In my group, the Weavers, I have committed to trying inquiry in Math. In the last month, I have been very close to giving up that goal. I was really not sure how I could possibly integrate inquiry into math. There are too many foundational skills that have to be mastered before students can move forward. This work is far too important for me to give up as much control as inquiry requires me to. Then I had a conversation with a math teacher in the district who is finding success with presenting her students with one cognitively challenging task and allowing them time to explore the problem with an understanding that students will not necessarily complete the task with success. Her focus is on the process that they go through as they collaborate in an attempt to find a solution to the task.
Talking to her made me realize that I don't have to approach inquiry in math as I might in science. Math inquiry can be truly 'mini,' focusing on a very small objective that I want the students to master. I decided to give it a shot.
My students have been learning about fractions. They have been composing and decomposing fractions, learning about equivalence, and learning about the reasons why we add and subtract them the way we do. I decided to try allowing my students the opportunity to "discover" how to multiply a fraction and a whole number.
We started by activating our schema around how we multiply. Collectively, we found four strategies for multiplying: by writing out our understanding that multiplication is x, counted y times (skip counting), creating equal groups, using arrays and by adding repeatedly. Once we made that list, I shared the objective with students: They will learn how to multiply a whole number and a fraction and apply a rule to these kinds of problems. We brainstormed three of these types of problems. Then I set them off to work together using any tools in the room to help them (pattern blocks, base 10 blocks, rulers, number lines, paper pencil.) They worked for about 15 minutes individually and in small groups of their choosing. At the end of the 15 minutes, some were still struggling with the concept of applying what they already knew to fraction multiplication while others had grasped that and had successfully created a rule. We debriefed the process and recorded our findings.
I was very encouraged by this process. I think it gave me hope that inquiry can be an essential element in students better understanding math. Overall, my students were more engaged in this process than if I had done a mini lesson on the rule then given them time to practice. I had a student, who generally struggles in math, ask me that if we use repeated addition to multiply, wouldn't we be able to use repeated subtraction to divide? I had another student think that she had found the rule, but could only apply it to two of our three problems. After much discussion, where I had to force myself to sit back and NOT say anything, she realized that, in fact, the rule did work, it just resulted in an improper fraction rather than the equivalent mixed number she came up with. She came upon this by asking questions of herself and her partner, not by having it delivered to her by her teacher (not that I didn't want to.)
I used to think that inquiry had to consist of complex, backward designed lessons. Now I know that I can approach inquiry as an innovative way to tweak my math workshop where I put the ownness of learning squarely on the shoulders of my students... AND they are totally capable of rising to that challenge. I also used to think that inquiry didn't have a place in math. Now I know that it fits into my beliefs about the way students best learn math.
My next step is to have students discover the process of multiplying two fractions in the same way. I want to see if I can replicate the success I had with this lesson. Another next step, or a missed step, is to assess this process to see how well my students understand how to multiply a fraction and a whole number.