# understanding

8 March 2021

## Multiplication Fact Fluency

Just how important is it to know your multiplication facts? There is confusion over the aim for number fluency and the means by which is it achieved. Let me explain. Many people learned their “times tables” by rote, sometimes under threat. This led to people know the right answer, but not necessarily knowing how to apply them. For example if they were asked what is five times three they could say 15, but given \$150 to split up between five people they would not know how to proceed. (Having said that, if it’s money involved people often do better.) I […]
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 […]
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 […]
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 […]
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 […]
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 […]
5 March 2018

## There are many good ways to teach mathematics

There are many good ways to teach mathematics and statistics Hiding in the bookshelves in the University of Otago Library, I wept as I read the sentence, “There are many good ways to raise children.”  As a mother of a baby with severe disabilities the burden to get it right weighed down on me. This statement told me to put down the burden. I could do things differently from other mothers, and none of us needed to be wrong. The same is true of teaching maths and stats – “There are many good ways to teach mathematics and statistics.” (Which […]
5 February 2018

## The Central Limit Theorem – with Dragons

To quote Willy Wonka, “A little magic now and then is relished by the best of men [and women].” Any frequent reader of this blog will know that I am of a pragmatic nature when it comes to using statistics. For most people the Central Limit Theorem can remain in the realms of magic. I have never taught it, though at times I have waved my hands past it. Students who want that sort of thing can read about it in their textbooks or look it up online. The New Zealand school curriculum does not include it, as I explained […]
11 January 2017

## Why people hate statistics

This summer/Christmas break it has been my pleasure to help a young woman who is struggling with statistics, and it has prompted me to ask people who teach postgraduate statistical methods – WTF are you doing? Louise (name changed) is a bright, hard-working young woman, who has finished an undergraduate degree at a prestigious university and is now doing a Masters degree at a different prestigious university, which is a long way from where I live and will remain nameless. I have been working through her lecture slides, past and future and attempting to develop in her some confidence that she […]
3 November 2016

## Talking in class: improving discussion in maths and stats classes

Maths is right or wrong – end of discussion  – or is it? In 1984 I was a tutor in Operations Research to second year university students. My own experience of being in tutorials at University had been less than inspiring, with tutors who seemed reserved and keen to give us the answers without too much talking. I wanted to do a good job. My induction included a training session for teaching assistants from throughout the university. Margaret was a very experienced educational developer and was very keen for us to get the students discussing. I tried to explain to her […]