Douglas McLeod, San Diego State University. I’m in the Department of Mathematical Sciences.
I have written about and talked about the affective domain as—as having three major components. Uh one area is the area of beliefs and in my work it’s typically related to beliefs about mathematics or beliefs about learning. And beliefs about mathematics might be um related to the functions that mathematics plays in society. Is it a useful subject or is it something that’s just an abstraction that we—that we talk about in abstract ways but don’t necessarily connect to—to real life. Um, is mathematics a set of rules that you follow or is mathematics a set of um ideas that you explore? And these kinds of beliefs really are deeply embedded in our culture and these kinds of beliefs about mathematics have a lot of impact on—they have a big impact on people’s affective reactions to mathematics. Then uh the second category after beliefs would be attitudes and these are related to liking or preferring one id—one approach to a problem over another. Maybe you like doing quantitative problems or some people don’t like to do quantitative problems. Some people like a challenge or a puzzle and some people don’t like to uh be challenged in that way. They want you to tell them the right answer. So these are attitudes or preference. And then the most intense is your e—emotional reactions and this is where a lot of my work has focused. Emotional reactions to um—particularly to mathematical problems and the—the emotional reactions typically involve some kind of uh physical change in the person. They get tense or their heart starts to beat faster. And in a mathematics class sometimes your heart starts to beat faster because you can see a solution to a problem and you’ve solved the challenge. But perhaps more often the heart starts to beat faster because you don’t see an answer and perhaps you’re getting frustrated. And so the teacher’s role in dealing with these emotional reactions is a very important one.
In the country—in the United States now, we have a lot of efforts to change and improve the mathematics curriculum and to find ways to help students understand mathematics more deeply. If a student believes that mathematics is a set of rules that you need to memorize then the student is not likely to be very interested in trying to understand the mathematics in a deeper way. So I think it’s really important that we try to consider the student’s beliefs about mathematics as we try to teach them mathematics. If we want them to understand then we have to see—they have to believe that mathematics is a subject that is—that it is possible to understand. And unfortunately in the past, mathematics was often taught as a set of rules to be learned, to be memorized, and so students got the impression justifiably that mathematics was not intended to be understood.
There—people have different ways of trying to address uh what it means to really understand mathematics. What I find useful is to have students demonstrate their understanding by showing connections between sets of ideas. And sometimes this is a connection between the symbols of mathematics and some real world objects. Uh if you’re going to talk about uh a division problem, for example, maybe you can take a set of objects and divide them. And so understanding comes from the connection between the symbolic representation of the mathematical ideas and the physical representation of the mathematical ideas. And that’s one way of talking about understanding. And then there are—when you get into higher-level mathematics, you may focus more on the justification that’s part of a mathematical argument, on the logical reasoning that’s involved in a mathematical argument. So at secondary or college level you would ask for more abstract justifications of some mathematical procedure or some mathematical concept.
Well, teachers are usually very good at understanding children’s affective reactions and that kind of understanding is often what pulls people into being teachers. So I think teachers are usually very good at interpreting students’ reactions. Being aware of the home environment, for example. I think that my uh emphasis has been on trying to help teachers um understand how student beliefs about mathematics may influence the way that they learn and that student beliefs about mathematics may cause them to be very frustrated and in fact have very strong emotional reactions against mathematics. And a—a teacher sometime has the opportunity to help a student see a difficult problem or a non-routine problem as a challenge or a puzzle that can be a source of great satisfaction to the student once the student solves the problem. And that I think is the part that teachers uh often don’t um think about. And—and this often because of the culture in which we live and which often parents and children agree that mathematics is something which the children aren’t really expected to understand. And how often we hear a parent say, “Well, I was never good in mathematics.” And that’s a—that’s the sort of context within the child can feel, “Well, I won’t be able to understand mathematics.” And so it’s the teacher’s role to try and encourage that student to make a sincere effort to understand the mathematics and to see that there are real rewards in solving challenging mathematical problems.
Well, I think—I try to help them become aware of their own affective reactions to mathematics and to let them reflect on how their affective reactions to the mathematics are going to color their students perceptions of the mathematics. So I try to model certain things. I try to be very interested and enthusiastic about their efforts to present creative solutions to mathematics. And I try to emphasize the—the rewards that come in being creative in mathematics. Now you have to structure this very carefully because often students come in with—to their pre-service teacher education classes with a lot of negative emotional baggage related to mathematics. But I try to model for them an enthusiasm for mathematics that a—that should, I hope, help them see how they can treat their students in the future and encourage that view of mathematics as a challenge that can be very satisfying.
Yes, there are similarities and I think the basic uh approach is the same. There’s a feeling that since secondary mathematics teachers have been successful in mathematics, that is the ones that have gone on to be math majors and then gone on to be math teachers. They’ve been successful in mathematics and they often are very good at being creative about mathematics. But that’s not always the case. There’s a range of um points of view or experiences. Some secondary teachers feel that they’ve been successful in mathematics because they could follow the rules and maybe they don’t even like puzzles very much. Other people are—are much more into solving puzzles or solving non-routine problems and making them—they respond positively to that challenge. So you get a range of responses both in secondary and in elementary teacher preparation. What’s different, of course, is that the problems have to be different uh because most elementary teachers can solve non-routine problems at a certain level but secondary teachers might find that they need a more advanced topic within which to propose a non-routine problem in order that they could have that satisfying feeling that comes when you solve a problem that’s really been challenging.
Well, certainly using um the right kind of mathematics for the students that you are teaching is crucial to help them have a positive response to mathematics and we find more and more emphasis now in curriculum on trying to devise curriculum that would be appealing to all groups of students. If you have students who have—come from a particular ethnic background, then you want to help them see that uh mathematics is useful to people like them. If you’re dealing with students that speak Spanish, certainly there’s a long—uh there are a lot of language issues that come up in terms of making the problems uh comfortable, making the problems appealing. And certainly you see a lot of this happening. I think that these are—are useful things to do. We want to be inclusive and we need to have all students participate in—in mathematics. In the past there was often a feeling that certain ethnic groups were good at mathematics and others weren’t. And this is really uh—we see these patterns of uh enthusiasm but I—I think they’re cultural patterns. Uh we haven’t done as good a job of encouraging all students to be uh participants in mathematics, for example, all through high school.
I think that in—in teaching mathematics we’ve often focused on the procedures in mathematics that can be practiced and memorized. And those procedures are important but that’s not all there is to mathematics. What’s really becoming more and more important in this technological age and this age of a global economy is to be able to solve all kinds of problems, including mathematical problems and to be able to solve sometimes quite complex problems using the technology that is available. And in order to solve complex problems you need to have a positive attitude about mathematics. So you need to work uh—we—we need to work in the schools on preparing students so that they can solve these complex problems. I think that the uh emphasis on mathematics as a discipline where students can contribute, they can build their knowledge, they don’t have to be told how to solve problems, I think that’s an important aspect of what goes on in classrooms now and it’s more important now then it was 50 years ago.
I like to think about the—the—both the mathematical knowledge that’s needed and the knowledge of children that is needed and that out of those two areas grows the knowledge of teaching. So you still—you need to have a knowledge of mathematics. You need to know the—the procedures, the algorithms, the arithmetic for the elementary school, for example, and the algebraic uh procedures, the geometric knowledge especially for the secondary school. And to all of that you have to add newer areas that weren’t emphasized when I was in school. Areas like probability, statistics. And in addition to that knowledge of the mathematics, the—you—you need to understand how children come to learn mathematics. That they need to build these ideas gradually based on their experience and on—uh based on their experience with objects and the real world to build the erythematic and geometric understandings that are important for uh solving problems that have a mathematical aspect to them.
No, I think you’ve uh—you’ve asked the important questions. I think that it’s—it’s very useful for teachers to keep in mind that mathematics is more then just a rules, that is involves a much more uh creative aspect then was apart of the traditional classroom for many of us. And so in order to help students approach uh mathematics as a discipline in which they can understand and solve problems, they need to try and um encourage students to see the positive aspects of uh coming to understand mathematics.