Wednesday, 5 April 2017

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…



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 

Monday, 3 April 2017

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.






Friday, 3 March 2017

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:


video


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!










Saturday, 18 February 2017

There is no miraculous new way to teach maths.



or.. Some things I learnt in 2016.



During 2016 I was fortunate to work on a project called:

reSolve: Maths by Inquiry

Keep an eye out for it - there is some good work behind this project.


This project is lead by the Australian Association of Mathematics Teachers and the Australian Academy of Science. It is based at the Academy in Canberra.


The main building at the Academy of Science, sometimes known as the Martian Embassy.

I learnt a lot over the year, much of which I will share with you over the course of time. But here are some of the "big learnings" that came through to me:

1. There is no miraculous new way to teach maths 

There is no magic pill, no silver bullet, no "deus ex machina", no program we can import "drag and drop" style into our schools that will save our PISA results or make us Top 5 in TIMMS. 

There is only striving for our best - our best efforts, our best examples, our best pedagogy, our best teaching practice. This is quality education.

2. If I do not give all students full access to the curriculum, then I am perpetuating discrimination and inequity.

There is a temptation to give some students a "watered down" version of the curriculum, to give an over simplified or "dumbed down" education in mathematics. This is not fair. It is not just. It is not equitable. 

Yes - it is difficult to engage all students. Yes - it is difficult to provide access to the full maths curriculum to all students. Yes - it is difficult to pitch lessons at a level that will challenge all students in the class.

But that it our job. It is why we choose to front up to the class each day. It is why we do not work in tele-marketing or data entry. We want to teach and we want to do it well.

3. We need to hear the voice of the child.

This has been simmering away in the back of my mind for a while now. To effectively engage students in learning, we need to give them a voice. We need to listen to their ideas and suggestions, we need to reflect on their observations and we need to examine their strategies to find new and powerful insights.

The great man himself, Peter Sullivan, told me, "99 times out of 100, the teacher needs to shut up. The other 1 time, the teacher needs to shut up."

This is very difficult for many teachers - don't we just love to fill the space with the sound of our own voices? But to stop and listen to the voice of the child - ah! there is deep learning to be had here.


Anyway, that's a few of the things I learnt in 2016.

The challenge is for me to take them into the classroom and see if I can actually do them.

I'll let you know how we get on.






Friday, 17 February 2017

I'M BACK!


Australian readers may appreciate the reference to a cultural icon - Aunty Jack.

Just when you thought it was safe, I have returned from my year off (well, working hard at the Academy of Science, but more of that later) and am back in the classroom.

Look out Year 2 - it is going to be a year full of maths.

So, as every teacher does, I have set a few goals and resolutions for the year:

1. To focus on pattern making and spatial reasoning - get the kids to make, create and extend patterns each day as a "start the day with a bang" kind of activity.

2. Be patient - this is a personal goal. Be patient with the kids. Be patient with colleagues. Be patient the system. Be patient with myself. Lots of opportunity for growth here.

3. Nurture relationships for learning - get alongside the strugglers and hear the story from their perspective. Open doors to growth by listening and finding those "things that make you go WOW!"

4. Get back to the bagpipes.

I'll let you know how we progress with each of these over the year. 









Wednesday, 9 December 2015

We apologise for the temporary suspension of services



Dear reader,

I am pleased to announce that I have I recently accepted a 12 month appointment as a Resource Writer with the "Mathematics by Inquiry" project being run by the Australian Academy of Science and the Australian Association of Mathematics Teachers.

As a result I will be away from the classroom and will not be able to post my ideas and reflections to this blog.

Thank you for your support over the last three years. Please stay tuned for some exciting developments from the "Mathematics by Inquiry" team. And please continue accessing the ideas I have already posted here on the blog.

All the best,

Bruce Ferrington

Wednesday, 18 November 2015

Maths in Science - an interview with Don Knuth




AN APOLOGY

In March this year I published interviews with some very prominent scientists, asking them about their experience of maths education and how they used mathematics in their field of science.

At that time I received a reply from the legendary computer scientist and mathematician Professor Don Knuth, a reply which I overlooked and failed to open. 

Today I received a polite e-mail from him wondering what had happened to his responses and why I had not acknowledged his participation in the "Maths in Science" project. When I checked my in-box, it was still sitting there from March and I had not opened it or read it.

I am deeply embarrassed and publicly offer my apology to Professor Knuth for my oversight. I have also replied to his e-mail and offered sincere apologies to him directly.

So please, read below the responses to my 10 questions provided so generously by Professor Knuth, Professor Emeritus at Stanford University, "the father of the analysis of algorithms", creator of several computer programming systems, creator of METAFONT and the author of "The Art of Computer Programming" - the bible for computer programmers everywhere.

 ---------------------------

Here is the note to the e-mail that Professor Knuth sent me in March this year. His kind comments make me feel even worse that I failed to open his e-mail:

Hi Bruce,

Ten answers are below!

One of the most important things for students to learn is how to ask good questions. You evidently have learned that well.

Best wishes, Don Knuth

---------------------------

1. Describe what maths lessons were like for you at school.

People of my generation (in Wisconsin, USA) learned multiplication tables in grade 2, fractions in grade 5, algebra in grade 9, two-dimensional geometry in grade 10, complex arithmetic in grade 11, three-dimensional geometry in grade 12. I came up with lots of questions that my teachers couldn't answer; so I spent most of my time thinking about other subjects (English, Latin, physics, chemistry, biology, music). But at home my father had a mechanical adding/multiplying machine, and I enjoyed playing with that. I spent hundreds of hours plotting the graphs of functions like 

$\sqrt{x+a} - \sqrt{x+b}$ for different values of $a$ and $b$, using colored pencils so that I could put several graphs on the same page.

2. Was the maths that you learned at school useful to you later in life?

Absolutely; I can't think anything from those classes that I have NOT used repeatedly! For example, the geometry classes not only taught me how to prove things rigorously, they also gave me the ideas needed to create the METAFONT language, with which many fonts of type have been designed; those fonts are now used by millions of people all over the world.

3. How good do you need to be at mental arithmetic to do calculations in your head?

I'm glad that I memorized multiplication tables up to 12x12. But I think going any further (like up to 99x99) would have been a waste of time. Calculations in my head are important only on problems that are fairly simple, or on problems that involve symbols instead of numbers. When I'm working on a research problem I generally begin by filling dozens of sheets of scratch paper with partial calculations. When I eventually get to a point where I can think about the problem while swimming, then I'm often ready to solve it.

4. Mathematics teaches us that you can put two things together to make a new thing. Is this important in what you do?

Complicated structures are made up of simple structures that are combined in simple ways. I think computer scientists understand this even better than mathematicians do, because we've learned how to represent many kinds of data inside a machine.

5. Mathematics is about finding patterns. Do you need to look for patterns, or exceptions to patterns, in your research?

Yes, I like to think that mathematics is in fact the science of patterns. The patterns that I work with daily are usually some regularities in relationships between objects, not between numbers. But numerical patterns are important too: Like the facts that:
 

$1=1^2$, $1+3=2^2$, $1+3+5=3^2$, $1+3+5+7=4^2$, etc., 
and that
$1^3=1^2$, $1^3+2^3=(1+2)^2$, $1^3+2^3+3^3=(1+2+3)^2$, etc.

6. Mathematics also teaches us about balance and equality. Is this idea useful in your research?

In the METAFONT language referred to earlier, we express the shape of the letter A by giving equations that should be satisfied by key points in the lines being drawn. "The left stem runs from the baseline, half a unit from the left edge of the enclosing box, up to the cap-height. Its slope equals the negative of the slope of the right stem." 

[Reference: Computer Modern Typefaces, page 369.]

7. Mathematics helps us to represent quantities and measurements numerically. Do you do this in your work?

In fact my program that draws the Greek letter $\pi$ actually uses the number 3.14159 in two places. [Computer Modern Typefaces, page 159.]

8. Is estimation good enough or do you need to measure things accurately?

A computer scientist must be especially careful, because tiny errors can easily be magnified --- with catastrophic consequences.

9. How do you use statistics to analyze your results?

Much of my work involves comparing different computer methods, to see which one is fastest. Basic statistics such as the maximum, mean, and median running time, together with the variance, are crucial in this analysis. More broadly, concepts of random numbers and probability are absolutely essential ingredients in most of the best computer methods known today.

10. Do you have any other insights to offer into how you use maths in your work?

For instance, when I brush my teeth I've got eight areas to cover, namely Left and right, upper and lower, inside and outside. It's most efficient to follow a "Hamiltonian path" or "Gray code":
  left upper outside
  right upper outside
  right upper inside
  left upper inside
  left lower inside
  right lower inside
  right lower outside
  left lower outside


__________________________________________


 Thank you so much Professor Knuth for answering these questions for me and for being a part of the "Maths in Science" project. You have given me, and hopefully many others, lots to reflect on, including great advice on dental hygiene.

And once again, I apologise for my error in not including your thoughts in the initial project back in March.