In an era that emphasizes the need for “deep understanding” in learning, it’s alarming that most of the students I meet requiring tutoring are totally reliant two ineffective strategies to find an answer – 1) guess and hope, or 2) don’t even try – that way you can’t get it wrong. Until recently my typical strategy would be to identify areas of misconceptions, and make students aware of what they needed to know to correct their knowledge of basic mathematical concepts . I stressed the importance of reasoning over guessing, and provided students with opportunities to practice and apply their new-found knowledge. I figured I was helping them acquire the “deep understanding” they had previously missed out on, and that this newly acquired knowledge would improve both their self-confidence and their motivation.
It didn’t work very well. Students were still discouraged, still wanted to guess the answer, and often previously covered material was forgotten. I often felt that my tutoring sessions didn’t meet my expectations. Although I asked questions to encourage the students to think, they still kept playing “Guess the answer”. When they couldn’t come up with an answer, I resorted to logically sequenced and scaffolded explanations – a one on one chalk-and-talk session that bored both of us. Any mathematical understanding gleaned through the session was painfully acquired and often short-lived.
This year has started differently. I found Malcolm Swan’s “Improving learning in mathematics: challenges and strategies” (available as a pdf document) , along with supplementary material from National Centre for Excellence in the teaching of Mathematics (national being UK). I stopped explaining. I gave them simple tasks (e.g. sort these decimal numbers in order) , and asked them to explain how they got their answer. If their reasoning as wrong (and it often was), I provided examples to challenge their reasoning. (If .125 is bigger than .2, because 125 is a bigger number than 2, then what about .200 – is it bigger than .2?) The really exciting part was that these students started to catch on quite quickly – cognitive conflict would kick in and these students started thinking for themselves to find solutions to problems they themselves had discovered. What’s more, they (and I) were enjoying the process.
Once decimals (and percentages as a by product) were mastered we moved on to sorting fractions of different denominators, but all with a numerator of 1. The commonly encountered erroneous thinking that the bigger the denominator, the bigger the fraction was present. So too was cognitive conflict when this reasoning was applied to his existing knowledge of the relative sizes of one half and one quarter. I could virtually hear the wheels in the student’s head start to turn – slowly at first, then faster as he corrected his reasoning and verified it for himself with further examples. I checked his understanding by asking him what would be the largest fraction possible to have with one as the numerator and a whole number as the denominator. (Old habits die hard – I slipped back into “guess what answer I’m fishing for mode” – in this case I was fishing for half.) His answer was better than mine – one – as in one oneth, which would be the same as the whole thing.
We started looking at comparing fractions where the numerator changed as well as the denominator, aided by an interactive flash program that lets you create fraction visualizations and compare them. When the period ended we were exploring the student’s theory that if the numerator AND the denominator of one fraction were bigger than the numerator and denominator of another fraction, then that fraction would be bigger than the fraction with the smaller numbers. The thinking is flawed, but it is mathematical thinking, something students struggling with maths normally avoid like the plague, and suggests the beginning of an exciting shift in this student’s attitude to maths. Even more exciting is the prospect that while he discovers and investigates exceptions to his theory ( 5/8 is not larger than 3/4), he will probably teach himself about common denominators.
Meanwhile I’m re-reading the booklet – there’s plenty of other practical suggestions in there besides cognitive dissonance.