Can You Justify That? - A Response to the Second Private Universe Project Workshop

I found the second workshop in the Private Universe Project series to be on target with the direction that math educators need to take. All too often, educators provide students with a problem, and they require students to solve the problem. While problem solving is a valuable skill, this video prompted me to think about the value of problem solving without the ability to understand an answer or justify an answer.

From my experience as a science teacher, I can tell that students do not understand why the answer to a scientific problem that involves math is what it is. They have trouble justifying an answer, and this leads to them coming up with solutions that make no sense. For example, I have seen students come up with lengths of times that are negative. How is this possible? Well, if students do not understand what they are looking for and do not have the ability to justify what they are doing then it is very easy to get an unreasonable solution. In essence, many students can do math, but they lack strong fundamental math thinking skills.

This video reassured me that the ability to think critically about mathematics can begin to develop at a young age. I remember when I was in school learning mathematics. Rarely was I asked to explain my answer or justify why my answer was correct. I was often expected to show my work, but this is not the same as proving that my answer is the only possible correct answer. This led me to have difficulty when I had to prove things in geometry. In fact, I remember saying on multiple occasions that I hated geometric proofs. However, in the video, students in the fourth grade were able to prove things by using inductive reasoning and breaking a problem down into cases. Honestly, I was amazed. In fact, it appeared that the fourth graders were able to prove their answers better than many high school students could or for that matter better than some teachers could.

The video made me think a lot about what students want and what is in their best interests. Students often want someone to model a problem for them so that they can copy the behavior over and over. However, this type of skill is not valued in industrialized nations like ours. High paying and well-respected jobs are likely to require students to think for themselves and be able to justify their thinking. Most people can do what they are told to do. However, it takes individuals with creative thinking skills and critical thinking skills to be able to form a justification for a solution. Thus, students who are taught mathematics in the way it was being taught in the video are likely to have a competitive advantage over students in schools where mathematics is taught in a traditional manner.

Another thing that I liked about the video was the fact that students were able to go beyond learning just the basic mathematical concepts. In school, we all learn about patterns, and in due time, we learn how to identify them. However, these students learned what patterns are, how to identify them, and how to justify an answer using patterns by the fourth grade. In my opinion, this is remarkable.

In addition, the level of engagement in the classroom was great. Not once did I see a student who was not engaged in the task at hand. Considering that getting everyone interested in a task is an extremely difficult endeavor, this fact alone indicates that the strategy of teaching in the video is an excellent one.

Lastly, I am shocked that students in the fourth grade were able to justify their answers to the four cube tower problem using almost the same logic as the logic used by teachers. Think about the level of education that a teacher has as compared to a fourth grader. Then, compound that with the fact that students’ minds are nowhere near being fully developed in the fourth grade, and one can see that what took place in the video was truly amazing. Thus, I am sold on the fact that math must be taught in such a way that students are given an opportunity to grapple with problems, discover ways to solve problems, and justify their solutions.