# Tidbits

A small collection of unrelated items.

• My future teachers see things differently than my readership. Readership was strongly in favor of pennies as parts of a dollar, with the dollar being the natural unit. My future teachers were strongly in favor of the penny as the natural unit, and the dollar being composed of pennies.
• Both ideas are correct.
• The results from my classroom ought to at least make us stop and think about the effectiveness of money as the go-to tool for explaining decimals.
• At least one of my students remembers the pennies/dollars conversation as one in which I came out in favor of a dollar being composed of quarters.
• I remember things differently.
• The morning after the temperature conversation I documented recently, it was even colder. I asked Griffin to guess the temperature, with the hint that it was below zero. He guessed -10. It was -7. He had no problem stating that his guess was 3 degrees too cold.
• I had a conversation with Sadie Estrella recently in which she made me wonder, What is the right amount of information for third graders to have about similar shapes?
• I have no idea what the answer to that is, and correspondingly I wish someone would write the geometry equivalent of Children’s Mathematics.
• Friday marks the first of several meetings of a Math Teaching Seminar I am leading with my colleagues that features readings and videos from Keith Devlin, Sal Khan, Dan Meyer, George Polya and Peg Smith (Five Practices, anyone?)
• The ALEKS developmental math curriculum includes (among many others) this topic: “Solving a rational equation that simplifies into a linear equation,” which seems entirely too specific to me and exemplifies what is broken in so much of developmental mathematics.
• My future elementary teachers think explicitly about patterns and struggle to think recursively.
• My College Algebra students think recursively and struggle to think explicitly.
• I do not understand why there are names for arithmetic and geometric sequences, but not for those that are described by a quadratic function on the natural numbers (except special ones like square and triangular numbers).
• If something is free, according to Tabitha, it cannot be described as an extreme case of cheap.

### 3 responses to “Tidbits”

1. 1) I love your conversations with your children. Mine’s only 2, but I’m looking forward to our own mini pizzas and guess the temperature games.

2) Thinking recursively, it is clear why arithmetic and geometric sequences take primacy. There are two basic operations on numbers, and the simplest recurrence relation uses one of the operations to combine a single input with a constant. No recurrence relation for (1,4,9,16,…) is as simple. Make up an appropriate notion of “hypergeometric sequence” and you’ll cover lots more sequences.

2. They’re usually called quadratic sequences. Why do they need a more arcane name? “Arithmetic” sequences is a terrible name, as there are many types of arithmetic. “Geometric” is even worse, as the sequences usually come up in contexts that have nothing to do with geometry. I prefer “linear” or “constant-difference” to “arithmetic” and “constant-ratio” to “geometric”. With this naming convention, quadratic sequences could also be called “constant second difference” but that is too much of a mouthful.