The Mathematical Elephant - really?

Some email responses from friends, colleagues and people wiser and more experienced than myself have suggested I've perhaps overplayed the importance of working with the "whole mathematical elephant" - that there are even more critical aspects of maths education we need to come to grips with. 

Mary Barnes suggested that more important than showing the whole elephant is "the need to evoke curiosity, surprise, amazement (or amusement)".  How many of our maths lessons involve these elements? It is by creating an environment for students to experience these things that they may then want to see more of the elephant.

How's this for curiosity, surprise and amazement?
Mind you - I prefer my elephants in the wild - I think they are even more surprising there.
http://commons.wikimedia.org/

Mary also suggests another key element is working out how to encourage ongoing, continuous effort: "But learning/doing maths also requires effort, and to make effort worthwhile there needs to be a payoff. For most kids (and adults) the satisfaction of solving a problem or understanding a new idea is not sufficient payoff for the effort they have to put in to get there." For some time now, I've been a strong advocate of real-life based mathematics, especially in a science context (can't help myself) - but another part of me says we also shouldn't totally give up on trying to share the conceptual, abstract joys of mathematics with students.  With respect to effort, my "mathematics is an elephant" metaphor makes me think of 22-month gestation period of an elephant - although I don't think that's going to help my student very much.

  

Ultrasound of 3 month elephant foetus (Whipsnade Zoo, UK)
http://www.zsl.org/zsl-whipsnade-zoo/news/jumbo-ultrasound-at-zsl-whipsnade-zoo,765,NS.html

I would argue though that by not showing the wider, connected view of mathematics, and its connectedness the rest of the world , we make the task seem so much more arduous, arbitrary and without direction.

Several people also highlighted the perennial student question about the relevance of mathematics: "What's the point of this? What is the value? And can't computers do all this anyway?". The other day as I was typing a simple mathematical equation in Microsoft Word 2010, I was stunned to see the program proceed to auto-correct by inserting the answer! There's going to be repercussions in the classroom when students realise just how much even their text editor can do! I can't think of a mathematical elephant response to this challenge ... yet - except that elephants can be very sneaky.

And last but not least, is the challenge of needing to build up skills, layer upon layer, year after year. Unlike other subjects, we can't just do a topic and move on - if a student misses a part of the mental construction, their building will be shaky indeed.

So putting it all together - I'll concede my mathematical elephant metaphor isn't the most useful pedagogically - but just perhaps it might help us think a little about how we approach our subject - and the many complex, fascinating and interconnected aspects of maths, education and students. Next time you go to grasp a trunk or an ear, don't forget there is a whole elephant that might be worth revealing in the context of the problem.

A vision for an online maths resource

Here's a scenario I dream of:

I'm about to teach syllabus topic X.Y.Z I visit my favorite open community site of fellow maths teachers, lets call it MyMaths(*) and there I can find, searchable by syllabus topic number, and/or topic descriptors, a manageable list of current, tried and tested, relevant online maths education resources. These resources include: deep links to activities at external online maths websites,  brief descriptions of key teaching ideas, activities, some GeoGebra files and Excel files, and even some links to digests of related maths education research. The resource attributes also describe the level of difficulty, and any support for students with special needs.

Once I find resources on MyMaths which I think may be interesting for my class, I'll bookmark them, and later, after I've used them for my class, I might return to MyMaths and rate them based on their suitability and level of difficulty - perhaps add in comments. If I find the resource is out of date, or not as expected, I'll flag it with a thumbs down - or at least suggest it needs to be re-evaluated.

When I find other resources I've found useful in class, I can go to MyMaths, and add the resource in for evaluation by my peers. I'm also following MyMaths on twitter - so I can track when new resources are added, or their status changes.

And since it is an open community site, the resources contributed to the site will be Creative Commons licensed (the contributor can choose the specific CC license - most likely by-attribution-non-commercial). There is an XML export option allowing users to import some or part of the MyMaths database. MyMaths is open to all mathematics educators, across all school systems - with a focus on linking to the Australian Curriculum outcomes. MyMaths is open in this way because it will allow critical mass - as opposed to restricting to one school system or one state jurisdiction. It's possible that AAMT and the state mathematics organisations might wish to be associated or support the site however, MyMaths functions as an open source project owned by the whole mathematics teaching community.

Is this an original idea? I seriously doubt it - and I see many sites across the Cloud approaching this vision - but I haven't yet found one that puts all the key ideas together. It's a project way too big for one person - and it would take a core group of involved mathematics teachers, supported by a wider group of interested maths teachers contributing and interacting regularly

The time for teachers to be working individually, each maintaining their own list of resources is surely reaching the end of the line. By pooling our knowledge, resources and experience, we could potentially have access to an incredible resource - and much more efficiently offer up to date teaching ideas to our classes. I suspect the lack of an efficiently organised resource goes some way to explaining why so many mathematics teachers just give up on online resources - instead going back to a folder of photocopied worksheets which they in turn collected from the generation of mathematics teachers before them.

If we had even just one mathematics teacher in each Australian school connecting into the MyMaths community - imagine what could be built.

What do you think? Would you be interested in working with a group to make something like this vision possible?  Or is there an existing project I could join in? Are the parts of the vision that are unrealistic - or should be modified to make it possible?

(*) Psuedonym used - no relation to existing 'MyMaths' businesses.

Mathematics is like an elephant

I'm coming to the conclusion that one of the biggest challenges in high school mathematics (and probably university mathematics too) is coming to grips with the fact that in so many different ways, mathematics is very much like an elephant.

Mathematics is like an elephant? Well yes - if you think about the story of the blind men and the elephant - depending on what part of the animal you feel, you get a very different idea of what an elephant is.  There are so many different aspects and representations in mathematics, that it's all too easy for both teachers and students to be so focused on the particulars of the trunk, the tusks, the ears or the tail - and fail to see the whole elephant.

 Based on Sophie Woods (1916), World Stories for Children

Another way in which mathematics is like an elephant is it can be a little terrifying to come to grips with. I remember going with my little 3 year old brother to zoo and he screamed blue murder when he saw the elephant. For some students, the experience of that elephant is like this amazing 1888 Japanese print of the blind men and the elephant story- which is just too special to even consider vandalising with cartoon bubbles onto....

http://commons.wikimedia.org/wiki/File:Blind_monks_examining_an_elephant.jpg

And maybe sometimes that's why as teachers we wrap those blindfolds on our students (and ourselves) and just hand the class a trunk or an ear to be examined. But the risk is, in the end, our students wonder why they are doing repeated exercises, year after year, on all these separate, unconnected body parts.

To work mathematically, you need to smell the whole elephant, hear its roar - and take pleasure in its beauty, strength and also its surprising grace and subtlety.  And if we don't want to scare the children? Well, who can resist a baby elephant?

Source: Matt Stanford (flickr)

In coming posts, I'll be considering other elephant aspects of mathematics, and what the elephant looks like when it's distributed in the Cloud....

Current Obsessions - Summer Break 2010/2011

A monthly update on where my head is at right now

New interests:

My brand spanking new DET Teacher Laptop (Lenovo Edge11 - DET image: T3)
Early access to my Christmas present:  The 1973 BBC series The Ascent of Man

Exploring new (for me) software: Adobe Captivate, Microsoft OneNote

Standards Based Grading - exploring the beta for ActiveGrade

Twitter

New thoughts: (new posts brewing...)

Mathematics is like an elephant

Living, learning and working in the cloud

Active interests:

Technology for education: one-to-one laptops, edmodo,

Professional Learning Network: Yammer, twitter, edmodo

On the back burner for now: 

moodle

Mathematics: GeoGebra, CAS, Ability grouping, Primary/Secondary transition

Real Maths :  See Conrad Wolfram and Dan Meyer

Science: Space, Genetics, Climate change

Open Textbook, Creative Commons

The Australian Curriculum

The Research-Practice divide

Public education

Restorative Practice, The Circle of Courage

Sharing out the bread crusts on a spherical loaf

Do you enjoy a crusty loaf of bread? Do you ever find yourself needing to share that loaf with family or friends to make sure everyone gets a fair share of the crust? An innocent enough problem posed in Professor Stewart's Cabinet of Mathematical Curiosities asks: If you had a spherical loaf of bread, and sliced it horizontally in slices of equal thickness, which slice would have the most crust on it?

Perhaps counter-intuitively, it turns out all the slices have the same amount of crust (*).  Even more surprising to me was finding out that proving this geometric fact was one of Archimedes most prized discoveries. So much so, he had a diagram of a sphere enclosed in a cylinder inscribed on his tomb. Which got me wondering - how is it as a relatively well educated person, I'm only just now finding out about this in my late forties? And how did I miss out the fun of exploring this unexpected relationship?

André Karwath CC-BY-SA-2.5
via Wikimedia Commons

Thinking back to my school days, solid 2D and 3D geometry was kind of - blah - not that interesting - lots of formulae to memorise - most of which I forgot later anyway. How much more exciting it was to discover that my favorite algebra could be used, courtesy of Descartes, to wrap around  those geometric lines and circles and turn them into something I could work with analytically.  But looking again now at Archimedes' methods of approaching circles, spheres and cylinders - through a sort of calculus lens without calculus tools, I'm left wondering why such wonders were never explored (or even just demonstrated) when I was a child at school?  Is it because by the time we teach and learn this topic (Year 9 & 10) - we've forgotten the joy of exploring these shapes - and are just on a roll of working through fairly meaningless formulae?

I'm thinking now there are many lessons that could be built exploring the Archimedean relationships between spheres and the cyclinder - from very hands on practical explorations in Year 9, to ways to approach calculus at both introductory (limits, intervals and slopes) and more advanced levels (using integration of circles of revolution).  Teach solid geometry this way and it will be both good fun and permanently etched in young minds.

While I might have to rework the lesson hook into something about chocolate shells, for me this problem will always be about the fight for equality of crunchy crusts.

(*) An excellent and not too technical explanation of the bread crust problem is presented at Math Central

Education in Australia: High Quality, Low Equity

A must read for everyone concerned about education and equity: PISA identifies challenges for Australian education.


Two sections in the summary stand out for me (emphasis mine):

Twenty-eight per cent of students in the lowest socioeconomic quartile were not achieving Level 2 in mathematical literacy, compared to five per cent of students in the highest socioeconomic quartile. Only six per cent of students in the lowest socioeconomic quartile achieved Level 5 or above, compared with 29 per cent of students in the highest socioeconomic quartile.

The performance of students with higher levels of socioeconomic background in reading literacy was one full proficiency level above that of students from lower levels of socioeconomic background, or the equivalent of nearly three years of schooling.

The different in our school sectors is described:

The effect of aggregated high levels of socioeconomic background can be seen in Australia’s school system, in which we have many children of parents with high socioeconomic backgrounds pooled into the independent school sector and to a lesser extent, the Catholic sector. The advantage that these schools have in terms of this pooling of resources is demonstrated by the fact that, after adjusting for student and school socioeconomic background, there are no significant differences between the results of students in government schools and those in independent schools. Of course, we do not live in a world where such adjustments are made, and so more must be done to address the level of resourcing in schools that the majority of Australian students attend.

Does it have to be this way?  Well not if you live in Canada or Finland. While we could argue that Finland is a very different society than Australia, can we dismiss the comparison to Canada so quickly?