As a point of reference, I was introduced to this tool by my mentor teacher. She used one of these activities for our Algebra 1 students and I really admired the dialogue it sparked amongst the students.
Here we see a student addressing a question that sparked tremendous classroom discussion.
For the sake of posting stuff on a regular basis, I will divide this discussion into two parts.
This quote from former CCNY Professor (now Mercy College Dean) Alfred Posamentier was recently brought to my attention:
“The point is to make math intrinsically interesting to children. We should not have to sell mathematics by pointing to its usefulness in other subject areas, which, of course, is real. Love for math will not come about by trying to convince a child that it happens to be a handy tool for life; it grows when a good teacher can draw out a child’s curiosity about how numbers and mathematical principles work. The very high percentage of adults who are unashamed to say that they are bad with math is a good indication of how maligned the subject is and how very little we were taught in school about the enchantment of numbers.”
You can read the whole letter, too: It’s quite interesting. Professor Posamentier is under the impression that we can get more students to like math (or at least not fear it) if we pique the interest of children in mathematics while they’re in elementary school. This sounds like a great idea and a sensible one too, but what methods can teachers use to motivate math learning in elementary schools, especially if the teacher is not well versed in math beyond K-12 school curriculum? What about for math educators in secondary schools? I would like to address some things that I believe can be done at both levels to stimulate students’ interest in mathematics or, at the very least, remove some of the stigma that math is a difficult subject.
Consider the sequence of numbers 0, 2, 4, 6, 8, 10, x. What is likely the value of x? Chances are you guessed 12; if so, why? Well for starters, we are looking at a sequence of numbers that seem to fit the pattern of multiples of two. Something in our intuition tells us that the next logical step in the sequence is 2*6. Our minds seem to have a pattern recognition mechanism that tells us what is likely to follow given a series or prior statements or facts. It is, after all, the next logical step. My experience with intuitive problem solving in mathematics is a positive one that sort of opened the door to interest in problems that require more critical and ingenious problem solving.
Thinking about mathematics was not always entirely contingent on a large collection of definitions, theorems, proofs, and the likes; mathematics grew out of a series of intuitive arguments that called for more rigor as time progressed and the subject branches evolved. Without going into research paper-like detail, many topics in mathematics arose from intuitive questions stemming from preexisting mathematical or scientific frameworks. I am not asking for math education to revert back to Euler’s proofs (or lack thereof) in Institutiones calculi differentialis nor am I asking for the reader to prove each theorem as an exercise a la Rudin, but I am asking for math educators, regardless of formal education, to dig down into the heart of the symbols and numbers and explain intuitively what is going on in a subject, how it is believed to have come about, and some of the neat results and consequences. Granted, rote tasks like learning elementary multiplication tables are hard to motivate, but one can easily go into a discussion on elementary multiplicative patterns in numbers, e.g. any number whose sum of digits is divisible by three is itself divisible by three. I must acknowledge that within the realm of intuitive thought, one ought to use different teaching methods to reach a wider learning-style audience, i.e. visual learners, hands-on learners, etc. The point is to digress from repetitive computations and lifeless introductions to new material that fail to stress key ideas and concepts in mathematics and venture into a world of inquiry and critical thought using the language of nature’s book.