# Abstract concepts vs. concrete examples for teaching math

April 26, 2008 3:24 PM Subscribe

A new study in Science claims that teaching math is better done by teaching the abstract concepts rather than using concrete examples. From an article by the study authors in Science Mag (requires subscription):

*If a goal of teaching mathematics is to produce knowledge that students can apply to multiple situations, then presenting mathematical concepts through generic instantiations, such as traditional symbolic notation, may be more effective than a series of "good examples." This is not to say that educational design should not incorporate contextualized examples. What we are suggesting is that grounding mathematics deeply in concrete contexts can potentially limit its applicability. Students might be better able to generalize mathematical concepts to various situations if the concepts have been introduced with the use of generic instantiations.*As if we could not make math any more abstract and inaccessible to the average student....

This reeks of bad science to me.

posted by caddis at 3:33 PM on April 26, 2008

This reeks of bad science to me.

posted by caddis at 3:33 PM on April 26, 2008

I did a study on ideas related to this for my undergrad psych project.

In my study, students were given examples that were equivalent mathematically, but differed (from one subject to the next) in how concrete the labels (given to the entities in the problem) were. They were also given varying instructions as to how to process these different examples before going on to complete the performance measures.

When people are given multiple examples of a principle, they will generalize the principle that underlies those examples if they explicitly compare the examples, trying to find correspondences between the solution paths for the problems. If you just give them a bunch of examples to study, the

The effect of abstract vs. concrete labeling was not measurable with my sample size, although the trend was that advantage given by abstract labels did not exist for students that had little math background, but was greater for students that had more of a math background.

posted by Jpfed at 3:46 PM on April 26, 2008 [5 favorites]

In my study, students were given examples that were equivalent mathematically, but differed (from one subject to the next) in how concrete the labels (given to the entities in the problem) were. They were also given varying instructions as to how to process these different examples before going on to complete the performance measures.

When people are given multiple examples of a principle, they will generalize the principle that underlies those examples if they explicitly compare the examples, trying to find correspondences between the solution paths for the problems. If you just give them a bunch of examples to study, the

*vast*majority (of my participants, at least) did not look for those correspondences unbidden. Participants that did not compare the examples did much more poorly on subsequent performance measures than participants that did.The effect of abstract vs. concrete labeling was not measurable with my sample size, although the trend was that advantage given by abstract labels did not exist for students that had little math background, but was greater for students that had more of a math background.

posted by Jpfed at 3:46 PM on April 26, 2008 [5 favorites]

Abstraction is not inaccessibility; usually it means quite the opposite. To abstract something is to derive general rules that are applicable to many instances. This makes a great deal of sense, and accords well with Paul Lockhart's point about how grinding boredom of mathematics education destroys the interest of students in learning the fascinating subject of mathematics.

posted by aeschenkarnos at 3:52 PM on April 26, 2008

posted by aeschenkarnos at 3:52 PM on April 26, 2008

That's interesting Jpfed. In the sciencemag article the authors say that for one group of students they asked them to explicitly compare the different elements of the concrete examples and write down the similarities. What they got was a bimodal distribution, with some (about 44 %) transferring the concepts well after the explicit comparison while the others were not helped.

posted by peacheater at 3:52 PM on April 26, 2008

posted by peacheater at 3:52 PM on April 26, 2008

peacheater-

It depends on whether the students are comparing superficial vs. deep aspects of the problem structure. Both my study and previous work suggests that there are many aspects of examples that can be compared- the entities that give rise to the problem (trains? water levels?), the setup of the solution, the individual steps from the setup to the conclusion, etc.

If people compare superficial aspects of the problem, noting that one problem is about balls drawn from an urn with replacement and another problem is about loaded dice, they will not go far. They have to explicitly compare solution paths to derive benefit.

There will be other variables at work, but this is one issue that has been treated previously in the literature, and borne out by my work.

posted by Jpfed at 3:57 PM on April 26, 2008

It depends on whether the students are comparing superficial vs. deep aspects of the problem structure. Both my study and previous work suggests that there are many aspects of examples that can be compared- the entities that give rise to the problem (trains? water levels?), the setup of the solution, the individual steps from the setup to the conclusion, etc.

If people compare superficial aspects of the problem, noting that one problem is about balls drawn from an urn with replacement and another problem is about loaded dice, they will not go far. They have to explicitly compare solution paths to derive benefit.

There will be other variables at work, but this is one issue that has been treated previously in the literature, and borne out by my work.

posted by Jpfed at 3:57 PM on April 26, 2008

aeschenkarnos-

Accessibility is different from generalizability. For example, I know from the various contexts in which I've seen the terms that groups, rings, and fields are very general concepts. But I don't

I think I would get them if someone said something like "here are some entities that are groups; notice how they all satisfy such and such properties."

Now would be an excellent time for someone to pop in with a quick derail and give some examples of groups, rings, and fields./hint

posted by Jpfed at 4:20 PM on April 26, 2008

Accessibility is different from generalizability. For example, I know from the various contexts in which I've seen the terms that groups, rings, and fields are very general concepts. But I don't

*get*them.I think I would get them if someone said something like "here are some entities that are groups; notice how they all satisfy such and such properties."

Now would be an excellent time for someone to pop in with a quick derail and give some examples of groups, rings, and fields./hint

posted by Jpfed at 4:20 PM on April 26, 2008

It's a shame that the articles on this go into so little detail about the actual experiment that was performed; the tests that were administered were extremely simple, and I think it offers far greater insight than the broad pronouncements and general examples in the popular press article. The conclusion is not that learning abstract concepts leads to better application in concrete situations, it's that learning abstract concepts leads to better application in abstract situations.

The problem, as presented in the "concrete" examples, was to learn addition modulo three. So there are three possible numbers 1,2,3, and if your addition is greater than three, than subtract three until you get down into the range of 1,2,3. These concrete examples used a pictogram for each number, and each pictogram clearly indicated that it represented 1, 2, or 3.

The "generic" problem replaced the pictograms with shapes, none of which would indicate a relationship to a number, so the problem was now to learn the rules of a group from abstract algebra. Likewise, the transfer problem similarly used pictures, none of which indicated a number, and therefore also represented the different problem of learning a commutative group of size three.

So when given the concrete examples, test subjects could easily use the arithmetic they know, the successfor function, counting on fingers, or whatever. However, these concrete problems are not necessarily isomorphic to the transfer and generic problems, as the concrete examples have another useful and familiar aspect to them, that of sequence and arithmetic.

The transfer problem and the "generic" problem were both more difficult, as you either had to learn the new specific rules of the group, or learn to map the integers to these abstract objects. There was significantly more learning going on when one learned the rules of the "generic" problem than when learning the "concrete" problem, so it's not surprising in the least that when presented with what amounts to an identical problem, those who were previously exposed to the "generic" idea could get it again.

It's a shame that the basics of this study are hidden behind a pay wall. And even for those who have access to the article, the real meat is hidden by yet another wall, in the supplementary material. The journals Science and Nature (and other closed journals) need to die or go open access, because this study is seriously open to misintrepretation, and letting people actually see the study would go a long way towards helping.

posted by Llama-Lime at 4:25 PM on April 26, 2008

The problem, as presented in the "concrete" examples, was to learn addition modulo three. So there are three possible numbers 1,2,3, and if your addition is greater than three, than subtract three until you get down into the range of 1,2,3. These concrete examples used a pictogram for each number, and each pictogram clearly indicated that it represented 1, 2, or 3.

The "generic" problem replaced the pictograms with shapes, none of which would indicate a relationship to a number, so the problem was now to learn the rules of a group from abstract algebra. Likewise, the transfer problem similarly used pictures, none of which indicated a number, and therefore also represented the different problem of learning a commutative group of size three.

So when given the concrete examples, test subjects could easily use the arithmetic they know, the successfor function, counting on fingers, or whatever. However, these concrete problems are not necessarily isomorphic to the transfer and generic problems, as the concrete examples have another useful and familiar aspect to them, that of sequence and arithmetic.

The transfer problem and the "generic" problem were both more difficult, as you either had to learn the new specific rules of the group, or learn to map the integers to these abstract objects. There was significantly more learning going on when one learned the rules of the "generic" problem than when learning the "concrete" problem, so it's not surprising in the least that when presented with what amounts to an identical problem, those who were previously exposed to the "generic" idea could get it again.

It's a shame that the basics of this study are hidden behind a pay wall. And even for those who have access to the article, the real meat is hidden by yet another wall, in the supplementary material. The journals Science and Nature (and other closed journals) need to die or go open access, because this study is seriously open to misintrepretation, and letting people actually see the study would go a long way towards helping.

posted by Llama-Lime at 4:25 PM on April 26, 2008

Also, the very best way to learn math is to simply look around you.

posted by Llama-Lime at 4:30 PM on April 26, 2008 [2 favorites]

posted by Llama-Lime at 4:30 PM on April 26, 2008 [2 favorites]

@Jpfed, honestly I think there's room for both kinds of teaching. While being given examples of groups, rings and fields is unlikely to be enough to get you to understand what they are, showing those examples to you and then clearly

posted by peacheater at 4:30 PM on April 26, 2008 [1 favorite]

*explaining*what makes each of those items groups, rings or fields might be. But I think they have a point when they claim that giving examples with no abstraction is not enough. I remember as a kid being given word problem after word problem and finding them tedious, but easy enough. But this was only after I had understood that if you have x groups of y things each, the total number of things is xy. I don't think I would have grasped this concept as well if I'd just been given the word problems alone.posted by peacheater at 4:30 PM on April 26, 2008 [1 favorite]

Jpfed: Good Math, Bad Math has a series of blog posts that are a good intro to group theory. start reading from the bottom ("What is Symmetry?") and work your way up.

Also, I'll note that there's no freaking way I would have understood groups without a few concrete examples. I could have stared at the rules that define a group all day and not have come up with a useful example on my own.

posted by jewzilla at 4:32 PM on April 26, 2008 [1 favorite]

Also, I'll note that there's no freaking way I would have understood groups without a few concrete examples. I could have stared at the rules that define a group all day and not have come up with a useful example on my own.

posted by jewzilla at 4:32 PM on April 26, 2008 [1 favorite]

Llama-lime, I agree with you that the "generic" problem required more learning than the "concrete" problem they gave. I think the authors' conclusion is interesting and quite possibly true but that the study needs more work.

posted by peacheater at 4:35 PM on April 26, 2008

posted by peacheater at 4:35 PM on April 26, 2008

apparently Captain Obvious works for Scientific magazine.

posted by proj08 at 4:47 PM on April 26, 2008

posted by proj08 at 4:47 PM on April 26, 2008

*As if we could not make math any more abstract and inaccessible to the average student....*

The point is, making things more 'concrete' does not make them more accessible, It just teaches them to solve those specific problems

*This reeks of bad science to me.*

Don't tell me you actually

*enjoyed*those story problems. I

*hated*them. I had no interest in learning math in 'real world scenarios'. Learning how to apply mathematical principles to real world situations is actually an important skill, but it was never and enjoyable part of math for me as a student. On the other hand, the abstract stuff was always fun.

On the other hand, it does seem like it would be a good idea to give kids some story problems (after they've learned the symbolic stuff) in order to figure out how to 'see' where the equations

*are*in that stuff.

And calling something 'bad science' because you disagree with it is pretty weak.

posted by delmoi at 4:49 PM on April 26, 2008

I have a hard time with abstract mathematical nomenclature. It wasn't until I learned on my own that groups dealt with symmetry and permutations that the basic concepts in groups clicked for me. The algebra class I had was entirely abstract and entirely inaccessible, because concrete examples and the ideas behind them were never communicated. I couldn't connect the ideas with anything discrete and tangible, so they never held firm for me.

posted by Blazecock Pileon at 4:53 PM on April 26, 2008

posted by Blazecock Pileon at 4:53 PM on April 26, 2008

actually, more like Captain Naive. The debate between abstract vs concrete has been totally beaten to death.

posted by proj08 at 4:55 PM on April 26, 2008

posted by proj08 at 4:55 PM on April 26, 2008

This brought to mind Judy S. DeLoache's experiments on symbolic thinking in young children. Here's a brief description of one of her experiments, teaching six- and seven-year olds to do subtraction problems that require borrowing. One group was taught using blocks designed to teach math to young children, and another group was taught using pencil and paper. The blocks group took three times as long to learn to do the problems, and one the children in these groups even opined that it would have been easier to learn with pencil and paper instead.

The argument has been made for a while that children find it harder to learn when math or arithmetic concepts are presented in terms of "If Jane has 1 apple and Tom has 2 apples, how many apples are there total?" instead of "1+2=3".

posted by needled at 5:18 PM on April 26, 2008 [1 favorite]

The argument has been made for a while that children find it harder to learn when math or arithmetic concepts are presented in terms of "If Jane has 1 apple and Tom has 2 apples, how many apples are there total?" instead of "1+2=3".

posted by needled at 5:18 PM on April 26, 2008 [1 favorite]

*Now would be an excellent time for someone to pop in with a quick derail and give some examples of groups, rings, and fields./hint*

Sure! A good example of an easy-to-understand group is the integers mod 12 under addition, aka "clock arithmetic". This is the set of integers from 0 to 11, but if you think of 0 as 12, then you can treat these elements as hours on a clock. So for instance, 1 + 2 = 3, and 2 + 2 = 4, but 8 + 6 = 2. Why? If you think in clock terms, 8 hours after 6 o'clock is 2 o'clock (ignoring AM and PM for now). So whenever you add two elements together to get something bigger than 11, you subtract 12 from it in order to make it "wrap around" into something that's between 0 and 11 (inclusive). Other examples: 6 + 6 = 0, 11 + 1 = 0, 10 + 10 = 8 (= 20 - 12), and 7 + 7 = 2. Filling out the whole damn addition table is left as an exercise. Or you could just try this on the integers mod 4, which is the set of integers from 0 to 3.

Why is this set a group under addition? It has an identity, 0, such that x + 0 = x = 0 + x, for each x in the set. Every element x has an inverse y, such that x + y = the identity (ie 0). For instance, 5 is the inverse of 7 because 7 + 5 = 0. Lastly, addition is associative: (a + b) + c = a + (b + c).

The whole mod thing is important to keep in mind. You can map any integer into the set of integers mod 12 just by dividing it by 12 and taking the remainder.

If you throw in multiplication, then you get a ring. But you don't get a field. You get a field if instead you use a prime number as the modulus. For instance, the integers mod 7, under addition and multiplication, form a field. This is a field because it forms a group under multiplication (*in addition* to forming a group under, uh, addition), with 1 the identity for multiplication.

If you can stand more long-winded explanations, I can give more examples.

posted by A dead Quaker at 5:45 PM on April 26, 2008 [4 favorites]

Are they kidding? I hope so.

Refer to 1963 and the "new math".

I was one of the subjects. Interesting. The "test" in my junior high school was carried out on the top 20% of the 8th grade. Went OK. Sort of.

Hurt our brains a lot and we were the "smart" kids.

I would not recommend it unless you are dealing with the top 10% of the class and you understand that it may/may not help the students.

posted by davebarnes at 6:59 PM on April 26, 2008

Refer to 1963 and the "new math".

I was one of the subjects. Interesting. The "test" in my junior high school was carried out on the top 20% of the 8th grade. Went OK. Sort of.

Hurt our brains a lot and we were the "smart" kids.

I would not recommend it unless you are dealing with the top 10% of the class and you understand that it may/may not help the students.

posted by davebarnes at 6:59 PM on April 26, 2008

Continuing A dead Quaker's example, for those with some knowledge of complex arithmetic:

For n=1,2,3..., there are n complex numbers z such that z^n=1. For n=2, these are 1 and -1 (1^2 = -1^2 = 1). For n=4, they are 1, i, -1, and -i. Take this latter case, and multiply any two of those elements together. You get a table like this:

Compare that to the table you get doing mod-4 addition, as A dead Quaker described:

You notice this table looks the same, except that (1,i,-1,-i) have been replaced by (0,1,2,3). So we started off by doing

It's easy to derive this result from Euler's formula, but if you look at it from a group-theoretic standpoint, you don't even need to. This is why groups are, as Jpfed stated, a very general contstruct. Anything you can say about one group is going to hold for the other, and this fact can be useful. For example, integer addition is much easier to compute than complex multiplication. That doesn't matter much here, but these techniques can be very beneficial when you start dealing with larger (even infinite) groups.

Now, being able to churn through the mechanics of modern algebra isn't tremendously useful in a field outside of pure math, but the sort of tricks you learn and intuitions you develop in the course of studying it can be useful all over the place. This is roughly the same dichotomy as is being discussed in the FPP, so I don't think this is much of a derail.

posted by 7segment at 7:30 PM on April 26, 2008 [2 favorites]

For n=1,2,3..., there are n complex numbers z such that z^n=1. For n=2, these are 1 and -1 (1^2 = -1^2 = 1). For n=4, they are 1, i, -1, and -i. Take this latter case, and multiply any two of those elements together. You get a table like this:

1 i -1 -i

_____________________

| | | | |

1 | 1 | i | -1 | -i |

|____|____|____|____|

| | | | |

i | i | -1 | -i | -1 |

|____|____|____|____|

| | | | |

-1 | -1 | -i | 1 | i |

|____|____|____|____|

| | | | |

-i | -i | 1 | i | -1 |

|____|____|____|____|

Compare that to the table you get doing mod-4 addition, as A dead Quaker described:

0 1 2 3

_____________________

| | | | |

0 | 0 | 1 | 2 | 3 |

|____|____|____|____|

| | | | |

1 | 1 | 2 | 3 | 2 |

|____|____|____|____|

| | | | |

2 | 2 | 3 | 0 | 1 |

|____|____|____|____|

| | | | |

3 | 3 | 0 | 1 | 2 |

|____|____|____|____|

You notice this table looks the same, except that (1,i,-1,-i) have been replaced by (0,1,2,3). So we started off by doing

**multiplication**over the 4th roots of unity, but it looks the same,*mutatis mutandis*, as doing mod-4**addition**over the integers [0..3]. The latter was shown to be a group, which means complex multiplication over these elements also forms a group*isomorphic*to it.It's easy to derive this result from Euler's formula, but if you look at it from a group-theoretic standpoint, you don't even need to. This is why groups are, as Jpfed stated, a very general contstruct. Anything you can say about one group is going to hold for the other, and this fact can be useful. For example, integer addition is much easier to compute than complex multiplication. That doesn't matter much here, but these techniques can be very beneficial when you start dealing with larger (even infinite) groups.

Now, being able to churn through the mechanics of modern algebra isn't tremendously useful in a field outside of pure math, but the sort of tricks you learn and intuitions you develop in the course of studying it can be useful all over the place. This is roughly the same dichotomy as is being discussed in the FPP, so I don't think this is much of a derail.

posted by 7segment at 7:30 PM on April 26, 2008 [2 favorites]

I had new math in 3rd grade! It was all groups, sets, and (at the time) weird stuff like that. I think PhD mathematicians were trying to teach the kinds of things they delt with in their jobs every day to 3rd graders.

They didn't give any practical examples of this highly abstract theory. I was considered good at it. I memorized all the concepts and could spit them back out, but I didn't really learn anything useful at all. It was a complete waste of a year of my life.

posted by eye of newt at 7:54 PM on April 26, 2008

They didn't give any practical examples of this highly abstract theory. I was considered good at it. I memorized all the concepts and could spit them back out, but I didn't really learn anything useful at all. It was a complete waste of a year of my life.

posted by eye of newt at 7:54 PM on April 26, 2008

*If you throw in multiplication, then you get a ring.*

Here, because we're talking about just one example, it's difficult to tell what specifically makes this set with these operations a ring.

Comparing examples is helps us determine the specific boundaries of the relevance of the features of an example- assuming that the set of examples we're comparing do not have consistencies not generally held in the overall population of problems to be dealt with. (In other words, the examples may encourage, through their misleading consistencies, to overfit to the learning set).

For example, both examples given (arithmetic mod n, and the nth roots of unity) are periodic structures. Is this a requirement for the suggestively-named "rings", or are e.g. the rationals (having identities for addition and multiplication, and having inverses for all combinations) a "ring" despite not having any such apparent periodicity?

posted by Jpfed at 8:21 PM on April 26, 2008

*needled said: This brought to mind Judy S. DeLoache's experiments on symbolic thinking in young children*

I've noticed in teaching my son (5) math that he really groks the concepts when we don't add extraneous items like blocks or grapes or trains, or whatever concrete examples were used when I was a kid trying to understand why they wanted me to calculate a train schedule.

Once we put a number line on the board, he totally got addition and subtraction. We're doing multiplication and division a little now because he's started to understand the concept that multiplication is really just addition with sets. Although, at the pace he's figuring math out, I have a feeling I'll have to find someone more math literate than me soon...he's got a pretty amazing grasp of concepts I've seen adults struggle to understand.

posted by dejah420 at 12:51 AM on April 27, 2008

*Don't tell me you actually enjoyed those story problems.*

They were my favorites. I then went on to study engineering in college.

I did not call this bad science because I disagree, but because their data set seems limited and weak.

posted by caddis at 9:45 AM on April 27, 2008

Apparently the sample for that study was college students. Many of whom are already likely to be Formal Operational Thinkers (i.e. able to use abstracts properly and translate them into sensible contexts) and in the top 30% of the population. In short, it's a sample consisting largely of those who can easily generalise from the abstract to the application, and who have enough education behind them that this isn't a problem.

What I want to know is if college students are learning on silly concrete examples why they are studying maths at college at all. And why a sample of college students in Ohio State University is relevant for year 8 classes. (Disclaimer: I haven't read the raw study - most of this is second hand from my girlfriend - a former professional teacher who is currently doing a postdoc in education research).

posted by Francis at 11:27 AM on April 27, 2008

What I want to know is if college students are learning on silly concrete examples why they are studying maths at college at all. And why a sample of college students in Ohio State University is relevant for year 8 classes. (Disclaimer: I haven't read the raw study - most of this is second hand from my girlfriend - a former professional teacher who is currently doing a postdoc in education research).

posted by Francis at 11:27 AM on April 27, 2008

My experience has been that concrete examples need to be exactly that: examples of the abstract concept, with the connections explicitly drawn. I think too often, in high school for instance, examples are overused to the point where what you learn is a mechanical process for solving a specific type of problem (this is a common complaint I here levelled towards math and physics).

On the other hand, my university experience with math and computer science courses has been much better: concepts are introduced first, and thoroughly fleshed out before specific problems or examples are provided.

I suppose some people may have an easier time inducing an abstract concept from an example, but I prefer to learn the concept first and use examples to solidify my understanding.

posted by zpaine at 3:38 PM on April 27, 2008

On the other hand, my university experience with math and computer science courses has been much better: concepts are introduced first, and thoroughly fleshed out before specific problems or examples are provided.

I suppose some people may have an easier time inducing an abstract concept from an example, but I prefer to learn the concept first and use examples to solidify my understanding.

posted by zpaine at 3:38 PM on April 27, 2008

Interesting discussion on this study going on at Matt Yglesias' blog.

posted by peacheater at 6:16 PM on April 27, 2008

posted by peacheater at 6:16 PM on April 27, 2008

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