Writing in the Times Educational Supplement in London (the TES), David Boorman questions plans to introduce a timed test of times tables to English schools.
Boorman’s first concern is that of ‘maths anxiety’. He worries that timing tests in this way will lead to more anxious students. I have no doubt that timed tests can make students anxious – although the evidence is hard to pin down – but I also see it as part of a teacher’s job to allay those fears and present such tests as a normal part of school life.
Boorman can’t see the need for a time limit, asking, “How long did Shakespeare take to write his plays? Does anyone care?”. I think this misses a key point: We want to test whether students have retained times tables as facts in their long term memory rather than checking whether they can work them out using working memory. If we ensure that time is limited then we are more likely to test the former than the latter. This is important because in higher level maths, students are rarely asked to simply work out a single multiplication. Instead, such skills will be embedded in a larger problem. Working memory is severely constrained and so it is important to free it up to focus on the problem in hand. If students memorise times tables then they release working memory capacity to focus on other aspects of the problem.
Boorman’s second point is a concern about a ‘times table check mindset’. He points out that many students might be able to find 5 x 7 but then be unsure of the answer to 7 x 5; something I am quite prepared to accept. But he uses this example to set time tables knowledge in opposition to an understanding of multiplication.
We often see this argument in the rhetoric about early mathematics education. And yet I have seen no convincing evidence to suggest that knowing facts somehow gets in the way of understanding. Yes, knowing times tables facts is not sufficient but I don’t think anyone ever claimed that it was. Ideally, students know their facts and also have a deeper appreciation of multiplication. In the case of the 5 x 7 example, the student would need to be taught about the commutative property of multiplication e.g. by considering a rotating rectangle. Why can’t we teach times tables and also teach commutativity?
Finally, Boorman demonstrates quite a limited view of why we want student to learn times tables. He complains that they are not useful in everyday life:
“I’m often told by friends and colleagues that they can help at the shops. However, I’m sceptical at best. Take six times eight, for example. When and why would one purchase 48 of an item? Of course, it could be six items priced at 8p each, but then what items are priced at 8p?”
Boorman’s colleagues have obviously got the wrong end of the stick. What is it about maths that makes people demand that every individual skill needs to have some mundane, everyday use? We don’t do that with other subjects. Nobody goes into a primary school class, observes a session of clay modelling and exclaims, “But when will students need to be able to do this at the supermarket!?” And nobody stops children from writing stories on the basis that they will never need to write stories in real-life; that only professional authors need to be able to write stories. And yet you hear these arguments about maths all the time.
Times tables are not particularly useful on their own and they are not intended to be. Maths is not a flat subject; it’s hierarchical with basic skills feeding into more complex skills. As we have seen, a facility with times tables frees up working memory to solve other aspects of a problem. For instance, an important skill in senior maths is to be able to factorise quadratic equations and this is much easier to do if you know your times tables. If students don’t know their tables then this limits their access to higher level maths. You can do all the investigations, critical thinking activities and genius hours you like but if students can’t do maths then you are effectively shutting the door on most STEM careers.
Perhaps most worrying of all, David Boorman is a lecturer in primary education and so he has the opportunity to promote these views about times tables to the next generation of primary school teachers.
Filling the pail 
“Perhaps most worrying of all, David Boorman is a lecturer in primary education”
Why am I not surprised…..
I am increasingly inclined to think that we will never sort out the mess of education in England until the government insists that trainee teachers are taught by psychology lecturers who actually have an understanding of what psychology research tells us, and are willing to convey that to their students. The resistance of Education lecturers to the evidence from research psychologists about systematic phonics instruction is a similar case to the ignorance shown by Boorman.
Psychology lecturers are also more likely to have a better grasp of the scientific method, and what actually constitutes good study design, because this is part of their training.
The problem in primary education lies in psychology itself and relates to psychology’s quest to be accepted as a science. See https://paceni.wordpress.com//?s=hirsch&search=Go
Um, right. Two references to Wittgenstein and the claim that “While it is clear that thinking would be impossible without a properly functioning brain, the claim that brains can think or that thinking takes place in the brain ought to be supported with scientific evidence. No such evidence exists.” Thanks, but I’ll stick to cognitive science.
Not just in England, friend…
“Of course, it could be six items priced at 8p each, but then what items are priced at 8p?”
Very few Items may be priced at 8p each but I can think of many which may be priced at £8.00 each, in which case knowing 6×8 would be very useful.
Reblogged this on kadir kozan.
My son was not naturally good at elementary math–we drilled times tables a lot and he got quite good (and of course understood the concept–that generally isn’t the hard part). The only thing I will concede about timed tests is that for very young children, their poor fine motor skills can hold them back with paper and pencil. However there are great free programs on-line for this. Which brings me to the fact that my son’s teacher (he’s in 10th grade) recently added Xtra Math, one of these basic math fact programs, for all the students! It only takes a few minutes a day. Obviously the teacher noted that the fact that most of the students were never actually DRILLED on this topic now means that there are a lot of high school students that struggle. I teach at a high school (not math) so I have known of this issue for many years. I was happy to see someone acknowledge it and then try to do something to remedy it. It isn’t hurting my own son either to review. Although he was once very fast with his facts, he has been allowed to use a calculator since and now uses it for things that he shouldn’t have to use it for–it has become a habit.
First Boorman probably knows very little about Shakespeare’s education.
http://www.shakespeare-online.com/biography/whatdidshkread.html
“At that time, as we have seen, boys usually went to the grammar school about six or at latest seven years of age, and entered at once upon the accidence. In his first year, therefore, Shakespeare would be occupied with the accidence and grammar. In his second year, with the elements of grammar, he would read some manual of short phrases and familiar dialogues, and these committed to memory would be colloquially employed in the work of the school; in his third year, if not before, he would take up Cato’s Maxims and Aesop’s Fables; in his fourth, while continuing the Fables, he would read the Eclogues of Mantuanus, parts of Ovid, some of Cicero’s Epistles, and probably one of his shorter treatises; ”
Anyone think there wasn’t pressure to memorize grammar and classic texts in that day?
Second the anxiety argument is just plain stupid. If it were a reason to stop doing things in education then that would include going to school itself and any testing or questioning by the teacher.
Third Boorman doesn’t even see how dumb is strawman 6 8p items is. If it doesn’t seem likely to find an 8p item for sale then why did you pick that example Mr Boorman? Pick a 7.99 one and round it up and suddenly it is handy to know 6 x 8/
Fourth, If Boorman really cared why we teach math in primary and secondary school he would look at data on what the needs are for people leaving high school. Some is easy to get. Universities math departments are quite vocal about what they perceive as a lowering of quality of math knowledge in university entrants.
It may well be that we don’t know which jobs will be good opportunities in 20 years but we can make a good bet that many good ones will involve a university education so looking at what universities want from high schools would give some clear data.
Having some like Boorman a professor teaching teachers is as bad as having Andrew Wakefield a professor at a medical school.
That would quickly generate outrage if it ever came close to happening (see https://www.thestar.com/news/gta/2015/07/31/u-of-t-scarborough-dean-resigns-after-vaccination-flap.html)
That’s right in medicine everyone involved up to the Dean would lose their jobs.
Why do so few care about garbage teacher educators? Why no headline “Professor of education spouts ignorant nonsense in response to times tables test”?
As a part answer to my question I offer that with education individuals who care can solve the problem for themselves with far less energy than it would take to change the system.
Parents who care can pick schools, complain about individual teachers, and hire tutors. In a competitive job market or university placement competition parents may even feel helping everyone would work against their interests.
It’s a form of tragedy of the commons. We would all be better off with a better educated populace. But with my resources I will be better off if I just educate those I have an interest in.
And twitter provides a perfect example. Parents will go to great lengths for their kids because they can make a difference.
http://www.theglobeandmail.com/news/toronto/toronto-area-parents-brave-the-cold-for-coveted-school-spots/article34393213/?utm_source=twitter.com&utm_medium=Referrer:+Social+Network+/+Media&utm_campaign=Shared+Web+Article+Links
But how much effect on their kids schooling could they have by changing a teacher educator by the time they have taken an interest.
To leave this on a positive note I’ll thank my Grade 5 teacher Mr Goertze who on finding his class didn’t know the times tables went about getting us up to speed with about 15 minutes a day for several weeks.
That is all it takes. Today any teacher or parent can follow great resources such as
Click to access TipSheet_Times.Tables-01.pdf
and get a student confident and proficient in a short time. Clearly Mr Boorman has never tried this.
I don’t like the title of the Jump pdf as it low balls the time it takes to commit things to long term memory. But there opening line will ring true to anyone who has done any math:
“Trying to do math without knowing your times tables is like trying to play the piano without
knowing the location of the notes on the keyboard.”
People who don’t know this don’t know math.
Stan, I won’t take exception to anyone’s interest in fluency. But are you not troubled by that attachment’s emphasis on this trick to memorize that multiplier and this other trick to memorize that other one? My experience is these tricks are fragile and result in students who expend their WM resources trying to recall which trick matches which multiplier. Yours?
I am not troubled by the tricks themselves. The first step is to be able to get the answer using mental arithmetic. Here tricks may be fine as long they are introduced at a pace that doesn’t create the confusion you describe.
The step of making recall automatic might include a phase where application of a trick is automatic but eventually becomes having the answer without any conscious effort.
My concern would be aiming for 5 days might lead to the fragility you describe. A more realistic goal might be a month of daily practice and a second month of weekly review.
I think sad that it is a debate whether students should learn the tables not which approach to learning is optimal.
I would defer to anyone who has a more scientifically tested approach.
“Here tricks may be fine as long they are introduced at a pace that doesn’t create the confusion you describe.”
I’m worried that an accumulation of tricks leads unavoidably to confusion.
Just to get a sense of where you’re coming from, how does this mnemonic strike you? Is it better or worse than the JUMP Math tricks?
Dan,
I promise you this collection of tricks delivered at a suitable pace does not unavoidably lead to confusion. But that is not a big claim.
Would you be concerned about just the one trick for the 6x tables? If not would simply slowing the process down remove your worry? Only one or two tricks at a time until these tricks are not required as long term memory is doing the work?
Also, the mnemonic seems inferior in at least three ways. It only works for one pair. There are at total of 5 tricks described in the jump pdf. (I am not counting commuting.) The mnemonic is also introduced with a lot of spurious information. The jump tricks are all based on number patterns so at some point might be mathematically interesting in themselves.
Agree that time-testing in this case does lead to where we want to go – that is, instant automatic recall of times tables – because this makes many other maths operations easier to do.
My take on this: Time for Tables? https://chemistrypoet.wordpress.com/2015/02/08/time-for-tables/ via @chemistrypoet
I don’t see this as an argument for timed tests so much as an argument for asking students to work out larger problems more often as an assessment of their fluency.
The trouble with trying to diagnose issues through a complex performance is that you can’t be sure what has gone wrong. If some students fail to solve the problem ‘fluently’ then how will you know that this is because they don’t know their times tables rather than because of some other issue?
I suspect detractors of timed tests would make a similar case. If a student fails to solve the timed times tables fluently, how can you know it’s because they don’t know their times tables rather than because of some other issue like anxiety leading to diminished working memory capacity, etc.? If an exogenous condition is necessary to see if students have their times tables in LTM, I wonder why we wouldn’t pick an exogenous condition we actually care about, like the ability work larger problems.
I think this maths anxiety thing is being overplayed. For some students, this might be an issue and that’s why it needs to be handled supportively. However, even you must recognise that a timed test of time tables knowledge is a much more direct test of times tables knowledge than solving problems that may or may not use this knowledge in an incidental way. If we follow your logic then we might also argue that problem solving might make kids anxious, further distancing us from the skill we want to assess.