Problems vs Exercises

This article also appears as Chapter 12 in my book “Math Education in the U.S.”

My high school algebra 2 class which I had in the fall of 1964, was notable for a number of things. One was learning how to solve word problems. Another was a theory that most problems we encountered in algebra class could be solved with arithmetic. Yet another was a girl named Fran who I had a crush on.

Fran professed to like neither algebra nor the class we were in, and found word problems difficult. On a day I had occasion to talk to her, I tried to explain my theory that algebra was like arithmetic but easier. Admittedly, my theory had a bit more to go. She appeared to show some interest, but she wasn’t interested. On another occasion I asked her to a football game, but she used the time-honored method of ditching someone by saying she would meet me there. When she didn’t show up, I resigned myself to the fact that she would remain uninterested in me, algebra, and any theories about the subject. (See Figure 1)

Figure 1. Fran

My theory of arithmetic vs. algebra grew from a realization I had when I was taking Algebra 1. It dawned on me one day that the problems that were difficult for me years ago when I was in elementary school were now incredibly easy using algebra. For example: $24 is 30% of what amount? In arithmetic this involved setting up a proportion while in algebra it translated directly to 24 = 0.3x, thus skipping the set-up of the ratio 24/x = 30/100. Similarly, it was now much easier to understand that an increase in cost by 25% of some amount could be represented as 1.25x.  What had been problems before were now exercises;being able to express quantities algebraically made it obvious what was going on. It seemed I was on to something, but I wasn’t quite sure what.

Henderson and Pingry

The issue of “problems vs. exercises” is one that has remained a part of education school catechism for many years. I first heard it during a discussion with the teacher of my math teaching methods class in education school. We had been talking about how Singaporean students obtain the highest score on an international math test (the TIMSS exam, given every four years). My teacher was quick to tell me that Singapore students have been successful on multiple choice, short answer tasks where you need to apply a known algorithm that has been practiced extensively. She stated that TIMSS and many of the other tests are focused on “exercises” rather than problems. “What happens,” she asked, “when we get off the ‘script’?”

This topic surfaces time and again, frequently appearing in theses and school papers written for education school classes. Since some of these papers show up on the internet I have read a few of them. The papers I’ve seen reference an article written by Henderson and Pingry (1953) which appeared in an annual report published by the National Council of Teachers of Mathematics. In it, Henderson and Pingry address the typical difficulties one encounters in differentiating between what is an exercise and what is a problem. As they state, exercises are the things you do when you’re applying algorithms or routine procedures you know. They point out, however, that this is relative, and what is a problem for second graders is an exercise for fifth graders. Given the changing nature of problem versus exercise, they identified three necessary conditions that define a “problem-for-a-particular-individual”:

 1.     The individual has a clearly defined goal of which he is consciously aware and whose attainment he desires.

2.     Blocking of the path toward the goal occurs, and the individual’s fixed patterns of behavior or habitual responses are not sufficient for removing the block.

3.     Deliberation takes place. The individual becomes aware of the problem, defines it more or less clearly, identifies various possible hypotheses (solutions), and tests these for feasibility.

The Henderson and Pingry article has piqued my interest not only because it addressed the issue of problems and problem-solving, but also because Henderson and Pingry were two of the three authors of the algebra textbooks I used in high school. (Aiken, Henderson and Pingry, 1960a and b). Having actually experienced the implementation of their theories as a student, I therefore had a bit more on the ground information than the casual author of education school papers or blogs, or rants on Substack.

The article by Henderson and Pingry (1953) is rife with familiar tunes. They echo the critics of textbook problems and state the advantages of “real life” or “real world” problems: i.e., real world problems have no definite question, but the student has to figure out what questions to ask at the outset, the student has to collect the data necessary to solve the problem, and a definite answer often is not possible. They argue that lacking such problems in mathematics courses, students will not likely become competent in solving them. (p. 234).

They repeat the criticism often heard today during debates on how to teach math, that problems in textbooks are frequently only a practice for a procedural problem solving method.  Such an approach, the critics argue, falls short of teaching math as a sense-making, problem-solving discipline. They are “inauthentic”. Today’s critics argue for math to be taught as it is practiced by professionals: problems first, gather data, and then generalizations and abstractions follow. And it would appear that Henderson and Pingry (1953) seem to be heading in that general direction:

There is considerable evidence that many mathematics teachers do not understand what problem-solving is. … One example of this is the manner in which many teachers teach the verbal problems of the algebra course. Many of the problems are catalogued into types such as “Mixture problems,” “coin problems,” “age problems,” and others. The teacher demonstrates to the student how to solve the type, and a list of problems of the type is then given to the students. The students do not experience problem-solving. Rather, they experience practice of applying a memorized technique. (p. 249)

This is where the similarity between Henderson and Pingry and today’s math education critics ends, however. First, and notably, the authors do not make the claim as many reformers and critics do, that students find these types of problems irrelevant and are therefore not motivated to try to solve them. In fact, they claim that such problems (i.e., mixtures, coin, age, work, distance/rate) –when not taught as memorized types and solutions–actually can be used for improving problem-solving ability.

If the teacher selects verbal problems carefully so as to be at the student’s level, and if he can get the students to identify themselves with these problems, then the verbal “problems” become real problems. (p. 234)

And this is, in fact, what they did in their algebra textbooks. They provided instruction on how to identify the data in a particular problem, what is being asked, and how this unknown quantity figures into the organization of data. For example, a typical mixture problem may ask: “How many ounces of 80% sulfuric acid solution must be added to 20 ounces of a 20% solution of the acid to make a 50% solution?” The authors show how to analyze and organize the information in the problem in order to solve it. If there are 20 ounces of 20% sulfuric acid, then the solution contains 4 ounces of sulfuric acid and 16 ounces of water. An unknown amount of an 80% acid solution is added. This would increase the sulfuric acid in the original container to 4 ounces plus x ounces times 0.8, or 4 + 0.8x. The total amount of solution would also be increased by x ounces or 20 + x  and the new acid solution could be represented as (4 + 0.8x)/(20 +x) which equals 0.5. And then, it is a simple matter to solve for x.

The problems that followed were similar to the original problem, which would likely cause critics to jump up and say “There, you see? They are just applying a memorized technique.” (Peter Liljedahl would call that “mimicking”.)

But as anyone knows who has learned a skill through initial imitation (or mimicking) of specific techniques, such as drawing, bowling, swimming, dancing and the like, watching something and doing it are two different things. What looks like it will be easy often is more challenging than it appears. So too with math. In the worked examples in my algebra textbooks, it wasn’t a simple matter of looking at it and saying “Oh, this number goes here, and that number goes there.” You had to think about what you were doing. What things remain equal? What changes? What are the relationships between the unknown quantities and what is known?

Henderson and Pingry’s approach was used to good effect in their algebra textbook. Problems of various types were appropriately scaffolded and each was different enough from the last so that they presented a challenge. That is, students could not simply plug numbers into a formula and get an answer. By varying the problems and increasing the difficulty, students remain challenged and problems remain problems. By the same token, there was enough repetition that students could master the basic technique before being presented with novel twists and challenges. (See Fig. 2)

Figure 2: (From Aiken, Henderson & Pingry, 1960b)

Despite the skillful use of scaffolding of problems, however, their algebra textbooks had a minimum of explanation for solving problems and would have benefitted from inclusion of more worked examples. This could have been accomplished without sacrificing student learning, or turning problems into exercises. I base this judgment on my experience in Algebra 1. My teacher, though very good at teaching procedures, admitted he was not good with word problems. Thus, he could offer little more help than what the book provided. He struggled with doing the problems as well as explaining what was going on. As a result, I was not proficient at solving word problems. I mention this because of a blanket assumption made about people like me which is “You would have understood math no matter how it was taught” or “You’re the exception”.

In fact, it was a combination of the book and my good fortune of having Miss Beck for Algebra 2, that made learning how to solve such problems a top priority. Miss Beck provided us with a system of diagrams – boxes—for solving mixture problems, which enabled us to organize the data and to enable us to see how to set up the equations to represent the quantities being mixed.

Miss Beck gave us a handout of about five problems of her own devising to supplement what was in the textbook. I did all but the last which was a bit different than the rest. It was something along the lines of: “A tank has a capacity of 10 gallons. When full, it contains 15% alcohol. How many gallons must be replaced by an 80% alcohol solution to give 10 gallons of 70% solution?”

I recall when Miss Beck showed us how to solve the problem. I saw that it followed the same pattern of organization of data as the other problems except for a subtraction step which had eluded me. With this realization came an elaboration of my theory of arithmetic vs. algebra. I understood then that the problems we had to solve could probably be solved without algebra, but algebra offered a more efficient and concise method for solving involved problems than solving by arithmetic.

Every Problem is Ultimately an Exercise

As a high school student, the power and utility of algebra in solving problems was abundantly evident, and the goal of attaining this efficiency seemed well worth pursuing. I liked the idea that “problems” could be reduced to what were ultimately “exercises”. But the trend of educational thinking is that standard mathematics textbook problems are too repetitive, too boring and “inauthentic”. Such a view fails to take into account that such problems are helping to form the building blocks of problem solving thinking, called “schemas”. Schemas allow people to gain additional knowledge by building on previous knowledge and skills. This is true of learning in general. A baby learns to use his hands to grasp and with that skill can then pick up objects. Picking up objects then is a schema that allows for other more complex tasks to be accomplished. The schema involved in solving math problems starts at a basic level, and ultimately can be built upon to non-routine types of problems–after much practice with many problems.

Most problems ultimately break down into basics and become exercises.  Students do best with very explicit instruction, starting with simple problems.  They then begin to develop the knowledge and skills to solve increasingly more difficult problems with novel twists. Without explicit instruction in problem solving, many just give up and don’t try the problems. Students benefit by seeing how to think about the problem before actually working it.  Imitation of procedure therefore becomes one of imitation of thinking. And later in life, most authentic problems that engineers and scientists work on—although more complex than what they worked on in school—are variants of the various problem types they have learned to solve in high school and later in college.[i] Totally new and novel mathematical problems that have never been seen nor solved are reserved for those involved in mathematical research and take years to solve—if ever. (Think of the four color map theorem, Fermat’s last theorem, and the twin prime conjecture.)

This is not something that the education establishment in general and education schools in particular like to focus on, however. For example, a paper by an ed school student (Yeo, 2007), categorizes various types of problems and establishes a set of problems that are in the “grey area” of higher order thinking and procedures. The ultimate goal, it seems, are to present problems that remain problems, that don’t break down into procedural steps. Yeo suggests that one way to get at this is to reinterpret Henderson and Pingry’s (1953) definition of what a problem is and extend it to open-ended and investigative problems. Thus, Yeo finds problems such as “Find patterns in power of 3” result in the stimulation of mathematical thinking as opposed to problems that can readily be broken down into procedural methods.

Problems such as the one above are not necessarily bad, but when presented without a sufficient learning base of prior knowledge and procedural skills, they do little to promote problem solving skills. It is as if the advocates of open-ended and investigative problems are saying that presenting students with non-routine, and open-ended problems on a constant basis form a “problem solving” schema. Furthermore, they view such problem solving schemes as independent of the mastery of basic types of problems that are learned by example and scaffolded to present more challenge. The thinking is that by giving students a constant does of challenging problems, not only are problem solving “schemas” being developed—so the theory goes —but also all the students in the class are in the same boat. That is, all students will be struggling and there won’t be those few who get it while others are left feeling inadequate. The danger in such thinking is that there is a converse to this theory that is usually ignored: that is, a steady diet of problems held beyond everyone’s reach may well result in students being in the same boat–one in which all are feeling lost and inadequate.

In fact, there is no schema for solving new and unseen and non-routine problems. In a paper on whether problem solving can be taught, Sweller, Clark and Kirschner (2010) state that problem solving cannot be taught independent of basic tools and basic thinking. Over time, these tools or schemas contribute to a repertoire of problem solving techniques. These tools are learned by working through examples. Once someone learns how to solve a particular category of problems and their variants, the person becomes much better at solving them and those problems ultimately become exercises. Turning problems into exercises is in fact the way in which problem solving skill is increased. Sweller et al, state that after a half century of effort, there is simply no evidence whatsoever that we can improve general problem solving skill so that people will become better at solving novel problems.

The standard problems of Henderson and Pingry, did present a challenge that students, with proper instruction and guidance, were able to meet. Many textbooks followed using the same techniques. Math reformers tend to criticize these type of standard textbook problems for not requiring mathematical thinking in the same way as the more advanced or non-routine problems. The problems of Henderson and Pingry and others do, in fact, require mathematical thinking. Students must exercise judgment and analytic skills in identifying how to use the data in the problem. Learning these skills allows students to solve more complex and demanding problems. The criticism that textbook problems don’t reflect “real mathematical thinking” confuses pedagogy (the practice of teaching) with epistemology (the theory and practice of knowledge) as pointed out by Kirshner, Sweller & Clark (2006). That is, novices do not and cannot think like experts.

The methods of the past often strike people as antiquated and ineffective. They are viewed in the same way that one looks at photos of students in old yearbooks from the 50’s and 60’s, or films of students from that era. I too am often amused at the formality of those times, and how some aspects of life have improved for the better. I have a strong feeling that many of us from that era received a far better education than many students today for whom problems will almost always remain problems.

Girls like Fran remain as poster children for that era and used as evidence by some that math as it was taught in the past was a failure for thousands of students. I don’t know how Fran did in the class, or how she ended up in life, but I do know that based on what I saw of my fellow students working problems at the board (which was the norm back then), most of them seemed to have a good understanding of the subject despite popular belief to the contrary. Although we were novices, we were becoming proficient at making exercises out of what previously had been problems.

References

Aiken, D. J., Henderson, K.B. & Pingry, R.E. (1960a). Algebra: Its Big Ideas and Basic Skills; Book I. McGraw Hill. New York.

Aiken, D. J., Henderson, K.B. & Pingry, R. E. (1960b). Algebra: Its Big Ideas and Basic Skills; Book II. McGraw Hill. New York.

Henderson, K.B., & Pingry, R.E. (1953). Problem Solving in Mathematics. In H. F. Fehr (Ed.), The Learning of Mathematics: Its Theory and Practice (pp. 228-270). Washington, D.C.; National Council of Teachers of Mathematics.

Kirschner, Paul A., Sweller, & J., Clark, R.. 2006. Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. Educational Psychologist, 42(2), 75-86. http://projects.ict.usc.edu/itgs/papers/Constructivism_KirschnerEtAl_EP_06.pdf

Sweller, John., Clark R., & Kirschner, P. 2010. Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics. Notices of the American Mathematical Society. Vol. 57., No. 10. November. http://www.ams.org/notices/201010/rtx101001303p.pdf

Yeo, Joseph B.W. 2007. Mathematical Tasks: Clarification, Classification and Choice of Suitable Tasks for Different Types of Learning and Assessment.  Technical Report ME2007.01. National Institute of Education, Nanyang Technological University, Singapore.

---