# pedagogy

11 May 2020

## Not all uses of equals signs are equal

The problem with equals signs The sign “=” was invented in 1557 by Robert Recorde a Welsh mathematician. He was tired of having to write out the phrase “is equal to” too often. Now we cannot imagine maths without an = sign. Using an equals sign correctly can be a challenge. Whenever an equals sign is used, it signifies that the expressions on either side are equal. A sequence of expressions separated by equals signs should all be equal to each other. For example: 4 + 5 = 3 × 3 = 21 – 12 However, the equals sign often […]
26 November 2019

## Creating and critiquing good mathematical tasks with variation theory

Variation Theory applied to teaching mathematics and statistics Highlights Careful selection of exercises can turn purposeful practice in maths into a task that also develops conceptual understanding. Poor, off-the-cuff or random selection of exercises can create barriers, feed misconceptions and at best miss out on opportunities for better learning. Using a framework of variation theory can help teachers examine and improve their practices and tasks, preferably collaboratively Spurious rules If students can learn a spurious rule for answering questions rather than the desired concept, they will grab it with both hands. In my class a student worked out that if […]
19 June 2019

## Fluency in maths

Fluency in language I can recite Latin verbs: the present tense of love is amo, amas, amat, amamus, amatis, amant. I recited them as I swam up and down the pool forty years ago: Amabo, amabis, amabit (breathe) amabimus, amabitis, amabunt (breathe). But if I were suddenly faced with an ancient Roman and had to express my affection, it would take a bit of thinking. I lack fluency in speaking Latin. When we are fluent in a language, we can respond and converse without having to think too hard. The language comes naturally, and we do not use up space […]
5 June 2019

## Achievable challenge in teaching maths

I like a good challenge I always choose the most difficult Sudoku puzzles. I like it best if I get really stumped and have to leave the puzzle and come back later. If I do manage to crack it, I feel a sense of achievement, and completion. From time to time I have tried “The most difficult sudoku” but have never managed to place more than one number. There isn’t a lot of fun in that. Fun exists in what is sometimes called “The Goldilocks zone” – not too easy, not too difficult, but just right. I have also seen […]
12 March 2019

## Multiplication facts or multiplicative thinking

We just want them to know their tables! It is a truth universally acknowledged by high school maths teachers that students need to be fluent in multiplication facts. (Apologies to Jane Austen) You can read more about this claim in my previous post: What Maths Teachers wish Year 9 students knew I have been thinking about why this is the case, what is so special about multiplication facts, and whether it is more an indicator of something else. Maths teachers like to teach algebra. Simplifying algebraic expression, and factorising quadratics are made much easier if one is at home with multiplication […]
26 February 2019

## What Maths Teachers wish Year 9 students knew

What do high school teachers want from their students when they arrive in Year 9? This is an important question. One of the biggest jumps in education in New Zealand is from primary/intermediate (years 1 to 8) to secondary (Years 9 to 13). In most cases children are taught by generalist teachers in primary/intermediate (which I will call primary from now on) and by specialist maths teachers at secondary school. Please be clear that this is NOT a criticism of Primary teachers. Primary teachers do an amazing job teaching such a wide range of subjects in a crowded curriculum to […]
11 February 2019

## Patterns, Mathematics and Statistics

Is mathematics really about patterns? Confession time: Just as I’ve never really thought of maths as beautiful, I’ve struggled to understand why people talk about maths being about patterns. For me maths is powerful and maths is about abstracting from reality to build models. So I thought I would explore the idea of patterns in mathematics. Number Numbers show patterns. The fact that all prime numbers over 3 occur on either side of a multiple of six, is pattern I find appealing. The digits of multiples of three sum to multiples of three. Multiples of eleven are pretty exciting with […]
13 November 2018

## Conceptual knowledge and procedural fluency when teaching maths and stats

Conceptual vs procedural when teaching maths and stats April 2008, Salt Lake City. It was my first NCTM conference and I was awed by the number of dedicated teachers of mathematics in one place. I had soaked in a pre-conference series about teaching statistics and my head was full of revolutionary ideas. I can’t remember the workshop I was attending but I declared that I saw no point in teaching students to calculate standard deviations by hand – and that I never did. The response was awesome! There was just about a stand-up battle between teachers who agreed with me […]
14 May 2018

## Spreadsheets, statistics, mathematics and computational thinking

We need to teach all our students how to design, create, test, debug and use spreadsheets. We need to teach this integrated with mathematics, statistics and computational thinking. Spreadsheets can be a valuable tool in many other subject areas including biology, physics, history and geography, thus facilitating integrated learning experiences. Spreadsheets are versatile and ubiquitous – and most have errors. A web search on “How many spreadsheets have errors?” gives alarming results. The commonly quoted figure is 88%. These spreadsheets with errors are not just little home spreadsheets for cataloguing your Lego collection or planning your next vacation. These spreadsheets […]
2 May 2018

## Why decimals are difficult

Why decimals are difficult Recently a couple of primary teachers admitted a little furtively to me that they “never got decimals”. It got me wondering about what was difficult about decimals. For people who “get” decimals, they are just another number, with the decimal point showing. Clearly this was not the case for all. So in true 21st century style I Googled it: “Why are decimals difficult” I got some wonderfully interesting results, one of which is a review paper by Hugues Lortie-Forgues, Jing Tian and Robert S. Siegler, entitled “Why is learning fraction and decimal arithmetic so difficult?”, which I draw […]