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About the Language We Use

by Mardi Gale
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I have often wondered how the language we use when teaching influences students’ learning of mathematics. I have heard it said that we need to make concepts “student friendly” so students will understand and remember what we mean. We’ve all heard those little “simplified” references: referring to a rhombus as a “diamond,” or “the alligator eats the larger number,” when we are referring to the “greater than” or “less than” symbols, or saying “minus” when we mean “negative” or “the opposite of,” and so on.[[{"fid":"237","view_mode":"blog_image_right","fields":{"format":"blog_image_right","field_file_image_alt_text[und][0][value]":"A math coach uses the correct terminology while conducting a fractions lesson with second graders."},"type":"media","link_text":null,"attributes":{"alt":"A math coach uses the correct terminology while conducting a fractions lesson with second graders.","height":"210","width":"300","style":"width: 300px; height: 210px; float: right; border-width: 1px; border-style: solid; margin: 1px 5px;","class":"media-element file-blog-image-right"}}]]

When I read Standard for Mathematical Practice Number 6: Attend to Precision, I am forced to reexamine what impact this “student friendly” language has on long-term understanding. I believe we can learn a lesson from our ELA colleagues about how we learn vocabulary. The most successful way to learn vocabulary is to see words, phrases, and ideas in context, used correctly and in more than one way.

When we say the “alligator eats the bigger number,” we are committing double faux pas. First of all, it is not an alligator or Ms. PacMan; it is a greater-than or less-than symbol. That’s what we should be calling it at all levels so students will refer to the symbol with precision from the start and will not have to relearn proper vocabulary later. And, yeah, our students can remember the correct words. Some primary teachers have told me their students cannot say (or remember) “trapezoid,” and yet these same students can say and remember “tyrannosaurus rex.”

Consider this:

Which is the biggest number? 4 or -7

We repeatedly hear this reference in classrooms. Yet we know that “biggest” is a function of size, not value. The proper question would be “Which number is of greater value?” If we refer to the relationship correctly, with precision, students will begin to recognize and remember the characteristics of that relationship.

[[{"fid":"238","view_mode":"blog_image_left","fields":{"format":"blog_image_left","field_file_image_alt_text[und][0][value]":"A fifth-grade teacher engages her class in \"number talk\" around a problem involving equivalent fractions."},"type":"media","link_text":null,"attributes":{"alt":"A fifth-grade teacher engages her class in \"number talk\" around a problem involving equivalent fractions.","height":"178","width":"300","style":"width: 300px; height: 178px; float: left; border-width: 1px; border-style: solid; margin: 1px 5px;","class":"media-element file-blog-image-left"}}]]And are we correcting students when they say that number is a “minus 7”? “Minus” is a reference to subtraction, which is a verb. “Negative” or “opposite of” are words that describe the value of the number. We know that the number is actually negative 7 or the opposite of 7. Do students? I used to ask students in my classrooms to listen carefully to what I said so I wouldn’t lapse into using the terms incorrectly. Talk about a way to get them to pay attention!

About two years ago, I read a short article by David Ginsburg in an Ed Week. He talked about the significance of the decimal point and the fact that we often don’t read decimal numbers with their place value. What do you say when you see 0.375? He makes the case that saying “zero point three seven five” is not helping our students learn about place value. Close your eyes and say that number correctly. When you say “three hundred seventy-five thousandths,” it usually brings up the picture of a fraction with the denominator (not “the bottom”) of 1,000. Insisting students say decimal numbers correctly goes a long way to help our students gain understanding of place value and fraction equivalence. Try it.

[[{"fid":"240","view_mode":"blog_image_right","fields":{"format":"blog_image_right","field_file_image_alt_text[und][0][value]":"High school students use precise vocabulary to describe a graph to partners who cannot see it."},"type":"media","link_text":null,"attributes":{"alt":"High school students use precise vocabulary to describe a graph to partners who cannot see it.","height":"219","width":"300","style":"margin: 1px 5px; border-width: 1px; border-style: solid; font-size: 13.0080003738403px; line-height: 20.0063037872314px; width: 300px; height: 219px; float: right;","class":"media-element file-blog-image-right"}}]]

While this is by no means an exhaustive list of the concepts we often refer to in “simplified” terms, these examples make it clear to me that we would get better results in conceptual understanding, and written responses of thinking and reasoning, if we use mathematical language with precision and in context.

A few other words of advice:

  1. Use the proper, precise naming of all polygons, such as trapezoid, rhombus, parallelogram, etc., in context and consistently so students begin to use them properly.
  2. Teach finding equivalent fractions and simplifying, not cross canceling. Students often forget when to cross cancel, and what the rules are that govern this short cut. Teach the distributive property so that factoring is done properly. If students do not have a clear understanding of why and how the concept of factoring is used to simplify, they may try to cancel incorrectly. Too often we see:
    [[{"fid":"228","view_mode":"blog_image_full_width","fields":{"format":"blog_image_full_width","field_file_image_alt_text[und][0][value]":""},"type":"media","link_text":null,"attributes":{"alt":"An image showing an incorrect simplification of (2x+3)/6","height":"180","width":"300","style":"width: 300px; height: 180px; border-width: 1px; border-style: solid;","class":"media-element file-blog-image-full-width"}}]]
  3. Teach finding equivalent ratios or ratio tables to solve proportions, not cross products. Cross products is an example of “answer getting” without meaning.
  4. When teaching order of operations, don’t use the little mnemonics. These little so-called memory devices misrepresent the actual order of operations. These little sayings always appear to be linear and confuse students when they try to apply them. No more “Please Excuse My Dear Aunt Sally.”
  5. When making comparisons or ordering quantities, be sure to refer to the value of the quantity, not the “larger,” “bigger,” or “smallest” numbers. Those are descriptions of size, not value, and value is what is being compared or ordered.
  6. Refer to simplifying expressions or fractions, not reducing. Once again, reducing refers to size not value.
  7. Remember, there is a difference between simplifying and evaluating an expression. Simplifying refers to performing all the indicated operations until there are no more that can be performed. Evaluating refers to substituting a value in place of the variable(s) in an expression or equation and then simplifying.

For expert perspectives and classroom examples, including those that address how precision of language can help students gain mastery of algebra concepts, see the Coherence in Algebra collection curated by Mardi.

Senior Research Associate, Science, Technology, Engineering, & Mathematics (STEM), WestEd

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