Learning math through inquiry
Richa Pandey
“A student who didn’t appear for the test shouldn’t be marked a zero as it would negatively affect the student as well as her entire group,” said a nine-year-old in my mathematics classroom. The incident took place when my fourth graders were engaged in a brainstorming session.
I work in a private school based in Vijayawada, Andhra Pradesh, as a mathematics facilitator. My work involves engaging 8-13 year-olds and facilitating their learning through inquiry. Inquiry-based teaching is student centric and involves investigating a given problem, making conjectures, and arriving at a solution. It relies on triggering the learner’s curiosity through thought-provoking questions.
Planning an inquiry-based session
My fourth grade learners had recently learnt symmetry and we were in the process of transitioning to the next unit. The next unit was part of statistics and included concepts like mean, median, and mode. ‘Averages’ as a concept is rich in terms of its practical applications ranging from average height, weight, age of a group to average salary of employees in an organization. There is no dearth of activities that one can design for this particular concept. Given that my school follows inquiry as an approach to teaching-learning, I planned the session involving extensive group discussion and brainstorming to initiate a conversation around the concept.
To ensure a smooth transition from symmetry (a concept in geometry) to averages (a concept in statistics), an exit test based on symmetry was conducted. Each learner was marked out of 20. They were then divided into four distinct groups.
On the day we planned to start learning averages, I shared the test results with the learners. Clear instructions were provided to the students to avoid any confusion. The learners had to sit in small circles according to their group numbers which were mentioned on their test papers. Each group was required to elect their group leader (through dialogue/negotiation/voting). The group leader was expected to analyze the performance of their entire group in the symmetry-based test. The leader had to represent the entire group in front of the facilitator.
Organizing an inquiry-based session
The first opening question for all the group leaders was, “How was your own performance in the test?” The leaders used different adjectives or phrases to describe their performance ranging from “excellent” to “I did well”. The next opening question was: “How did your group perform?” The leaders were expected to use only words and no numbers to answer this question. Again, different phrases were used including “they did well”, “they could have done better” and “they did well but could do better”.
This question was extended further as leaders were asked to describe their group’s performance using numbers. As soon as this question was asked, one group leader started revealing the marks of different learners in the group. She was immediately reminded of the instruction that was given at the beginning of the activity, i.e., “the marks of different group members of your group should not be revealed to the other groups”. She modified her response and said, “My group members did well but they could have done better.” A similar response was given by the next group leader. They didn’t know how to use numbers to describe the performance without revealing the scores of individual members of the group.
The discussion took an interesting turn when the third group leader said, “My group performed well, no one scored less than 10 or more than 18.” The fourth group leader immediately picked up this new way of explaining the performance of the learners and rushed back to confirm her group members’ scores. She came back with more confidence and a more appropriate analysis of her group’s performance. At the end of this round of discussion, I asked all the group members to explain their group’s performance using the sentence, “In my group, no one scored less than ___ or more than ___. Then I revealed to the learners that they just used a new concept called “range”. Interestingly, this concept wouldn’t be introduced at this point if not for the curious nature of the learners.
Extending an inquiry-based session
In the third round of discussions, group leaders were expected to describe their group’s performance using only one number. Five minutes were given to all the groups to discuss and decide what that number should be. The groups had to come to a consensus about this number and each member in the group had a right to question the number selected by the group. After a good five minutes of discussion, all the leaders came back with a number that represented their group. This time, the responses were quite diverse. One group leader wanted the highest score of the group to represent the group while the other group leader was not sure what that number could be. There were two groups that had the same strategy, i.e., they wanted to “mix” the scores of all the group members. When asked about the meaning of mixing, they showed me the number they had calculated on a piece of paper. It was the sum of all the scores in their group.
At this point, I asked all the groups, “Is it fair to choose the highest score of the group to represent the group’s score? The groups unanimously rejected the idea as it seemed unfair. On the same grounds they rejected the idea of using the lowest score to represent the group’s score. They seemed to prefer the “mixing of scores” strategy. So, I asked all the groups to find the total score of their group. Group 1 had scored 65, Group 2 had scored 73, Group 3 had scored 93 and group 4 had scored 78.
Now, I asked all the groups a common question, “Which group in the class performed the best?” The groups were inclined to say Group 3 when the leader of the fourth group objected by saying that, “Group 3 has six members but my group has only five members. They have scored 93 out of 120 but we have scored 78 out of 100.” This was the moment that any facilitator craves for. I couldn’t stop smiling. Then I asked all of them to rethink the strategy and come up with a more fair way to represent the score of the group. Immediately one group leader said, “We can subtract 20 from the total score as well as obtained score for their group,” (a common misconception about fractions). I asked her to think whether 5/10 was the same as (5-1)/(10-1). She was able to notice the problem. As I was convincing her to rethink her strategy, another student said “average”. A different student said “percentage”. Before I could take up any of these responses, it was time for their next class so I left them with their own questions to think about.
Problem + Solution = Another Problem
The next day we began the session by recalling what we had discussed previously. They reminded me of averages and percentages as the possible strategies. Now, percentage has been discussed with this group of learners briefly in relation to decimals and fractions but this group doesn’t know how to calculate percentage yet. They have never used average as a concept before. So, it was a surprise for me as well. But it gave me a good point for starting a conversation around mean. I asked them to share their total score among the group members equally and come up with a number. They followed.
By the end of the session, we were able to resolve the problem by finding the average score of each group. However, there was one more problem. Group 2 had six members but one of those members had not written the test. The group leader said, “She has just joined the school and it would be unfair to test her knowledge on the concept she hasn’t learnt”. All the groups agreed. The next question was, “Should we mark her zero?” The groups were confused. They didn’t want to test her. They didn’t want to give her zero. One member firmly said, “If we give her zero it would affect her negatively and would affect the group’s score too.” As the groups were brainstorming to solve this problem, the bell rang and they were left with yet another problem to think about before the next class.
The author is a mathematics facilitator at AIMEE International School, Vijayawada. She is interested in exploring different ways of making learning an active process for young learners. She can be reached at richapandey735@gmail.com.
