Discovery learning in math: Exercises versus problems Part I
Barry Garelick 7.22.09
What is Discovery Learning?
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 only 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’?”
Those are important questions, but I will argue in this article the following points:
“Aha” experiences and discoveries can and do occur when students are given explicit instructions as well as when working exercises; and 2) Procedural fluency does not exclude conceptual knowledge—it leads ultimately to conceptual understanding and the two are key for applying mathematics to complex problems.
I’m not against asking students to discover solutions to novel and challenging problems—the experience can be quite powerful, but only under the right conditions. A quick analogy may be useful here. Suppose a person who knows how to drive automatic transmission cars travels to a city and is forced to rent a car with a standard transmission—stick shift with clutch. The person in charge of rentals gives our hero a basic 15 minute course, but he has no opportunity to practice before heading out. In addition to this lack of skill in driving a standard transmission, the city is new to him, so he needs to rely on a map to get to where he needs to go. The attention he must pay to street names and road signs is now eclipsed by the more immediate task of learning how to operate the vehicle. But now suppose that prior to his trip he is given instruction in and ample opportunity to practice driving standard transmission cars. With proper training and guidance, he can start off on quiet streets to get the feel of how to coordinate clutch with shifting, working up to more challenging situations like stopping and starting on hills. Over time, as he accumulates the necessary knowledge, and practice, he’ll need less and less support and will be able to drive solo. There will still be problems that he has to figure out, like driving in bumper to bumper traffic that requires starting, slowing, downshifting, and so forth, but eventually, he will be able to handle new situations with ease. Now, given the task of driving in a strange city, he will be able to focus all of his attention on navigating through new streets (having already achieved driving mastery of the vehicle that will take him where he needs to go).
Whether in driving, math, or any other undertaking that requires knowledge and skill, the more expertise one accumulates, the more one can depart from the script and successfully take on novel problems. It’s essential that at each step, students have the tools, guidance, and opportunities to practice what they learn. It is also essential that problems be well posed. Open-ended, vague, and/or ill-posed problems do not lend themselves to any particular mathematical approach or solution, nor do they generalize to other, future problems. As a result, the challenge is in figuring out what they mean—not in figuring out the math. Well-posed problems that push students to apply their knowledge to novel situations would do much more to develop their mathematical thinking.
To make this discussion more concrete, let’s take a look at three math problems. The first is a discovery-type problem that the target students do not have the necessary knowledge to solve. The second is an ill-posed problem. The third, in contrast, is a well-posed problem that relies upon prior knowledge and is mathematically meaningful.
The first problem comes from the first-year textbook of the Interactive Math Program series (IMP, 1997) and is given to students who have just started algebra. These students have had limited exposure to systems of linear equations—for example, two equations with two unknowns.
“You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in all possible combinations of two: bales 1 and 2, bales 1 and 3, bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4, and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90, and 91. Your initial task is to find out how much each bale weights. There may be more than one possible solution; if this is so, find out what all of the solutions are and explain how you know.” (IMP, 1997; p. 27)
Students who are just beginning algebra do not have the prerequisites to solve this problem efficiently. They would probably have to use “guess and check,” a method that might result in the right answer, but that is not likely to deepen students’ understanding. A key observation necessary to solve the problem efficiently is that no two bales are equal in weight because no two of the sums provided in the problem are equal. Such a relationship is not easy for beginning algebra students to see, and is only one of several different kinds of reasoning required to solve such a problem.
How would beginning algebra students with little foundation in systems of linear equations and mathematical reasoning feel when confronted with such a problem? David Klein, a mathematics professor from California State University at Northridge, (personal communication with author, October 29, 2008) commenting on this problem, said, “It is an annoying problem and has little educational value. If I had been given such problems at that age, I think that I would have hated math.”
Why such strong words? Because it is unlikely that guess-and-check will provide any insight that can be transferred to other problems or result in a deeper understanding of mathematics. Meanwhile, the foundational knowledge that comes with mastery of different types of algebraic problems over time has not been learned.
2. Our second example is not a discovery-type; rather, it is an ill-posed problem that can be interpreted many ways and, as a result, is not educational. This problem comes from the “Ten-Minute Math” section of a teacher’s guide for TERC’s “Investigations in Number, Data, and Space” for fourth grade. (Russell et al., 2008) In this particular activity, students decompose numbers in an exercise that is ultimately designed to get students to think beyond place value. The guide explains that decomposing numbers “is more than just naming the number in each place. It includes understanding, for example that while 335 is 3 hundreds, 3 tens, and 5 ones, it is also 2 hundreds, 13 tens, and 5 ones” (Russell et al., 2008; p. 27). It then proceeds with the following instructions (for the teacher) and questions:
Step 1 Write or say a number. Write a number on the board (or say it and have students write it.) For example:
Step 2 Ask: "How many groups of _______ (10, 100, 1,000, etc.) are in the number? For example, ask students how many groups of hundreds are in 1,835. If students think that eight is the only answer, ask them to consider a context such as money.
If this were money, how many hundred dollar bills would we have if we had $1,835?
Establish with students that there are 18 hundreds in 1,835.
The problem is poorly worded and constructed given the answer the authors seek, and I feel sorry for the dedicated student who tries to make sense of it. Students who have an understanding of place value and see the 8 in 1,835 as representing 8 hundreds are now confronted with conflicting information: there is more than one answer, and 8 is not the answer being sought. They are then guided to make a “discovery” that there are 18 hundreds by making a connection that if there are 18 hundred dollar bills contained in $1,835, then there are 18 hundreds in 1,835. Those are two different statements. Worse, the latter is mathematically incorrect in the context of the question asked. Since the previous “Ten-Minute Math” focused on decimals, students may reason—correctly—that although there are 18 hundred dollar bills in $1,835, there are actually 18.35 hundreds contained in 1,835. Asking how many hundreds are in 1,835 is a division problem (1,835/100), but the activity calls it a place value problem, and the result is an incorrect answer. If students are not already profoundly confused by all this, they will be soon: the activity then asks them to “make up five different combinations of place values [sic] that equal 1,835: 15 hundreds + 33 tens + 5 ones; 16 hundreds + 23 tens + 5 ones; and so on.”
While the problem may result in students thinking of different answers, it does not encourage mathematical thinking, does not push students to further their knowledge of mathematics, it incorrectly characterizes place value and in so doing, it confuses more than enlightens.
3. Our third example offers a sharp contrast to the other two. This problem comes from the fourth grade textbook in the series called Primary Mathematics from Singapore. (Primary Mathematics, Standards Edition, 2008). It is well posed and requires students to apply their prior knowledge.
“What is the value of the digit 8 in each of the following?
a) 72,845 b) 80,375 c) 901,982 d) 810,034 e) 9,648,000 f) 8,162,000”
Students cannot escape the lesson about place value since they cannot simply note where the 8s are, they must know what the various positions of the 8s mean. Preceding this problem in the Singapore text are other problems that introduce the concept of a number being a representation of the sum of smaller components of that number by virtue of place value; i.e. 1,269 can be expressed as 1,000 + 200 + 60 + 9.
Similarly, students are asked to express written out numbers, such as ninety thousand ninety, using numerals in the standard form (i.e., 90,090). They are also asked to write numbers in numeral form, such as 805,620, in words.
In short, students are asked no ambiguous questions, and the underlying concept of place value is indicated clearly via examples that can be applied directly to problems. By the time students reach the problem asking for the value of “8” in the various numbers, they have a working knowledge of what the numbers in various positions represent. This problem pushes them to apply that knowledge, thereby revealing any confusion they may have and also providing enough guidance for them to see that the position of the number dictates its value.
Advocates of complex problems that get students “off the script” may think this problem is not challenging enough. After all, any discovery students make is inherent in the presentation of the problem and the solution clearly comes from work that the students have just completed. But as anyone recalls from the early days of having to learn something new, it feels a whole lot different answering questions on your own, even after having received the explanation. In fact, such experience constitutes discovery. So I have to ask, what is wrong with acquiring incremental amounts of knowledge through well-posed problems? It is, after all, much more efficient than discovery-type problems that require Herculean sense-making efforts and leave most floundering for a solution, without a clear sense of whether they are right or wrong.
Contrast the approach of the first two examples with the third problem. This problem is well posed and it requires students to connect what they have just learned about place value to this new application. Instruction that uses such problems isn’t “handing it to the student.” To the contrary, it’s providing the support and guidance that students need to grow. I see it as a staging; a way to get students to apply easier problems to solve harder ones, and a way for procedural fluency to lead to understanding. I am not alone in such thinking. According to a study by Liping Ma, Chinese teachers interweave conceptual and procedural knowledge of mathematics. They believe that “a conceptual understanding is never separate from the corresponding procedures where understanding ‘lives.’ ” (Ma, 1999) The key is for problems to be carefully sequenced such that they incrementally increase in difficulty and require students to use their knowledge in new ways—and that’s the key to making a meaningful discovery.
Related article One Step Ahead of the Train Wreck
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