So far, we have examined a handful of resources available to math teachers whose goal is to teach students not only foundational math concepts, but also critical problem solving skills. First and foremost, the significance of these (or any) resources stems from the distinction between instrumental and relational understanding, initially discussed in the first blog post. As stated before, instrumental understanding relies on memorization and formulaic application, whereas relational understanding addresses why and how a given formula or procedure works. As teachers seeking to increase relational understanding among students, we must look closely at ways in which we can improve teaching strategies and engage students in mathematical discovery.

WolframAlpha and GeoGebra are examples of technology resources that can pique student interest in math and serve as a tool for uncovering math properties and procedures. It is interesting that technology can be a means of increasing relational understanding, because technology is often abused by students who mindlessly feed numbers into a calculator or Google-search the answers to their homework. However, in these two examples, students utilize technology by manipulating and researching their ideas while problem-solving. WolframAlpha provides real data to students learning how to make bar graphs and pie charts. Using GeoGebra, students might begin to see the connection between equilateral and isosceles triangles or the comparison of rhombi and squares.

*Nix the Tricks* offers insight into many of the math shortcuts (instrumental teaching techniques) that fail to offer comprehensive understanding of a given math concept. Interestingly enough, many of the same “tricks” that have been used in classrooms for decades are still being taught today (see this post for examples). Fortunately, *Nix the Tricks* offers alternative teaching methods to the shortcuts it highlights, helping teachers to reduce instrumental teaching in the classroom. That being stated, it is equally important to note that shortcuts themselves are not bad, it is when they are the sole focus of instruction that they become problematic. By teaching foundational concepts first, shortcuts can aid in memory or test-taking strategies.

All in all, these topics should have two major consequences on teaching math in the classroom:

- Tricks and shortcuts should take a back seat to explanation of the larger mathematical framework of a concept.
- Technology can serve as a means of linking student investigation and interest with mathematical discovery.

This requires math teachers to plan lessons intentionally, and, at times, utilize teaching methods outside of the teaching norm. Above all, this requires teachers to realize that they must teach students to think for themselves, because it is that singular skill that will carry students beyond the classroom.