This unit is being written specifically to be used with my third grade class. I proposed to use subtraction as the basis for this curriculum on problem solving because the unit needed to be grounded in some specific mathematical topic or area. Therefore, I decided to focus on subtraction -- more particularly subtraction with renaming. My interest in this area comes primarily from my experiences in teaching math to third and fourth graders. It has surprised me over the years that even the very best math students I have taught still have trouble, or are confused by subtraction with renaming. The topic is usually taught in third grade but continues to confound students long after they first encounter it. While students seem to love and anticipate getting to multiplication and learning times tables, subtraction seems to be regarded as dull and laborious. For the most part I have found that whenever the math lessons find their way back to subtraction with renaming there is a kind of lapse in memory and students usually look askance at the problem until they get a quick refresher on how to do the example. This, I must suppose, is a clear indication that the concept is not really understood. If this is the case, then why and how can that be changed?
For many of the ideas about teaching subtraction that are presented in this unit I have relied on my reading of
Knowing and Teaching Elementary Mathematics
by Liping Ma, especially the chapter concerning subtraction and renaming. I have also used
About Teaching Mathematics: A K-8 Resource,
by Marilyn Burns. Burns is a well respected author and former classroom teacher who has written widely on math instruction. She has also authored a group of popular classroom resources. A series of books published by Heinemann:
Children’s Mathematics: Cognitively Guided Instruction, Thinking Mathematically
and
Making Sense: Teaching Mathematics with Understanding
were also extremely helpful. These books are the work of a group from the National Center for Research in Mathematics. The purpose of the group is to promote the teaching of math in elementary schools.
I would like to begin with a conclusion gathered from the research presented in the readings -- namely, that children who have constructed mathematical knowledge are adept at using and retaining what they have learned. This seems to most of us a plausible statement, but how do we achieve this in the classroom? The focus of instruction must change from an emphasis on calculations to one that includes the recognition that doing calculations is ultimately only a tool for problem solving.
The Ma book is a presents a fascinating comparison between the mathematical knowledge and performance of American and Chinese elementary teachers.1 What seems clear in the Ma book is that the major difference between the American and Chinese teachers was the difference between procedural understanding and conceptual understanding. This is not to say that neither side was totally superior or inferior, however, the Chinese teachers seemed to be more comfortable teaching math. For the majority of American teachers teaching subtraction with renaming was more procedural. Students learn to go to the next column and cross out and add a one in front of a number but they have little understanding of what and why they are doing it. I suggest that the reason for the emphasis on the procedural is two-fold. First, I don’t think that most teachers understand how the math curriculum develops below and above their grade. Most of my colleagues in the lower grades are not familiar with what types of questions are on the Connecticut Mastery test. Likewise most of us in third and fourth grade have little concrete knowledge of what is taught in fifth and sixth grade math. The Chinese teachers seemed to understand the continuum that made up what they were teaching and they had a firm idea of where students were coming from and where they needed to go. Secondly, it seemed that the Chinese teachers had time to think over what they were teaching. I almost felt as though they must be teaching only math (which is the case for many of them) because where did they get the time to do all this contemplating about their lessons and making modifications to suit their students’ needs. I have to say that since most of us teach with a set curriculum and pacing chart, there is not a lot of time to expand lessons. Indeed most of the examples cited in the books are laboratory situations where the math programs are experimental. The emphasis on package curriculums is one added reason we may tend to be so procedural in teaching math. However, the curricula could develop the reasoning if it chose to do!
General Remarks on Problem Solving
When computational competence was the main goal, a procedural emphasis may have been effective. As Burns points out one of the reasons students have trouble solving problems is that the problems they are given usually follow a lesson either on some mathematical operation be it addition, subtraction, multiplication, or division.2 Then a series of word problems to be answered by using the operation just taught would be given. The problem wasn’t really a problem it was just another way to practice the procedure. The students knew that if they had just studied division, then these word problems would probably be solved in the same way. In reality a problem does not come with its solution tied to it. The person trying to solve the problem has to understand it and try to figure out the best way to solve it. In my readings I ran into once again the four steps advocated by George Polya of Stanford University in his book
How to Solve It.3
The four steps are:
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Understanding the problem
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Devising a plan
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Carrying out the plan
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Looking back
In some of the readings there is a suggestion against teaching a kind of formulaic way to solve problems. While I understand the desire to make problem solving uninhibited in terms of the number and types of solutions possible for a given problem, I also know that my students do not have the profundity of language and experience and may need some helpful suggestions. I’m not beyond offering up a set of strategies that I have seen in one form or another in a few textbooks. The following are taken from Marilyn Burns:
Problem-Solving Strategies
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- Look for a pattern
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- Make a Table
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- Make an organized list
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- Act it out
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- Draw a picture
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- Use objects
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- Guess and check
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- Work backward
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- Write an equation
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- Solve a simpler (or similar) problem
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- Make a model 4
While these are useful strategies for students to be familiar with, many like Burns do not recommend the teaching of word clues like “altogether” means add, or “how many more” means subtract. I have found these clues give students a place to start when they first read a word problem. Burns seems to feel that this only leads to giving students tricks to finding the answers. It also makes students think that the right answer is all that is important. They don’t care about how they got the answer as long as they have it. I agree with her that these clues should not be overemphasized otherwise students will not develop their own problem solving abilities.
