Friday, 25 September 2015

Why do we teach multiple ways of doing the same thing?

This morning I got a message from a colleague on Twitter.

@JasonGraham99 asked:

Why are there so many different math strategies that exist today that (didn't)? exist when I grew up? And more importantly why do we need the different strategies - a Devil's advocate question…sorry but I wonder why. Sometimes we teach 5 different ways to get to the same answer (lattice, partial products etc..) Why? - again Devil’s advoc) But I am serious. Does having multiple ways of learning math confuse learners? Or open minds?

This is our last day of Term 3 - but we work up to the last bell of the day so I am keen to give this some thought.

Or better still, get the kids to do the thinking.

Here is what my Grade 2 class had to say about why we learn to do the same thing in multiple ways:

So we can make things easier. Say everyone’s thought of one way but someone thinks of a new way.

Having different ways is what we need to do because in maths you need different techniques.

You’ll learn more.

You don’t want to just get the same answer everytime.

Because it’s really fun to have different ways. It can make it easier for little kids. 

So people get more knowledge.

If you don’t know one strategy you can go on to the next one. One maths question can help you with another one.

Pictures can be used to help you if you can’t read numbers.

There could be simpler ways so we need to keep looking for them.

And then I asked them if having multiple methods was confusing:

I think it would be really confusing if you had lots of different answers.

It gets confusing if the same question is asked in lots of different ways.

Yes and no. You know that having lots of answers is good because if you don't get one then you might get another.

It’s confusing when there are no pictures.

It’s confusing if the extra ways of doing it are harder or have more steps.

And finally I asked if they thought if multiple strategies actually helped open their minds:

Depends on how hard the question is.

Having more ideas of different ways of doing things can help you do other things.

Knowing how to do a little sum can help you in future learn how to do a big sum. A little mouse can one day help a big lion.

If one way is easy for you to do it opens your mind up and helps you understand other questions.

It’s helping your brain because your brain gets smarter.

It helps because you see those different ways because you can choose which one to do. If you look at a question and you’re stuck on it you could choose one of those ways.

I think it’s confusing because you don’t know which one is the best. One way could take too long.

I was impressed with what they had to say. It was certainly an interesting question for me to ask them.

I also have a few ideas of my own:

1. Golf

I always like to compare life to a game of golf. The more clubs I have in my bag, the better equipped I am for the many problems and hazards I will face out there. I might be pretty good with the driver off the tee, but the driver will be no use to me when I end up in the sand. The more clubs I have, the better chance I have.

I think the same is true for maths. If I know multiple strategies, I can select the most appropriate for the context. Do I want a quick method? - use mental calculation.
Do I want a detailed result to many decimal points? - use a calculator.
Do I want to show my thinking to someone else? - use a picture or diagram.

2. Why?

Many of us will remember being taught a method or process in mathematics, such as for multiplication or division, but few of us will have been shown WHY we do it that way. This teaches us to be "superstitious" mathematicians - we do it this way just because we always have. 

Learning multiple strategies for an operation without explaining the WHY and developing the connections simply compounds confusion for kids. I agree with Jason's question - it is confusing - if there is no explanation. 

But it is also liberating and "mind-opening" when students are able to make connections, to join up their learning and find relationships. And they can do this if we are able to help them understand a variety of different strategies and how they are related.

If a student understands the distributive law, they can appreciate a number of different methods of multiplication. Vertical algorithms, lattice method, arrays - all have a link to the distributive law. However, memorising a list of different strategies is not much help by itself -  it really does get back to knowing WHY we do something. 

3. Fun

Finally, it can be fun to know different ways to do the same thing. It can be boring doing the same thing all the time - variety of the spice of life.

So I would encourage teachers and students to play with maths, to have fun and find new ways to do things just because they can. 

Hopefully this makes some sense and answers a few of those questions from Jason.

And what a great way to finish the term!

Now I've got to get home and dust off the golf clubs...

Thursday, 27 August 2015

Patterns That Grow

As Marilyn Burns says, pattern is the password of mathematics.

I like playing with patterns. I use them to get the kids interested in finding relationships.

So I gave them this provocation:

Here's a pattern - 1, 5, 9, 13….

Based on this, can you tell me if 21 is going to be in this pattern? And then will 45 be in it as well?

So we got the blocks out and started making some patterns. Here is what the kids came up with:

"The pattern makes a cross shape. And 21 makes the same shape so it must be in the pattern."

"And 45 can make a really big cross too."

From talking as a group we were able to work out that yes, indeed, 45 is going to be in our pattern.

In fact one student pointed out that it was like the 4x pattern (4, 8, 12, 16…) but just one more.

It was a good starting point. The conversation was never going to end there.

What other patterns can you make? Can you explain your pattern using numbers? Can you tell me what the next number will be without making the model of it?

Here's a nice pattern - just like the cross but missing a leg.
CAn you see a link to the 3x pattern?

This one adds on a leg and goes 3D.
And now we were thinking about the 5x pattern.
Is Grade 2 too young to start talking about y = 5x + 1?

We had a good talk about this one. Is there a step missing somewhere? Should there be something between 1 and 8? Is 1 part of the pattern?

Another pattern based on squares - ah ha! Square numbers!

And then taking the square into the third dimension!

This is not the end of work with patterns for the year. We will revisit the concept many times but I feel that the kids have a good grounding now in identifying, making and explaining patterns.

And we had a lot of fun.

Wednesday, 26 August 2015

Messing With Their Heads

I know - it's not something they teach you in your training. I know it is abuse of trust. I know I shouldn't be advocating this.


I wanted to see what would happen when I showed my Grade 2 kids this on the white board:

I had just spent Saturday at the Canberra Maths Association conference where I presented a workshop on multiplication. I am interested in kids understanding what multiplication is all about - not just in memorising parrot-fashion their times tables (although memorisation is important, it sure has helped me in life, but it needs to be backed up with understanding.)


I put the "2x tables" as above in front of the kids.

And they were happy to start reading them out, chanting as they went.


About half way down, their confidence started to waiver and the mumblings grew.

By the time we got to the end, with me loudly proclaiming that 12 x 2 = 122, I had very few followers.

And some insolent 8 year olds began to suggest that I had made a few mistakes!


Anyway, I got them to help me correct my errors, which they did:

And order was restored to the universe.

Why did I do this, when I knew that many of my students were uncertain and lacked confidence in multiplication?

Why did I deliberately give them wrong information and incorrect answers that would confuse them?

Well, I was hoping that they would be able to use the knowledge they had to find the mistakes, correct them and re-establish the pattern that they knew underlies the 2x tables.

And they did.

And hopefully no-one went home and told their parents that Mr Ferrington is hopeless at maths and doesn't even know that 3 x 2 = 6.

Wednesday, 17 June 2015

Nets of 3D Objects

I don't think my kids have done much with nets before. It's important to be able to represent 3D objects in this way.

I was keen to see what they could do so we got out some equipment to have a play.

Here are a few of the 3D objects we looked at and some of the drawings the kids did:


Suggestion #1 - like a rectangle with a square off the side.

Suggestion #2 - Like a cross

Suggestion #3 - lots of square bits

Suggestion #4 - A cross made up of 6 smaller squares 

So we had the general idea that the net of a cube is made up of lots of squares - just not too sure about the details.


Suggestion #1 - it's a triangle with some stripes

Suggestion #2 - it's got more than 1 triangle

Suggestion #3 - think I've seen one before with a square and triangles coming off it

So we don't really have a very good idea about this one. Lots of opportunity to learn here.


Suggestion #1 - it's got 2 circle shapes on top of each other

Suggestion #2 - It's definitely got 2 circles somewhere

Suggestion #3 - It's got circles AND rectangles

Suggestion #4 - yep, I've seen this one before

As teachers, we learn so much from the "fails" of our students. The "correct" responses are fine but they don't in themselves give us much insight into the mathematical thinking of the students.

BUT the errors, the mistakes, the ones that aren't quite right - they are the ones that tell us so much about what our kids are thinking and how they "see" mathematics.