Although I teach various levels of high school math, I keep two sets of magnetic fraction bars on my whiteboards. In the beginning of each year, I often grab them as I am explaining Algebraic operations that happen to involve fractions. I switched from quick sketches of pies to bars this year when I realized my students had shifted with most of the country under the pervasive influence of the Common Core State Standards (CCSS).
One would think that by high school, students would have fractions operations nailed down; but my experience is that only about a third are fluent. Most of the students in my Algebra 3 class are there because they had a rough year in Algebra 2…mostly because of weak fraction skills, as I explain below. I do not think weak fraction skills are the fault of the elementary teachers who have followed the carefully constructed path the CCSS writers designed. But somewhere along the line, the ball gets dropped; and I believe when math teachers truly understand the long-term repercussions, their instruction will be decisively impacted.
Fraction skills are have been identified as primary predictors of success in Algebra*. Fraction sense is built in grades 3-7. Sometimes 8th grade teachers struggle to cover the breadth of 8th grade standards and may find themselves avoiding the messiness of fraction operations by having students convert them to decimals. Well meaning teachers may provide students with tricks to avoid the cognitive burden of remembering how to make sense of fractions while they are learning about functions and expanding equation-solving skills. High school teachers may throw their hands up and refuse to remedy the gaps, “We don’t have time for that. They should have learned that years ago.”
I am not convinced.
When students bypass fraction operations by using decimals, they hit a wall in Algebra 2 because algebraic rational expressions cannot be converted to decimals. The effort required to master long-neglected basic fraction operations in the context of rational functions requires more effort than many students have the time or willingness to invest. Large numbers of students become disaffected and write themselves off as being not good at math. As a result, higher levels of math become inaccessible to them; and the majority of the new careers associated with 21 Century technology are beyond their reach. This is a socioeconomic justice issue.
Even if a student has his or her heart set on a trade, no student should complete a high school math class without proficiently working with fractions. With today’s learning technology, teachers should not be letting their students or themselves off the hook.
Here is how I ensure my students have every opportunity to build fraction skills as we progress through the year.
- I assign a high-point mastery quiz to identify weaknesses in fraction skills the first week of school. I provide personal feedback and video links. Students stop by before/after school or lunch for retakes. The cycle is repeated as necessary on an individual basis, and I communicate progress with parents. As the students see their grades rise with their skill level, they appreciate this gamification process. I show my Algebra 1 students the example above and remind Algebra 3 of where they got lost to increase motivation.
- I incorporate fractions with most new skills learned and include them often in homework problems and assessments. I emphasize why we don’t just turn them all into decimals. When I address specific questions that students have, I reach for the fraction bars as a tangible reference to their prior learning (anchor theory). In Algebra 3, I make sure the fractions they work with are heinous before they reach the rational functions unit.
- I hit A.REI.3 hard with fractions because that standard is important in math and in science. “Literal equations” (solving for a variable in terms of other variables) offer opportunities to for students to generalize their understanding of fraction operations.
Of course I cannot promise their teacher next year that my students will not have forgotten some things about fractions, but I am confident each one is much stronger. This year, while doing a little research for STEM integration, I saw large numbers of Algebra 3 students shift their declared majors from non-STEM (reported in August) to STEM (reported in January). They have the skills now, so I get to cheer them on.
*Siegler, R., Duncan, G. J., Davis, P. E., Duckworth, K., Claessen, A., Engel, M., Susperreguy, M., Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23: 691-697. doi:10.1177/0956797612440101