Above all else, mathematics is a way of thinking. Eric Robinson, the director of COMPASS*–an NSF-sponsored program to implement new secondary mathematics curricula– talks about the real-world advantages that students gain by learning to think with math concepts.
Why was COMPASS established?
In our efforts to improve the mathematics experience for pre-college students, there has been a realization that change needed to occur in all areas of the classroom: the way math was taught, what math was taught, what students were expected to be doing, what students were expected to get out of their mathematical experience, and how students were assessed.
So, our purpose is to provide support and advice to schools and districts as they implement mathematics education reform, and particularly to help them consider and implement mathematics curricula that reflect this new approach to mathematics education. COMPASS focuses on five high school, multi-year curricula that were developed with funds from the National Science Foundation and which support the vision of school mathematics set forth by the National Council of Teachers of Mathematics.
In what ways have the goals of mathematics education changed?
For one thing, we live in a new age. A strong understanding of mathematics that goes beyond computation is not just for future college math majors in today’s world. It is important for all school students. Society is much more dependent on mathematics. Math affects almost every career–sometimes subtly and sometimes very apparently.
Let’s look back a bit at the Industrial Age. If you worked on an assembly line, you were given a certain process and you just did the process over and over again. The world is not like that anymore. What we need, in the business world as well as in the professional world of mathematicians, scientists and engineers, is people who able to deal with open-ended situations, problems that are non-routine, and problems that aren’t very well formulated. People have to be creative about their solutions, and draw on a variety of different sources to solve problems.
The mathematics curriculum needs to train students to do those things. It’s not good enough to be able to solve just template problems. It’s creative stuff. It’s in a world that is changing almost every 24 hours in terms of what tools we have to work with and, what information and how quickly we get that information.
Maybe we should clarify what is meant by the term “mathematics”?
Well, there are several ways to talk about what mathematics is.
Certainly, it is a collection of facts. For example, all isosceles triangles have equal base angles.
It also contains a collection of operations, algorithms and procedures where, if you follow a certain sequence of computational steps, you get a guaranteed answer.
But to me, as a mathematician, far more important than any of those is that mathematics is a way of thinking. It’s a method of inquiry.
More than once I’ve heard students say “I don’t understand what they want me to do, but tell me what to do and I’ll do it.” They might have had the ability to do the procedures or memorize the facts, but they didn’t have the ability to see where their facts were important, when to do the procedures or what the procedures meant.
To me, that’s the antithesis of mathematics. The whole idea of doing mathematics is figuring out how to approach a problem. And then, of course, you have to have the ability to do some procedures. We do have to know how to do the computations—sometimes facilitated by the use of technology. But being able to do computations alone doesn’t mean we know mathematics.
Doing mathematics includes lots of things. It can include looking at specific examples, trying to find patterns or making comparisons to situations that we already understand. It includes making conjectures. It includes figuring out how simpler situations work and seeing if we can then generalize to more complicated situations. It includes being able to abstract mathematical properties out of real situations. And it includes being able to reason logically.
Real mathematics requires all of these things. It requires mucking along, as well as knowing immediately what to do in some cases. But I know with too many students there’s been a tendency to look at a problem and then, if there isn’t an understanding of what to do immediately, they skip the problem and ask the teacher the next day. We’re not helping kids build in a sense of persistence or build in a sense of exploration that they’re going to need to have to solve problems in the real world.
Is there other evidence that indicated a need for change?
Other evidence includes the Second International Mathematics and Science Study and the Third International Mathematics and Science Study. Both of them showed clearly that, internationally, we weren’t competing well.
There was also the problem that students were leaving the study of mathematics in droves. As soon as students reached a level where mathematics was no longer required, half of them would quit. Then, the next year, half of the ones who were left would quit. And so on.
That trend of course is exactly contrary to increased need for mathematical ability in society. We were funneling everybody into mathematics courses at the beginning of school and getting a very, very small residue of those who would pursue it beyond high school.
For a whole variety of reasons we had to do better. Some of these reasons I have already mentioned. Others include the need to update the mathematical content of our courses and incorporate the use of new technology that allows us to examine mathematical concepts on a deeper level as well as utilize mathematical tools that remain primarily theoretical without the use of technological computational power. And so early in the 1990’s the National Science Foundation put out a call for proposals to update the mathematics curriculum, incorporating not only a more complete approach to learning mathematics, but also based on the knowledge we had gained about how students learn and effective ways to teach.
And the result was?
The result was the development of new comprehensive curriculum materials at elementary, secondary and high school levels.
At the time, at the high school level, by and large across the country schools were committed to an algebra, geometry, advanced algebra, pre-calculus sequence. But when you look at how math is used in the real world, very often real world problems come without the content nicely separated like that.
A problem often can involve algebra and geometry and maybe some probability.
So, with the NSF support, out came high school level programs of three-year length or four-year length in which algebra, geometry and so on were intermixed.
These programs focus on mathematics as a process of reasoning, of thinking. They develop concepts so that there’s an understanding of why, rather than simply being able to compute. Students develop a deeper mathematical understanding. And also they develop a deeper way to apply mathematics, such in as multi-step problems. These programs require students to synthesize ideas. And they require students to use math concepts and skills in places other than at the end of the chapter in which they’re discussed.
What else was different about these materials?
For a long time mathematics educators adopted what I might loosely call the Euclidean approach. If you go back to Euclid’s Elements, which was produced about 300 BC, the organization of the information was first, a statement, and then, a rationalization for it. So you had the answer first. Then, you got an understanding of why it worked.
Math education up through the early ’80s was very much in the same spirit. Present some mathematical concepts, maybe some definitions, present the justification for those, present certain formulas and then the explanation of those formulas, do some examples of problems, and then practice with problems that were similar to the problems done in the text.
While this way may be an efficient way to present known results, it is not the best way to engage students in mathematical thinking. What we’ve learned is that it’s more effective if you don’t present quite so much to the student up front. Now, we try to present the questions before we give the answers so you get the kids’ attention and interest. You get them hooked on pursuing understanding. And then you let them figure out how to get an answer. And when the students come up with the approach, they not only have a much better understanding, they retain their understanding better, too.
Retention, by the way, has long been a major problem in US math education. That’s why a lot of topics in the US repeated year after year after year, because there was no retention of what was done. Now we have found that we can really improve retention, by putting the development of mathematics in a context that has some meaning for students. Very often, that takes the form of a real world situation. Real world context can be a wonderful way to provide meaning and mental “glue” for mathematical ideas and concepts.
What about the idea that some students will not be as good at developing inquiry skills, and will fall behind?
Well, it’s true that some students will go further than others. But what we’ve seen is that every student can improve his ability to do mathematics and learn mathematics with this approach. Whereas with earlier approaches, students who did not grasp every new definition or procedure as it was presented to them would not be able to keep up at all with subsequent material.
If the curriculum is engaging, students will come along and will use each problem as an opportunity to exercise their ability to think. Their study of mathematics becomes a process of learning what they can do, not what they can’t do.
What’s supposed to happen as a result of having these new curricula?
Well, I think, personally, it would be wonderful if these five new curricula were adopted in every school across the country. But that is not just a simple matter of changing textbooks. It often requires a lengthy process, because it involves having teachers and other stakeholders really look at their beliefs and assumptions about what kind of mathematical education they want their high school students to get. It requires significant understanding by all concerned of the aspects and ramifications of such change. And it requires district administrative support for professional development of teachers who want to improve the educational experiences of their students in mathematics. Systemic change is not easy.
Of course these five programs are not the only answer–but they represent five different models that demonstrate an approach math education in the way we’ve been talking about
Some schools are going to transition in slower ways and will more gradually change their curriculum. But generally I do think we are seeing a continuous motion in the direction suggested by the NCTM standards and successful programs in other countries.
Anything else you’d like to say?
Just that I am very encouraged when I look at all the activities at improving mathematics education—including the valuable resource and promise of the Futures Channel. Because, to my mind, we are getting to the root of what mathematics is. In the world that our children are growing up into, we do need clear thinkers, we do need critical thinkers, we do need creative thinkers. And what better place than to develop those skills than in a mathematics classroom?
*COMPASS: Curricular Options in Mathematics Programs for All Secondary Students. http://www.ithaca.edu/compass/