Tag Archives: Multiplication tables

The Logbook

logbook cover

Recently I spent a quite fascinating afternoon looking at an old school logbook.  It wasn’t until the 1870 Education Act that education became universal, and it used to be mandatory for schools to keep a record of events.  The Headteacher would make an entry at least once a week; judging by this particular logbook, dating from the latter years of the nineteenth century, the Head would have a lot of discretion about how this requirement would be met.

logbook (1)

Over a period of thirty years or so the job changed hands a few times, and some incumbents wrote just a single line – sometimes simply “Nothing important happened this week”.

logbook (3a)

Later Heads wrote more, and as the book filled up over the years they would regularly be writing a page or more.

The school was in a village  with a population of 500 or so near Banbury, around halfway between London to Birmingham, and the book covers the period from the mid-1880s to 1906.  Typically, roll numbers were around 75 with an infants class and another class for older children.

It was a rural community and children were often away from school helping with duties like potato-picking and harvesting, and others I’ve never heard of – “leasing”, and “birdminding”.   A consequence of the 1870 Act was that the authorities were pretty strict about attendance, with visits from the attendance officer and the attendance registers being audited frequently.  Later Heads would state the percentage attendance for both classes every week, but even so clearly had the authority to use some discretion (on one occasion the Head decided not to open school on the day Barnum and Bailey’s circus came to town).

It wasn’t just the attendance officer that the Head had to worry about.  He himself needed to join classes to check on progress; the Rector visited regularly, and the Government Inspector came as well, perhaps once a year.   I was a little surprised to note that often the reports of the Head and the Inspector would often give mathematics (more precisely, arithmetic) a low profile, being subsumed within “Basic” studies. Greater priority might be given, particularly in the Infants, to Handwriting, Singing, Needlework, or Recitation.

logbook (4)

(“…. The knowledge of the elementary subjects is good on the whole, but Arithmetic is weak in the fifth and seventh standards.  Geography is good, History fair, and Needlework is well done.”

Ever since I began teaching we’ve been told that in the olden days every child knew their multiplication tables.   It’s not true!logbook (5a)

(“Standard III want great attention in their arithmetic tables not well known.”)

There’s another widespread belief – that children in the past were impeccably behaved, and that today’s society, and teachers in particular, have allowed standards of behaviour to plummet. The 1890s Head had plenty of non-impeccable pupils. In a school of just 75 or so, half a dozen are named week after week and several others less frequently.  Not all of them were boys – Minnie W seems to have been a real problem, being excluded from class time after time.  Her brother? / cousin? Reginald is pretty well as bad, while Oliver G “can’t be left for a moment without getting into mischief”. One senses a grim smirk on the next page when Oliver falls off a prohibited wall and breaks his leg – but a year later he “is just as bad as before he broke his leg”.

John J was another regular offender, with a particular habit of “molesting the girls on their way to school”.

I was shocked by one incident, when I read that Ernest L and Clement W (another relation to Minnie and Reginald!) attacked their teacher on her way home. I’ve never heard of such an incident, and I hope the teacher was satisfied that sending offenders home and making them apologise dealt adequately with the matter.

logbook (13b)

(“Two boys, Ernest L and Clement W, waylaid their teacher on her way home and stoned her – troublesome boys but the first is an imbecile and dangerous.  The correspondent asked that he might be sent home and the other to apologise.”)

Indeed, and contrary to what one might have expected, corporal punishment seems to have been rare. In 300 pages I found only one direct mention, when John J “an excessively bad boy … at last had a stripe this Friday afternoon”.  From another source I find that boys in their early teens would routinely receive fierce punishment (birching, or hard labour) for stealing items worth just a few pence, so if physical punishment at the school was indeed as rare as it seems then that does indeed surprise me.

No doubt the teachers breathed sighs of relief when Oliver and John and Reginald left school for the last time, probably at the age of 13.  Little did anyone know that several of those happy, carefree, mischievous boys had fewer than fifteen years left to look forward to.  This tiny village of just a few hundred sent 86 men to fight in the Great War, and no fewer than 25 never returned.  Reginald and Clement died on the Somme within a few months of each other; to the unimaginable grief of their parents both lost an elder brother as well.

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The Orchestra and The Farmyard

I used to do courses for classroom assistants who didn’t have a maths qualification.  We’d do a short course at the start of September.  Most had spent their lives failing in mathematics lessons, and the night before the course started some wouldn’t sleep for worry – they’d cheerfully have spent the morning with their dentist rather than with me.  But when St Peter asks me why he should let me into Heaven, I’ll tell him that there are a few hundred people in the world who by the end of the course were crying with laughter that they’d enjoyed themselves so much with activities like The Orchestra and The Farmyard.  

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The Orchestra is quite simply the most fun you can ever have with a group in a maths lesson.  Use it with a small group (fun), a whole class (terrific) or the entire school (brilliant).

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Appoint about a quarter of your group to be the 2s.  Their job is that whenever you count a multiple of 2 they have to stand up and sit straight down again.

So they have to go:

1   –   2 (stand / sit)   –   3   –   4 (stand / sit)   –   5   –   6 (stand / sit)   –   7   – 8 (stand / sit) – ….

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Another section are the 3s.  Their rôle is:

1   –   2   –   3 (stand / sit)   –   4   –   5   –   6  (stand / sit)   –   7   –   8   –   9 (stand / sit)    –   10 – ….

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And the 4s will go:

1   –   2   –   3   –   4 (stand / sit)   –   5   –   6   –   7   – 8 (stand / sit) –   9   –   10   –   11   –   12  (stand / sit) – ….

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The final group has the easy job, so you can reserve it for those with lots of energy.  They’re the 1s, and they stand / sit for every number.

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Personally, I like to allow a little time to let each section rehearse on their own, but as soon as you’re ready you can conduct the whole performance – you make the count, and all groups perform their contributions simultaneously, no doubt with great cheers on the occasions when everyone has to stand / sit at the same time.

You’ll find all sorts of ways of implement The Orchestra in your own style, but if you want a really unforgettable experience then use the Farmyard variation – so the 2s might be chickens and have to “Cluck Cluck” each time they stand / sit, and the 3s might be donkeys, who go “Ee-aw”, ….

And while it may be the most wonderful fun, The Orchestra offers much more than that.  You’ll get – and indeed feel – some quite profound insights into factors, multiples, and the like.

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Fizz-Buzz

Everyone knows Fizz-Buzz, but since I’m putting up a number of counting games I might as well include it as a reminder.  It’s another counting game you can use with a group of any size.

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I start by introducing Fizz, which requires us to say the counting words – but replace any multiple of 3 by “Fizz”.

So the count goes:

1,   2,   Fizz,   4,   5,   Fizz,   7,   8,   Fizz,   10,   ….

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Buzz is similar, but you have to say “Buzz” whenever you reach a multiple of 5:

 1,   2,   3,   4,   Buzz,   6,   7,   8,   9,   Buzz,   11,   12,   ….

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And obviously the next stage is to try Fizz-Buzz, which demands:

 1,   2,   Fizz,   4,   Buzz,   Fizz,   7,   8,   Fizz,   Buzz,   11,   Fizz,   13,   14,   Fizz-Buzz!,   16,   17,   ….

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There are of course a hundred variations – change the multiples, start from a different number, start from a high number and count backwards ….

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Find The Factors – the blog of Iva Sallay

One of the blogs I follow is Find The Factors.  It’s very different from this one.  For a start, it’s frighteningly efficient, and appears on a daily basis – not like this one, which happens now and again when I get around to it.

Iva Sallay gives us a new puzzle every day, where you’re given a few entries in a multiplication square and have to figure out the multipliers heading the columns and rows.  So clearly we’re in an area highlighting multiplication facts – in the puzzle below the 42 in the top row has to be the product of 6 and 7 – but we can’t immediately do anything with this because we don’t know if  it the 42 comes from 6×7 or 7×6.  And the 20 will offer more problems, since we’ll have to discover whether it comes from the product of 2 and 10, or 4 and 5; even then, we must decide on the orientation.

A Find The Factors puzzle will be full of challenges like this, and the beauty of the blog is that they’re graded from level 1 all the way to the most challenging level 6.  The puzzle below is level 4:

FIND THE FACTORS

 

This puzzle is number 177, and you’ll usually get some facts about the puzzle number – its factorisation, and in this case, you’re also given a magic square made up of prime numbers; the magic total is 177.

The obvious comparison is to Sudoku puzzles.  I rarely do Sudoku; I usually find I get so far and then have no option other than guessing between several possibilities until I find the right one.  Just occasionally I’ve had to do this with Find The Factors, but generally speaking even at level 6 I can eventually reason the whole thing out.  Mind you, sometimes I have to work hard, but that’s surely the whole idea!

I’ve shown one of Iva’s 1-12 puzzles, but she also gives 1-10 puzzles as well, and you can have a week’s worth of puzzles to download in Excel format.

You can find Iva’s blog at   http://findthefactors.wordpress.com

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I do the puzzles for my own enjoyment, but my pupils would find them daunting and indeed too time-consuming.  However, the basic idea is one I’m happy to challenge them with, so I may offer them a 4×4 square constructed on the same theme.  Once they’d tackled one of those, we’re likely to use the interactive version on the NRICH website  (http://nrich.maths.org/7382)  where you can find a variety of challenges all based on the same starting point).

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True / False Cards

I use these a lot.  They’re not threatening, they’re fun, and they’re hugely informative.  The set I use most often is a simple set of a couple of dozen cards, each with a

multiplication statement, e.g. 6×6=36.  Some of the statements are true, but some, such as 8×8=63, are false.

True False C

I ask children to sort them into three categories, true, false, and don’t know or not sure.  What’s particularly revealing is when children tell you why they put cards into the false pile.  One child will say 8×3=25 can’t be true because the correct answer must be even; another will say it’s because neither number is 5.

It’s such a valuable activity that I’ve got plenty of other sets as well, with different numbers and operations.

True False D

Using Multiplication Strips

I’m guessing, but I imagine that if you’re reading this there’s a good chance that you see a multiplication square as a beautifully structured array, logical and full of intriguing patterns, some known and some waiting to be discovered.  For most of the children I work with the situation is very different.   For them a tables square can be an intimidating affair, rather like one of those medieval maps of a largely unexplored continent where only the outer fringes are known and the inner areas are largely unfamiliar – “Here be dragons” indeed. In the last couple of years I’ve made much less use of a multiplication square; instead I’ve very often used a set of multiplication strips – a tables square cut into ten separate strips, one for each multiplication table.  The strips are much less daunting and far more accessible, and they introduce a tactile and exploratory aspect. Give a child a multiplication square and they’ll look at it – and that’s about it.  Give them a set of multiplication strips and they’ll pick them up and start to do something. Perhaps they’ll put them together and reassemble the tables square:

Multiplication strips, photo 1a

or often they’ll start to pair them up or sort them into groups.  Sometimes they’ll sort them into two groups, the odd-numbered tables, and the even-numbered. Straight away there are interesting discoveries to be made.    Choose one of the even-table strips.  Can you find some odd numbers on it?  No, they’re all even.  So what do you expect if we look at the odd-table strips?  Here’s an observation that tends to come as a complete surprise – three times as many of the numbers within a tables square are even as odd, and there are no odd numbers to be found anywhere within any even table.

More often, the initial sort puts strips into doubling pairs, x3 with x6, and x5 with x10.  Immediately all sorts of questions arise.  If you pair x2 with x4, then what do you do with the x8, and the x1?  So in fact some of your sorting puts the strips into families rather than pairs.  Once you’ve got the x1, x2, x4, x8 family isn’t it tempting to wonder if there’s not a further member waiting to be included – so you can incorporate the x16, and … ? And when you sort into pairs, what about those strips which don’t have a partner, like the x7 ?  Perhaps we really ought to have a x14 strip?  Or could we have a x1½  strip to provide an alternative partner for the x3 ?   Of course none of these questions arise every time, but all of them are questions which have arisen with my pupils, and I find it useful to have some empty strips on hand. There’s no need actually to write out the full x14 table onto the strip, simply labelling it x14 makes the point perfectly well.  

Multiplication strips, photo 2a

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