The Password of Mathematics is "Pattern"

We've done a lot of inquiry-based activities in maths over the last year. It has been fascinating to watch the kids (and their teacher) develop their understanding of mathematical thinking.

But as I reflected back on what we have been doing, I started to wonder if my inquiry maths journey was more than just a series of fun activities. Shouldn't inquiry go a bit deeper?

Well, I had made an effort to link our maths to what our unit of inquiry was that we were investigating. We are a PYP school (International Baccalaureate Primary Years Programme) and we have 6 units of inquiry each year. Making authentic links is central to true transdisciplinary learning. 

But is this enough?

And what about those left-over bits of content that don't fit authentically into any of the 6 units of inquiry? You still have to teach them. We often refer to them as the "stand alone" bits of the curriculum.

Or for those who don't teach PYP, how would you approach teaching maths as an inquiry? Where would you start? Where would you go?

Numbers Patterns - A Reflection

Last week, we launched into an inquiry into number patterns. It was something we had to "cover" (don't you hate that expression - sounds like you're making a bed or tiling a bathroom floor rather than inspiring children to learn...).

Here's where we went on our journey.

Tuning In

We started with the provocation - "The Password of Mathematics is Pattern" - taken from one of my favourite books: "The I Hate Mathematics Book" by Marilyn Burns.

What did this statement mean? Does it apply for ALL mathematics? And what type of patterns are we talking about? And what is a pattern?

This was a very interesting discussion.

We followed up with the kids creating 10 of their own patterns. They shared their patterns with a partner. The partner was asked to add the next number to the pattern and explain what was happening.

Here's a few patterns they came up with. See if you can find the next number:

1, 2, 4, 8, 16, .....

8, 20, 32, 44, .....

2, 6, 7, 21, 22, 66, 67, ......

100, 50, 25, 12.5, 6.25, ......

50 000, 5 000, 500, 50, 5, .....


The debrief was interesting. Here's a few of the comments:


"Can this have 2 answers?"

"I know these two are the same but I don't know what they're called."

"Is it a pattern if you add and multiply?"

"It's a pattern if the change is regular."

"What does it look like?"

We also found one of my all-time favourite movies on Youtube:

Donald in Mathmagic Land - a 1959 Donald Duck classic. I remember watching this (several times) when I was in high school on a 16mm movie projector. 

Finding Out

Next step was to find out a bit more about patterns. Some independent research uncovered several different types of patterns. We found out about:

• arithmetic sequences
• geometric sequences
• triangular numbers
• square numbers
• cubed numbers
• Fibonacci sequence

Students were able to share what they found out about each type of number pattern and discussed how the patterns looked. Mathisfun.com came in handy here.

Sorting Out

Now we knew a bit about different sorts of patterns, the next step was to see if we could sort out a few examples. I provided about 20 examples of different patterns, some written numerically, some as pictures or diagrams. 

Using our 6 groups of patterns that we had identified, we proceeded to spend the next hour arguing about what went where and why. Some were easy - others less so.

Here is how it looked:

It's a bit hard to see properly but there were 22 patterns that the kids had to place in one of 6 groups.

There's a diamond-shaped pattern that is in the "Geometric Sequence" group that proved difficult.

Going Further

There are some great clips on Youtube that we had a look at, to see what other people had learnt about patterns.

Two favourites were:

Amazing Number Patterns - this we great - we watched it a few times and paused the video frequently to discuss the patterns that were presented

Nature by Numbers - an awesome visual exploration of patterns by Etereae Studios 

Then we grabbed the i-pads and headed outside to photograph any patterns that we found. Finally, we imported the photos into the Comicbook app and annotated our comic strip. This provided an opportunity for the kids to demonstrate their understanding of patterns.

Here's some examples of these:

Reflection

This inquiry cycle was conducted over 7 days. After it was concluded, I had a few questions for myself:

• Did we actually go any further than just doing a series of fun activities?
• Did the final activity actually provide the opportunity to show an understanding of a diversity of number patterns?
• If "patterns" are the "password of mathematics" then shouldn't this inquiry run for a bit longer than 7 days? In fact, shouldn't it underline everything we do throughout the year?
• And why is it being taught outside the rest of the inquiries that we are pursuing in the classroom? 
• And where were the kids in the process? Had I pushed them in directions that I chose for them rather than letting them have a voice in the process?

Ahhh....teaching is learning!

Solution Fluency with Lee Crockett

I am fortunate to be spending the day in a workshop in Canberra with Lee Crockett, global educational leader and author of "Literacy Is Not Enough".

Davey Crovckett - no relation

His model of 21st Century Fluencies considers the following 5 fluencies:

• Solution Fluency
• Creativity Fluency
• Collaboration Fluency
• Media Fluency
• Information Fluency

SO, we were considering Solution Fluency, which Lee has divided into 6 steps, or phases, in a cycle:

• Define
• Discover
• Dream
• Design
• Deliver
• Debrief

Then we were presented with the task - build the tallest tower you can using newspaper and tape. You've probably done this with kids before.

Immediately, we grabbed for the paper and tape, totally forgot about the 6 D's, and built a tower. 

I did try to suggest to the gathered 30 or so teachers that maybe we could all collaborate, to produce one big tower, rather than several small ones. Got a few laughs - and 3 people came and joined their group to ours.

Our group very cleverly headed up the stairs to get a height advantage,

defining "tallest" as meaning "height above sea level".

Here's out final tower - something went wrong at the top.

Here's another finished product - not a bad effort.

In the final "debrief", it was interesting to see how quickly a group of educated teachers had abandoned a framework that they had been presented only minutes before. Little effort had been made to define, discover or dream, some design had happened but most of the time was spent on the "deliver".

Interesting.

What does that mean for my classroom?

What Is My Area?

Wait - before you say anything - I know your body is 3-dimensional and area is a measurement of two dimensions - so I probably should say, "What is the area of my silhouette?"

So...

What is the area of my silhouette?

In the past week we have measured height and width, talked about Vitruvian Man, combined our heights, compared them with other classes.

Time to have a look at comparing areas.

My chief interest here is to look at how the kids are going to approach this question. I want to see if they can come up with any short-cuts that will speed up the calculations, so they don't have to cover their entire body in 1cm grid paper and count out each square.

Look what they did!

First job was to draw an outline of the body. It was important to get this accurate - what do you think of this attempt?

"It's never going to be accurate!" said one student. "Even with the best artist in the world doing the outline, you couldn't make it 100% because the arm is round not flat."

                            

Once the outline was done, most of the kids got busy covering their shape with 100 squares, though not all of them recognised that they had 100 squares in them or that they were equal to 100 square cms. But we got there in the end.

We had a good discussion at the end about accuracy (again) and about those missing gaps - would that make your "estimate" under or over the actual area?

This one was interesting. I had been harping on about finding out ways to make it easier than counting up each individual square cm. The picture isn't great but you might be able to see that they have divided their body in half, deciding that the body is basically symmetrical and you only need to do half of it, then double your result.

This one shows that the student has calculated each individual finger in an attempt to produce an accurate answer. When questioned about their methodology, for example what you do about the bit not covered by the 10 block, there was some talk of fractions and halves and how they were added in to the calculation. I wasn't entirely convinced but there was some thought put into improving the level of accuracy.

Realising that the 100 square was too big for the arms, this group has opted to use 10 blocks and has drawn them in individually - a much more accurate method.

 And this was a fascinating example but again the photo is disappointing. This group decided to measure around the outline with a piece of string. The string was then reorganised into a rectangle and this was used to calculate an area. Even though the resulting rectangle was obviously several times larger than the body shape, which neatly inside it, they maintained that their measurement was accurate because they had been very careful to get the string the exact length.

So how big is a body?

Well, we got measures of between 4500 and 5500 square centimetres. There were some closer to 6000 square cms and even one of 8500 square cms - not sure where they went with this one.

And then much discussion followed about the inaccuracy of this method and how much better it would be if someone invented a machine where you could lie down and the machine could go over you and measure you in a three-dimensional way. This would be so much more accurate.

Had to break it too them - they already exist.

Top Math Movies 2012

So it's Oscars time.

And everyone is getting excited about who is going to win.

Well, I'd like to introduce a new award - 

Best Math Movie of the Year  

And here they are, in descending order:

5. Clint struggling with basic geometry.

4. Yep - it's the number after 39.

3. A basic numerical operation - the reciprocal of multiplying

2. Dividing by zero does it every time.

Look - I get an "E" on my calculator!

1. This year's winner of the Oscar for 

"Best Math Movie of the Year 2012"

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And while you're thinking about movies, check out this blog:

http://spikedmath.com/401.html

It's a math movie pictogram quiz!

Da Vinci and the Vitruvian Man

You've seen the picture before, on posters, t-shirts, in books and movies. It's pretty famous. I even used it last week in this blog!

It's known as the "Vitruvian Man." Da Vinci did the drawing and made some notes based on the ideas of Vitruvius, who may sound like a famous volcano near Naples but who was in fact a famous architect in Rome. He had some ideas about the proportions of the ideal man. Da Vinci was able to use these to draw this diagram inscribed inside a square and a circle.

According to Wikipedia, my first port of call when I can't read Leonardo's unintelligible scrawl, the notes by Da Vinci around the picture are a summary of the key ratios of the perfectly proportioned human body:

• a palm is four fingers
• a foot is four palms
• a cubit is six palms
• four cubits make a man
• a pace is four cubits
• a man is 24 palms
• the length of the outspread arms is equal to the height of a man
• from the hairline to the bottom of the chin is one-tenth of the height of a man
• from below the chin to the top of the head is one-eighth of the height of a man
• from above the chest to the top of the head is one-sixth of the height of a man
• from above the chest to the hairline is one-seventh of the height of a man.
• the maximum width of the shoulders is a quarter of the height of a man.
• from the breasts to the top of the head is a quarter of the height of a man.
• the distance from the elbow to the tip of the hand is a quarter of the height of a man.
• the distance from the elbow to the armpit is one-eighth of the height of a man.
• the length of the hand is one-tenth of the height of a man.
• the root of the penis is at half the height of a man.
• the foot is one-seventh of the height of a man.
• from below the foot to below the knee is a quarter of the height of a man.
• from below the knee to the root of the penis is a quarter of the height of a man.
• the distances from the below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one-third of the face.

   These measurements are amazing. I could share most of them with my class at school but there might be a few giggles about using the penis as a point of reference so I might hold off on that one.

   Anyway, a useful insight into the mind of a genius. Maybe if I worked backwards, I could get the kids to discover some of these relationships for themselves....

Kindergarten Number Patterns

Working with our buddies

Every second Friday we get together with our kindergarten buddies. It is a great chance for us to socialise with the younger kids in the school and to develop a bit of leadership.

And it is fun.

Finding patterns

In her book "The I Hate Mathematics Book", Marilyn Burns revealed a great secret and mystery...

"The password of mathematics is pattern." (pg 6)

And it's true, isn't it? If you can find a pattern, you are half way to finding a solution.

Today, the kindergarteners were finishing off some inquiry into patterns. They had used a great variety of objects to build their patterns - plastic toys, feathers, hats, blocks. 

    

We were able to sit with them and help them record their learning, writing down their patterns and having them explain what was being repeated.

It would be really cool to get the kindy buddies outside and hunt for some patterns out there in the big wide world. I wonder what they would find?

Four young men stepping out into the world to discover some mathematics.

Any patterns out there?

How Wide is Your Class?

In Year 6 we are currently inquiring into "Who We Are", a unit of inquiry that is giving us lots of room to work with measurement.

Last week we explored "How tall is your class?" Today we decided to think laterally - and see how wide our class is.

What did da Vinci say?

Seen this picture before? It is known as the "Vitruvian Man", drawn by da Vinci based on the work of Vitruvius, a Roman architect. (This picture is fascinating - a quick google gave me enough data to write a separate post - stay tuned!)

Anyway, one of the points of this diagram is to show that the height of a correctly proportioned man is equal to the span of his arms.

Is that true? And would it be true of everyone in our class? And would it be true of the other Year 6 classes?

We already knew how tall we were from our previous investigation.

Time to get inquiring...

So how wide are we?

Using a similar method to last time, we first measured each other individually and added together our measurements.

Then we went outside and lay down on the oval again - I know it looks like the kids are just lying there in the sun but really they are flat out doing some hard maths.

Interestingly we found that the distance from finger tip to finger tip was less than when we lay end to end.

Year 6 working hard again

Total height = 3714cm

Total width = 3686cm

Not a big difference but enough to ask a few questions.

Why are the two numbers different?

Was da Vinci wrong?

Were we wrong/inaccurate?

How would we test our accuracy?

Is there a reasonable margin for error?

Will it change over time?

So many questions!

Where will we go next?