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. I believe that students’ difficulty in solving new problems is more likely to be because they do not have the requisite knowledge and/or mastery of skills—not because they were given explicit instruction and homework exercises.

Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still gelling students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.” One teacher with whom I spoke summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?”

Discovery learning in math: Exercises versus problems (pdf file)

by Barry Garelick

THIRD EDUCATION GROUP REVIEW / ESSAYS, Vol.5, No.2, 2009

make them struggle

education professors: students must struggle

KUMON: "work that can be easily completed"

handing it to the student

## 18 comments:

Hey Katherine, I think that you have a really strong point here. Especially in Mathematics which we've really conditioned kids to hate by making it the 'hard' subject in school. As a Math and Spanish teacher my whole life, I work so hard to build confidence and show eager kids that they can be little math whizzes and Spanish speakers too. Sometimes that takes 50 mere exercise problems that they can get right before working on their problem solving!

wow!

Good for you!

"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."

Kids who know math can look at a problem and apply a straightforward, mathematical solution. For kids who know squat, EVERYTHING is a problem that has to be figured out. That is not a good thing. The route to complex problem solving does not bypass mastery of the basics.

"I believe that students’ difficulty in solving new problems is more likely to be because they do not have the requisite knowledge and/or mastery of skills—not because they were given explicit instruction and homework exercises."

Of course. They confuse mastery of basic skills with rote learning. It's shows complete ignorance of math.

“What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure?"

I'm going to start humming tunes from The Music Man. Forget scales and practice. Just think, men!

“What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure?"The same thing that happens when you put a kid who's been raised on TERC & Connected Math. Most people are not going to exit high school able to come up with complex mathematical solution in unfamiliar environments.

The difference between the TERC kid and the Saxon or Singapore Math Kid is that the latter is going to be able to come up with a solution when he's trying to figure out if he can afford the mortgage the guy from Countrywide wants to sell him while the TERC kid is going to be stuck trying to "generate a procedure."

The problem is that educators know very little math. They don't know how it's used in real life.

One of my specialties is curved surfaces for geometric modeling. I write software for shape design and analysis. I once had to write a routine that would find the intersection of two tensor product polynomial surfaces. This has to be done algorithmically, and there are lots of methods given in the literature. It would be stupid to ignore those solutions just to discover my own. I'm not proud. I'm more than willing to copy what someone else has done, so I studied the literature. In fact, if you write a technical journal article, you better show that you have a full grasp of and reference to all other key articles and books. If you ignore the literature and (re)discover a technique, then it either won't get published or it will get trashed by your colleagues. Ignorance is not treated lightly. Prior art reigns supreme. (knowledge and skills)

However, nobody had a solution that met my need for speed of calculation. I had to create my own solution, but I don't start from scratch, and I don't use some sort of pattern recognition or critical thinking to find a solution. I use my toolbox of mastered skills. First, I have a fast way to convert each polynomial surface into a large set of triangles. All I needed next was to find a very fast way to determine if any two triangles intersect. I don't "discover" a solution. I look at the problem and see how my toolbox of mastered (rote) skills can be applied; vectors, dot products, cross products, parametric equations, different forms for defining plane equations, and matrices.

These skills don't exist in some sort of rote or out-of-context space. They have a meaning and a use. Two independent vectors define a plane. If I take the cross product of the two vectors, I have a new vector that is perpendicular to that plane. I didn't discover that. I was taught that.

Don't educators understand that being creative mathematically requires a whole lot of basic, mastered skills? The larger your tookbox of mastered math skills, the more creative you will be.

I don't know why they have this rote or script hangup. They just don't have enough understanding of math to know if anything they do is correct or incorrect.

It would be stupid to ignore those solutions just to discover my own. I'm not proud. I'm more than willing to copy what someone else has done, so I studied the literature.I develop financial service business applications. The FIRST thing you learn as a programmer is steal EVERYTHING that has aleardy been written. If someone else has devolped a procedure, use it and move on the the next problem. Life is too short for discovery programming. You need to spend your time on the issues that don't yet have a solution.

And the more tools (basic skills) you have in your toolbox, the faster your development cycle.

One of my specialties is curved surfaces for geometric modeling. I write software for shape design and analysis.I wonder just how one would go about discovering this.

I still feel like I'm missing something. I don't fully understand what's going on in their heads.

“What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?”

What 'script' are they talking about here? The script to do the lattice method? Obviously, they are not talking about basic skills here.

There are two issues; basic skills, and the application of the basic skills to problem solving. Once an Everyday Math student selects his/her own method for multiplying numbers, it becomes his/her own script. If a student selects to use a calculator, then that becomes the script.

So what kind of problems are they talking about and what kind of script are they referring to? Even in "traditional" math, there are no scripts. Actually, I remember some sort of box charts I was supposed to use for mixtures, but that is the closest thing to a script I know of for problem solving.

Discovery learning is a separate issue from problem solving.

Many difficulties that students have relate to inflexibility and lack of mastery of their basic skills. When a teacher sees that a student has trouble tackling a new problem, he/she might think that it's a matter of problem solving and that the student is stuck on a script. The script might be a problem, but the solution is not better problem solving skills. The solution is mastery of the basic skills so their script always works.

Educators have hangups about scripts, but they don't seem to make any distinctions. If you discover a new script, then it's OK to use it? When people are confronted with "complex" or "novel" problems, is the goal to apply basic scripts (math skills) in new ways, or is it to come up with a new script?

How can you have a constructive discussion when you don't have a clue to what they are talking about. Educators also seem to be clueless about all of the work that has been done in the areas of mathematical problem solving and systems analysis. The are stuck in their own little world. They wouldn't even know what to Google. So much for 21st Century Skills.

Help! Has the Third Education group's website gone down? I'm unable to find Barry's pdf, or even their site.

I still feel like I'm missing something. I don't fully understand what's going on in their heads.They think that students are not able to apply the basics to more complex problems because one has to be taught how to do so. They hold that traditional math classes may teach students how to do something like solve a "work problem" (John can mow a lawn in 30 minutes working at a constant rate, and Dave can mow the same lawn in 20 minutes. How long to mow the lawn if both work together?) Such problems are generalizable to a whole class of problems, but some hold that they limit students to solve problems that look exactly like these problems. I.e., they don't see it as a step to what Polya calls "look for a similar problem" or "look for a simpler version of the problem" as Allison talks about in a different post. I bring this out later in the article on discovery learning.

Unfortunately you can no longer get that article because Third Education Review was forced to shut down.

yup - it's gone

I can email copies of Barry's article (I hope!)

cijohn @ verizon.net

Barry - is that OK?

I have the 22-page version.

That's fine with me.

"...but some hold that they limit students to solve problems that look exactly like these problems."

It all depends on how you do it. You don't just stop after one or two problems. You continue to do variations of those problems until you understand the generalized class of problems. But what do they propose instead, some sort of magic thinking process?

In terms of problem solving, I like to provide many more details, tools, and tricks than just drawing pictures and thinking about related problems. I like to talk about governing equations.

When kids see terms like speed and distance, they should think of the D=RT governing equation. You can even generalize that to any problem that uses rate and time, like the work problem. You can have a governing equation of

Amount = SUM(rate(i) * time(i))

where (i) refers to the ith person or rate object.

This will work best if students understand units or dimensional analysis.

If one student mows a lawn in 30 minutes and another mows the lawn in 20 minutes, then students have to learn that they can divide 1 lawn by 30 minutes to get a rate. The prerequisite skill is that they have to master the ability to maintain proper units in an equation. You don't try to learn about units while tackling complex word problems.

Another skill is the ability to see what happens when you have different variations of the governing equation. You could have multiple amounts, multiple rates, and multiple times. Under what circumstances can the rates be combined or kept separate? You can methodically examine all variations of the governing equation. You let the math (equation) show you what you need to know. For example, with the governing equation above, what would the equation simplify to for the case when the two kids work together to mow one lawn?

I'm big on governing equations and letting units help you when you get confused. With a governing equation, it's not some sort of magic thinking process or some ability to find another problem that is similar.

I'm also big on more specific advice to problem solving. If a student has a problem that relates to geometry (distances and angles), I tell them to look for the right triangles. If they don't see right triangles, create them. Other kinds of triangles can be split into two right triangles.

Next, find the distances and angles for these right triangles. If they have two pieces of information (lengths or angles) about a right triangle, they can find the rest of the information about the triangle (in most cases). This is important for those who have trig. Students need to master the skill of looking at a right triangle with a length (10") and an angle (60 degrees) and say something like: "side b is 10*cos(60). This has to be automatic. You don't try to develop this skill while you are figuring out how to solve complex word problems.

There are very specific skills that make problem solving easier. If they are combined with a general approach to solving problems using governing equations, then that is a very powerful approach.

Let me just say that I did not get a lot of this with my traditional math education. I can see why some educators might say:

"...but some hold that they limit students to solve problems that look exactly like these problems."

But what, exactly, are they proposing instead?

My reaction is that they are going in the wrong direction.

So Barry, how do they propose to teach students how to solve any type of word problem?

You may not have gotten a lot of this with traditional math education but I'm sure you learned D = R*T and the general form for work/rate problems. In the article on discovery learning, I use the work problems as an example of what the governing principle is underlying that problem and how it is difficult for students to see underlying principles in problems that don't "look like the problems they learned from". I reference Willingham's account of inflexible learning, and he advises that it takes practice and experience. I think showing students how the underlying or governing equations work in a variety of problems helps in that regard, and I would agree with you that in the classes I had, there could have been more effort there.

You ask how "they" proposed to teach students how to solve any type of word problem? By giving them problems which they haven't had prior experience to solve, based on the theory that students will learn what they need to learn in order to solve it. Last year I sent you a sample problem about paper airplanes and an investigation some middle schools students were to do, coming up with all sorts of classifications of "flies longest" "flies smoothest" etc, without any trainng in even the basic D = R*T formula. That's the kind of thing "they" think works: authentic problem solving and the theory that experience teaches you what you need to learn.

Hi,

Would it be possible to get a copy of Barry's article?

Thanks.

The article is now posted at EdNew.org in two parts.

Part 1 is here.

Part 2 is here.

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