I’ll admit it: I was skeptical of how “creative-thinking” strategies in math would go over with my 4th grade enrichment students. I see these students just once a week in pull-out groups, so every lesson counts. And I was nervous that this one might be a complete disaster.
My students usually solve math problems to practice a specific skill. Then we discuss the different ways they approached the problem, helping them understand the skill more deeply. This lesson, however, called for the students to write their own math problems. How would my students react when I put them in charge of their own learning?
I knew that, if it was successful, this lesson would get right to the heart of Common Core Mathematical Practice Standard #3, helping my students learn to “construct viable arguments and critique the reasoning of others.”
Our opening problem: When does 1 + 1 = 24?
(Can you solve it?)
The students tried to substitute numbers to answer this problem. For example, one suggested, “When you change one of the 1’s to 23, then the problem would equal 24 because 1 + 23 = 24.” But this strategy didn’t work. If one 1 was a substitute for 23, the other 1 would be, too—and the answer would now be 46.
To help scaffold the instruction for the students, I provided them guiding hints, like “Think about how you can use this problem to explain something that happens in everyday life.”
They were not able to independently solve this first model problem. But their eyes lit up when I shared a possible answer: Imagine “1" actually means “one dozen eggs” so that 1 + 1 equals 24 eggs.
They “got it” now, they told me. Drawing on this practice experience, I asked them to construct their own creative-thinking math questions.
Example One
Student Question: When does 7 + 60 = 8? Student Answer: When you add 60 minutes to seven o'clock and get eight o'clock.
Example Two
Student Question: What two numbers equal 136 when you add them and equal 24 when you subtract them? Student A: 80 and 56
Example Three
Student Question: When does adding 1+ 6 = 1111120? Student Answer: In the Golomb computer programming language, 1111120 is what you get when you add the encoding of a quotient of 1 (10) to the encoding of a quotient of 6 (1111110).
This lesson showed me that creative thinking had a lot to offer to my enriched math instruction. Here are some benefits that I discovered from challenging my students to think creatively about math:
• This is more than an activity; it is an assessment. I learned about whether students could apply higher-level thinking skills to the math lesson. Some students relied more heavily on the model math problem in constructing their own, while others were able to go even further with the concept of constructing an “it’s not what it seems” math problem.
• Analyzing math problems naturally improves argumentation. I witnessed my students explaining the logic of their math problems to the class. Their peers could explain why the problem made sense. And on the contrary, students could explain why some of their classmates’ reasoning was flawed.
• Teachers learn about students’ interests and experiences. I gained insight into my students’ personal interests from this lesson. For example, the student who created the math problem for example #3 later shared with me that he and several other students in my group attended a workshop on programming. This led me to the realization that linking lessons to technological examples might motivate this group.
• Students can work at their individual level. This open-ended lesson allowed students to work at their own level because they had to write questions that they could teach to the class. The students had to be able to explain the concept to the teacher and their classmates. In essence, this lesson naturally differentiated the instruction for the students.
• Students teach the teacher. Before this lesson I knew absolutely nothing about computer programming. In fact, I had to Google the problem myself to make sure it made sense. So I also learned something new from my students from this lesson.
Although this lesson made me nervous, I’m so glad I taught it—I learned so much about students’ progress toward Mathematical Practice Standard #3. I went on to challenge these students to craft children’s stories that taught a particular math concept—and learned even more about what each student did and did not understand.
If you incorporate creative thinking in your math classes, I’d love to learn from your experiences and techniques. Please feel free to share your story in the comments below.
