Problem:
Take away any 5 straws (or sticks) and leave 6 triangles.
https://woodcraft.org.uk/sites/default/files/Match%20stick%20puzzles.pdf
We posed the above stated problem to the sixth, seventh, and eight grade students at Norma Rose Point School.
One of the strengths of the problem was that there were multiple solutions to the problem. There were also multiple approaches to solving the problem. In addition, this problem required no prior experience in problem-solving in order to find the solutions. In other words, this problem was accessible to a wide range of learners with minimal instructions. I think that this problem was also fairly easy to pose, both verbally and physically manipulating the straws/sticks. Because this problem was simple to pose both verbally and visually, I think that it was somewhat easier for language learners to understand the problem. As soon as we finished posing the problem, the majority of the students started looking for solutions. Right after posing the problem, students seemed to dive into rearranging the sticks.
There appeared to be a flow as students went about manipulating the sticks. The flow was temporarily interrupted by frequent shouts after finding solutions. Occasionally, these shouts appeared to distract some students from concentrating on what they were doing. Often times, some students walked over and looked at other students’ solutions. We were hoping for some sort of verbal interaction to occur between students who found a solution and those who went to look at a solution. For the most part, it was just a visual glance of the solution and nothing more than that.
In relation to language learners, this problem’s structure was decontextualized/simplified as much as possible (removal of extraneous information), almost appearing to be like a routine problem. Despite this, there were instances where language learners had difficulties understanding the stripped down version of the problem. In these cases, without our request for other students’ help, their classmates stepped in and assisted the students who could not understand the problem. In these instances, the assisting students appeared to translate our instructions both verbally and visually. After getting help from their classmates, language learners were able to successfully show/tell us (rephrase) that they have understood the problem and went about solving the problem, just like the other students.
One of the drawbacks about this problem was that it appeared to be a routine problem that was detached from real life situation. Nonetheless, this problem can be considered as non-routine partly because it had multiple entry points and numerous methods of finding valid solutions. Every student was able to find at least one solution and many other permutations of this solution.
The materials that were used to solve the problem both strengthened and weakened the approaches to finding solutions. We used different sized/colored plastic straws and two different types (small colored and large uncolored) of Popsicle sticks. One of the weaknesses of providing these manipulatives for solving the problem was that many students were quickly able to devise plans (not necessarily bad) to come up with valid solutions. That is, they were able to come up with strategies to solve the problem just by moving several sticks (after removing and holding on to five sticks), almost like their own algorithms. These strategies (sort of like looking for patterns or guess and check) produced only certain types of solutions (rotations of the same answer). Those students who employed these routines struggled to come up with different solutions. Many of the students, who were using these routines, were able to notice the rotational nature of the solutions, but couldn’t move beyond and appeared to get stuck. So these routines masked the possibilities of finding alternative solutions. Thus, some of the students were unable to devise alternative solution methods using these manipulatives.
I think that the majority of the students came up with multiple solutions on their own over the course of the mathematics fair. Although our intention was for students to work in pairs of two to arrive at solutions, it didn’t quite work out as planned for the most part. It appeared as if students preferred to work on their own, partly because we provided sufficient number of straws and Popsicle sticks. Even on instances when students started out in pairs, they quickly decided to work on their own. We noticed that group decision-making was hard for this problem. When thinking about our intentions, it was unreasonable to expect students to collaborate on moving a straw or a stick as a team. It was quicker to try multiple moves individually rather than making decisions as a team to make the moves. These students were too energetic and wanted to find the solutions. Despite many students preferring to work individually on this problem, many of these students not only shared their solutions with their peers, but they also explained about their methods of finding their solutions to us and to their friends.
Occasionally, we noticed one or two students talking to themselves about which straw or stick to move/remove. Some of these students were explicitly thinking aloud and explaining their reasoning to their peers, which I believe offered some sort of guidance to other students gathered around their table.
We made several modifications to our problem. Initially, we wanted to pose the following problem:
Take away any six matches (sticks/straws) to leave only three triangles
But we decided not to pose this problem as we thought that there was only one solution and one way of arriving at the solution. I think that our decision to not use this problem was partly based on targeting a wide range of problem solvers. Our objective was such that every problem solver would at least find one solution, if not all the solutions. We wanted every problem solver coming to our station to be a successful problem solver when they left our station. We were glad to have chosen a problem that enabled every problem solver to be successful at our station in the mathematics fair.
During the fair, we made several changes to the problem on the fly. We added additional constraints, such as each triangle should have no more than three sticks/straws. This constraint prevented the students from coming up with solutions that counted triangles (big ones) with more than three sticks/straws. We also discounted some solutions that had open-ended sticks/straws (i.e. when one end of the stick/straw was not touching another stick/straw.) I think that these changes enhanced the problem-solving strategies of some students and made the problem a little more interesting for others.
If we were to follow up with a student who claimed that she had found 14 solutions to our problem, we would have asked this student to show us one solution at a time and explain their thinking that this is indeed a solution. We would also challenge this student such that their solution might be rotation of another solution. The challenge would primarily be to create doubt in the mind of this problem solver so that this student really (re)think through the solution.
If we were to pose this problem to secondary students, it would probably be a precursor to a modified (but related) problem. In the modified problem, students would have to use logical reasoning to predict, generalize, and extend their understanding of the problem to other areas of mathematics. For example, going from solving a puzzle with physical manipulatives to finding abstract algebraic formulas. For instance, we could add another layer of triangles at the base (four more). We could then modify the problem such that when five sticks (or any predetermined number of sticks) are removed, then exploring how many different ways of finding a maximum or minimum number of remaining triangles. We could further complicate this modified problem by asking if there is a pattern in the minimum or maximum number of triangles as more layers of triangles are added to the base.