I think relevancy is something all teachers think about– after all, an extremely common question is, “So when am I going to use this in real life?” (As far as math goes, my answer is, “You might not use this exact formula/concept, but learning about all aspects of math gives you the ability to be a more logical, creative, and reasonable thinker– math is a different way of seeing the world”). Nonetheless, many concepts provide opportunity for a more satisfying answer to such a question before it’s even asked. Here are some examples.
While Mrs. D had a planning period on Tuesday, I snuck in to observe another teacher’s “fundamental” section. They were focused on the aspects of central tendency: mean, median, mode, and range. Mrs. B (the other teacher) used arbitrary numbers for her examples of data sets until she noticed some of the students losing their focus. She noted, “In fact, I actually use mean when calculating your homework scores. They’re a weekly average: I add up the grades for all of the homework given in the week, and divide by the number of assignments.” A few ears perked up, glad to finally have some insight on the mysterious formula that produced their scores. “Who wouldn’t mind if I shared their homework scores anonymously to show you the process?” Nearly every hand went up, excited to be the subject of attention.
The example went as follows. A student was given five homework assignments in the previous week. He had received an 82, an 85, a 93, a 90, and a 0 because he didn’t turn in Friday’s homework. (Here Mrs. B asked a student to calculate the mean). The result was a 70. Looking at the average of just the four days when he DID turn in his assignment, his score was 87.5 (rounded to 88). By skipping a single night of homework, his weekly grade went from a B+ to a C-. The students glanced at each other with a slightly worried and awed look.
The story has a happy ending, though. The student in question brought in his homework on Monday. The maximum points he could receive was 80 because the policy entails a 20% deduction for each day late. He got the maximum score– 80. A second student calculated his new weekly homework average: an 86, or a B. There were three morals to this story:
- Missing your homework means that your grades diminish quickly.
- Late homework is far better than no homework.
- “Mean” is used in real life by real, familiar people.
Later that afternoon, Mrs. D’s students practiced multiplying whole numbers by percents/decimals/fractions (they have been developing fluency in interpreting each of the forms and moving between them). Mrs. D first asked her students to turn to a partner and brainstorm situations where they would have to multiply by a percentage. When they reconvened a minute or so later, there was a long list that included figuring out how much an item cost during a sale, tipping a waiter or waitress, and sales tax.
After a brief review of what sales tax is and the current rate for this state (6%), the students were asked to solve the first problem on their whiteboards. Mrs. D knew one of her students had bought a skateboard the week before. She asked him how much it was at the store ($79.95). The students were then asked to calculate the price he paid at the register. They quickly realized it was a multi-step problem. They had to convert 6% into decimal form (which was fairly automatic for many). After multiplying 79.95 by .06, they recognized that the answer was only ~4.80: in other words, it was only the sales tax itself, and needed to be added to the original price. Mrs. D asked volunteers to share an item they had recently bought or planned on purchasing for a few more practice problems. As in Mrs. B’s class, the students were eager to share a personally relevant item.
Finally, one boy raised his hand and said, “My dad lives in another state where there is NO sales tax. Why don’t all states just make it that easy? Wouldn’t people be more likely to make purchases?” This led to a really great discussion on different types of taxes, why they’re implemented by the state and country, and how different states set different tax rates in each particular category.
Mrs. D slightly shifted the focus towards interest and the similarities and differences with taxes. She asked how many students had a savings account at the bank– about half raised their hand– and by multiplying a reasonable amount ($100) by a hypothetical rate (8%: high, but with a purpose), they discovered that leaving saving money in an account ultimately meant that interest worked in their favor.
After a few more examples, Mrs. D asked a child she knew was really into cars what type of vehicle he wanted to purchase and to give an estimate of the cost. His only guidelines were that it had to be appropriate for a high schooler to drive and similarly affordable. He picked a Honda Civic, estimating that an older used one would cost around $6000. Mrs. D explained that when most people purchase a car, they take out a loan for the price of the car and then pay it back with interest over a few years. She asked the students to work in pairs to figure out what the actual cost of the Civic would be if the student took out a $6000 loan with an 8% rate over 3 years. The caveat was discovering that the result of 6000*.08 had to be multiplied by 3 to account for each year of payment. The students were astounded to discover that in order to purchase a reasonable $6000 car, they would have to pay an additional $1440 in interest alone. Other students pointed out that this was just a foundational cost, too: there was still car insurance, gas, and repairs to consider. *Note: technically this isn’t the precise formula for finding the accumulation of interest on a loan, but Mrs. D’s version gave a fairly accurate estimate of the responsibility for taking out a loan.*
The period ended with further points about how banks might offer you various rates depending on your credit score (and that this in itself was calculated based on how well you managed your finances) and how a second option might be to finance directly with the dealer, though the rate would probably be higher than that of a bank. I was also able to share that the lesson was extremely relevant to my situation, as I’m about to purchase and finance a reasonably priced car.
It was a rich economic discussion that allowed the students to practice the content in a meaningful way, consider how relevant the material was to their everyday lives, and begin to think about how important their financial decisions would be in the near future.
These lessons reminded me of a phenomenal workshop I attended at the National Council of Teachers of Mathematics conference in April. Two educators from New Jersey created a database of Common Core-aligned word problems related to topics of social justice called Socially Conscious Math. The questions are meant to invoke thoughtful conversations surrounding issues such as gender and racial inequalities, the impact of poverty, environmental conservation, public health, historical context and more– all through the lens of mathematics. Here’s a sample:
6. Equality: The entire world has a population of 7 billion people. The US has a population of 314 million people.
6a. Find the percentage of the world that lives in the US.
6b. The entire world has a population of 9 million in jail. The US has 2.2 million people in jail. Find the percentage of the world’s prisoners that are in the US.
6c. Compare your two figures, the US population to the whole world population and the US prison population to the whole world prison population. What are your thoughts about these two ratios and why they differ?
In their presentation, the founders Deborah Gordon-Goodrich and Gary Kaufman shared guidelines for using their problems (and similar socially-applicable ones) most effectively in the classroom.
- Clearly convey that these are the numbers, not your opinions.
- Encourage students to recognize that their viewpoints are relative: for example, “cheap” and “expensive” are subjective adjectives for monetary values.
- Allow 20 or more minutes per problem in order to structure time for the accompanying conversation. (If a problem takes significantly less time, you may need to challenge your students to think more deeply and more critically).
Additionally, the classroom culture needs to be such that the students can easily have a meaningful and respectful conversation on a topic that may not be freely discussed in many settings. This requires that students feel safe to express their thoughts and opinions to you (as their teacher) and to their peers. Thus these problems should be implemented only after strong routines and expectations are in place.
Finally, Deborah and Gary note that their database is just a starting point. They hope classroom teachers (and students) will begin to write their own problems based on the plethora of numbers represented in nearly all current topics.
I yearn to teach tolerance and acceptance in my classroom. I want my students to be able to use what they learn (in math and in other subjects) to change their world. All too often, I think that math is left out of this process as it’s not considered a subject where students can freely express their beliefs and ideas. Socially Conscious Math– and even simpler applications like in Mrs. B and Mrs. D’s classes– are paving the way to change this.