Algebra Students: Vocabulary Is Important In Math Class, Too! Ask If You Are Confused About Meanings

I recently read a blog from a math teacher whose main area of concern is elementary math and literacy. I had to laugh at the example he used--why do we use the word "borrow" in subtraction when we don't intend to give it back? This is a point well-taken. We tend, in mathematics, to use terms from everyday speech, but in mathematics, these terms often have very different meanings. This can be and often is a source of confusion for many Algebra students; and, unfortunately, students often don't even realize they have confused the meanings of a term, and the teacher doesn't catch it is until too late. By "too late" I mean that the mistake has been practiced and has become ingrained in your brain as a fact. These mistakes are difficult to fix. It is better to avoid these than try to fix them.

One term from your elementary days that still causes students difficulty is the term "value"--especially with respect to fractions. If I asked you, "Is 3/4 > 1/2" what you say? You and almost everyone else would say "Yes." In reality, the answer is "not necessarily." The problem here is that fraction symbols do not actually have a VALUE until you know the "of what." Is 3/4 of an inch greater than 1/2 of a foot? Of course not, you say? Why not? You just told me 3/4 > 1/2.

Now that you are thinking a little bit more about it, you realize that fractions can only be compared IF they are fractions of the SAME THING. So why have math books had homework sections with instructions to "compare the fractions?" Because someone assumed you knew that fractions can only be compared if they are of the same thing, and they left out an important part of the instructions. "Compare these fractions on a number line" or "Assume these fractions are of they same thing" would be appropriate directions. On a number line, 3/4 and 1/2 represent parts of the same size unit. We math teachers tend to assume that every student is picturing the same thing or understanding a definition the same way that we intend; but you and I both know this isn't always true.

In Algebra, there are two huge examples of extremely important concepts that students often get confused with their everyday meanings--or at the least, cannot really explain what the math meaning actually is. These two concepts are: "solve" as in solve an equation and "factor" as in factor this expression.

You know what it means to solve a puzzle or to solve a problem you are having making free throws in basketball; but what does it mean to solve an equation? To find the answer, you say? How do you know when you have an answer? It works? What does that mean? Very few Algebra students can actually say in words--with any real understanding--that to solve an equation means to find values for the variables that make the equation TRUE.

You know that washing hands is an important "factor" in slowing the spread of disease; but how does this apply to factoring an expression like a^2 - ab? The mathematics meaning of "factor" is totally different from the everyday meaning. In Algebra, to factor means to "re-write as multiplication." What? Well, a^2 - ab in factored form is a(a-b) since when you multiply a and (a - b) you get a^2 - ab.

All of mathematics--not just Algebra--is full of these terms with different meanings in the everyday world than in the mathematical world. For your own success, you must always memorize math definitions immediately, practice these definitions, and even discuss with yourself and with your teacher the differences in meanings. It is OK--in fact, important--to know that a term has several different meanings. It is equally important that you understand each of the meanings and know when to use which meaning.

If you get confused, or are ever in doubt, ASK YOUR TEACHER! It is the teacher's responsibility to teach you. We math teachers are not perfect human beings, even if we like to think we are. We often ASSUME more than we should. DO NOT BE AFRAID TO ASK QUESTIONS. That is your responsibility. Ours is to answer your questions.

Shirley Slick, "The Slick Tips Lady," is a retired high school math teacher and tutor with degrees in Mathematics and Psychology and additional training in brain-based learning/teaching. Her goals: (1) to help parents help their children with math, (2) to help eliminate the horrendous Algebra failure rate, and (3) to inform the general public about problematic issues related to the field of education. For your free copy of "10 Slick Tips for Improving Your Child's Study Habits," visit her website at http://myslicktips.com/

Math Teachers - Cautions With Vocabulary! Your Mental Images May Not Match - Never Assume They Know!

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!

Algebra Students: Vocabulary Is Important In Math Class, Too! Ask If You Are Confused About Meanings

I recently read a blog from a math teacher whose main area of concern is elementary math and literacy. I had to laugh at the example he used--why do we use the word "borrow" in subtraction when we don't intend to give it back? This is a point well-taken. We tend, in mathematics, to use terms from everyday speech, but in mathematics, these terms often have very different meanings. This can be and often is a source of confusion for many Algebra students; and, unfortunately, students often don't even realize they have confused the meanings of a term, and the teacher doesn't catch it is until too late. By "too late" I mean that the mistake has been practiced and has become ingrained in your brain as a fact. These mistakes are difficult to fix. It is better to avoid these than try to fix them.

One term from your elementary days that still causes students difficulty is the term "value"--especially with respect to fractions. If I asked you, "Is 3/4 > 1/2" what you say? You and almost everyone else would say "Yes." In reality, the answer is "not necessarily." The problem here is that fraction symbols do not actually have a VALUE until you know the "of what." Is 3/4 of an inch greater than 1/2 of a foot? Of course not, you say? Why not? You just told me 3/4 > 1/2.

Now that you are thinking a little bit more about it, you realize that fractions can only be compared IF they are fractions of the SAME THING. So why have math books had homework sections with instructions to "compare the fractions?" Because someone assumed you knew that fractions can only be compared if they are of the same thing, and they left out an important part of the instructions. "Compare these fractions on a number line" or "Assume these fractions are of they same thing" would be appropriate directions. On a number line, 3/4 and 1/2 represent parts of the same size unit. We math teachers tend to assume that every student is picturing the same thing or understanding a definition the same way that we intend; but you and I both know this isn't always true.

In Algebra, there are two huge examples of extremely important concepts that students often get confused with their everyday meanings--or at the least, cannot really explain what the math meaning actually is. These two concepts are: "solve" as in solve an equation and "factor" as in factor this expression.

You know what it means to solve a puzzle or to solve a problem you are having making free throws in basketball; but what does it mean to solve an equation? To find the answer, you say? How do you know when you have an answer? It works? What does that mean? Very few Algebra students can actually say in words--with any real understanding--that to solve an equation means to find values for the variables that make the equation TRUE.

You know that washing hands is an important "factor" in slowing the spread of disease; but how does this apply to factoring an expression like a^2 - ab? The mathematics meaning of "factor" is totally different from the everyday meaning. In Algebra, to factor means to "re-write as multiplication." What? Well, a^2 - ab in factored form is a(a-b) since when you multiply a and (a - b) you get a^2 - ab.

All of mathematics--not just Algebra--is full of these terms with different meanings in the everyday world than in the mathematical world. For your own success, you must always memorize math definitions immediately, practice these definitions, and even discuss with yourself and with your teacher the differences in meanings. It is OK--in fact, important--to know that a term has several different meanings. It is equally important that you understand each of the meanings and know when to use which meaning.

If you get confused, or are ever in doubt, ASK YOUR TEACHER! It is the teacher's responsibility to teach you. We math teachers are not perfect human beings, even if we like to think we are. We often ASSUME more than we should. DO NOT BE AFRAID TO ASK QUESTIONS. That is your responsibility. Ours is to answer your questions.