Oh boy - If I had a dollar for every person who has complained about not understanding long division in school - Well, I would have at least $23 by now.
Long division seems to be that thing that defines our school maths experience. For those of you who 'got it', you felt good, gained confidence and progressed. For those who didn't quite 'get it', you felt slightly worried, unsure whether you would be able to understand the next bit (and hoping long division was not pivotal to future learning). For those who never understood a bit of it - we knew we were screwed.
Let's revisit it, and see what all the fuss was about.
But first, have a go at the problem below and try to use long division. Don't worry if you have forgotten, just have a go - all that knowledge is sitting in your brain ready for action. Just begin, write down the numbers, put it in the format you remember. It'll come back.
The problem is a famous research question.
An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed?
I've spent the last few weeks working through some data I collected on learners' beliefs about quick learning. Here is a brief summary:
Learners tend to believe that 'understanding' occurs in a single moment, usually in response to listening or watching a tutor demonstrate or explain some aspect of numeracy. They say things like, "Some people just 'get it', but I usually don't", or "I love it when I get it". This notion of 'getting it' permeates their thinking around mathematics and numeracy. And it's all bad.
If a learner believes that understanding happens in a 'moment', then when they do not 'get it', they may begin to doubt their ability to understand it at all. Often, they will ask the tutor to repeat the content, example or demonstration, or they may just ask the tutor to 'show us again'. They hope, and expect, to 'just get it' and believe that they ought to be able to do so. When they cannot (and see others getting it) they often use this as evidence of inability.
The truth is that understanding can on occasion happen quickly, but only when the groundwork for understanding has been laid. Understanding is a 'process', not a 'state'. It takes time and effort. Often it emerges from an extended period of confusion. True mathematical understanding develops over time, not in response to someone telling you something. Yes, there are moments of insight, but these are the conscious outcome of temporal subconscious processes.
If learners believe that understanding ought to happen quickly, then they are set up for failure and negative affective responses. For example, if Kelly believes that she should understand the concept of ratios as the tutor tells her (in that moment), and she doesn't, then she may believe that she has a mathematics problem. She may then give up, and reaffirm her belief that she is no good at mathematics.
Add to this, the learners lack of strategic learning repertoires, and their subsequent reliance on tutors, and it becomes clear that for these learners, listening to the tutor and hoping they 'get it' is the only option. This means learners who do not 'get it' have to ask the teacher effectively limiting their learning opportunities to periods of tutor instruction. In an adult numeracy classroom in NZ, tutor responses to learners questions are abysmally small. Learners don't ask for the tutor to repeat information because it reveals to the rest that they don't understand. And that noisy learner? You know the one that asks all the questions, all the time? Well that learner isn't asking the right questions, and yet tends to dominate the tutor. We have the conditions for a perfect storm.
Finally, and to the point. How many tutors talk about the 'aha' moment. In particular, how we feel good about our roles when learners suddenly 'get it'. We may in fact be mythologizing a negative meme. The 'aha' moment is a passive response to a usually accidental delivery of content. Instead, we should be talking about the learner we motivated to go home, and spend hour after difficult hour, working on that confusing concept until they finally began to make sense of it. That would be a real inspiration.
Mathematical understanding is the result of hard work, time, effort and often confusion. Learners who think that this experience means they are dumb, are not going to persist for long. It's a bit like a hopeful marathon runner who interprets discomfort as a signal that they are no good, - because all the good runners do it so easily.
Do you think understanding happens quickly? Or gradually over time?
Does it make a difference?
Girls are consistently out performing boys across all areas of education. The advantage that boys had in mathematics has largely vanished, and girls have always enjoyed an advantage in the areas of language.
However, one area that girls do have difficulty with is at the higher levels of education, particularly high-achieving girls. What seems to be happening is that girls achieve well, and expect to achieve well. However, the constant success may not develop the tenacity, or grit, that is required at higher levels of education, particularly in maths. Hence, when the learner fails, and they are not used to it, it hurts. Often it damages, or collapses, the learners self-belief. Some of these learners have not experienced 'fighting out of the hole', and instead feel defeated and as thought they have reached their limit. Many brilliant people quit.
Here is a sentence from Carroll Dweck that should get your attention, "What we found was that bright girls didn’t cope at all well with this confusion. In fact, the higher the girl’s IQ, the worse she did."
Contrast this with the average achieving learner who has scratched and clawed their way through, constantly fighting to stay in the game. They may have developed resilience, persistence and grit -which count for so much at higher levels of education.
Of particular note, is the impact of beliefs on how learners interpret and respond to failure. Your beliefs as a parent will strongly influence your children, positively or negatively.
Anyway, the link below is to a great article by Carol Dweck. If you are a parent of daughters, it is well worth a look.