What’s so improper about fractions? Perscriptivism and language socialization at Math Corps – Stephen Chrisomalis
Stephen Chrisomalis’s study is about Math Corps, a math program in Detroit, that help low achieving math students. Math Corps had a specific way of talking about math and using “mathematician” vocabulary. His study did not show a significant cognitive growth, but just the idea of math as it’s own language is powerful. Chrisomalis has inspired me to finally write what I have been thinking about lately. We use much language in our classrooms, but do we even take notice of what math language we are passing along? Are we helping students grow in their mathematical language along with their skills? Are we pass along tricks, rhymes, and chants, that don’t actually help with conceptual skills?
“Keep, change, flip.” “Cross multiply.” “Rise over run.” “Line up the decimal.” “FOIL.” “Three point five.” Eleven over nine.” “Cancel.” “Add on both sides of the equal sign.”
These are some of the things I hear my middle school student say as we move through concepts. I have a very hard time getting them to unsay it nor can I can get them to explain what it means. Many of those use the sayings no matter what concept they are learning. I want to tackle a couple of these because they are ones that I have stopped using or just mention briefly so that they can align their learning with their future math teachers.
“Keep, change, flip.” This one took me the longest time to understand. I was an inexperienced 6th grade math teacher who didn’t understand the concept. I just knew that it was something we did when dividing fractions. I asked around and finally got the answer as to why “keep, change, flip”. Even though I have the answer, I don’t use the phrase in my class ever. We talk about multiplying by the reciprocal but only after looking at many problems and coming up with the pattern. I don’t show them the work, but at least I have an understanding and don’t use the phrase anymore.
“Cross multiply.” If your students are like mine, they do this every time there are two fractions even if there is an addition sign between the two fractions.
- When comparing fractions or ratios, there are two things that I talk about with my students: 1) create like denominators or 2) make/imagine the pieces that you are drawing. This creates conceptual understanding instead of students thinking that there is a trick to comparing fractions.
- When solving proportions, use inverse operations. Teach and use algebraic skills. Remember, the fraction bar is the division operation.
“Rise over run.” Or even the slope formula are two thins I no long address in my class. The idea of rise over run makes sense on a graph, and slope formula is great for two points. I just find that too many students lose the conceptual connection it has with rate of change. Keeping to the concept of change in your dependent variable (y-values) compared to the change in your independent variable (x-values), students keep the connection between rate of change and slope. What I have discovered since making the change to talking about change in y over change in x, students have better linear graphing skills and better skills at finding the equation of a line. Even when it comes down to graphing linear functions, we don’t talk about using the y-intercept or (x_1 , y_1) and slope until they can see it in the function. I force students to use the x and y intercepts to graph and change in y over change in x to find slope. There is always that one or two students who see the connection and we no longer have to calculate but just graph.
“Line up the decimal.” When should you and when shouldn’t you? We should really be instilling in students place values. By the time they are middle school most students are able to decipher the places values greater than the units. But going to the right of the decimal gets hazy. Knowing place values, students are better able to work with decimals and operations. Also as we know, lining up place values is only for addition and subtraction, teaching them place values will teach conceptual understanding of when we multiply and divide decimal numbers.
“Three point five.” Eleven over nine.” Teach them and enforce place values. This enforces the fluency between decimals, fractions, and percents. Too many students see numbers like 40%, 5/8 and 0.4 as different numbers. Fluency and ability to move between the forms come from knowing and understanding place values.
“FOIL.” When multiplying two binomials, this works great, but what are students to do when there is a polynomial as one of the factors? FOIL gets confusing with polynomials, and students miss terms. Using the area model concept, students separate out each polynomial into its terms and multiply. This brings back the elementary concept of using the area model for multiplying multi-digit numbers and allows students to see the connection between elementary and algebraic concepts.