# Author Archive: Paul Hartzer

## Deriving Euler’s Identity

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…

## The Natural Base e: Thoughts on Teaching

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…

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## The Problem with Problems

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…

## Some Thoughts on Teaching Mathematics

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…

## Dividing and remainders

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…

Mathematics

## Converting Between Bases

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…

Computer Programming

## Is Zero a Factor of Zero?

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…

## Number values and multiple representations

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…

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## Speaking English vs Speaking Math

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…

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## Three and a half methods for finding square roots

The easiest way to find a square root in this technological age is to use a calculator. That’s a fine method if what you want to do is simply calculate a square root. However, if what you want to do…

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## De Gua and the Pythagoreans

The Pythagorean Theorem states that, given a right triangle, the areas of squares placed along the two legs will have the same area as a square placed on the hypotenuse. This is normally written as $$a^2 + b^2 = c^2$$,…

## Pascal, Pacioli, Probability, and Problem-Based Learning

I’m currently reading Howard Eves’s Great Moments in Mathematics After 1650 (1983, Mathematical Association of America), a chronological collection of lectures. The first lecture in this volume (the second of two) is on the development of probability as a formal field of…