In my early days of teaching--a long time ago--I treated math vocabulary just the same way vocabulary has always be treated in English class. I gave definition and spelling quizzes. The complaints were loud and frequent: "This isn't English class." My favorite was: "What does this have to do with math?" In spite of the complaints, my students knew how to PROPERLY pronounce and spell "commutative," and they had a good understanding of the definition. With all of the time pressures of No Child Left Behind, I, like so many other teachers, was forced to let go of this focus on definitions and spelling. Students went right back to spelling and pronouncing commutative as "communitive" and their understanding of the meaning went right out the window. Commuting to work and back is hardly the same thing as communing with nature, but they never got the concept of the importance of pronunciation, spelling, and definitions.
I recently read a teacher's blog by David Ginsburg in which he was making the point that we math teachers need to work harder on language fluency--especially when we are using terms that have very different meanings in our mathematics class than they have in everyday life. His focus is on elementary school; and my favorite of his examples deals with the word "borrow" in subtraction. Why should we say borrow when it is never going to be returned as the everyday meaning of borrow implies?
The very same issues happen in upper level math. In Algebra, factoring is an extremely important concept. Our students are frequently given instructions like: Factor x^2 - 3x - 28
I know from many years of both teaching and tutoring Algebra that even when students can actually correctly factor this expression, they cannot tell you in words what it actually means "to factor." Similar to Pavlov's dogs salivating to the sound of a bell, many of our students know that when they see the word factor beside a trinomial like x^2 - 3x - 28 they are supposed to write (x - 7)(x + 4). But what does the word "factor" actually mean to them?
They are familiar with this use of the term: (1) washing hands is an important factor in slowing the spread of disease, or (2) teens' failure to use protected sex is a major factor in teen pregnancies. To our students, "factor" implies a reason or a cause. What does that have to do with x^2 - 3x - 28?
Note: In mathematics, to factor means to re-write as multiplication. x^2 - 3x - 28 becomes (x - 7)(x + 4) when written as a multiplication problem.
Another example from Algebra involves another extremely important concept: Solve. We know what it means to solve a puzzle or to solve ones own troubles. How does "put it together," as with a puzzle apply to: Solve the equation x^2 - 3x = 28? Find your own answers sort of applies, but what is an answer to an equation? How do we know when we have an answer to an equation? We simply do not do a good enough job of drilling in the concept that an answer "makes the equation TRUE." We say it once or twice and from then on ASSUME they understand.
In Geometry, we start using protractors to measures angles in degrees. Does that have anything to do with temperature? Are you certain that all of your students are picturing in their minds the same thing for one degree that you are, or do you just assume they are?
On of my biggest frustrations with missing understanding is with the use of the word "value" with respect to fractions. We say that reducing a fraction does not change its "value" or we tell our students to compare 3/4 and 1/2 as if fractions themselves have "value." THEY DO NOT HAVE VALUE--YET. We cannot compare 3/4 and 1/2 until we know 3/4 "of what" and 1/2 "of what." I can almost guarantee you that 3/4 of the money in my pocket is LESS than 1/2 of the money in your pocket. We do not stress enough to our students that to know if 3/4 > 1/2, we must first know that they deal with THE SAME THING. Even on a number line, the reason that 3/4 > 1/2 is because BOTH are fractions of THE SAME UNIT. However, 3/4 of an inch and 1/2 of a foot CANNOT be written 3/4 > 1/2!
I strongly suspect that if you asked Algebra students, "Is 3/4 always greater than 1/2?" the vast majority, if not all, would say "yes." Homework instructions never say "compare these two fractions from a number line" or "assuming these fractions are of the same item, compare things fractions."
We math teachers must be constantly checking for understanding about what students are picturing with terms and definitions. Be honest with them about the possibilities for misunderstanding. If we don't catch these early, the misunderstanding can become practiced and learned and become almost impossible to repair.
Never ASSUME they are picturing what you are. Never ASSUME they know what you think they know. ALWAYS VERIFY!
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