# Achievable challenge in teaching maths

8 May 2019
19 June 2019

## 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 similar ideas called “achievable challenge” and “the zone of proximal development”. I like the idea of achievable challenge.

## The right level of challenge

Achievable challenge is personal. My adult learners enjoy maths when I can set the work at the right level. For them it needs to be only just challenging – more towards the achievable end of achievable challenge. This is because they lack self-efficacy with regards to mathematics from years of not being taught mathematics in a way that worked for them. Possibly the greatest motivator is success – past success gives self-efficacy, which encourages the prospect of future success. My learners have blossomed as they have experienced success in a subject they had previously not succeeded in. There is a virtuous cycle in which success develops self-efficacy which in turn leads to more resilience in the face of challenge, leading to greater success. We need to help our students stay in this virtuous cycle.

The challenge for classroom teachers is finding tasks that provide achievable challenge for all their students. Sometimes I wonder if this is an unachievable challenge, which causes moral exhaustion for teachers feeling unable to meet the diverse needs of all their learners.

## Variation

One of the big ideas of statistics is that of variation. There is variation in all areas of life. Scientific experiments are designed for some aspects of variation in order to explore other areas of variation. You can see a video about variation here:

When we teach we are confronted with a wide range of students with variation over multiple dimensions. These dimensions include but are not limited to motivation, self-efficacy, prior knowledge of the relevant content, cultural understanding, health, mental health, stereotype threat, resilience in the face of challenge, mental processing speed and time available.

Board games, card games, computer games, sport all include challenge, success and loss. I have recently been playing Pandemic, which is a co-operative board game where the players play together against the game to save the world from disease. It has the perfect level of achievable challenge for me and we win about one in every four games. As I have high self-efficacy in this sphere, I almost prefer it when we fail. If it were too easy, it would not be fun.

In contrast, when things are too difficult they cease to be fun. One year in a fit of delusion I joined a social basketball team. I practised and practised and practised, but I did not get much better. The team was more competitive than I had anticipated, and I spent most of the season on the bench or in brief spurts on the court. One time I did manage to get a goal, everyone was so surprised even the opposing team clapped. I suspect I was in the wrong level of team for me to experience achievable challenge. I failed too often for it to be fun, and I often felt I was letting the side down. My self-efficacy in the realm of basketball was low and went lower. I decided running was a better sport for me.

## Games and achievable challenge

When people are having fun, they are more likely to continue doing what they are doing. This is why games can be powerful way to help people learn things such as basic maths facts. The best games can cater to a wide range of people. For example our Fraction Action game teaches about adding fractions and equivalent fractions. However, the graphics are designed such that a child who can add by counting can play. (I tried it out on my 5 year-old grandson). It also has a level of challenge and choice that engages people who are adept at adding fractions. Using a game like this in a family or classroom enables practice and conceptual development at multiple levels. You can find out more about Fraction Action here: Fraction Action game

## Growth Mindset and productive struggle

There is interest currently in developing a growth mindset around learning, and specifically learning mathematics, with Jo Boaler and Carol Dweck as two of the main researchers and proponents. As with all good ideas and research, the theory can sometimes be misapplied with counter-productive effects.

I was very keen to get my adult students to experience productive struggle, telling them that this will help them to learn. However, I failed to consider their history and feelings around mathematics. They were not ready for any struggle at all, until they had been successful. I have gradually increased the challenge and given students choice, which has further developed confidence. Several of them also had a perfectionist streak and once they started getting 100% in quizzes, nothing less would satisfy. We spent a bit of time looking at why this was the case and getting them to accept that getting things wrong is a step towards learning. They still see it as failure though.

## Achievable challenge for teachers

Earlier I wondered if teachers have a task that is an unachievable challenge. It depends on how much we are concerned about the individual learners. With a university class of 800 students I could not be concerned with each individual student but endeavoured to provide every opportunity for them to learn and get the help they needed. When marking exams I would remind myself that I was not failing them, but rather assigning a grade to the level of understanding they displayed. I felt satisfaction that most of the students learnt most of the material.

Now I have a class of ten adult mathematics learners. Now it is personal. I think about each one and their history, and design tasks that will suit them. I rejoice in their success and worry over their concerns. It is both demanding and rewarding and within my range of achievable challenge. I am learning to cope with productive struggle in my teaching rather than perceived instant success which may not lead to long-term success.

This is what makes teaching what it is.