I’m trying a few new things this year in math. I will try to summarize how each week goes throughout the year and highlight successes and failures.
This week I really tried to introduce the honors freshmen to “real” math. That is, some basic proofs, how generalize things in math and expose them to some difficult questions that are easy to approach, but difficult or surprising in their answers. But all of it was done in a way that is accessible to the students and with high levels of participation.
I really like when a student shares a thought and I repeat their thinking outloud and ask if the others understand, not necessarily agree, but understand. This seems to really get them thinking and communicating.
Some of the questions we explored were, Is 0.999… less than one or equal to one, why/how we know the square root of two is irrational (we actually did the proof in class, carefully), is zero odd or even, why can’t we divide by zero?
There is also a challenge question posted, in two parts. Part 1: Given that a and x are natural numbers, and a is less than or equal to x, and x is greater than 1, could the following number be prime: (x! + a)?
Part 2: If x = 99, how many values of a would be composite.
The purpose of all of it was to challenge their thinking, hopefully incite some curiosity and promote deeper understanding.
One thing I really wanted them to understand is that rational numbers could be expressed as a ratio of integers. The old way I would’ve quizzed them on this knowledge would be to say, Write the following numbers as fractions. The better question is, Express the following as a ratio of integers, as that addresses the definitions of rational and irrational numbers.
The number zero was a little difficult for some, but since we discussed that division is best thought of as a question, the denominator times what is the numerator?, that went well for most. For example, instead of reading 8/2 as “eight divided by two,” it is better to ask, “two times what is eight?”
This is most effective when showing why you cannot divide by zero.
I was pleasantly surprised by the show of knowledge and understanding when I asked them, on their quiz, to express the square root of two as a ratio of integers. Many said that the square root of two could not be expressed as a ratio of integers, that’s why it is irrational.
The depth of that understanding is of little consequence really, but it is a big victory that they have taken something that they always memorized in the past, and now truly grasp.
How well this translates to them owning their learning will remain to be seen, but I think we’re off to a good start.