Why early success in maths can be a curse
I have been tutoring for over 10 years. Throughout that time I have observed different types of students. The most common type of student I see is the one who has been struggling with some maths concepts for years and therefore has some (or many) deep conceptual gaps in their knowledge. These students are the ‘Traditional’ students most of us think of when we think about tutoring.
However, there is another key group of students that I support. Let’s call them ‘Cliff’ students. These are students who tend to be labelled as smart or bright. They can often breeze through maths in primary school by focusing on following processes and doing calculations in their heads. Success in one year is followed by success in the next.
Then suddenly BOOM! Seemingly out of nowhere, test results start to drop. 80s become 70s, 70s become 60s…. This tends to happen around year 7 to 9.
Early success is a curse
In many ways these students are victims of their own success. In junior years these students learned their times tables, and perform basic arithmetic without putting pen to paper.
a)7 x 5? b)103 + 4302? c)52/4?
All a piece of cake. (For those playing along at home the correct answers are 35, 4405, and 13 😊)
With only 1 or 2 steps involved in the calculation, these students can handle all of it in their heads. Why bother wasting time writing things down when you can reliably get the answer without it? Economists would be happy with this level of productivity.
However, fast forward to high school and now problems start to integrate multiple concepts. Solving these now require 2, 3 or more steps of working. Additionally, abstract thinking in the form of algebra is introduced to complicate things further. Solve for x:
Therefore, in one step students go from an environment focused on process-level thinking which can be done in 1 or 2 steps to another involving abstract concepts requiring 3 or more steps.
Overloading our working memories
“Most people can easily remember 6 digit numbers. Very few can recall 14 digit ones.”
Our brains have an amazing but limited capacity. Here is a quick game you can play to test yours out:
Get a friend to write down a random number, look at it for 30 seconds, and then recall it from memory. Start with a 4 digit number and increase by 1 digit each time.
Most people can easily remember 6 digit numbers. Very few can recall 14 digit ones. The reason is that remembering these numbers takes effort from our working memory. At some point our brains run out of puff.
The same is true when solving maths problems. Most people can perform 1 or 2 steps in their head but doing 3, 4 or more is too much. It overloads our working memory. Have you ever felt frustrated when thinking through a mental problem, where the answers feel just out of reach? That’s your working memory being overloaded.
Solving maths problems in our heads is like lifting objects with one-hand tied behind your back — it’s possible but makes everything harder.
Fortunately we can reduce the strain on our brains by letting the page do some of the work. By working through problems we don’t have to hold so much information in our brains. The benefits are less frustration, fewer errors and better results.
Practicing bad habits
“Practice is a double-edged sword. While spending hours practicing can improve knowledge and skills, practicing the wrong things entrenches bad habits.”
People get increasingly good at the things they practice. Studies have shown that deliberate practice is a critical factor in gaining mastery. In his book ‘Outliers’ Malcolm Gladwell took the idea of deliberate practice further and popularised the notion that it takes 10,000 hours of practice to become an expert.
Yet practice is a double-edged sword. While spending hours practising can improve knowledge and skills, practising the wrong things entrenches bad habits.
And so it is with these Cliff Students. In an education system that only rewards the final answer, it is more expedient for a student to simply solve problems in his or her head, with no working needed. Over hundreds of hours these students have been indoctrinated to believe that they don’t need to work through problems.
The result? They develop an unhelpful mindset.
When these students get to high school and the problems get complicated they fall back on the same strategies that worked before. However, these new concepts and integrated questions provide more mental weight: more load for the working memory. Now they can no longer hold all the steps in their head and errors creep in.
The cost of not working problems out
In tests these students get dinged twice, first for getting the answer wrong, and second they lose out on partial marks for working: 0 out of 3.
Now that these bright sparks can no longer hold all that information in their heads, new emotions begin to enter the maths experience - frustration and upset. Moreover, something more sinister starts to occur. These smart kids have often been called smart all their lives. A large part of their identify can be wrapped up in this persona. Good test scores are evidence that indeed they are ‘smart’. Good scores lead to praise. Praise leads to pride.
Now the wheel has turned and this virtuous cycle quickly turns vicious.
Setting children up for success in maths
So what can be done to avoid children wandering down the wrong path?
1. Parents: Encourage process as well as outcomes
Parents, you can begin this at home early by offering praise when your child shows working on their homework. If they come back with homework or a test with the correct answers but no working, inform them that they have only done half the job and to redo it with the working shown. And yes, this won’t make you all that popular but it will start building the right mindset.
2. Teachers: Award marks for working and the answer
Teachers, you can also build these mindsets by awarding marks for both process and working. While this tends to happen in high school, this is something that primary school teachers should be adding to their repertoire.
For example on a geometry problem, to find the area of a square most tests provide just 1 mark for the correct answer (and hopefully units). Instead 2 marks could be awarded: 1 mark for the correct answer and 1 mark for the working.
Don’t get fooled by children simply getting the right answer. To achieve long-term success children need to develop not just conceptual knowledge but the skills to work through challenges when they get tough. It starts by realising not everything can be done in your head.
Jesse Whelan is the Founder and Director of Learning at Sandbox Learning Australia. He is passionate about helping children maximise learning by using effective long term strategies.
Do you, your child or someone you know need maths help? At Sandbox Learning Australia we use the science of learning to deliver maths coaching that is both personalised and effective.
If you are in Sydney please contact us to arrange a free baseline assessment of your child’s strengths and weaknesses.
