence (correct) or an instruction to go forth and compute (incorrect). Students were given either direct instruction in the procedure to follow (add up the left hand side and then subtract the known term on the right from that sum), or given a purely conceptual instruction with no explicit steps.

What is really interesting is the result: both groups did roughly as well on post-instructional assessment -- but only on the problems that used addition, and so were exactly analogous to the problem they'd been instructed on. The post-test, however, also included problems that required the same principle, but used multiplication instead:

\[ 3 \times 2 \times 3 = \_\_ \times 3\]

In these problems, the children who had been taught a procedure "followed" it by doing the multiplication on the left and then subtracting the known "term" on the right from the product! Around 40% of the children who had been taught the principle underlying the problem successfully transferred knowledge to the unfamiliar setting, compared to 10% of the "procedure" group.

The remainder of the study is interesting too: their results indicate that teaching concept-plus-procedure actually undercuts transfer: basically if you teach a concept and then immediately teach a procedure for it, students' conceptual understanding gets washed out by the procedural knowledge.

Consider the Gebra

Today's Times features an oped which is sure to excite a swarm of comment. The author, a retired political scientist at one of the senior CUNY schools named Andrew Hacker, asks "Is algebra necessary", in the sense of "is the sine qua non status of algebra in our elementary-to-collegiate education tracking justified". His response is decidedly negative.

Now, normally I'd sarcastically begin my response with "There's only one thing wrong with this..." and then proceed to let the author know why he should leave and go feed penguins or something. Normally, this kind of attack begins from some deeply flawed assumptions, and it just gets worse from there. This piece, though, isn't that. So instead of sarcastically pointing out the one little thing the author got wrong, I think it's useful to mention the only thing the author got right: there is a deep and low-level problem with how we teach and learn school mathematics, and in particular with what role we expect mathematics instruction to play in a student's larger academic trajectory.