MTHEMATICAL THINKING

Mathematical Thinking


James Dunlap

C&I 431

November 27, 2001



Most teachers of mathematics would agree that when they teach, they would like for their students to not only learn the material behind what they are teaching, but also to learn to “think mathematically” even though some teachers who claim this might be hard pressed to define the term. I will, for the purpose of this discussion, define mathematical thinking as a cognitive approach to a problem that is both logical and mathematically sound. This definition allows us to approach it in a way that is conducive to
solving mathematics problems while not restricting us (or the students) to saying that there is only one correct solution or that we must use the shortest and quickest method possible.

Most people are aware that, years ago, Socrates taught his students by asking very leading questions that would ultimately lead them to a discovery of the knowledge that they sought. This method is used sometimes today and is called a “Socratic seminar.” While I will not attempt to argue that all classrooms should change and only use the Socratic method of questioning students, I do believe that we should incorporate this into classrooms without allowing it to cloud our vision as to what our ultimate goals are. I will also not attempt to persuade anyone as to what the rest of the goals should be, other than that we should be teaching our students to think mathematically.

Probably the most important question to ask is why we should be so concerned with this issue. Throughout the years, as we watch the fields of science, business, engineering, etc., we constantly see people who rise above all others. Many times, this is credited to them thinking “outside the box.” Many great discoveries were made because people were willing to challenge the norm of their time and question things as they were. Galileo was sentenced to house arrest because he wanted people to believe (rightly so) that the Sun – not the Earth – is the center of the solar system. Columbus ignored the beliefs of his day that the world was flat and set out to find a western route to India. While Columbus was not necessarily thinking mathematically, many great discoveries in the field have been made because people were unwilling to look at things as everyone else does. They knew intuitively that what they were setting out to prove was correct. Which brings us to a very important point about the teaching of mathematical thinking.

There are many views about how mathematical knowledge is constructed and a teacher’s view on this directly relates to how that teacher would teach the students to think mathematically. For example, if the educator takes a positivist or rationalist approach (Dossey 1992) to mathematics education, they will find most of what I will say to be useless to their view. These educators would probably say that mathematics is simply true and is not a matter for discourse. They would also argue that mathematics is not intuitive. I, on the other hand, believe that mathematics is very intuitive and disagree that there would be no use of the approaches about which I will talk if the teacher were to believe a particular epistemology to be true. These methods can work in any view of the truth of mathematics.

Lakatos (1976) presents an extremely idealistic view of the mathematically thinking classroom where the students discuss Euler’s formula in an intuitive and open manner. (Euler’s formula states that for a polyhedron where f = the number of faces, v = the number of vertices, and e = the number of edges.) The students and the teacher discuss in an open forum a proof that the teacher has devised. Students may freely interpose definitions, counterexamples, and ideas onto the teacher’s proof. All ideas are at least accepted and discussed with the teacher taking a back seat in the discussion and helping out when the discussion comes to a point where the students need a push. The intuition of the discussion is seen by students sporadically throwing out ideas that seem to have little mathematical basis, followed by the discussion that develops the mathematics behind the assumption. The class as a whole discusses what is valid and what is not by trying to more specifically define the idea of a polyhedron.

I say that this is an extremely idealistic view. The reality of the situation is that we are constantly being asked to teach more material in the same or, in some cases, a smaller amount of time. It would have taken a transmissive teacher about twenty minutes to slap the equation on the board, do a few examples, answer student questions and move on. This discussion, however, took at least two whole days of class time. This alone makes this an unrealistic approach to use at all times. I qualify this by saying that the previous statement applies primarily to students of today. If students were trained from the beginning to think mathematically, then discussions of this type would probably move quicker as they are in that mode of thought. The other issue relates directly to this in that students as a whole simply do not make statements as those of the students that Lakatos presents. They have been trained not to question anything that we say and accept every word that proceeds out of our mouth as gospel. (How many teachers of mathematics have made a simple arithmetic mistake in the first step of a problem, only to have a student point it out after filling up three board sections and spending ten minutes trying to figure out what went wrong? Think matrices.) There must be a point at which we start if we can only find it.

This was exactly the question that was being discussed and researched two and three decades ago. Problem solving is a buzzword that has been flying around the field of mathematics education for at least that long and was being discussed in other areas before this. In the early 70s, research was being conducted using problem solving in other fields, but it was not until later in the decade that this research reached into the field of mathematics (Kantowski 1981). There are issues today about exactly what problem solving is, but at least one of these can be found as far back as the early 80s. Problem solving was defined as solving real (application) problems or non-routine problems. A non-routine problem is one that requires the solver to apply mathematical thinking to find an algorithm different than what they learned in class. It requires the student to create a method that is unique to the problem by piecing together parts of similar problems and algorithms. Problem solving requires that the teacher not only consider the solution, but the method of reaching said solution as well. Typically, there is more than one method and/or solution to a problem. Problems such as these promote mathematical thinking in that students must synthesize information and make intuitive leaps as to what methods will work and what will not.

Problem posing (Moses et al. 1990) is slightly different, but still a valid tool for the teaching of mathematical thinking. Moses talks of several ways that we can foster the creative thinking of students using problem posing. First, modify problems from the textbook. By changing the problems found in most textbooks, we can modify them to be more challenging and require the students to modify algorithms and create dynamic mathematics. Second, use questions that have multiple answers. Problems that have only one correct answer and one (obvious) method of reaching a solution do not foster creative mathematical thinking as students can just apply the algorithm that they already know. Moses provides further ideas in the way of allowing students to choose their problems, not timing the solving of these problems, and working through some problems with the students. These ideas would tend to foster the creative mathematical thinking of our students. I always work harder and enjoy solving a problem that interests me than one that I was assigned by an arbitrary person who was putting my education on autopilot. Even giving the students an illusion of choice can satisfy this criterion. By giving them multiple problems from which to choose, their interest in the problem they select to solve can be multiplied as they feel they have some control. By not timing them, they will be allowed to develop the solution at their own pace. Teachers are taught in universities that students learn differently (Gardner’s multiple intelligences theory), so why do we sometimes feel that students must learn at the same rate? By allowing the students flexibility of time, we foster the creation of mathematical thinking. Finally, modeling the process could never be wrong.

Steffe (1990) presents an alternative way of presenting a common problem that would provoke mathematical thinking in students in the way that it is presented. Steffe takes the problem of finding the number of handshakes that would occur if n people shook every other person’s hand and allows the students to construct their own method for solving this problem. Then, he presents various problems that could be solved using the same method or formula. This method is extremely valid and would be ideal if used in conjunction with problem solving and problem posing. The primary difference in the results of this method is that while problem solving and posing strive to get students to realize that there are different methods for solving a given problem and that one of these methods is not necessarily correct, Steffe’s method desires to have students relate an already existing base of knowledge and algorithms into a new problem type. This also promotes mathematical thinking in that students should not look at every problem being posed as new, but should relate the problem to a preexisting method or algorithm and apply the algorithm if it works or create a new one if it does not.

The primary problem with teaching students to think mathematically is related to the concept already presented of students learning at different rates. In order for any of these methods to be truly effective, the problem has to be outside of the solver’s schema (Mayer 1982). A schema is an area of knowledge. Unless the problem is outside a person’s sphere of previously discovered facts, then it is not a challenge. If we were to give a minimal area with a given perimeter problem to a calculus instructor it would not be a challenge as that instructor teaches the derivative process of solving these very problems each year. If, on the other hand, we were to give the same problem to a pre-calculus student, the problem becomes relevant to the teaching of mathematical thinking as the student must devise a method of solving the problem as it is outside the student’s accepted body of knowledge. Even if the method of the student becomes one of trial and error, it is an acceptable, though somewhat time consuming, method of solving the problem and is valid.

While this example may seem convoluted, consider that each student sitting within a given classroom has a different body of knowledge and probably a different understanding of the knowledge that they share. This is evident when a teacher assigns grades at the end of a grading period. There are always students who get A’s, but there are also students who do not achieve as well as the teacher would have liked or expected. While it is not strictly true, many times the high achieving students have a different schema than the lower achieving ones. It is, therefore, necessary to provide a variety of problems for the students to solve so that all students will be challenged and add to their ability to think mathematically.

Now that I have given a better understanding of what is meant by mathematical thinking, I would like to revisit the question of why it is important to teach the students to think mathematically. I am sure that most mathematics teachers have been asked the age-old question, “When am I ever going to use this?” (If you haven’t, then maybe you haven’t been teaching long enough.) I had been asked this question so many times that I bought a poster that is now displayed on my wall to answer that very question, and students are constantly admiring it. (More to see what fields they don’t want to go into than to check those that interest them.) This question is also asked at times that many teachers have difficulty answering it and sometimes make one up or give them an answer that will likely always be outside of the student’s schema. (I always tell my students that matrices are used in 3D game programming, which is true.) As a mathematics educator, I can honestly say that I have never used calculus to solve a problem that I came across in my normal, ordinary, everyday life. I have never used geometric or trigonometric properties around the house. So why do we teach the topics that we do? Mathematics is taught to teach students to think (Schoenfeld 1982). While I have not used geometric properties at home yet, I have used some of the principles behind geometric proofs. They are structured and logical, and that, to a miraculous extent, is useful in many areas. Even when writing a paper, a person must be logical and structured. I would argue that completing a proof is not an algorithm, but rather, a way of thinking mathematically. Proofs require a certain level of mathematical knowledge to perform, but unless a person can think in a logical manner, they cannot complete a proof from scratch. I see this phenomenon with my geometry students on a yearly basis. There are certain students who just cannot seem to think logically. I believe that these students have been done an injustice. The system is currently implemented in a way that is not conducive to the teaching of mathematical thinking throughout all levels. Too many times, we expect students to regurgitate an algorithm that they were taught previously in the year. It is for these reasons that we must implement a strategy for teaching them to think mathematically. I would disagree with anyone who says that mathematics is just a set of algorithms and rules. Instead, I believe that mathematics is a way of thinking.

For those who are still not convinced, perhaps a quote from the standards is in order. “Secondary school students need to develop increased abilities in justifying claims, proving conjectures, and using symbols in reasoning” (NCTM 2000). Apparently, the NCTM believes that today’s students do not currently possess these skills. While I believe that the standards should only be considered as a guideline and not a cure-all, I do believe that this is an extremely important statement that has direct relevance to the topic at hand.

The problem remains, however, of how to implement this process and whether or not it actually works. To test this, I tried posing some questions to one of my geometry classes. To make the results more evident, I chose the lowest achieving class. Throughout the year, I have occasionally posed questions to them, to most of which I refused to tell them the answer. At first, I received only a struggle from them. One student went so far as to tell me to “just give us the answer.” As the year progressed, however, it became easier to have discussions with them, and they were more willing to input ideas. They have begun to question other theorems and “facts” that I put on the board and often will logically leap to an idea that appears later in the book. They ask probing questions concerning what will happen if particular conditions are met, such as if the legs of a trapezoid are congruent, then will the angles also be congruent. While some might mark these questions as being normal, I have never before had a class that asks the type of questions that these students are, and neither of my other two classes this year ask these questions. They seem to be interested in being asked to consider a particular problem. (Today, they started considering the question of why a point that is located on a circle with a radius of 2 will travel twice as far when rotated along a surface when it is placed to be concentric with a circle with a radius of 4 than it does by itself even though the circle still only turns one rotation.)

It is also important to consider another alternative that is a widespread educational opportunity that is too often overlooked and underused. The use of technology can extremely aid a teacher in the classroom. It is relatively to use a program such as The Geometer’s Sketchpad to create a diagram that can be used to illustrate a geometric theorem or postulate to the classroom. Through the use of technology, students can also be allowed to investigate some of these concepts on their own and draw their own conclusions.

Research seems to sometimes disagree on the effectiveness of technology in the classroom. Honey writes that the information gathered in research during the last several years has actually shown that while many studies have shown that the use of technology usually results in more students gaining more points between pro-testing and post-testing, however the results were biased by teacher bias and the technology used. She discusses that the primary concern in many of the cases that she studied were that the technology was an appendage that was secondary to the classroom teaching and was not created as a major part of the school. She also discusses a case in which using technology was an integral part of the school system and its curriculum. It was a basic part of the school improvement plan.

It is obvious to me that many students are being done an injustice in their education. When I have to struggle with my students to get them to begin to think mathematically on their own, there is a problem that needs to be addressed. Students should not be introduced to this concept as sophomores and juniors in high school. Many of my students will not continue to college, and their formal education will end upon graduation, so if we do not teach them this extremely important concept, who will?



References



Dossey, J. A. (1992). The nature of mathematics: Its role and its influence. In Grouws, Douglas A. (Ed.), Handbook of research on mathematics teaching and learning (pp. 39-47). New York: Macmillan Publishing Company.



Honey, M., McMillan, K., & Carrigg, F. (1999). Perspectives on Technology and Education Research: Lessons from the Past and Present. [2002, March 26].



Kantowski, M. G. (1981). Problem solving. In Fennema, Elizabeth (Ed.), Mathematics education research: Implications for the 80’s (pp. 111-124). Reston, VA: National Council of Teachers of Mathematics.



Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.



Mayer, R. E. (1982). The psychology of mathematical problem solving. In F. K. Lester & J. Garofalo (Ed.), Mathematical problem solving: Issues in research (pp. 1-10). Philadelphia: The Franklin Institute Press.



Moses, B., Bjork, E., & Goldenberg, E. P. (1990). Beyond problem solving: Problem posing. In T. J. Cooney (Ed.), Teaching and learning mathematics in the 1990s (pp. 82-91). Reston, VA: National Council of Teachers of Mathematics.



National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.



Schoenfeld, A.H. (1982). Problem-solving research and mathematics education. . In F. K. Lester, & J. Garofalo (Ed.), Mathematical problem solving: Issues in research (pp. 1-10). Philadelphia: The Franklin Institute Press.



Steffe, Leslie P. (1990). Adaptive mathematics teaching. In Cooney, Thomas J. (Ed.), Teaching and learning mathematics in the 1990s (pp. 41-51). Reston, VA: National Council of Teachers of Mathematics.