**The Advantage of Abstract Examples in Learning Math**

Kaminsky, J. et al.

*Science.*(2008)

When learning new mathematical concepts, for example the concept of mathematical groups of three objects, is it better to start with a concrete example or with a generic representation? This is the question Kaminsky and colleagues at Ohio State University investigated in their 2008 article.

80 college students were divided in four groups, one of which received an abstract introduction while the others received

*one, two or three*concrete examples (one of which is illustrated below as "Concrete A").

The other two concrete examples involved pizza slices and tennis balls, and were presented in a like manner. All concrete, and the one generic, representations were taught together with multiple-choice exercises. Thus, which method was used for teaching counts here as

**the independent variable.**

**The dependent variable was transfer:**whether or not participants could answer multiple-choice questions (isomorphic to the questions used during learning of the examples) regarding the exercise below:

(This picture is actually from a different Kaminsky article,

"Do Children Need Concrete Instantiations to Learn an Abstract Concept?",

but she refers to it in the present article.)

**The results**were that just learning the one generic illustration gave superior learning to learning one, two, or three concrete examples.

Kaminsky et al. then conducted three more experiments. In

**experiment two**, students were explicitly helped to find the analogies between the concrete examples. This did not affect transfer.

In

**experiment three**, students were asked to reflect on any similarities in the concrete examples and write their reflections down. The results were bi-modal. About half the group achieved high transfer, while half performed as before.

In

**experiment four**, half the students were taught only the generic representation, and half were given first a concrete example and then the generic representation. Transfer was higher for both groups compared to the students in the previous experiments, but even here the one generic representation gave better transfer than concrete-then-generic.

The results of experiment 1 and 4 are illustrated in the figure below:

**Criticisms:**critics have argued that Kaminsky's concrete examples were not concrete enough, which is strange because they seem to be precisely the kinds of examples math teachers use all the time. A more relevant criticism is that the transfer test (the children's game) was abstract rather than concrete, so it's no wonder that the students who learned the generic representation performed better. I find this criticism to be quite important. However, I think that it is reasonable to predict that results would have been very similar on a more concrete transfer test. Other critics have questioned whether results from college students are generalizable to children. The answer seems to be: yes. Kaminsky did a very similar study with sixth-graders and here are the results from that study:

"Do Children Need Concrete Instantiations to Learn an Abstract Concept?"

Kaminsky et al., available here

**So what does this imply for teachers?**Kaminsky et al. argue that giving generic representations is superior to giving concrete examples to introduce new concepts and methods. However, it is conceivable that transfer would be even better if the sequence was generic-then-concrete. The main idea seems to be that introducing concrete examples before the generic representation somehow locks the students into a restricted way of looking at the concept. Also, it may happen that the extraneous information in concrete examples distract students from focusing on the core concept. Most teachers have experienced that concrete representations help students understand a problem or concept, as is supported in the Kaminsky study on children. But we need to be careful and evaluate not just the direct and immediate gains in understanding, but also more long term mastery and transfer.

The original article, published in Science, is available here.

**Challenge:**Help me create a

*good*task which tests abstract vs. concrete and I'll test it, on 40 students or so, and do all the rigorous stats and everything. I actually need to do an experiment anyways for my psychology course-work and this is the most interesting idea I've had so far.

I read about this before I started blogging, and to me it seemed full of holes. I wrote a long response then in email, and then a shorter one on my blog after Ben Blum-Smith wrote about it.

ReplyDeleteThe link says the post is missing on your blog. Can you please re-link?

ReplyDeleteHowever, I checked out Ben Blum-Smith's posts and I have to say: I disagree. Yes, the examples are very poorly worded. But because learning took place successfully and quickly, how does the wording matter?

And yes, the examples are not as concrete as we get with well-crafted wcydwt lessons. But the transfer task wasn't very concrete either, and in any case - do we have ANY reasons to believe that MORE concrete would give better results?

And yes, concrete examples are important in order to connect mathematics to students other experiences - but Kaminsky doesn't say to get rid of concrete altogether. Start with generic, then do whatever you think is best - is the message I get.

Hi Julia - just skimmed your post and read this last comment. Will give the post a fuller read soon. In the meantime -

ReplyDeleteMy objection isn't that in the concrete condition, the lesson was poorly worded, or not concrete enough. It's that the way the study "equalized instruction across conditions" made it irrelevant to judging the (appropriate) use of concrete situations in teaching, because the structure of the lesson was at its heart an abstract structure. A good concrete lesson isn't just "more concrete" or "better worded," it's completely built around

usingthe students' preexisting knowledge and intuitions about the concrete situation to develop the relevant mathematics. (This is what wcydwt is all about.) Meanwhile, the lessons in the study actually seemed designed todivorcestudents' knowledge and intuition about the concrete situation from the math they were supposed to do. I wrote a followup post in which I discuss this in a whole lot more detail.Ben, using student pre-existing knowledge is what I mean by a good concrete lesson. But there must be a generalization taking place for mathematical knowledge to be constructed and transfer to take place. Look at Dan's recent "how many points do you need to make a parabola" for example. What students will take away from that lesson, if he doesn't follow it up with a mathematical motivation or proof, is that "parabolas need three points because otherwise I don't know if the ball goes in the basket". Will they be able to apply this to other situations with parabolas? Is that what we call conceptual understanding?

ReplyDeleteI think Kaminsky would say that it would have been better to start with a general parabola and show that it needs three points, and only later illustrate with the basketball analogy.

Of course it's possible that good wcydwt-types of concrete learning experiences build conceptual understanding and promote transfer better than generic/abstract representations. But where is the research that supports this idea? I wasn't able to find it but if you know of any I'd be grateful to see it.

Without having put too much time into this discussion I would just like to make a small comment.

ReplyDeleteIt seems to me that Ben's criticism is valid. There is often a significant gap between what is tested and what is supposed to be the lesson learned. That math teachers have a concensus about what concrete examples are tells us more about the context created by math books than about what should really be counted as concrete/abstract.

I am still sceptical though to the Idea of bringing out abstract concepts via concrete examples. Of course you can use clock arithmetics as a concrete example of a group of order 12, but do we really want students to think about how they count on a watch when doing calculations in Z/mod(12), or to try and think about a new time system when doing calculations in Z/mod(5)?

Shouldnt it be the other way around, that becoming familiar with and better at doing calculations in finite abeian groups should help students in doing clock arithmetics, or counting on a calendar or whatever? The best way of teaching the arithmetic of finite abelian groups should be to simply state the rules of the arithmetic and then start doing some simple calculations to illustrate how everything is supposed to work. After doing this it might of course be desirable to let the students see how effective this kind of computations can be in concrete examples i.e. real situations in the real world.

Let mathematics be mathematics, not only at university level but at all levels of education.

/Thomas

Thomas, point taken. So what we can conclude is do is that if you're going to do it concrete, do it well. But I'm still surprised by the difference in learning and transfer.

ReplyDeleteHi Julia, yes, I think that's the take-home.

ReplyDeleteI started to write a very long reply yesterday, and finished it today intending to write more about this in the future on my blog and link here when I do. I didn't notice your last comment till I finished so not sure it's still a propos, but since I spent so much time on it I'm just gonna post it. It will be a few comments b/c it's too long to be just 1. Sorry for the one-step-behind thing.

*****

Okay now I have read everything thoroughly.

I think it will help to separate the conversation about the Kaminski article from the conversation about the issues the article raises.

I started to write down my thoughts on the issues the article raises first, because I sort of feel like they're the elephant in the room if we just go into a conversation about the article on its own merits. But then I spent 2 hours and wasn't done, so I'm just going to give some brief thoughts. I plan to blog about this and I'll link here when I do, so you should get a trackback. It'll probably not be for a few months.

I think that conversation around the topic we're dealing with has a strong tendency to get stuck. Some people want to say "beginning concrete is better," some people want to say "beginning abstract is better." Both camps have in our minds very specific (good or bad) images of the classroom - down to the manner and order in which things are happening - that motivate our stance, but these specific images are not the subject of the conversation - instead, the broad question "concrete vs. abstract." We tend to talk in circles until these images get moved from the background to the foreground of the conversation and dealt with in detail.

This is what I intend to blog about at a future date. In the meantime let me just say that I think the concrete vs. abstract distinction needs a lot more thought. In particular, from a teaching point of view I think it's very important to regard the distinction as

ReplyDeleterelative to the learner. Functions and groups might be characterized as abstract ideas by Kaminski et. al. but to people who have a reasonable level of mathematical development (you, me, them...) they are pretty concrete, in the sense that we perceive them as real things, have intuitions about their behavior, and can work with them flexibly, and possibly use them for purposes not originally intended, just like something we picked up off the table. I don't know very much about category theory, so categories and functors are still pretty abstract to me, though not as abstract as they were before I studied topology and worked a bunch with the fundamental group, and then realized that it is a (concrete example of a) functor. Last year I wrote in somewhat more detail about this idea about concreteness being relative to the learner here. I suspect that if you think about the last time you satisfyingly learned a new concept, your learning began when you were able to relate that concept to things that were already concretefor youin the sense I've been describing. (Let me know if this isn't true.) I was reading an article today on Japanese teachers' shared conception of what good mathematics education looks like (the first article in the latest JRME), and one of the principles of Japanese math teaching according to this article is that a good lesson should build on students' previous knowledge. This is basically the same point.Part of the reason I think you've had trouble finding research that supports the idea that "wcydwt-types of concrete learning experiences build conceptual understanding and promote transfer better than generic/abstract representations" is that I have only ever heard this question (is concrete or abstract better?) asked by people who want to show that abstract is better. I don't think people who are into things like wcydwt are, in general, interested in the question at this broad a level because they are too concerned with the specific form of the lesson. Dan Meyer is an example. He's the very dude who coined the term "wcydwt," and yet he's also spent the last two months doing a weekly series on problems that are embedded in "real life"

ReplyDeletebadly. Nothing he loves more than a good concrete-context-based problem; nothing he hates more than a bad one.Now to the article. If you evaluate it on its own merits, apart from any thoughts you have about whether its finding is correct, I really think there's nothing to do but to conclude that if it is claiming to show that it is better to begin with an abstract than a concrete example as a general principle, it has demonstrably shown no such thing. Maybe I'm wrong, but I feel so confident of this that I am going to attempt a mathematical proof. Let me know what you think.

ReplyDeleteDefinition: A "good concrete lesson" is a lesson in which students' knowledge and intuition about a concrete situation is used explicitly and in a natural way to bring out key mathematical features and ideas.

Definition: A "bad concrete lesson" is a lesson in which a mathematical idea is embedded in a concrete situation, but the lesson sends the message that students should not use their knowledge and common sense about the situation in reasoning about the mathematical idea.

(See Dan's introduction to his series on bad contexts for further elucidation of this second definition.)

Lemma: a) None of the concrete lessons in the study are good concrete lessons in the above sense. b) All the concrete lessons in the study are bad concrete lessons in the above sense. (See the post I linked in the previous comment for my argument why this is true.)

Do you object to the lemma or consider it insufficiently demonstrated? If not:

Postulate: It would be reasonable to expect substantial differences in transfer from good vs. from bad concrete lessons, as defined above.

Do you object to the postulate? (I think this is self-evident, but that's what they said about the parallel postulate ;) If not:

Theorem: The study's results do not apply to good concrete lessons.

Proof: According to the postulate, good and bad concrete lessons can be reasonably expected to have different transfer outcomes. It follows that results obtained regarding the transfer outcomes of bad concrete lessons do not have implications regarding the transfer outcomes of good concrete lessons. According to the lemma, the Kaminski study tested only bad concrete lessons and no good concrete lessons. It follows that the study's results regarding concrete lessons are limited in scope to bad concrete lessons. By the above, such results do not have implications for good concrete lessons. QED

If you question the postulate or consider the lemma insufficiently demonstrated, let's have that conversation. Short of that, this looks pretty airtight to me - what do you think?

Ben, The first half of my week is incredibly hectic but I will reply with a challenge or two asap.

ReplyDeleteBen,

ReplyDeleteI find it a difficult to organize my reply so I think I'll just answer you point by point.

First, I don't think the concrete vs abstract necessarily runs as deep into other issues as you say. For me, a more important distinction is between creative investigatory work on one hand, and on the other hand routine application of rules handed down from teacher or textbook. Whether the investigations or exercises use concrete or abstract is only important to me from a practical point of view: which one really helps students best learn and transfer that knowledge, while at the same time understanding and developing appreciation of the nature of mathematics.

So if there is an elephant in the room, I don't see it.

Also, I don't think that the only people asking "concrete vs abstract" are the people on the abstract side. My limited experience of these types of discussions (I blog because I have no one to talk to about this irl) is that on one hand we have the concretists saying "it's OBVIOUS that conrete is better, and in fact lets institute all these awards and prizes for teachers who innovate and use laboratory materials in mathematics" and on the other hand the skeptics who say "wait! what evidence is there in favor of either side?" I've yet to meet or hear from someone siding with "abstract".

Regarding the Kaminsky article, I agree with every part of your proof that Kaminsky does not show that abstract is better than good concrete.

ReplyDeleteHowever, I would argue that a mathematical style proof is inappropriate in such a situation.

Kaminsky's article can best be understood as a study on human cognition, and should therefore be analyzed and evaluated according to the standards of cognitive science such as cognitive psychology.

In psychology, the kinds of problems that occur in Kaminsky's study are pretty standard. We have the difficult generalization due to restrictions in sample, operationalized variables, etc. In this case, we can agree that the variable "concrete or abstract" was operationalized in such a way that it didn't do justice to concrete.

That's definitely grounds for criticism, but not for a dismissal.

Compare with a classic experiment by Craik and Tulving on levels of processing: subjects who consider the meaning of words remember them better than subjects who just judge whether the words were written in capitals or if they rhyme with some other words. Such experiments are the staple of cognitive psychology, even though we could certainly argue that Craik and Tulving fail at showing that people normally learn better by considering the meaning of what they are learning. After all, how often are we trying to memorize lists of words?

Never the less, many similar experiments, and a logically consistent theory of memory, support the idea that processing meaning enhances encoding.

This is how I judge the Kaminsky article as well: it points in a certain direction, but needs a consistent theory and more supporting research (which addresses the problems of this article) to be fully convincing. We're not there yet, maybe never will be, and that's fine. Right now, what Kaminsky has done for me is sparked my own interest in experimenting with abstract and concrete with my students.