By guest author Tom Clark
Parents and educators have begun abandoning the middle school configuration, generally opting for a K–8 format, and new research suggests that the way we organize grades does matter. When this age group is gathered by the hundreds and educated separately, both behavior and learning suffer. Of course, the controversy continues.
But what does all of this have to do with teaching Mathematics? Changing arithmetic thinking to algebraic thinking is a critical dynamic in the study of Mathematics. And, once we are comfortable that students are competent in Arithmetic (generally around the 6th grade), we should immediately begin revisiting arithmetic procedures with the understanding that many of those algorithms will change in Algebra. Let me explain.
The word “arithmetic” comes from the Greek arithmos meaning “numbers.” That’s primarily what arithmetic is all about, isn’t it? Elementary students concentrate on discovering the concept of “how many” and learning to count, to quantify a group of objects.
They develop a “number sense,” growing to the point where they can just look at a small group of objects and know, without counting, how many there are in the group. They learn how to operate on these numbers, discovering what it means to add, subtract, multiply, and divide. As a result, they realize that the counting numbers aren’t enough.
To be able to perform various operations consistently and with confidence, they need to expand their group of numbers to include the number zero, the negative numbers, fractions, and decimals. Further, they explore the relationships between the numbers, coming to an understanding that, for example, .75 and 3/4 actually represent the same amount.
Underlying all of this, however, is a strong inclination to begin making everything mechanical. It’s all so concrete that all you really have to do is memorize a procedure. A classic example is learning how to divide fractions. I’m sure you remember that, right? Just turn the second fraction upside down and change division to multiplication.
But why? Oh, don’t worry about that. “Ours is not to reason why. Just invert and multiply.” Is it any wonder that students become less interested in math, even to the point of beginning to distrust it, in the upper elementary grades?
If I may wax philosophical for a moment, what are we really trying to teach our students, regardless of the subject? What is our goal in education? We are striving to prepare our students to educate themselves throughout their lives. We want them to become analytical and critical thinkers, so they can be lifelong learners.
And how do we accomplish that? We can’t just teach them to do something without understanding why. We must teach them to think about what they’re doing. That is why these middle school years are so critical educationally. This is a time for reflection and change in the way students think.
You see, the word “algebra” comes from the Arabic al-jabr, which means to “bring back and reunite” (originally having to do with bone-setting). And how does that relate to the subject of Algebra? Well, just think of the biggest equation you can. There is an x in it. Something is missing. Your job is to search the universe, find that number, and “bring it back and reunite it” with the equation to make a true statement. That is a very different way of thinking, compared to the ordinary activities of Arithmetic.
You know, I think I can actually fix this country, mathematically, if I can just convince all middle school math teachers that their job is not to “review” step-by-step arithmetic algorithms. Their job is to “re-teach” Arithmetic, making sure students understand why those arithmetic procedures work and how they are going to be applied in an Algebra setting.
The fact is, there are things you learn in Arithmetic that simply won’t work in Algebra. For instance, in Arithmetic, what are we supposed to remember when we see parentheses in a problem? We know we are supposed to “do what is in the parentheses first.” Well, guess what. That doesn’t always work in an Algebra problem. When you have an x inside the parentheses, you can’t do anything. Then what do we do? The middle school teacher should have revisited that guideline and explored alternatives, such as using the Distributive Property, to expand the expression, getting rid of the parentheses in the process. The student will then be able to handle this situation when it comes up in an algebraic setting.
As your students approach their middle school years, however you define that, and you begin to think about your goals in their education, please give considerable attention to materials, resources, and teaching strategies which encourage your students to really think about what they are doing. Encourage them to ask “why.” Encourage them to dig deep, to truly understand. Encourage them to use those middle school years to change from arithmetic thinking to algebraic thinking.
After all, even if the specific content of the course seems irrelevant to our lives at the moment, the learning of that content should focus on developing and honing the analytical and critical thinking skills necessary to continue to educate ourselves throughout our lives. Those are life skills which are much more to be valued then memorizing rules, tricks, and shortcuts.
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