I finally managed to read the original text of that Science article. You know, the one that raised hackles?

It addresses an issue I’ve troubled myself over before.

For all of the naysaying about the validity of the study, it rings true to me.

How do I get my kids to see the connection between these two problems:

1) You have a piece of string that is 17 inches long. You cut it into four pieces. How long is each piece?

2) The base of a dog house is 17 square feet. If the dog house is four feet wide, how long is it?

Completely different physical concepts – exactly the same abstraction. How do I effect the transfer?

#### The Article

One of the studies in the paper looked at this. For students that learned from either one or two concrete examples, there was almost no transfer of knowledge to a new scenario. For students that were present with 3 distinct cases, there was some improvement over the other two groups.

The group that learned the rules in an abstract symbolic fashion first transferred the knowledge far more easily than either of the other groups.

A further study (not reported in the Times article) addressed a common teaching strategy: What happens if you teach them the abstraction, but after you give them an example to give them an idea of what it’s about?

It turns out that the group that had a concrete example first had more difficulty transferring the knowledge to a new domain. Something about the concrete connection made it difficult to envision new situations. (The article didn’t make this direct comparison, but it appears that, from the data, that the concrete/general methodology is still a step above just teaching pure concrete examples.)

A further study addressed one of my personal teaching techniques: the compare and contrast. Students were given two different concrete representations, and then asked to find commonalities between the two. This technique actually did yield some results. Bimodal results. Some students (44%) actually learned how to transfer the knowledge, while others (51%) did not.

#### Impact

So, how does this inform my teaching?

Maybe not much. Because of our spiral model of standards, most of my students will have seen most of the material before, presented with concrete examples. I’m not painting on a blank canvas – I’m adding to 8 or 9 years of work by vastly differing painters before me. I don’t get to present an idea in the abstract, and only then make connections to concrete examples.

On the other hand, the idea of transfer is critical to the gaining of abstraction. I think part of my summer curriculum (busy as it will be with a brand new subject to teach next year) will be to examine every concept for transferability, and focus on that.

I’m not sure this is the right answer. There could be flaws in the research that I’m not aware of. (It certainly has more rigor than most education research I’ve seen, though.) I may be drawing the wrong conclusions. But if I want my students to be strong in their later education, they’ll need to be able to transfer what the get in my room to something else.

I need to make sure that can happen.

## { 4 } Comments

What happens if you teach them the abstraction, but after you give them an example to give them an idea of what it’s about?Was that the one with the measuring cups thing? That example would confuse the hell out of anybody.

What annoys me generally (about this study and similar others) is the “magic bullet” aspect; that every topic in mathematics should be approached in the same way. I find the effect of various strategies varies wildly based on the the topic being taught and the context.

If you’re asking about the stacked styrofoam cups problem, no.

If you’re asking about the measuring cups that are empty, 1/3 full, or 2/3 full, then yes that’s the example used in the article.

I’m not sure I see a silver bullet here. I do see a lot of room to research how transfer of abstraction works.

I also think that, since I view math as a collection of abstractions, that this is something I need to pay attention to in my own teaching. I mean that not in a prescriptive way, but rather as a facet to pay attention to in my reflections on the effectiveness of a lesson.

There’s another, closely related skill that they need – being able to distill a complicated problem into simple examples.

As an able Mathematician, if I forget a result or come across a new problem, I can observe what’s important, produce a simple example, and explore that – then apply the results to my complex problem. That holds for university-level Maths, but it also extends to checking basic power rules with powers 2, for example.

Being able to simplify a problem, and then apply the results to the original question, is for me one of the signs that a student will ‘make it’ in Maths. It’s something that many people simply never get the hang of – how do you try to teach it?

Yeah, I mean the 1/3 full, 2/3 full and so forth. I mean, if you’re going to do a study like that, you ought to at least use a

goodexample.Overall, I admit I might be reading between the lines a little too much. It’s just most things I’ve read on the topic come predisposed with a “everything must be

thisway” message.Also fun: a teacher at our school used to teach welding. He put two problems on a test: given a hiker going 4 miles north and 3 miles east, how far is he from base camp? And later: given this joint with these sizes (4 inches and 3 inches) how long should the brace be? Essentially everyone got the second problem right and the first wrong, even though it’s exactly the same problem. (He even had pictures for both of them, so it wasn’t a reading issue.)

Of course, there was more than just an abstraction issue here; as the students explained, the second problem “wasn’t math”.

You should try that sometime (put the exact same problem twice on a test and see how students do).