In today’s math classrooms, we all want our students to perform the thinking while solving problems, right? Oftentimes, however, the teacher is ultimately the one doing the thinking in hopes that their students “catch on” and understand the teacher’s processes to solve problems. In addition, textbooks often stifle the thinking for students by providing follow-up questions helping to lead the student to a certain strategy or formula. Dan Meyer agrees that textbooks often pave a smooth, straight path to an answer without allowing students to do the thinking (Meyer, 2010). So, how can we fix this in all math classrooms? Pay attention to the details within the question!
Here is an example from a recent lesson conducted in a 7th grade math classroom. The original problem the students were trying to tackle was, “How tall would a tower of a million pennies be?” The textbook offered the following guiding questions:
- How many pennies does it take to build a tower that is one centimeter tall? Use the tools provided by your teacher to answer this question.
- “I have an idea!” Carol said. “If I know how tall a tower of one hundred pennies would be, maybe that can help me figure out how tall a tower of one million pennies would be.”
- Work with your team to figure out how tall a tower of one hundred pennies would be. Can you find more than one way to figure this out? Be sure that each member of your team is prepared to explain your team’s reasoning to the class.
Here, students are encouraged to think about a tower that is one centimeter tall or one hundred pennies tall. While guidance might be useful after some productive struggle, such constraints may inhibit student thinking, creativity and wonderment.
A small adjustment was made by the teacher I recently observed in a 7th grade classroom. She simply posed the question, “How tall is a million pennies?” After some think time, she asked the class, “What tools do you need to solve this problem?” Students immediately shared that they would need a ruler and some pennies to get started.
Once students had the materials they requested, some teams began measuring a single penny. They used proportional reasoning to then figure out that 1 penny = 1 mm; so 1 million pennies must equal 1 million mm. This allowed the teacher to ask follow-up questions to promote more thinking. “Are you sure one penny equals 1 millimeter? Try measuring 5 pennies.” Students began to notice that 5 pennies does not, in fact, equal 5 millimeters. Some teams determined that 5 pennies was equal to 1.2 millimeters and then used proportional reasoning to conclude that if 5 pennies equals 1.2 millimeters, then 1 million pennies was 240,000 millimeters.
Still, other teams determined that about 7 pennies was one centimeter and concluded then that 70 pennies must equal 10 centimeters and so on.
This was fascinating as it allowed the students to do the thinking and use their background knowledge in proportional reasoning to arrive at a solution. Some answered in millimeters, some in centimeters, and others in inches. To close the lesson for the day, the teacher discussed the mathematical practice standard of attending to precision and how not measuring accurately with a small amount of pennies can largely throw off our calculation about the height of 1 million pennies.
The main goal for this lesson was to use proportional reasoning. After listening to the students discuss and justify their thinking, it was fair to say that all students used proportional reasoning by measuring a smaller amount of pennies and scaling up to calculate how tall a tower of a million pennies would be. Given the opportunity, the students thought about their own ways to tackle the problem!
Are you ready to take on this challenge and allow the students to perform the thinking in your next math lesson by asking an open question? Give it a shot! You might be surprised at what they think of!