OPEN-ENDED INQUIRY LEADS TO UNEXPECTED LEARNING IN MATHEMATICS
Written by Stephanie Bentivoglio - 3rd/4th grade teacher
Some students want to share 36 crackers equally. How many ways can you show this?
I posed this question to my students recently in our Patterns & Modeling class. At first glance, you might assume that a part of the question is missing. Many people are used to looking at math as a single equation to be answered correctly. When mathematics is viewed in this way, it limits the possibilities of what can truly be accomplished when thinking critically about the patterns and problems we observe in our world. Let’s take a look at how Sycamore’s 3rd grade class tackled this cracker challenge.
The students initial reaction was, as expected, “How many students?” My response to that was “How many students could share 36 crackers equally?” At that, the students got to work! For tools, they had dry erase tables and counters (fake coins substituting as crackers). When I began the activity, the overarching goal was to introduce the concept of multiplication as the addition of equal groups. It is normal for students in 3rd grade to come into the year with a wide range of knowledge:
Some students are not yet familiar with multiplication
Some students understand that they can solve problems with equal groups using repeated addition
Some have already begun to learn their 1-12 multiplication facts
With so much variance, one may wonder how to ensure they are engaging all of their students and pushing each of them to grow. Jo Boaler, Professor of Mathematics at Stanford University and creator of Youcubed, has been developing mathematical frameworks to provide activities that can create interest in all students in a classroom. They are tasks that are accessible to everyone but can be extended in many open-ended ways. We can examine how our 3rd graders approached the Youcubed cracker problem to get a better understanding of what this actually looks like in practice.
One of the students remarked right away, “Well, if we had 36 students then each student can get one cracker.” In response, another student at the table asked “Okay, so what happens then if each student gets two crackers?” They began to grab the counters and put them into groups of 2. Counting by 2’s until they got to 36, the student found that they had 18 groups. They added “18 groups of 2” to their notes, and without prompting began to make groups of 3 counters.
Another student took a different approach. They gathered 36 crackers into a pile, and then drew little people on their dry erase boards to represent students. They started with a random number of students, and began passing them crackers one by one from the pile. When the student didn’t have enough crackers to share equally, they added or removed a person and then redistributed the crackers. When they got to 6 groups of 6, the student remarked “Oh, wait, this is multiplication. I know I can write this as 6 x 6.”
At another table, a pair of students were having a deep discussion. I walked over to listen in, and one of the students asked me “are we able to break the crackers in half? If so, then 72 students could each get half of a cracker.” I responded, “Why not try it! Can you break it apart further?” Although they had not yet been introduced to fractions, using a topic that was relatable like sharing food helped them to better understand the idea of breaking it in half.
On their whiteboard, they drew out what appeared to be half a cracker. When prompted to break it apart further, they wondered what that might look like. The three of us put our heads together to draw it out, and I let them know that the name for it is ¼, or a quarter. One of the students remarked, “I doubled 36 to get 72. So I think we should double 72 to see how many students can share this.” I encouraged them to try it, and after a few minutes they found that 144 students could share ¼ of a cracker if they had 36 crackers in total.
By the end of the activity, all of our students had discovered the math facts that lead to 36. This was what I had hoped for. In addition to that, the students collaborated to come up with surprising results that I had not expected! They were able to extend their thinking further by using their doubling theory until they finally reached “1,152 students could each get 1/32 of a cracker.”
To debrief, the students came up in pairs to share their findings on the board while I copied them onto chart paper.
In this activity, all of the students were exposed to new strategies, ways of thinking, and previously unseen mathematical work. Whenever we introduce a new skill to our classroom, whether it’s multiplication, division, fractions, or geometry, it is always beneficial to start out with a task that is open-ended in this way. Not only does it make the students feel capable and confident in what they can do together, but it provides me as their teacher with helpful information on their individual learning needs. As with any subject, different contexts call for different methods: sometimes small groups are preferred, and other times it is better to work individually with a student. But open-ended tasks like these that challenge them to collaborate as a whole group encourage high levels of achievement and critical thinking among all of the students. With different entry points to a problem, each student has the opportunity to be successful and to push their thinking.
Everyone can do math, regardless of prior attainment. When students are allowed to explore through strategies and patterns that make sense to them in a meaningful way, they can excel through potentially challenging problems. This type of flexible thinking creates a strong growth mindset and an understanding that challenges create situations that lead to the greatest learning.
For more information on Youcubed and tasks like these, you can check out their website here: https://www.youcubed.org/