Not understanding mathematics can be extremely frustrating for students. As a teacher, figuring out how to help students understand mathematics can be just as frustrating. My primary go-to resource for these situations is Teaching Student-Centered Mathematics, by John Van de Walle et. al. because it’s all about focusing on big ideas and helping students make sense of the math they’re learning in a conceptual way.
Recently, I was asked to model a lesson for a 6th grade class who was having difficulty working with percents. So, I turned to my go-to resource, and during planning, I realized that I didn’t know anything about these students other than that they were struggling with percents. So, I couldn’t assume anything. I ended up creating three separate lessons and combined them into 1.
First, I handed groups of students a set of Percent Cards and Circle Graph Cards. Their task was to match the percent with the corresponding circle graph. As students were working on this, I heard groups reasoning about how they were matching the cards. Many started with benchmarks of 25%, 50%, and 75%, while others started with the smallest (10%) and matching it to the graph with the smallest wedge. As groups finished, they were asked to find pairs, using the matches they made, that totaled 100%. Once finished, a discussion about their process for completing these tasks revealed a solid understanding of percent as representing a part of a whole.
Now to shake their world up a bit. I asked them to leave their cards because they would be using them again shortly. I introduced these Percent circles and asked them what they were. A brief discussion revealed some misconceptions. Some students said they were fractions, others said they were wholes because nothing was shaded. I altered my planned line of questioning to questions that eventually led to a common understanding of what fractions were and how the pictures of the fraction (percent) circles really showed wholes and parts (fractions).
Their next task was to match their cards with the equivalent fraction circle. This was incredibly eye-opening. Groups began to notice that some percent card matches could fit with multiple fraction circles (50% could be matched with the halves, quarters, eighths, and tenths). Thirds and eighths were the last to be matched. But their reasoning didn’t disappoint. One group noticed that the percents ending in .5 all belonged with the eighths because they were too small to be thirds (the other percents with decimals).
Students were eager to share their thoughts about what they learned about fraction circles and percents:
- Fractions and percents are the same because the pieces look the same.
- 1/4 is the same as 25% and 2/8
- I don’t get why the eighths end in .5.
- The percents all can be fractions.
- 1/8 is 12.5% because it’s half of 25%
Finally, I asked students to solve a percent problem (now that they’ve all realized that fractions and percents can be used interchangeably). I gave them the m & m problem from this set of percent problem cards. The only direction I gave was that they had to solve the problem using some representation of the percent in the problem before they wrote any numbers.
My bag of M&M’s had 30 candies inside. 40% of the candies were brown. How many brown candies is that?
While this was problematic at first, students looked at their fraction circles and percent cards and realized they could use four of the tenths since each tenth was the same as 10%. Most students needed just one “least helpful” question to get on the right track: Where do the 30 m & m’s belong in your representation?
Most groups were able to make sense and persevere to solve the problem correctly, and explain why they “shared the 30 m & m’s equally among the ten tenths in the fraction circle” and why they “only looked at four of the tenths because that’s the same as 40%.”
My beliefs that were reinforced with this lesson:
- We can’t assume understanding from correct answers alone. We need to listen to students reason through problematic situations.
- Students really want to share their thinking when they realize that someone is really interested in hearing it.
- Students crave understanding. They really want to make sense.
- Procedures are important, but not at the expense of understanding.
- Empowering students by allowing them to build their own understanding and allowing them to make connections allows students to feel comfortable taking risks in problem solving.
Please take a look at Jenise Sexton’s recent blog about percents with 7th grade students for some fantastic ideas about students using number lines and double number lines to solve percent problems. It’s SWEET!