I have recently had 2 discussions that have been interesting to me, that I would like to share. The first one was with my class about number sense. Another was with a concerned parents about me giving students “the steps to solve problems.” The reason I am grouping both of these together is because they made me reflect upon the idea of why/how do I teach mathematics.

The first discussion was with one of my Algebra classes on a day where we were discussing solving exponential equations. As we were investigating a problem where the common base was neither of the original bases, I had one student who verbalized how to solve the problem and identified the common base that we should use. Another students responded, “how did she do that so quickly?” My response was that the other student had very good number sense, to which the student responded “what the heck is number sense?????” I explained that number sense is the ability to understand how number work together, and that there is no lesson in any textbook that is titled number sense. I describe that the development of number sense happens over long periods of time, and that students who do not “abuse” calculators have better number sense. The curious student then followed up with “how do I get more number sense?” I didn’t know how to respond…. Is it past the point where this student can develop it. This students is an “honors” level student, shouldn’t they have number sense? What can I do as an Algebra 1 teacher to help my students develop number sense? I do not have the answer to these questions. If you do, or would like to discuss more about it, please let me know. It is not often in my class I don’t have the solution to a problem, but to be honest, I am a bit confused about how at the Algebra 1 level I am to BEGIN developing number sense.

The other recent discussion was with a concerned parent. We were discussing what can be done so that the student could have a better understanding and a stronger grasp of materials. During our conversation the parent ask if I was “giving students the steps to solve the problems.” When I responded NO, the parent seemed quite perplexed. They mention how their child NEEDED the steps to solve the problems. In my opinion it is more important that students understand the RULES of mathematics and understand broad topics that can be applied to many situation, but how do I convey this to a parent. I quickly grab the textbook and picked two problems that were right next to each other in the textbook and showed the parent how one problem takes 3 steps to solve and the problem next to it takes 5 steps. Then I explained that there are other ways to solve those problems that were different than they ways I showed them, so how could I give STEPS to these types of problems? After showing the parent these problems and discussing it a little more they finally saw my perspective. As a math teacher I want my students to be PROBLEM SOLVERS, not robots that follow a bunch of steps. I want my students to understand a handful of broad topics that they can apply to many situations to solve problems. That’s why we learn math….to solve problems!!!

In the past year or so I have spent a lot of time reflecting on my teaching techniques and process. I am always trying to improve my instruction, and make my students better problem solvers. What do students need to succeed in Algebra 1 and beyond? Is their prior education preparing them? What can I do to make up for some of their lack of skills? These are some of the question I am always pondering!

Sometimes, when I quickly solved an arithmetic problem in my head, somebody would say something like, “Ha! You’re a matheatician so of course, you can do it.” But that’s not right. I personally know several brilliant mathematicians who can’t do arithmetic in their heads to save their lives. Being very good at numbers isn’t necessary to be an excellent mathematician. What mathematicians need are exceptional analytical and problem solving skills, ingenuity, creativity and hard work.

I was formerly a professional magician but I had also done “Lightning Calculations” routines like mentally multiplying two 5-digit numbers and determining the day of any date. So, I’m familiar with this kind of thing.

Having said that, to become good at numbers, you need a lot of practice doing mental calculations. But of course, you also need good methods or tricks to practice with. After all, as the mathematician Cauchy proved when he outdone a 13 year-old savant (the kid was asked to calculate I think the sum of the first 83 fourt powers) that a great method beats natural talent. While the kid was doing a very fast brute-force calculations in his head, Cauchy simply solved it in his head by using the power-sum formula.

If one is serious about learning quick reckoning, then he should find all the available mental math techniques, algebra tricks and even calculus tricks that he can and practice them regularly.

Reading recreational math books can also help to further develop one’s appreciation of math since those books look at math from different perspectives which a textbook alone can’t offer. Some of the best were written by Martin Gardner, Ian Stewart, Clifford Pickover, to name a few.

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