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March 06, 2017

Multiplication tables with children who never learnt it

There are children who are able to hold their fort without knowing the multiplication tables in the calculations in Mathematics. Some are quick at addition, some remember specific tables and extrapolate quickly. They survive till 9th grade where powerful mathematical concepts are introduced with milder calculations and are able to generally manage.

However, for most children for whom the number sense has not landed and are slow with calculations, the lack of knowing the tables is a serious hindrance in survival and any possibility of enjoying mathematics.

We took up the challenge of identifying 7th, 8th (and 9th) graders who did not know their tables. We worked with the children in identifying what they already knew - generally the 2, 5 tables. For a couple of children, the short-cut for the 9 tables was quick and they were comfortable with using that quickly. Some have the hand rules for 6x6 onwards, but it appears too slow for practical use. The table tricks were kludgy and not used e.g. some knew methods to write down the 7 tables, but it was too easy and not practiced enough to be usable in action and certainly not at random.

The common issues for children were 3x6 to 3x9, 4 tables, 6x6 to 6x9, 7x7 to 7x9, 8x8 and 8x9. I looked at a demo of the Vaughn Cube. Now, the demo is public domain and looking at the video it is easy to figure out how it works. They have given enough in the video (unknowingly?) to figure out not just how it works, but work out the details of the method. The rest of the post is about our Indian hack of the same and what we did with it.

The idea is simple, replace mathematical facts with objects that you need to remember in a 3-D room. The sounds t(1), n(2), m(3), r(4), l(5), ch(6), ka(7), f(8), p(9) are memorized in simple ways. An advantage of an object between two number is that it both 3x7 = 7x3 are the same. Iya one of the children who had a lot of difficulty with learning the tables announced that other than the 8 and 9 tables he knew everything. When I drew a pic of the 9 tables on his request he suddenly realized 9x3 = 3x9. Hey Anna, 9x3 is the same as 3x9. Then he noticed that all the other objects were also familiar, this makes the table so much easier he added. I had attempted to convey the same to him earlier and when I shared his Aha moment with his teacher she remarked that she had told him as much many times. I guess, it doesn't matter how many times someone tells me something, it only matters when I get it.

What we did with it. 
1: Game: Figure out as many of these combinations by watching the video. As it gets tedious, use the skip forward and back feature to make it a little interesting.
2: Exercise: Practice with what you know and find out what is missing.
3: Project: Make a mini project in Scratch that shows the objects as you type in the questions.
4: Practice, Practice, Practice
5: Project: Make a game that tests rigor by asking tables at random, build on it to time yourself. By now I have quite a few kids who want us to test them because they have gotten so good at it.

Some notes, 
- With some practice, the children who had given up on the tables in the past were able to remember tables.

- However, these are mathematical facts and it lacks a logical framework that children can work with when things go wrong. e.g. if the child remembers the wrong object it can be an absurd result.
- I'm still sticking with when all else fails (flash cards, logic of knowing 5 tables, knowing squares and adding and subtracting) perhaps give visualization a try.

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