Category: Mathematics
-
[mathjax]Some version of this question keeps getting asked on the internet. What is \(8\div2(2+2)\)? Some background The strength of mathematical notation is at the intersection of clarity and simplicity. We could be completely clear, which would leave us writing details we don’t need and make it hard to read. We could oversimplify, and lose clarity.…
-
[mathjax]Euler’s Identity has been called “the most beautiful equation” in mathematics. It neatly encapsulates five key values and three operators into a true equation: \[e^{\pi i} – 1 = 0\] But why is it true? In this entry, I’m going to take it apart. Fully understanding the equation involves looking at various parts of algebra…
-
[mathjax]In “Burn Math Class”, Jason Wilkes spends quite a few pages deriving the value of \(e\). I did not notice him at any point mentioning compound interest. Since we’re currently wrapping up the chapter on exponential functions and logarithms in the Algebra II classes I’m teaching, I was already thinking about the best way to…
-
[mathjax]I’m currently reading “Burn Math Class,” and it’s got me thinking about language. Yesterday, I saw an item about teaching students why cancelling works in this case: \[5 + 3 – 3\] but not in this case: \[5 + 3 = 5 – 3\] The conclusion that the students were led to is that the…
-
This morning, I was reading the NCTM blog, and the subject was on students struggling with systems of linear inequalities. First, as background: I don’t have any difficulty with systems of linear inequalities, and I don’t remember ever being taught such things (although I may have, I just don’t remember). But then I get two…
-
[mathjax]As a high school teacher, I struggle routinely with getting students to understand that \(x/0\) is undefined. Students don’t seem to understand that division with a remainder is incomplete. I have long attributed this to the way that division is taught in the elementary school years. For instance, \(17\div 5 = 3R2\) is considered perfectly…
-
I was working through the November problems for NCTM’s Mathematics Teacher. There’s a problem on converting between bases, which led to me developing a new-to-me method. What I was taught I started by using the method I’d been taught by my computer programming teachers: Identify the value of each place Divide by the highest value,…
-
[mathjax]Generally speaking, if \(a \times b = c\), then \(a\) and \(b\) are factors of \(c\). This concept appears at the secondary level in two contexts: The factors of positive integers, and the factors of a polynomial. If we limit the domain and range to be positive integers, for instance, the factors of 7 are…
-
[mathjax]One of the questions my mind keeps returning to is: What is a number? I don’t mean this in a high-level set theory way. I’m not talking about aleph-numbers or other sorts of concepts. I’m restricting my thoughts here to the sort of numbers that are the domain of high school mathematics, nothing trickier than…
-
[mathjax]One of the challenges that I see with students learning mathematics is their confusion with what qualifies as the content of mathematics and the language of mathematics. In a famous and enduring article, “Relational Understanding and Instrumental Understanding”, Richard Skemp pointed out that teaching concepts instead of procedures will be difficult if students think that…