Non-blog life (otherwise known as life) has eased up enough to get back to posting, and there’s quite a backlog of topical, “must do today” posts, plus a requested TNDOT. Usually, if I don’t get to a topical post quickly then I just drop the idea. Some recent media stuff has sufficiently annoyed me, however, that I’ll pretend it’s all still topical and I’ll post anyway.

First, another quick word about Tony Gardiner, who unexpectedly died last month. Tony’s death has affected me deeply, particularly given that I never met him, modulo attending a memorable lecture, and given that it was for only a couple years that we conversed, by email and through this blog. But I had begun to realise, and more so with his death, how much Tony had been guiding me, by explicit, admirably blunt, advice, and much more by example. I realised that I had started to compose blog posts with a “What would Tony think?” voice in my head. And Tony’s example is unparalleled. The amount that Tony contributed to mathematics education, for decades, is simply phenomenal, much of it dirt cheap or free. Tony had a missionary dedication to the mathematics education community, and he had a wisdom about mathematics education second to none. I know how fortunate I am to have Tony’s voice in my head. It will always be there.

On with the post: multiplication tables. Or, more accurately, “multiplication facts”. Last week, Greg Ashman had a short bit on a *Conversation* article, which had appeared a few days earlier. By two University of Sydney maths ed academics, the article is titled,

*What are ‘multiplication facts’? Why are they essential to your child’s success in maths?*

Greg is hesitantly positive on the article. After noting the debilitating anti-tables madness of Boaler and her minions, Greg writes,

*It is therefore pleasing to see a new article in The Conversation promoting the benefits of students learning their times tables. The authors make the inside baseball point of preferring the term ‘multiplication facts’ to ‘times table,’ include some throat clearing about supposed ‘problem solving skills’ and generally overcomplicate the process of learning times tables. Overall, I am not a fan.*

*However, this is an article by mainstream mathematics education lecturers — the type of people who, until recently, would disdain the learning of times tables. And yet it is an article that advocates practice of times tables. So, I think we can see it as a sign of progress.*

Perhaps we can see it as a sign of progress. But perhaps instead, or at least also, Greg should be even less than not a fan.

After a quick “this is important” introduction, the *Conversation *article begins by explaining the meaning of “multiplication facts”:

*Multiplication facts typically describe the answers to multiplication sums up to 10×10. *

Uh, “sums”? And these multiplication facts “describe the answers to” these “sums”, and only “typically”? Not always? Whatever. On to the second sentence:

*Sums up to 10×10 are called “facts” as it is expected they can be easily and quickly recalled. *

No, sums are things, not facts. And facts are called facts because they are facts: easy recollection is neither here nor there.

*You may recall learning multiplication facts in school from a list of times tables.*

Yes. And we made it to the end of the first paragraph.

OK, so I’m being a little nitpicky. But the language is extraordinarily careless and clumsy, particularly for education academics, and particularly for an article attempting to justify a change from clear, accepted and way-old nomenclature. Which begins in the second paragraph:

*The shift from “times tables” to “multiplication facts” is not just about language.*

Well, no, the shift is just about language. The reason for the shift, however, is not:

* It stems from teachers wanting children to see how multiplication facts can be used to solve a variety of problems beyond the finite times table format.*

Of course in the old, Dickensian days all we did, for years, was the times tables. In their “finite times tables format”.

*For example, if you learned your times tables in school (which typically went up to 12×12 and no further), you might be stumped by being asked to solve 15×8 off the top of your head.*

Yeah, you might be. Probably not, but you might be.

*In contrast, we hope today’s students can use their multiplication facts knowledge to quickly see how 15×8 is equivalent to 10×8 plus 5×8.*

Hope is a wonderful thing. And “in contrast” to what, exactly? Kids thirty years ago wouldn’t have thought to consider the tens and the ones separately? Or, which might have given your argument a fighting chance, they wouldn’t have thought to first break up 15 = 3 x 5 or 8 = 2 x 4 or similar? Nonsense. But perhaps the contrast is that now the kiddies are supposed to use the idiotically amorphous term “equivalent” to refer to equality. And I’m really trying to give up on the nitpicking, but “their multiplication facts knowledge”? How about “their knowledge of multiplication facts”, or simply “multiplication facts”? Or nothing?

*The shift in terminology also means we are encouraging students to think about the connections between facts. *

Yeah, unlike presenting the kids a sequence of numbers in arithmetic progression. What were we thinking?

*For example, when presented only in separate tables, it is tricky to see how 4×3 and 3×4 are directly connected.*

If only someone had thought of throwing everything together in one big table.

The authors then go on to tell us how “mathematics education has changed”:

*In today’s mathematics classrooms, teachers still focus on developing students’ mathematical accuracy and fast recall of essential facts, including multiplication facts.*

Sure. As if Peter the Not So Great shilling ACARA’s proficiencies is indicative whatsoever of what is or is not learned in today’s mathematics classrooms.

*But we also focus on developing essential problem-solving skills. This helps students form connections between concepts, and learn how to reason through a variety of real-world mathematical tasks.*

All nonsense, and all entirely irrelevant to mastering the purely mathematical concepts, facts and techniques of multiplication.

*By the end of primary school, it is expected students will know multiplication facts up to 10×10 and can recall the related division fact (for example, 10×9=90, therefore 90÷10=9).*

Well, by the end of Year 4, according to the Australian Curriculum, but “expected” is probably overstating the reality, even for the end of Year 6. And for guys hellbent on having kids making “connections”, it’s a little odd for them to note that the “facts” used to go to 12 x 12, that they now go to 10 x 10, but then be totally silent on the (insane) reason for this change.

The rest of the article is devoted to emphasising the importance of “multiplicative thinking”, and giving some friendly parental advice, strategies and games for practising “multiplicative facts” with the kiddies. Chanting wasn’t listed, presumably because memorising without the Semantic Seal of Approval is verboten; it doesn’t matter if the strategy works, only if the strategy is Good. This second part of the article is not all that bad, but that’s about the most that can be said for it.

For all that, Greg is probably right: having a couple maths ed academics going out of their way to write on the importance of learning “multiplication facts” is far from usual and thus is some sign of progress. But not a lot of progress. The term “multiplication facts” is an incredibly awkward furgle, which rolls off the tongue and down the throat. And it is a damaging furgle, almost inevitably leading teachers to embark upon a mish mosh of half-lessons. The tables might get properly learned through it all, but it’d be foolish to bet on it. Words matter, and “multiplication facts” are the perversely wrong words. The tables won’t properly be turning until the tables are again referred to, and presented as, what they are: tables.

I am an advocate of going up to 13 and that’s what we’re doing at my place. Why stop at 12 when there’s a juicy prime number next door? However, 12 is a convention and for that reason, I can tolerate it. Going only as far as ten is really stupid, in my opinion.

Thanks, Greg. I don’t think 12 is simply a convention. I think there’s a good argument for stopping at 12 (and of course no good argument for stopping at 10). I’m puzzled why your school goes to 13, but I’m not against it, or going further.

A few years ago I had a great exchange with the Sovereign Hill schools people. My daughter’s primary school typically does the SH camp, and at an assembly the returning students reported about “learning” the 16/17/18 times tables. I inquired, and the SH people dug. It all came from the Irish catholic textbooks used in the early 19th century, the tables of which went up to 19 (with 20, of course, being a gimme). An 1837 textbook had this, but the 1857 edition had reduced it to 12.

Obviously, times tables need to be memorized at least up to 9 because the multiplication algorithm requires knowledge of all products of single-digit numbers. But yeah, it doesn’t mean we need to stop at 10, and like Marty I don’t think 12 is simply a convention. A lot of things in our culture use base 12 (and sometimes base 20, which might explain Marty’s Irish textbook example), so I think it makes sense to go up to 12.

Greg’s 13 sounds like Spinal Tap’s amp to me though. The fact that 13 is prime is precisely why it’s not much used as a base.

Thanks, Marc. It’s astonishing how many teachers do not agree with your “obviously”. On the 12, it’s not simply that we use 12 and its multiples in lots of contexts, although that is plenty sufficient argument for the tables going to 12. The point is that we use 12 a lot because 12, and then its multiples, have lots of factors; that’s the directly mathematical argument for wanting the 12-table at your fingertips. I also agree with your thoughts on 13; the primeness of 13 is a good argument for stopping at 12, except for the not worthless novelty value.

Yes, the fact that 12 is highly composite is also why it’s a common base (superior to base 10 in my opinion, as I always tell my students).

I wonder whether even a very simple and common fraction like one third not having a finite decimal development leads to misconceptions in learners about the nature of numbers (for example, to the idea that one third is “not an exact number”) and to delays and difficulties in learning mathematics, difficulties that may now be even further exacerbated by the ubiquity of calculators. I’m not sure how we could test that, but I think it would be an interesting research question in mathematics education.

I don’t think the clumsiness of 1/3 in decimal form is intrinsically that big a problem. But I think it is, in fact, a huge problem, because of the current massive overemphasis on decimal form, and the flipside denigration of fractional form, as the meaning of numbers. The decimal fetishists have been a disaster.

Don’t forget that 12 used to be the basis of currency in Oz as well as some of the Imperial weights and measures. So 12 times tables made a great deal of sense then. Still have 24 hour clock and dozen eggs. Not sure on 10 or 12 either way.

By the way – I though times tables weren’t compulsory in the Australian Curriculum but based on previous efforts trying to find something, I’m not really tempted to try?

Define “compulsory”.

Hi JJ. Someone called Simon replied to you, but then his comment disappeared. In any case, in the AC the tables are begun in Year 3 and are completed, up to 10, in year 4 (AC9M4A02): see the curriculum in semi-human form, here.

recall and demonstrate proficiency with multiplication facts up to 10 x 10 and related division facts; extend and apply facts to develop efficient mental strategies for computation with larger numbers without a calculatorThat does not imply that primary teachers regard the proper memorisation of the tables as either important or compulsory.

Three comments.

“Multiplication facts typically describe the answers to multiplication sums up to 10×10.” Recently I have seen questions that asked students to solve the equation 10×10. I have seen this terminology used several times by an organisation that produces a popular on-line product.

In the four government 7-10 schools in Bendigo there is a concerted effort on multiplication tables.

My main school experience started in Year 3. I was expected to know my multiplication tables up to 13×13 to get into the school. Guess how I spent the summer!

Terry, why don’t you name the source of “solve the equation 10 x 10”?

I just choose not to name the source. However, I was pleased that one of my Year 7 students noticed it.

Nonsense. You didn’t “just” choose to not name the source. Why did you choose to not name the source?

Dunno

Sure.

‘Multiplication facts’ is an awkward term without any doubts. Sounds like people trying to masquerade simple – I think it is appropriate to use this word here- fact that children must learn multiplication tables.

I’m guessing that most readers/commenters here are secondary teachers, but if there are any primary teachers or education academics here, I have a question:

How similar, or what are the similarities between the “reading wars” and the “times tables wars”?

To an outsider, they seem to have very similar structures and very similar “key players”. I would be curious to know if this is the reality in the primary school classroom.

I think, unfortunately, the few primary teachers who might read this blog may also be too nervous to comment. The maths ed academics are, of course, too proper and professional to take part in the lowly discussions on this blog.

I found the term mathematical fact an attempt to Humpty Dumpty the discussion so they have a phrase that can mean whatever they want when they want.

As such it is a useless phrase as no one else can know what it will mean in the next sentence.

Btw what is the correct phrase to describe that 10×10 is the same number as 100? Is this a property of integers?

And what would accurately be described as a mathematical fact using normal definitions of each word?

Thanks, Stan. Yes, the looseness of “multiplication fact” [they don’t use “mathematical fact”] is a deliberate ploy. Good point.

10 x 10 and 100 are equal. No need to work harder, and it is a mistake to do so.

A fact is a true statement. A mathematical fact is a true statement about mathematics. For example, “Jo Boaler is a menace” is a fact, and 2 + 3 = 5 is a mathematical fact.

More specifically, 2 x 3 = 6 is a multiplication fact, although “multiplicative fact” is probably better English. Is 6 ÷ 3 = 2 a multiplicative fact? I would argue it is, but the authors would call it (only?) a “division fact”, a phrase they use in the article. Which demonstrates the intrinsic meaninglessness and uselessness of the terminology.

It’s a “divisive fact”. Sorry, couldn’t resist.

It would seem that the Maths Ed folk have, yet again, bastardised a sensible idea. This time the idea is

“Drill has long been recognized as an essential component of instruction in the basic facts. Practice is necessary to develop immediate recall.”

bastardised with ‘times tables’, with the word “times” replaced by ‘multiplication’ for good measure (probably because it sounds more ‘mathematical’ than ‘times’). This is a common ploy (in many walks of life) – to make something old seem new or fresh and then accrue the various benefits that come with that.

There is a 1999 Honours thesis (in primary school education) titled “Automatic Recall of Multiplication Facts and Number Sense” (the link is not provided for reasons that are obvious if you read Edith Cowan University’s Copyright Warning on page 2) which contains the following statement:

“Multiplication number facts will be defined as the 121 multiplication facts from (0 x 0) to (10 x 10), commonly described as the ‘times tables’.”

No reference is given for this definition. The definition no doubt pre-dates 1999.

The phrase “Times tables” is used 6 times, Multiplication facts” is used 56 times and “multiplication number facts” is used 3 times.

And 6 x 56 x 3 = 18 x 56 = ….

Thanks, BiB, and thanks for the caution. However, I don’t read the copyright warning as forbidding linking to the thesis (and it’d be way weird to so forbid it). So, for those interested, the thesis is here.

I wouldn’t presume that “multiplication facts” generally has the restricted meaning used in the thesis, but I also don’t care enough to try to find out.

= 20 x 56 – 2 x 56 = 1120 – 112 = 1008 🙂 It’s only the two times tables!

In Ancient Egypt, they knew how to multiply any two integers together using only multiplying by 2 and addition.

That might be it: ACARA might be making a beeline for 2000 BC.

We should be so lucky.

Hi Marty, I haven’t seen anyone refer to multiplication as repeated addition. I think is so important that I get my students to articulate 8 times 7 as 8 lots of 7. I get them to draw an 8cm by 7 cm rectangle in a 10mm grid book. They track across the rectangle with a transparent ruler and say their 7 times table multiplication facts while seeing the number of sq cm boxes increase. In a similar way, they track down through the rectangle and say their 8 times table multiplication facts. The commutative property of multiplication is readily established. The construction of other rectangles with area 56 sq cm leads naturally to developing the factors of 56. I think this approach enhances the development of number skills.