By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.

I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.

Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have one class and a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.

In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”—to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.

To set this in context, it is important to understand an underlying belief espoused in my school of education: i.e., there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

As someone who learned math largely though mere exercises and who now creatively applies math at work, I have to question this thinking.

Discovery learning in math: Exercises versus problems

by Barry Garelick

Me, too.

## 12 comments:

The assumption is that they are sincere in their attempts to understand the problem of math education. Perhaps they are sincere, but their lack of content knowledge prevents them from thinking outside of their ed school box. That would be ironic.

Actually, I find many to be very arrogant. They are protecting their turf. Process trumps content. But everything falls apart when you get to the details. Look at any publisher's multiple math series for 7th and 8th grades. It's not a different approach to the same material. It's different (less) material. The assumption is that their way is better. they're talking about apples and oranges.

At best, one could argue that a slower approach is better than a faster approach, but if you look carefully at the sort of series they push (Everyday Math, CMP, IMP), you will see that the goal is not to prepare kids for top end STEM careers. The content is different.

The content is different because they expect that what's missing will be discovered. I'm only slightly joking.

But there IS a difference between problems and exercises, and the difference Barry noted is even an excellent description. The problem is their denigration of exericses, and failure to understand the connection, not their taxonomy.

This is right: there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation.

The problem is that they are 100% wrong on how you get from one to the other.

Competent math teachers and professors know that you need to do the EXERCISES to automaticity before you can tackle problems. Once every way of looking at the basic problem, the primitive parts, the procedural parts, the definitions, etc. is known to you, then you have a full toolbox.

You can then tackle a real problem set with real problems where you can't see the immediate procedural parts, because you can start working from the definitions and procedures you know and move into that new space. They have no idea you need automaticity, because either they've forgotten what it took to get there themselves (the case for many profs) or they never did.

Their problem is they've got just enough truth in their explanation to hoodwink the unsuspecting, even if it's themselves. They have no idea th

When I talk with many educators about math, they think that they are using a better approach to the same material. Perhaps they can get away with this in the lower grades, but the difference really shows up in 7th and 8th grades.

Our school used CMP for years and claimed that it taught problem solving and critical thinking. (Whether it really did is another issue.) Unfortunately, it did not cover enough material for students to make the transition to geometry as a freshman in high school. I don't think they expect students to discover the missing material. They just don't like the material. They don't like the idea of factoring polynomials. So we had K-8 educators deciding that it was not important to make sure that the K-8 math curriculum matched up with the curriculum in high school. This finally changed when enough parents and high school teachers complained.

Now we have a proper sequence of pre-algebra and algebra, and nobody is complaining about a lack of problem solving skills. The K-6 supporters of Everyday Math, however, still haul out those justifications.

I'm a math specialist at an elementary school and we are using Singapore Math. It does do an outstanding job of scaffolding and the exercises do lead to the development of serious problem solving skills. Because we ability group our kids I can see where the read difference comes in with the need for exercises vs. just throwing problems at kids. There ARE kids who have the ability to learn in the constructivist manner- the same kinds of bright kids who are natural learners and were able to become readers with whole language- without explicit instruction. In fact, I see a parallel between the new math programs and the whole language philosophy in terms of the misguided desire to make everything fun and the disdain for direct instruction. In my experience, most kids need and WANT direct instruction and the lower the kids are performing the more explicit your instruction needs to be.

>>>I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly.

Sure, but student led discovery is much more time consuming than guided discovery, especially if the person is not interested in discovering new material or the relationships between the new material and the known or doesn't have the background necessary to do so.

The CogAT people only recommend discovery for those in Stanines 7-9. http://www.riverpub.com/products/cogAt/pdf/cogATshort.pdf

The rest are thought to need more support. This is of course in conflict with the ed school movement of full inclusion based on age and whole class teach to the bottom.

>> As someone who learned math largely though mere exercises and who now creatively applies math at work, I have to question this thinking.

Where did your problem solving skills come from? Mine came from life, but math was the real discipline builder - independent study of Dolciani from gr. 7-11, with the exception of Geo. Dolciani builds logic and problem solving skills unlike the Gr. 7-12 math curriculum used in my district, which is a rote algorithm memorization curriculum.

It seems the whole idea of student-led discovery is to slow down the math and make sure no child gets ahead, combined with a steady supply of paying customers in the community colleges. I have yet to see any scheme where a child can demonstrate mastery and move on.

"...but student led discovery is much more time consuming than guided discovery..."

Exactly, if it's done right. I've harped about this for years. You can always trade breadth for depth, but their pet curricula are a mile wide and an inch deep. How does that work? It doesn't. Discovery and problem solving are only covers for lower expectations all around. I finally figured this out when the curriculum head at my son's old private school told me that Everyday Math (versus Singapore Math) was better for "their mix of kids". The people in charge were intimidated by Singapore Math, but they had to have pedigogical and "mathy" cover for lower expectations. They talk about discovery and critical thinking because those are things you can't easily quantify. It's all quite convenient.

--Dolciani builds logic and problem solving skills unlike the Gr. 7-12 math curriculum used in my district, which is a rote algorithm memorization curriculum.

Can you tell me what you're talking about, specifically? What is a "rote algorithm" curriculum in Algebra 1, Algebra 2, Trig and Precalc?

I used Dolciani Alg 2, structure and method 20 odd years ago. It was decent (though I still have no idea why anyone raves about it), and filled with *exercises* of the exact kind that I support: exercises that "build skill" by forcing you to do something until it's automatic. There are no problem solving skills until you've got those exercises down, and Dolciani does the exercises. Not exactly constructivist student centered stuff. So what does a rote curriculum look like to you? And what does a student centered constructivist one look like?

But to the bigger picture, a student centered discovery learning system is meaningless unless it defines WHAT THEY WILL LEARN.

Most student discovery is garbage because they will learn nothing. If you are very lucky, and you're supposed to be deriving the area under the curve from limits, but all you manage to do is learn to interpolate between points, at least you've gotten somewhere. Most student discovery won't lead to any insight, won't lead to any clarity, because they won't have discovered anything. There's a reason it took several thousand years of math to get to where we are now.

And it isn't just time consuming for the student. Done right, it is for the teacher, too. It's FAR MORE time consuming because you must carefully intricately delicately constantly adjust their discovery to make sure it goes in the right direction.

I'd like to comment on Allison's previous comment regarding the fact that there is in fact a difference between exercises and problems. Absolutely correct, and also correct that the ed school professors exploit the dichotomy to the advantage of whatever argument they're trying to make.

The trickery and treachery of their argument is when they use it to "prove" that the students in Asia who outperform the US on the international math test TIMSS are nothing better than performing monkeys. They maintain that the questions on TIMSS are not "problems" but rather "exercises" and that the students have been trained to know the procedure to come up with the answer.

This is simplistic. The problems on TIMSS are many times multi-step problems which require students to have mastered some level of "exercises" in order to analyze what needs to be done to solve them. Thus, even a 2nd grade problem as simple as "Mary gives the grocer a $5 bill for a loaf of bredand receives $3.67 back in change. How much would three loaves of bread cost?" While some may scoff and write this off as "inauthentic" math, it does require the student to be able to put together concepts sequentially. You have to solve one problem to get to the other; it requires mastery and understanding of subtraction and multiplication.

But the purveyors of the "exercise vs problem" dichotomy maintain that these kids are just performing monkeys and that therefore this test proves nothing. A REAL test would have the open-ended, ill-posed problems that they like to call "real world". They use this as an excuse -- stating the in the US, we are striving to teach students "authentic" and "real world" math. Thus, if US students do poorly, it's because they don't waste time on "exercises" when it's really the "problems" that matter.

I think it was SteveH on KTM who said something to the effect, that a true expert gets to the point when many problems break down into "exercises". That is, the more skilled you are at your subject, the easier it is to see a problem as a sequence of exercise-like steps in order to solve it.

Also, as you point out, the step from getting to "mere exercises" to "problem solving" takes much practice and some time. It is not something that is taught directly. This is also a misconception in the education world. That the pedagogy must mimic what the professional does. Thus science classes should require students to do "inquiry based" projects with minimal guidance. And similarly with math, a problem-based learning approach will force the students to learn the tools necessary to solve the problem in a sort of "just in time" approach to learning. The "just in time" approach is a business inventory model that does not convey to learning. Connected Math Program is based on this model. A look at a CMP lesson briefly might convince the casual observer that it's good because all the concepts are there. Yes, it's all there, but not in any sequence that will do students good.

But the epistemological approach is the one taken. To do math, you have to think like a mathematician, so give the students problems that are unfamiliar because they are going to have to learn how to apply prior knowledge to new situations.

Coherent and sequential.

These concepts are desperately lacking in most math curricula, and very few teachers could possibly have the time and skill to put them into discovery learning for a classroom of 15, much less 30.

We haven't spoken enough about the lack of coherence and lack of sequence.

Your paper on Everyday Math addressed it, but it's true of every textbook out there.

Someone bright once asked me why, "in one sentence" even the private schools with essentially infinite resources had such terrible math programs (the idea being that this eliminates "low expectations" from the immediate answer). My best answer was

they believe education theories too strongly based on their adult experience.

How would you phrase it?

btw, barry, do you have an email address where I can write you? or would you drop me a line and send it to me? My profile in blogger has mine.

I think both views are wrong.

Correctly constructed exercises are scaled in difficulty, each requiring more thought than the last and building upon what is already understood WHILE firmly connecting new "tweaks" to the old foundation. Exercises and drills are different. The problem with exercises without problem solving is that they become no better than drill, which has a place, certainly, but is the development of speed, not though. The problem with most American "problem solving" approaches is that, even if successfully tackled, they seem too disconnected from the topic at hand and do not contribute to further development of mathematical sophistication.

Anonymous:

I don' know what "both views" are in your note. The ed school view is that word problems themselves are viewed as exercises. I think this is how you are viewing what you call American "problem solving" approaches. There are degrees of difficulty in word problems as well. The word problems found on Singapore's exams, for example, are fairly complex and beyond the reach of many US students. But the ed school view is that even those problems are "exercises" and don't contribute to "further development of mathematical sophistication" as you put it. In fact, it takes much practice and time to reach the level of sophistication you speak of. It does not come about by giving students problems for which they have had little or no prior knowledge as is the current mode of thinking in ed schools.

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