I, along with many others, attended Professor Malcolm Swan’s session on teaching mathematics at the Melbourne Graduate School of Education last week, as part of the Dean’s Lecture Series. Malcolm is the Professor of Mathematics Education at the University of Nottingham and was one of the principal movers of the Shell Centre and its series of excellent resources for mathematics education. I am still using the materials from their publications Space and Number and Algebra and Graphing, which came out in 1985.
A copy of his PowerPoint slides and a downloadable audio stream are available at the website I’ve linked above.
He started by reviewing what we know about the transmission model of teaching mathematics:
*XXX teaching : Xplanation, Xample, Xercise (which I fear is still very much prevalent)
*Content/procedures are ‘covered’
*Learning is an individual activity based on listening to the teacher and imitating (templating)
*Teaching is linear and text-dominated
He then contrasted this list with the one for a collaborative learning/intellectual challenging mathematics classroom:
*network of ideas
*students are challenged through prompting-thinking questions that offer cognitive dissonance
*recognition of misunderstandings, making these explicit and learning from them (effective feedback)
Readers of this blog should also find THIS list familiar!!
He mentioned some great ideas for the thinking-prompt questions that could engender some great student discourse and also elicit some misconceptions for the teacher to work with:
For example: Hand out numbers on cards to groups of students (eg.square root of 2, four-fifths, cube root of 8, negative 7 etc) and ask students to write down everything you know about, or can find out about, this number
He spoke about the need for teachers to deliberately plan for students to move from a passive role in maths classes to being more active learners. What types of learning do we value in mathematics? Do the learning activities we use reflect these values?
Malcolm also talked about how fluency is essential for some things in maths (eg.multiplication facts, knowing how to solve an equation) and that these forms of knowledge and types of skill can be practised on one’s own. Interpretations of concepts and transferring concepts across multiple representations, however, need to be a product of discourse. This is an important point for what we set as homework.
There were a few interesting ‘compare and contrast’ lists he used in his presentation.
One of these was the tensions/conflicts list:
Content Coverage versus Reflection and creativity
Convergence on important Openness of investigations
theorems (learn this) (Explore this)
Illustrative Applications Real life problem solving
(Use a particular process) (Use any effective process)
Another list was the Effectiveness of Teaching Approaches:
More Effective Approaches
1.Offer challenge before help
2.Listen before intervening
3.When students can’t explain, don’t let them off the hook
4.Elicit interpretations and methods
5.Discuss ways of working
Less Effective Approaches
1.Offer help before challenge
2.Intervene before listening
3.When students can’t explain, explain for them
4.Elicit facts and answers
5.Tell students what to do
I think it is well worth a look at his PowerPoint slides.