Initially, I found it challenging to apply T-GEM to mathematics as I found it hard to picture, but after completing this activity, it seems to be very similar to many of the activities I already strive to do with my students. A challenging concept for my students in secondary math is creating and understanding equations for lines and quadratic polynomials using graphs or scenario details. This challenge has been identified through concept pre-assessments, the nature of their questions throughout their work, and continued struggles on summative assessments. They struggle to make the connection between the information they are given and the algebraic, abstract representation.
Generating: For the generating phase, I would have students brainstorm what they already think they know about equations with regards to what they mean and how they can be used. Students will randomly select equations from a centralized equation bank to explore using Desmos (available either online or as an app). Working in pairs, students will develop a set of rules for creating equations based on their explorations of manipulating equations within Desmos.
Evaluating: Once students have developed their guidelines, pairs of pairs will be joined into groups of 4, in which students will compare their initial hypotheses and justify their perspectives. Ideally, reflection would occur as students need to explain how they arrived at particular conclusions. At this point, scenario-based problems will be introduced to the original pairs, expanding upon the initial work with base equations. Students now need to determine if the rules they established in their initial phases apply appropriately to their new scenarios. If they are not able to use their rules to accurately create an equation to represent the scenario and use it to solve problems, they will need to identify the gaps and determine what adjustments need to be made. Desmos will continue to be the technology tool at this level, as students are able to easily test, adjust, and visualize their inputs.
Modifying: In their pairs, students will reevaluate their list of rules for equations, taking into consideration their initial hypotheses, their discussions with peers, and their testing of their hypotheses. As a class, we would come back together for a large class discussion to compile their ideas into a community-based understanding.
This process could be further expanded to include different types of polynomials. For example, students initially working with equations of lines could then generate, evaluate, and modify new hypotheses regarding quadratics, based on their work with linear relationships. Subsequently, work with quadratics could be further cycled to work with cubics, then quartics and other higher order polynomials, as appropriate.
By having students use Desmos to work with the different parameters of the equations, they are able to actually experiment to see the effects of changes, rather than simply being told to memorize, for example, that the c value of a quadratic equation determines the vertical position of the graph but doesn’t directly affect its shape when a and b stay constant.
I believe that many mathematical principles and concepts can be approached using similar strategies to those used for scientific principles and concepts, and aim to include them when possible in my teaching. One challenge I have in senior math is the perception of teachers regarding the comparable value of experimentation and hands-on math in the university-bound courses as compared to the middle school or college/workplace-bound courses. I am often met with resistance from colleagues who don’t believe there is a place for experimental or hands-on learning in the higher-level university-bound courses, and that such activities are frivolous at that level. Do you feel that there is value in hands-on math learning for the senior level university-bound math students?
Khan, S. (2007). Model-based inquiries in chemistry. Science Education, 91(6), 877-905
Khan, S. (2010). New pedagogies for teaching with computer simulations. Journal of Science Education and Technology, 20(3), 215-232.