You are currently browsing the tag archive for the ‘math’ tag.

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 🙂

Education in Victoria is succeeding in some areas, but failing in many others – we have new school buildings, but they are overflowing and the school rolls climbing so quickly that teachers and principals have no bandwidth left for improving educational outcomes. Schools are adapting to technology, but failing to handle the wide range of ability and rates of learning our kids have. The system is not flexible enough to handle the needs of low achievers and high achievers in specific areas.

Every couple of months there is a new study showing how badly Australia is doing compared to other countries in areas such as Math. We know there are approaches that have worked elsewhere but we seem unwilling or unable to change and adopt them.

Homeschooling is an important right for Victorians – in many cases it is the only way to solve problems with bullying, with low achieving students and with high achieving students. Homeschooling is a rising demographic which serves as an important barometer of how well our schools are serving students and parents.

If the government understands this, then it will understand the value in Homeschooling, and will preserve that right as a legal option, and keep the current registration regulations intact. Homeschooling also serves the Dept of Education – it relieves pressure form a strained system, and gives a flexible way of educating students who are not well served by schools. It is part of the solution, not part of the problem.

It is so important that the Department of Education _listen_ to Homeschoolers, not try to tell them how to educate, or punish them – rather use it as important feedback. I was surprised to find many ex-teachers among Homeschool parents, and other parents had studied education theories in some depth. Homeschoolers as a rule are those who value education highly – they are pro-education, not anti-education.

I can say personally, it is heart rending to make the decision to take your child from school, you would only do it if there was a real problem to solve, or a clear benefit in doing so.

My son has just turned 13yo, he is by some accounts gifted – but the reality is simply that he was read to a lot from an early age, and had some opportunities for books and learning and discussion, and excelled because of normal healthy genes and a supportive environment. He has attended public schools in inner Melbourne for 3 years, the other years being Homeschooled – so I have some basis on which to compare the good and bad of each approach.

Early this year he was accepted into the SEAL program at a very good new school in inner Melbourne. The SEAL program is great for many kids because they immediately skip a year and jump ahead closer to their current level. I’m in favor of the SEAL program, its a good thing – but its not the complete answer. In my sons case he was repeating material he had done a couple years earlier in Math, so the homework was ‘busy work’.

I tried him out on year 10 questions and he worked through them well, so it seemed he was at that level. I asked the teacher if he could work ahead in Math, then asked the year coordinator and finally the deputy principal – and was surprised that this request was politely ignored in every case. At first I was angry, but then I realized that they probably just saw my request as “more work” for them, and they are already straining to keep up with massive expansion in student numbers. The roll is growing at a massive rate, and I think this is why they just don’t have bandwidth to gather a real focus on learning outcomes, let alone catering flexibly to students who fall outside the norm.

As an aside, there are ways to teach and learn math that are vastly better for all students than the approach we have in most Australian schools now. You don’t have to invent new methods, they are tried and work well overseas – you can read about Jo Boaler, ProofSchool, MathCircles, KhanAcademy, AoPS.com, Australian Math Competition etc. You can read any review of our current math texts by university mathematicians, or look at any comparison study with other countries to know we are doing it badly. The system needs to be flexible enough to accommodate and experiment with these new methods. Its not the curricula per-se, it is the way its communicated – it is not visual enough, it is too topic-centric and should be more problem-centric, it is not interactively explored.

Id like to see schools adopt these approaches – but right now they are too busy handling roll growth alone, and in moving from paper books to ipads.

This means the only solution, for now, is to Homeschool your child if they excel in Math – school is a hostile environment towards learning math deeply.

We need to change the way we think about Homeschooling – it is valuable for mainstream education in Australia, it is a place to see how new methods work and take the needed risks in new approaches to learning. It is a pressure valve for a school system experiencing the stress of rapid growth, and it is the only way to accommodate that small minority of students who will not excel at schools, no matter how good those schools become in future.

To this end I propose that the Victorian Government / Department of Education Victoria consider supporting Homeschooling in the following practical ways :

- Preserve the current lite-touch Homeschool registration regulations in Victoria [ realise that making regulations tighter will likely result in mass non-registration ]
- Fund a fulltime Homeschool liaison specialist educator in DETV [ to support homeschoolers, not police them ! ]
- Establish an open registry of public school events that Homeschoolers can join in with
- Fund several masters/phd opt-in studies on Homeschool education approaches and attainments
- Fund Math and Science specific programs for both schools and homeschoolers, eg: [ alternative curriculum materials, such as AoPS.com books, Math Circles and Robot workshops ]
- Establish a yearly tax deduction for extra costs associated with homeschooling your child [ taken from the money that homeschooling saves the government on schooling ]

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.

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 ?

Just added a color palette to GridMaths, so it now has rectangles, lines, ellipses in few simple colors and weights. The idea is to help make diagrams clearer, and help with Venn diagrams and other cases where you want to group things together.

Heres a grid sheet comparing fractions : 2/3 and 4/5 … then we show the product and sum using the same visual representation.

If you look carefully, you may notice that I ‘abuse’ the grid in that each grid square is 1/5 high and 1/6 wide – so a 1 unit x 1 unit square is actually 6×5 grid squares. I think this is a legal abuse, in that we often graph things with different x and y axis scales.

The alternative is to have a resizeable non-square grid.. which I think creates too much complexity for not much gain. The philosophy of GridMaths is “keep it simple”, so you can do the basics quickly and easily. I used to think more features = more power = better, but after my son showed me the Minecraft game, and all the cool things people have built with it, I changed my mind about this. I kept thinking.. why is Minecraft so popular, when you can do all that and more in Blender ? Blender is a superb free 3D modelling package, which like all powerful modelling packages takes a while to become proficient at. So, it dawned on me that Minecraft is so brilliant because it brings down the barrier to entry, and makes the 80% of things you need to do to make a world, really easy and quick for everyone.

So I’m adding the most useful things to GridMaths in a way that keeps it really simple. Its not an algebra system, it most likely wont have handwriting recognition… but it should be a really fast way for math teachers to make Math diagrams and for students to do a wide range of worked Math problems, replacing grid paper for that 80% of tasks and adding some nice features. Its kind of like an infinite supply of grid paper, that weights nothing and can be erased and replayed, and saved for later use And.. you can step forward and back thru your edits and change stuff, and students can step thru a worked problem. And you have counting beans and … :-]