What is the place of facts in mathematics?

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Friday Night Reflections

Last Friday evening I caught up with a few friends who I hadn't had a chance to spend any quality time with recently.

After we had solved the world's problems and analysed the youth of today, one of my friends came out with a very provocative statement:

"I think 'facts' are given too much importance in science," he said, or words to that effect. 

Now, let me say that the gathered assembly that totalled 4 people comprised science or maths educators (x4), well educated science graduates (x3), current or former president's of state professional associations (x2), and of course the mandatory partridge in a pear tree. You can use a Venn diagram to solve that one kids.

So this provocative statement about 'facts' soon generated some passionate discussion.

And three days later I am still thinking about it. 

And I think I agree - 'facts' ARE given too much importance:

• not because I am pushing some sort of anti-intellectual glorification of ignorance agenda

• not because of any Trump-esque view that 'facts' are subjective (although context and interpretation are always critical in understanding facts)

• not because I am advocating a free-form content-less maths curriculum

BUT (and here is my big BUT)

• because I believe that 'facts' - and this includes 'maths facts' - are just a starting point. 

What are these 'facts' of which you speak?

When I am talking about 'maths facts', these are a few of the things to which I am referring:

• addition and subtraction facts
• times tables facts
• formulae
• processes and operations
• arithmetic
• symbols and representations
• notation

These things are often presented as "being the mathematics". When you Google for an image of "mathematics" you get chalk boards (do people still use them??) covered in squiggly notation and complex formulae. These are the things that are considered by many to "be" mathematics.

I don't think they are.

Just as an umlaut isn't the essence of German language, so too a square root sign isn't mathematics.

I think we have lost sight of the purpose of mathematics - to make life better/simpler/more beautiful. This purpose has been obscured by......'facts'.

So what then IS the place of facts in mathematics?

Let me back pedal a little to explain my position:

• yes - it is useful to know the 'facts' - it is more efficient to know them rather than having to invent them each time I want to use them. 
• yes - I still teach them to the kids in my class (where 'teach' means explore, play, investigate, pull apart and reconstruct - as well as memorise)
• yes - a component of assessment needs to consider the mastery of 'facts' and 'skills'.

BUT (and here is my big BUT again)

1. These maths 'facts' are just the starting point. 

They are not the end in themselves. If my assessment is purely based on a student's ability to crunch numbers, recall formulae and perform calculations, then it is a pretty thin type of assessment.

2. 'Facts' need to be useful to make sense of a context. 

And the context is useful to make sense of the facts. 

Here is a fact - water boils at 100 degrees celsius.

Ah - except when...

3. There are bigger things than 'facts'

There are some 'big ideas' that are at the heart of education, things that we need to explore and consider on a daily basis. Things such as:

• connection
• equality
• consistency
• representation
• form
• function
• cause and effect
• (insert your favourite concepts here)

These are the things that I am hoping to get my 8 year olds to explore and dissect in the short time that they are with me. It is these things that I want them to know about. These are things that I want them to start analysing - using 'facts', but developing understanding.

Thanks to my PLN and the provocation last Friday night. Hopefully some of this makes sense - or at least provokes some thinking.

Symmetrical Patterns

One of the ideas the kids have been playing with in our pattern making sessions is "symmetry".

I want to show you a few examples of what they have come up with. This is pretty visual - I think the images speak volumes:

It might start quite simply...

...but it doesn't take long before we start to get more complex ideas.

And then we can head into the 3rd dimension.

And someone always has to go over the top...

But just making pretty patterns - do the kids really understand anything about symmetry? Have they learnt any real maths? Or are they just having fun?

So then, I asked them, what is symmetry?

And the kids told me:

- It is about balance.

- I look for the line of equal.

- I know if I put a bit here, I have to put the same bit on the other end.

Insightful 7 year olds. Love it. Pretty sure they have learnt something about maths today.

Patterns! Patterns! Patterns!

This year we have started each morning with patterns.

The kids come in and we have about 15 minutes of making patterns - sometimes I nominate a feature to explore, other times the kids get to make their own. Sometimes they work in groups, sometimes they work alone. 

It always starts off looking like this...

Yesterday after 15 minutes it looked like this...

Busy weren't they? 

Let's look a bit closer at what the kids produced...

A nice mix of materials. 

I do like the red and blue "zipper" pattern made from diamonds and trapezia.

More nice things including an interesting spiral arrangement. 

Love what can happen with a hexagon.

This looks like a screen shot from "Space Invaders" or Galaga"

So much happening here:

1,1 patterns

2,1 patterns

1, 1, 1 patterns

3, 1, 1 patterns

3, 1 patterns

This 15 minutes each day has become significant for my class for several reasons:

1. It is fun - it encourages the kids to get to school on time. They know if they turn up 10 minutes late they will miss something that they enjoy.

2. They are learning from the minute they walk in the door - at the start of the year, there were students who couldn't make a simple pattern AB repeat. Now they are making much more complicated patterns for themselves or building on a pattern started by someone else.

3. It is connected - when we make patterns, we are using language, we are learning social skills, we are developing mathematical understanding, we are building concepts that will apply across all subjects, such as shape, growth, prediction, expectation, observation, generalisation, perspective, form, and so much more. 

Have a go - see what your kids can do.

Polygons and Non-polygons

We are moving in to looking at 2D shapes. Our conversation started with a bit of a provocation.

My colleague and I drew up a chart on the white board and we had a stack of pictures of different shapes. Without saying anything, we started placing our shapes into either category - polygons or non-polygons.

We ended up with this:

The idea was to get the kids to think about what we had used as criteria for making decisions about placing items into categories. 

...and the ideas started coming. 


Here is our conversation and the ideas from the kids:

Question: So what is a "polygon"?

Piper – It’s probably a shape.

Sam – They’re shapes that don’t have any round bits on them. All of the lines are straight.

Sarah – They have straight sides.

Josiah – The oval is not a polygon because it has no straight sides.

Andrew – Polygons have straight and non polygons have curved sides.

Rupert – Some of them have four or five sides.

Liam – They have 3 or more sides.

Carys – They can’t have 2 sides; it would have to be curved.There are only two points on the straight line to get joined up so you need a curve.

Alexander – If a side is not joined up then it is not a polygon.

Andrew – The sides of a polygon don’t have to be the same size.
Molly – The sides can’t cross over.

So we distilled it down to one statement:

A polygon is a 2D closed shape with 3 or more straight sides that do not cross over each other.

Happy with that?

Well, then we started throwing out a few more examples and asked the kids to place them in the appropriate group.

One of our very perceptive students was given this picture and sat for a significant time pondering where to place it:

I was confused. Why was she hesitating?

I soon found out.

"Well," she said, "It has most of the things. It is a 2D shape, the lines all join up so it is a closed shape. They don't cross over. It has more than 2 sides. The only thing it doesn't have is straight sides. So it has 4 out 5 things for a polygon."

Nice thinking Abbey!

This raised the important point: being a polygon means you have to have all 5 criteria:
1. 2D shape
2. Closed - no gaps
3. Straight sides
4. 3 or more sides
5. No crossing over

If you fail one of these things, you are not a polygon.

So I went looking for examples that were "almost" polygons: 4 out of 5 of these things but only missing one of them.

Like this one:

straight sides - check

3 or more sides - check

closed shape - check

no crossing over - check

2D shape - uh-oh!

straight sides - check

no crossing over - check

closed shape - check

2D shape - nope

3 or more sides - nope

I wonder what other shapes we could find that are "almost" polygons? I think this question helps use to examine shapes using our definition and make appropriate decisions when classifying shapes. 

And the kids went off to search for new shapes to look at...

Creativity in Maths

Is it possible to re-think our approach to maths?

Here is a video put together with help from my son and one of his friends. We did this a while ago but the sentiment still holds true. Hope you like it - it's my first attempt at video…

The quality of the video is a bit dodgy but Blogger has limits on the size of files you can upload. If you want a better version, send me an email and we can work something out.

bruce.ferrington@radford.act.edu.au 


I have also now posted this video on Youtube:

https://www.youtube.com/watch?v=aULT8bU9-_o

Hope you enjoy it!

Number Patterns

I have been giving my kids a challenge each morning when they arrive at school - make a pattern. Sometimes I have told them what sort of pattern I would like them to make. Other times it has been free choice.

I put out some materials - blocks, dice, shapes, counters, beads etc - and watch what happens. I will show you some of these things in the next few days.

But today I put out some number cards. These are cards I made up with the digits 0-9 on them. I asked the kids to make a number pattern.

Whenever we do a task like this, there are a few key individuals who take a lead and other students tend to follow their interpretation of the problem. I am keen to discourage this. I would like to see all students thinking for themselves and not relying on other people to tell them how to interpret life.

Anyway, here are some samples of what we first came up with:

Some numbers with patterns in them...

Not quite was I was expecting. Even this creative version:

We had a conversation about this - stopped working and considered the nature of the task. (This is a common conversation - I often pause the activity after about 10 minutes and refer the students back to the original request - what do you need to do?)

Were these really number patterns - or were they numbers with patterns in them? Is there a difference? How do you define a "number pattern"?

This got a few brains recalibrated and so we got bak to work.

And here is what we got after another 10 minutes:

Ah-ha! Counting up by 100's.

And counting up by 10's.

And then I saw this one:

3 5 7 9

2 4 6 8

1 3 5 7

"Stop!" I said. "Don't go any further!"

I asked the class to look at the pattern.

What would the next number be?

I put it up on the whiteboard and asked students to add their thoughts during the day.

I also intervened/interferred and put the numbers in reverse order to that which their creator had placed them (because when she explained it to me, she started with the bottom row and moved upwards to the top row.)

Here is some of the solutions that the students gave:

So, what IS the next number in the pattern?

How would you explain what is going on? 

Are we adding by 1111? Or are we following an odd/even pattern between each row? Or are we doing consecutive numbers in the digits considered vertically?

Ask your class - they may have another explanation.

Good luck.

World Maths Day - Pattern Hunt

A few days ago I got a tweet from @prashanigamage wondering about ideas for World Maths Day. I suggested a "pattern hunt" - she suggested students write a book celebrating what they know about number.

I've put the two together - we have made a book about patterns we found in the school.

We started with our pattern hunt. Here are a few we found:

Here is an i-movie we made from photos we took in the playground:

We had lots of fun - so much fun that we ended up making a book about the patterns we found.

Here are a few excerpts from the book:

One of our climbing things has white, red and yellow balls on it.

We have some artwork in the playground.

This is a worm with consonant phonemes on it.

Our landscapers were creative with the placement of trees and seats.

Also practical, now that those trees have grown a bit and give us shade.

Our fencing has a regular number of bars on it.

Thanks again to @prashanigamage for the inspiration.

Our focus on patterns continues!