What makes for a mathematical culture in the mathematics

classroom?

MAWA, Bunbury, 2004

Anthony HarradineDirector

Noel Baker Centre for School Mathematics

•A teacher (of Mathematics among other things) for 20 years.

•Presently Director of The Baker Centre.

•My base is Prince Alfred College.

•‘My words’ come from my work and the work of my colleagues at the ‘spirit pen’ face, lots of reading, thinking and much PD work with teachers. They may or may not resonant with you.

My Background.

•I agreed and then found out I was following a talk on SEX!

•I said to my wife - “I’ll pull out.”

•She said “You can match her show.”

The shock of following Clio!

Her suggestion.

•Show people some images and see how they react.

•Try to ensure they are warming images to start with though.

•I first tried this in NZ.

Culture-how can you gauge it?Some expert advice.

I used this image ….

and ….

Feel free to react as your inner feelings drive you.

Now it is your turn.

Eddie Everywhere

‘Eddiocentric’ view of Australia.

Next image …..

Next image …..

If we are to define in some way what is meant by ‘a mathematical culture’, then we must first define ‘culture’.

So, some views ….

Some views on culture

•Matthew Arnold - mid 1800’s

•‘High culture’ -

“contact with the best which has been thought and said in the world”

(Arnold, 1869)

•Culture involved characteristics such as: beauty, intelligence and perfection.

•He said culture has its origins “in the love of perfection”

Arnoldian view of culture.

•Raymond Williams - mid 1900’s

•‘Ordinary culture’, his work moved the thinking of culture from ‘High Culture’ to the view that culture was the

“lived experience of the everyday”.

(Gray and McGuigan, 1958)

William’s view of culture.

•He suggested that every society has it own:

•shape

•purpose and

•meaning.

William’s view of culture (cont.)

“Every human society expresses these, in institutions, and in arts and learning.”

William’s view of culture (cont.)

“The making of society is the finding of common meanings and directions, and its growth is an active debate and amendment under pressure of experience, contact and discovery, ….”

William’s view of culture (cont.)

“…it (society) is made and remade in every individual mind.”

“The making of a mind is first the slow learning of shapes, purposes, and meanings, so that work, observation and communication are possible.”

“Then, second, but equal in importance, is the testing of these in experience, the making of new observations, comparisons and meaning.”

William’s view of culture (cont.)

• Culture has two aspects:

1. “The known meanings and directions, which its members are trained to ….”

2. “the new observations and meanings, which are offered and tested.”

• “These are the ordinary processes of …and the human mind”.

• The learning and the creative effort.

William’s view of culture (cont.)

“ …man is an animal suspended in webs of significance he himself has spun, I take culture to be those webs, …

Clifford Gertz (1973)

Gertz’s image

Is the culture of ‘mathematics’ better described as‘High Culture’ or

‘Ordinary Culture’?

•What did the PISA and TIMSS have to say?

[Recall Steve Thornton’s fine presentation last year.]

Is the culture of school mathematics, in general,

better described as‘High Culture’ or

‘Ordinary Culture’?

Where did the ‘ordinary culture’ go?

Work of mathematicians

Polished publications of mathematicians

Text-books

Students

What does the curriculum intend?

Why is it so hard to achieve?

Are we caught between two cultures?

Is assessment really the sticking point?

How many students see mathematics as a ‘passing

phase’ that will be over when school is over?

What about the classrooms of other subject areas?, eg.

English?

Has the culture of English classrooms changed? Why?

What do our students feel?

These were some of the questions that I asked when

attempting to make a plan for the on going development of ME and two faculties I have administered.

I did not like some of the answers I came up with.

Humour and culture.

“Gags are one of the great pillars of common culture, but they’re one of the first things to get lost in translation -- and if you can’t share a joke, it’s hard to have shared culture.”(Source unknown.)

•So jokes give us some insight into how people (often those from outside the culture) perceive a culture!

So, some humour from the society at large!

•A professor is someone who …………

And more …..

talks in someone else’s sleep.

A mad person walks into a room and goes up to a person and says in a loud and scary tone, “I will differentiate you I will integrate you”.

The person is unperturbed and replies,

“You don’t scare me – I am e^x.”

How do we help students to understand the culture of

mathematics?

1. Understand it ourselves - climb the ‘webs of significance’.

2. Reflect it in our behaviour.

3. Provide them with opportunities to, in William’s language,

‘learn and be creative’.

Understanding it ourselves.• I am sure I did not understand it, but am

beginning to now.

• Access to good ‘mathematical’ material that works you mathematically.

• Desire to become meta-cognitive.

See the hard hitting article:

Corwin, Rebecca B. Doing Mathematics Together: Creating a Mathematical Culture. Arithmetic Teacher, NCTM, 1993. (Available on the WWW).

Understanding it ourselves - continued.

• Enter ‘nrich’ - one of many, but one of the best. For students and teachers.

www.nrich.maths.org.uk

•Monthly:

•Problems•Articles•Games

•Investigations

•Huge archive of the same dating back to November 1996.

•Variety of levels of content and challenge, hints provided sometimes and students solutions published.

The nrich summary.

•To laugh, cry, smash things.

•To fail miserably time and time again.

•To be intimidated.

•To learn so much about thinking and learning processes.

•For it to take a long time.

•To be infected and changed forever.

•To measure your success by how you rise after having fallen - it is a part of the culture.

Be prepared:

€

12 ×13 ×14 ×15 ×16 ×17 ×18 × ........ × 99

• How many zeros are on the end of the number:

• How many zeros are on the end of these numbers:

€

12 ×13 ×14 ×15 ×16 ×17 ×18 × ........ ×19

€

12 ×13 ×14 ×15 ×16 ×17 ×18 × ........ ×19 × 20

€

n − 5n + 3

Consider all the possible values of:

How many of these values are integers?

How to start?

• Take any two multiples of three and add them, is the sum divisible by three? Why?

• Take any two multiples of three and subtract them, is the difference divisible by three? Why?

3

€

n + 3 must divide n − 5

⇒ n+ 3 must divide n + 3 −8

⇒ n+ 3 must divide n + 3 and−8

€

⇒ n + 3 = −8,1,8,−1,−2, 4,2,-4

⇒ n = −11,−2,5,−4,−5, 1,-1,-7

•Bridging questions are important. They are the ones we ask ourselves.

•They help students to understand the structures that are importance.

Bridging questions.

Are you a differentiable function? Because I'd like to be tangent to your curves!

I am equivalent to the Empty Set when you are not with me.

Pick up lines.

Combinatorists do itas many ways as they can.

Algebraists do itin groups.

Pure mathematicians do itrigorously.

Topologists do iton rubber sheets.

How they do ‘it’.

Dynamicists do itchaotically.

Statisticians do itrandomly.

or a little more specifically …..

How they do ‘it’.

•Cantor went on and on and on and on …..

•Fermat tried to do it in the margin.

•Galois did it the night before.

•Möbius always does it on the same side.

•Markov does it in chains.

•Turing did it but couldn't decide if he'd finished.

How they do ‘it’.

•Replacement units

•Contained experiences that required ‘web climbing’ by the teachers and students.

•They contain Stendusers!

•We have been at it for 8 years.

•We have been as consistent as we could be (without becoming insane).

Developing a faculty-wide approach.

•Lets fold some paper.

Stendusers.

Stendusers.

•Hard, but we were so far one way, we knew any movement would not be too much.

•We do not know what the balance is.

•We are still a slave to the high stakes exams, but not like we once were.

•The benefits are great.

Balancing the cultures.

•Effected how I see mathematics and the structures within it.

•This, in turn, effected how I saw learning can/should take place.

•I saw pathways (logical ones) that could seriously help student learning.

•I became more analytical.

Other effects of climbing the web.

Bottom up vs. top down design.

Novices…need a tool that is designed from their bottom-up perspective…and can develop in various ways into a full professional tool (not vice versa).

Rolf Biehler, 1995

•Are based on basic concepts but are amazingly powerful.

•Recall the ‘integer’ problem.

•Assist you to reason and most importantly focus on the reasoning.

Novice tools.

Let’s make a plot.

•What were the fundamental concepts we used?

•Separation and stacking.

•Let’s consider this a little more - enter Tinkerplots.

Can you make an argument?

Go to Tinkerplots

•Eng_Aust_90.tp

•mawa.tp

A ‘fundamental’ approach.

•Climbing the mathematical webs like we did not before (us and the kids).

•Building learning from the bottom up, rather than the top down.

•Focussing on the use of simple tools so the students can focus on reasoning and then moving to more ‘professional tools’.

Summary.