Who needs numbers anymore?
This week a student dropped in to a support session looking for help with written arithmetic. This is more noteworthy than it should be. Whilst all our young learners need help with arithmetic, most of them believe such “primary school stuff” to be beneath them. Besides, they say, there are barely any questions on that in the exam anyway. They’re right. Even in the new “rigorous” GCSE only one out of three papers are non-calculator. However, he was preparing not for his GCSE maths exam, but for an armed forces aptitude test. After a quick flick through his practise book I began to wish all our students had to sit this test. If passing it were a prerequisite to studying for the GCSE, I bet the GCSE pass rate would be a lot higher.
I was enjoying showing him how to multiply decimals when the student provided yet another pleasant surprise. His go to method for multiplication turned out to be the column, rather than the grid method. Why was this a pleasant surprise? Once I saw that he was able to multiply in quick and efficient steps, I felt instantly reassured. I was certain I could teach him anything else he needed to know. As a teacher in secondary school, I rarely met a student who could use the column method. When I did, though, they were usually among the quickest learners.
I understand why the grid method is taught in primary (and secondary) schools today. To someone unfamiliar with either approach it looks clearer on the page. It’s proponents state that it is more “intuitive” and provides students with a deeper understanding of what is happening when you multiply. It also has the advantage of being a bit newer, and everyone likes to be doing something new. But basically, it’s easier. Like “chunking,” (its equivalent in the world of division) it is also painfully slow and requires only a weak grasp of place value.
In my last two years of primary school I had an ancient maths teacher who made us spend the best part of a lesson working through pages of sums in silence. There were multiplications in there over which my students today would go on strike. Huge ones, with multiple steps involving four and five-digit numbers! I derived immense benefit from it. It takes a lot of repetition to achieve fluency in a complex method. Solving problems at speed makes us better at them. But you cannot solve long multiplications at speed using the grid method. Beyond three and two-digit numbers it is just too clunky. It is a gimmicky method, designed to get a student through simpler questions and allow the teacher to tick off that section of the curriculum. Lip service is paid to “learning other methods later,” but from what I saw in secondary school that is rarely realised.
Most of the people who observe and rate lessons today would issue a judgement of “needing improvement” to my primary school maths teacher. Modern commentators on education decry such practises as a waste of time in the age of the smart phone. But I am convinced the ease with which I later picked up more advanced mathematical topics is directly related to my fluency in written arithmetic. I see evidence of it every day in the students I teach. Our older adult learners arrive confident in long multiplication and short division, having had it drilled into them in their early years of school. They have often never heard of stem and leaf diagrams or probability trees, and yet they pick up these supposedly more advanced concepts with remarkable ease. Meanwhile, our young learners have been prepared by well-meaning professionals for a world in which everyone has a calculator on their phone. They didn’t waste time drilling written arithmetic, they gave it a quick pass, got it right once and barely practised it again. They moved on quickly to the skills and concepts that matter in the modern world. But they never mastered them. They come to us having spent five years learning about probability, yet still they cannot understand it. Within months, the more numerate adults (for whom this is often brand new) have overtaken them.
Our assessment as it stands emphasises knowledge that is the tip of the iceberg. The huge body beneath the water is the ability to apply precise algorithmic steps – and that comes with confidence in number work. You cannot teach it once and move on, terrified of going back to something you’ve already covered because you won’t be showing progress. It needs to be practised over and over, and then constantly maintained. It’s the key to unlocking everything else. It just doesn’t look very exciting to the man at the back of the room with a tick sheet on his clipboard. In FE we are as obsessed as everyone else with showing progress in lessons. Ever with one eye on our retention figures, we are even more squeamish than schools about making students do things they don’t want to. Woe betide you are caught spending a term making grade 3 learners practise how to add, subtract, multiply and divide. “Why are you teaching them primary school topics?” is a question that is asked far more readily than “Why on earth can’t they do this?” So, we continue plastering over the cracks, ignoring the fact that the whole edifice is built on shaky foundations. Even if we get it passed by the building inspector (a grade 4 in this thinly stretched analogy) we know it won’t hold up in the long run.
Amidst the recent talk of scrapping GCSEs, I would like to propose instead adding a new compulsory qualification. Make everyone sit and pass the armed forces aptitude test before they are even allowed near an algebraic expression. Call it the GCSE readiness test. Make them renew it in year nine before starting the KS4 curriculum. Make schools and colleges give numeracy the importance it deserves, not as a set of antiquated methods rendered obsolete by modern technology, but as the key to unlocking everything else we want our students to learn in their maths lessons.