Math Fact Fuency

Nicky Morgan, the British Minister or Education, recently announced that "all pupils must know, by heart, their times tables up to 12 x 12". Apart from the use of the archaic term "times tables", which one would expect from a Tory, the whole issue of fact fluency, to give it its current term, is a really interesting one.

I have always believed that fact recall, but up to 10 x 10, is an important part of the mathematization process children go through. It's the math equivalent of being able to spell words, but the facts are by no means the "basics" of mathematics. The basics are everything that is included in the field of numeracy; being able to count, to recognize number patterns, to subitize, to see numerical relationships and so on. Remembering the math facts makes math easier and more efficient.

Memory, remembering things, is a crucial part of education, but it is pretty useless when we memorize things with absolutely no understanding of what we are memorizing. Memorize this list of words; Arun, Ouse, Rother, Stour, Medway, Darnet, Mole and Wey. Now use any of these words during a conversation you have with someone  over the next few days.

If we are going to require students to remember their math facts they must understand what they mean. The multiplication facts for example can mean 'groups of' as in 5 groups of 4 people are 20 people. They can mean area as in a carpet 5 yards by 4 yards has an area of 20 square yards. They can also mean the muliplicative comparison as in "I have 20 Hotwheel cars which is 4 times as many as you have if you have 5".  Each of these concepts of multiplication is different but each can be solved with recalling the fact 4 x 5. Or is  it 5 x 4?

Talking of which, when you see the fact written 4 x 5 do you read it as 4 groups of 5, or four 5 times? I asked my grad class this the other day and half saw it one way and half the other way.

For a great read on the topic of fluency read Jo Boalers incredible article Fluency Without Fear which includes a very relevant criticism of EngageNY's approach to fluency.

The names above, by the way, are the Rivers that flow out of Southeast England.

Maths Should Be Fun?

For some reason we seem to be going through a phase in our culture where math has to be fun. Judging by the incredible number of websites that have sprung up with mindless activities that you can play on-line and be subjected to a barrage of ads, or buy at great cost the business world has discovered there's money to be had with people's current dissatisfaction with public education. One such company IXL seems to appear everywhere and has probably the most useless activities of any to be found on-line.

Of course math should not be the incredible tedium it once was and still is in some places but neither should it be fun. It should be CAPTIVATING, INTERESTING, MEANINGFUL, RELEVANT and above all else, IT SHOULD MAKE SENSE. If everything in life was fun we would not know what fun was. Fun is something we have to relax and enjoy ourselves when we are not working. Learning math is work, it is something that, hopefully, is all of the things in CAPS above but it cannot be fun; not all the time. There are some things like muliplication facts that just have to be learned.

There are times of course when what we learn in math can lead to fun, When we are playing games that involve math knowledge and understanding such as cards, board games and many of the tech-based games that children play. Have you ever thought about how much math there is involved in playing Wii or the DS games? (Wii Play Together). There are also some great Apps for the Ipad2 that help children learn math facts and concepts in a motivating way.

OK, so there are some fun games for helping children remember their math facts but these should only be used after children have developed strategies for remembering their facts. Here's my current favorite on-line math activity. It's called Arcademics. Have fun, er, I mean, be captivated and try to
remember your facts.

Number Patterns in Problem Solving.

This appeared on my Facebook page today. I get lots of these with the invitation to solve it. I'm not very good at FB so I never like to click on anything I think is going to get me into trouble. Who knows what is lurking the other side of a smple mouse click?

Anyway this is a really neat little problem that very eloquently demonstrates the power of pattern in learning math. As I have said many times before the identification of a numercial pattern changes everything in terms of trying to remember facts as well as more higher order thinking activities such as genuine problem solving. Most algebraic problems require the application of a pattern of some sort.

So what does the idea of pattern have to do with this little FB treat? First there is a pattern to each of the descending series of numbers; at least until the 3 is reached. They descend by 1 eachg time. The numbers on the right also descend but by different amounts, 14, 12, 10. extending these patterns, if there was a 4 on the left the number on the right would be 12 ( 8 less than 20), then would come 3 on the left and 6 on the right (6 less than 12). Now look at th relationship between each of the pairs of numbers and you get 8x7=56. 7x6 = 42, 6x5=30, 5x4=20, 4x3=12, and 3x2=6. Each number descends by 1. I could be wrong, of course, and completely missed the trick, if there is one, to get a differerent answer.

I'm not sure if one needs to be a genius to solve this. I would expect any 4th grade student to be able to solve this if they have been taught that math is the science of pattern and not a random collection of facts and ideas.

Aaah, Multiplication Facts

There are few things mathematical guaranteed to stir the emotions more than the multiplication facts. Over the centuries just about everything has been tried to help students, round about the age of ten, memorize or remember their multiplication facts. Traditionally, they have been presented in the format in the image to the left. For example for the 'fives' the 5s came first followed by x1,x2,x3,x4 etc. This format is still frequently used which is really unfortunate because it would be so much more efficient if the 1x,  2x, 3x, 4x, etc came first (as in 1 x 5 = 5). This would allow students to use the repeated addition concept of multiplication to learn the sequence of the multiples of 5 more efficiently and effectively as in one 5 is 5, two 5s are 10, 3 5s are 15 and so on. Here an additional 5 is being added each time; something that does not work when the 5 is presented before the x1, x2 etc.

More recently, we have started using the multiplication square as a way of learning and remembering the facts. This method has the added advantage that each fact makes an array (rectangle or square)which helps the student visualize the fact as they are learning it. For example 5 x 4 can be visualized as a rectangle with side 5 and 4. Square numbers can also be idenfied as squares such as 36 or 6 x 6. Prime numbers make only one array (e.g. 13 only makes 1 x 13) but that's another story.

Today, in my Teaching Math to ELL students class we interviewed students from different countries around the world to find out about the way they learned math. Interestingly, most learned their multiplication facts to 9 x 9 while some learned to 10 x 10 and one student from Saudi Arabia learned up to 19 x 19. No-one seemed to learn to 12 x 12.

10 x 10 is the usual limit for remembering the facts but I always have wondered why it was, traditionally, 12 x 12. Perhaps it was because 12 figures so large in our Western culture (12 Apostles, 12 months in a year, 12 hours in a day, 12 in a dozen, twelfth day of Christmas, 12 inches in a foot etc). Perhaps there is a more logical reason? I can't find an answer on Google, yet.