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I was discussing with other parents online, about the merits of various school Math multiplication approaches.

In the “Japanese Line method” you draw lines for each digit, and then count the intersections and add them up.

In the “Box-Method” you draw areas for each sub-product, and in “Longhand Multiplication” you just use the square grid to keep track in a table, without use of a diagram.

My main issue with the Line Method is that for problems like 97*98, you’ll be drawing a lot of lines .. its a lot of work even for 64×34 :

I also like the idea that Box Method starts with a physical / visual intuition – you can start children counting rows and columns, then progress to counting areas such as “How many 1ft square tiles do I need to tile the 3’x7′ kitchen floor ?”

Then you can discuss the distributive law a(b+c) and (a+b)(c+d) and explain the method behind longhand multiplication .. and then go on and show some handy tricks, such as using negative numbers to make less work, vis :

The above example, which does the same problem in 2 different ways, shows the kind of approach Jo Boaler recommends, called “Number Sense” – where students are encouraged to find their own way of doing a math problem. The idea is that there are many valid ways to do it, and struggling to find your own way is a good thing [ even if you fail ], and students tend to engage more, and certainly recall more when they have actively discussed with other students.

The same Box Method diagram can lead onward naturally to algebra and quadratics.. and even the beginnings of calculus 🙂

Just made a couple of videos explaining multiplication in a visual way, using the ‘Box Method’

A **MathCircle** is where a group of young people get together and work on some interesting Math-related topics introduced by a mentor.

The topics are sometimes problems like you find on ArtOfProblemSolving.com or Math Competitions [ vis AMC Sample Questions ]. They can cover quirky topics that aren’t normally seen in the school curriculum, such as Catalan Numbers.

The idea sounds geeky, and is un-apologetically so, yet has the potential to engage students who might be bored with the traditional material. It preps interested students for careers in science / finance / medicine / engineering and helps the school do well in math and science competitions. Math Circle meets can also be a lot of fun.

### Why

**Question** : “Why do we need more math.. we already do that in school, right?”

**Answer** : “yeah..but you don’t learn enough Tennis in PE class to become a club player, and you don’t get enough Instrument practice in Music class to get into music school .. you really need to train, to play in an ensemble and have dedicated practice for that.”

MathCircle takes the same approach of **intensive practice** which you find in Basketball practice, Dance club, Swim meet or musical instrument group … except focused on math-related skills.

### Practice with Duration + Intensity

One important aspect that both Math problem solving and programming share is the **sustained concentration** on one task – in my opinion we have gone too far in the direction of byte size chunks of learning. Sometimes you need to chew on things for a while.

We can look at how Basketball and Music are learnt, and apply what works to math. Training sessions often last for 2 hours, and this is accepted as a social norm – it is well understood it takes time to get into the zone.. repeat the basics, introduce a new skill, practice it, and integrate and perfect it over time thru a range of scenarios.

Its also not about the one or two elite players – the whole team improves and transfer skills around and offer peer-support thru the shared activity. It can help develop individual qualities that are useful for success in other areas of life, namely ‘character’.

### With Code

MathCircle is something most parents haven’t heard of .. but teaching young people how to **program** is an easy sell – it has had a lot of positive marketing over the last year or so, and its a clear pathway to a good salary.

I also think that writing small programs is a great way to introduce and discover math ideas .. its tactile, interactive, hands on, iterative, experimental. Your also working with the real concepts – a lot of educational apps and games seem to win in terms of engaging and entertaining, but lose in terms of conveying deeper ideas. When your making a program you are really tinkering under the hood with the engine, not just zooming around the racetrack. So a MathCircle with an emphasis on making your own programs to investigate math topics, and using tools like Geogebra, might work really well.

### Example Topics

A MathCircle which has a code-things-up emphasis, we could call a **MathCodeCircle**.

Here are some topics that might be covered in such a MathCodeCircle :

- find prime number factors, use for solving lcm/gcf problems
- pong game variant – balls bounce around and collide
- adding waves together – beats, square waves
- simulate jumpy stock prices, compare with compound interest
- planets orbiting / solar system simulator
- circle inversion using GeoGebra

Most of these would be developed in javascript, and run in the browser – using the canvas api to render 2D graphics – its real programming.

### More

Some links if your interested :

- What is a Math Circle
- Circle In a Box – how to build and run a Math Circle
- MathCircles – Washington University vid
- San Francisco MathCircle – vid
- Introduction to Programming – Khan Academy

I get the feeling a lot of people are surprised when they find out you can _understand_ math.. it isn’t just a series of random facts and some Rube Goldberg machinery.

I think this problem must start quite early in school, and get enforced so often it becomes a belief system. For example its rare that the distributive rule is related to areas of rectangles.. but that’s such a good visual explanation, it should be the default way of introducing the idea :

After a couple of these diagrams you can mention it works for any rectangle with sides of length a+b and c+d , namely : (a+b)(c+d)=ac+ad+bc+bd.

Maybe with tablet computers we can make Math more intuitive, when good visualizations can be seen by most students.

I wonder if teachers are so constrained to teach the points of the curriculum, handle admin tasks and control the class, that there is no time left for cultivating Math ‘understanding’ ? But surely its faster to learn / teach by understanding?

If you never get that little rush of endorphin from understanding, I could see how Math would be very boring and random.. because its not really math then, its something else ( and that’s not good, is it Precious? Not at all.. no it isn’t.. Precioussss… hmm… ghollum ! )

Some comments in response to a video TED talk that Conrad Wolfram gave called “Teaching Kids real math with computers”

I did enjoy Wolframs video, but I think its too tempting to take away computations.. in doing so we risk losing the math understanding behind them. I just think we need more of both : more understanding and more facility with actual practical problem solving.

Technology can help to explain math better. I just found this video today, which contains a superb intro to Bezier curves, among other things. I think this would be pretty engaging to high school students, be a motivation for them to learn calculus.

One of the things I wanted with GridMaths is to take some of the pressure off long working out but also have the student solve the problem for themselves. Its easier to line up numbers in long multiplication and you tend to make less typos. By design, GridMaths is _not_ a calculator, there are other apps that do that well.

I’m currently working on a hybrid approach where you can use ASCIIMathML to get good looking math expressions, which should open GridMaths up to wider use at high-school level. Maybe we can make worked problems less of a chore and more about understanding … but still keep the facility and practice with computations at a good level.

I can remember an awful lot of fiddling around with sharpening pencils and erasers at school.. push-pencils solved that, but I must have bought 15 different compasses :] Once a person can write legibly, maybe the emphasis should segway from paper to electronic tools, but still keep the ability to do the constructions, structure the essay, think critically, type well, organise computations etc.

I recently saw a report from one school where they moved to 1-to-1 tablets, where they saw savings of ~20k/month in stationary costs.. maybe exaggerated and an affluent school, but paper is kind of expensive at volume.

Its certainly a lot quicker to make a GeoGebra construction on a tablet, than to do it with ruler, compass, dividers, protractor and pencil … you can concentrate on the concepts and specifics, rather than paper-management. Then the diagram you construct is malleable, you can drag and interact… that’s a big innovation, and helps understanding.

I wonder what ways software can help take the load off teachers, so the mechanics are easier for them, and they can spend more time teaching face to face.

I’m working on adding nicer text and math symbol support in GridMaths, so formulas will look like they should.

LaTex is the ‘gold standard’ but I’m thinking of using a simpler alternative which gives most of the benefit, namely ASCIIMathML. I think its quite a nice format, something that covers most of the things a highschool Maths student will encounter, with a reasonably predictable and short syntax.

I’m experimenting with the best way for GridMath UI to enter math expressions, so it may be that you don’t normally see the text representation, but it may be exposed when you need to edit. I’ll use MathJax to render this, so it may be that you can edit and enter LaTex directly if you are a power user and know it well.

In my googling, I happened upon this video ahowing how to do long-multiplication of polynomials.. and thought I’d try the same problem using the Grid or Box method :

For some students it might bridge the gap and give them a helpful intermediate step so they see it as the same kind of thing as multiplying integers. I would also try out the box approach to introduce multiplying mixed numbers with fraction parts, and numbers with decimals, and so on – as it relates back to the earlier understanding of multiplication as area of a rectangle.

So GridMaths.com is in open beta, now works reasonably well on iPad and recent desktop browsers. Android browser support coming sometime soon.

Heres a pic my 9yo created while testing things out on the iPad.. with the obligatory battle scene [ dad vs browser quirks ? ] :

Here’s a couple of sheets on how I like to present long multiplication…

Firstly a concrete rectangle where you can actually count the squares to satisfy yourself its right…

Then move to a more compact form, which still shows the rectangle grid, but not to scale..

This shows that lots of digits should not induce panic.. the same systematic approach works [ which is why computers can do multiplication so well ].

I think having the box grid is a nice way to remember where all the pieces come from [ single digit products ]. Also I think it really helps to use the blank grid spaces, rather than fill in every 0.

The lattice method is slightly more compact, but I think this box approach reminds students of whats really going on.

Thanks for all the emails and encouragement so far as I build this.

Enjoy, and let me know how your using GridMaths.

My son is working on some LCM word problems on Khanacademy. LCM is short for “Least Common Multiple”, this KA vid explains it pretty well.

Basically the idea is to take two different measuring rods of two different lengths, start at the same place and keep measuring out lengths of each until the ends match up exactly : the first time this happens is at the LCM.

Easier to see than explain, especially if you get kids to experiment by putting rulers end to end, its kind of a nice little discovery, and the kind of open exploration that gives you the cool math-buzz when you discover something yourself.

I had fun doing a couple of these in GridMaths…

One nice thing is you can go step by step and see the rulers being added as they chase each other, with the ends not matching.. each team trying to win the race, each one getting ahead for a while..until the grand finale when… ahh, its a dead heat ! :]

I guess the potential with a software tool, rather than paper, is that you could potentially have Cuisenaire measuring rods of any length [ a set of the first 50 primes would be a nice grab bag of tricks ] … and you could get LCM for larger numbers by just laying down more grid and scrolling to the right. Its early days, so GridMaths doesn’t have this feature yet.. but I really like the fact that its open to doing things like this.

Prime Cuisenaire Rods anyone ?