Structured Problem Solving

This week in class we discussed structured problem solving in math. We used the example of the growing dots problem. While completing this problem, we were told we must visualize what is happen and draw a diagram. This forced us to solve the problem in a structured way with visuals. Without these instructions many of us may have completed the problem mathematically instead of solving is using diagrams. However even though we were told to use visualizes and diagrams there were still several different approaches to the solution. Two of which can be seen in figures 1 and 2. One is the approach I took of a growing squares and since a square has 4 corners it grows by four dots every minute. The second approach was one that viewed the four corners as orbital’s that grew out by one dot each minute. Both of these approaches come to the same solution but they way the visually problem solved to reach the solution was different. As I’ve said in previous blogs, I think it is very important for teachers to realize that students can approach the same problem in many different ways, and in face teachers should encourage this creativity, as this is what leads to higher order thinking.

figure 1

figure 2

            During this class we also had Diana’s presentation on grade 10 applied math. For her activity we were to solve a problem that was different for each group and can be made based on students’ interests. My group completed a problem about choosing a banquet hall for the school athletic banquet. The solution we came up can be seen in figure 3. This activity is a creative way to show students real life situations when they would need to know how to use substitution or elimination. It was also good that we were not restricted to one method and we could choose if we wanted to use substitution or elimination. We chose substitution as that it was we were all most comfortable with, however we then saw how easy it would have been to use elimination with our two equations, and this may have been the method a grade 10 applied student would have chosen. This is an example of non structured problem solving, which is sometimes more beneficial than structured problem solving as it can lead to more creativity.

figure 3

Using Student's Work As A Mathematical Resource

This week in class we discussed materials and resources in a math classroom. This is something that may be obvious is a school that has access to things such as computers, smartboards and the finances to support a technological math classroom, but this is not always the case. Something that I think many teachers and students don't think about is using themselves as a resource. Students can be used as the primary resource in a classroom through things such as collaboration and presentations of hands on actives with each other. They have the ability to create, draw, write and research and then their work can be used as resources. If you allow students to be creative and come up with their own approach to a problem in mathematics, chances are you will have several different solutions to the same problem. As a teacher you could then use these different solutions to teach how these different approaches work and the math behind them. This allows for collaboration, creativity and higher level thinking in the classroom as well as student centered learning. Using students’ examples to teach would be a great way to use your students as a resource in the classroom, especially if you’re in a school that may not have access to many resources.

            This week I also had my presentation on the grade 10 academic mathematics course. In this presentation we (Ela and I) did an overview of the whole course looking at each major strand: Quadratic Relations, Trigonometry and Analytic Geometry. We then each did an activity that can be used in this course. I did my activity for the Quadratic Relations unit focusing on the transformations of a parabola. In this activity I use a student-centered approach while incorporating technology. In this activity 8 different groups of students would each be given 4 equations representing one a transformation. For example group one would be given 4 equations of a parabola shifted up, group 2 would be given 4 equations representing a parabola shifted down, group 3 would be given 4 equations representing a parabola shifted right etc. The eight groups would be given one of the following transformations: shift up, shift down, shift right, shift left, stretch and compression.  As a teacher you may want to strategically choose these groups to ensure that each group has 1-2 strong students depending on the class size. From here the groups would graph their functions by hand using a table of values and discover the pattern between their graphs. From here each group will present their findings. This is where you can incorporate technology. I used Desmos, which is an online graphing calculator. Using this technology you can either have each group put their graphs onto the same grid so everyone can visually see what each transformation would look like in comparison to a non-transformed parabola. This can be seen in figure one.

Figure one: Desmos Screenshot

 From here students would get into new groups where one student from each of the eight groups form a new group. Each of these new groups would then have an ‘expert’ of each transformation. They will then be given a list of parabolas that contain several transformations of the form y=a(x-d)²+h. They will then come up with a general solution for what each parameter represents and share these findings with the class through a gallery walk and/or class discussion. Desmos also has a parabola transformation software on it that can be used for this part where you adjust each parameter to see how the parabola changes. Other forms of technology can be used throughout this activity such as graphing calculators, a smartboard and gismos. This activity is a student centered lesson which uses the student’s work as resources as well as technology if it is available, but the technology is not required is just there for visual assistance. This activity also allows students to collaborate and explore.

figure two: student graphing their transformation on desmos (student: Ela)