Marty Ross wrote an opinion piece in yesterday’s Education Age : How Maths Became the Sum of Many Failings.
In this article, he rails against:
- Text books
- The training of teachers of mathematics
- The mathematics curriculum
I would like to respond to a number of points made in this article.
I have stated elsewhere that textbooks should not define the mathematics curriculum of a school. They are just one resource available to teachers and students. If the text IS the syllabus then this is, indeed, a problem in my opinion. Too many mathematics syllabi are still comprised of a list of content and the exercise(s) from the text that address this content. It is my belief that teachers determine what concepts, ideas and knowledge are important for students to learn (albeit within the constraints of either statewide or, to come, national standards) then formulate the assessment that will determine if students have learnt these things, then determine a set of learning activities that specifically and deliberately targets these understandings. A text is a good source of questions that can be used to assess students’ formative understandings…but only if the questions chosen are those that address the desired understandings. This is the job of the teacher; a teacher who knows what they want their students to learn and understand, who knows what misunderstandings were evident in the classroom and who wishes to address those misconceptions in order to improve learning.
Of course, in order for a teacher of mathematics to do the above, they need to know their subject well and the associated pedagogy of teaching mathematics. This is where teacher training comes in – or ongoing professional learning that addresses pedagogical issues. Too many student teachers will repeat the behaviours of their own teachers and these behaviours may be less than the ideal. It is difficult. It is becoming increasingly harder to find school placements for student teachers. Student teachers rely on their supervising teachers to assess them so are unwilling to go against any advice offered. Consequently, student teachers may replicate their supervising teachers’ methodologies rather than try out something else associated with what they might have learnt in college/uni. If these methodologies are ones that aren’t conducive to a greater understanding of mathematics, the cycle goes on.
Mathematics is all about reason. It’s about ideas. I have blogged previously about what follows but it’s worth re-stating it here, I think.
There is a poem by Peter Hooper which is partly given below:
Poetry isn’t in my words
It’s in the direction I’m pointing
If you can’t understand that
And if you’re appalled at the journey
Stick to the guided tours
Perhaps teaching and learning are much like this – the journey is the real education; the content merely the vehicle by which we explore the landscape. In what direction do we point our students with the what, the why and the how of teaching? From W.S. Anglin: ”Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness where the explorers often get lost”
Do we sometimes forget about the journey in our desire to get to the destination?
The teaching of ideas should be the journey in the teaching of mathematics. These ideas are taught through the vehicle of content topics that develop mathematical thinking and behaviours (acting like a mathematician) as well as a set of mathematical skills.
Mathematical thinking is about looking for patterns, testing these for the circumstances under which they prevail (proof) and abstracting them in order to generalise and predict. In order to inculcate such thinking in our students, teachers should strive to build on connections, coherence, creation, collaboration and conversations. Understanding mathematics is the destination. Assessment then addresses these understandings deliberately.
Teaching for understanding involves thinking – deep, sustained thinking that leads to students being able to construct an abstraction in their minds that actually makes sense of many distinct pieces of knowledge.
Instead of asking the question “What topics do we need to cover?”,we need to aim to develop processes and approaches which result from asking “What are the powerful ideas and processes that we believe are important for young people to learn in mathematics ?”
Our goal is then to develop syllabi around desired understandings – the big ideas – then ask “What information and what experiences do we engage the students in, in order for them to develop these understandings?”
The focus of assessment is in terms of the further question “If students understand this idea, how can they demonstrate it?”
The task of the teacher is not to put knowledge where it does not exist, but rather lead the mind’s eye so that it might see for itself. Our goal should be to provide learning opportunities within the curriculum that develop the students’ capacity to construct their own understandings and deliberately plan lessons that offer, support and develop rich and authentic thinking.
Research has shown that students who solve difficult problems on their own — without the help of other students or teachers — often gain a better understanding of mathematics concepts.
The learning comes with the struggle.
Let’s endeavour to give students opportunities to struggle through a problem, and refrain from directly telling them how to solve it. We can support and encourage risk-taking and the making of mistakes as a natural part of learning. We can emphasise understanding as the goal rather than just looking at results. We can encourage students to reflect on their learning and consider how they learn best in order to improve future learning. We can encourage students to be creative and open-minded in their thinking and consider multiple perspectives and develop alternative pathways to solution.
It’s a question of being more mindful about our purpose in the teaching of school mathematics.
This idea of purpose is an important one that should guide all teaching. Proof need not be the dry and largely esoteric horror people may recall from their own schooling. The capacity for absolute proof is unique to mathematics. It is important. But it can be as simple as asking “how do you know?”. In the same vein, the use of the calculator, including the CAS models, can assist mathematical understanding and free up teachers and students to focus on big ideas and deep thinking, instead of spending large amounts of time doing calculations. It all comes back to purpose. What is the purpose of the learning activity? If it is to do arithmetic and develop quick thinking in numeracy, then a calculator isn’t the appropriate tool. If, on the other hand, the purpose is to investigate continuity and differentiability of functions, then a calculator can enable these concepts to be ‘seen’ and understood a lot better without the ‘distraction’ of numerous calculations that have the effect of providing intellectual ‘white noise’ and distract the learner from the concepts.
I certainly agree with Marty’s final paragraph: if mathematics curricula are not written in the right ‘spirit’ that reflect the heart and soul of the learning enterprise we call mathematics, then students will not enjoy mathematics nor will they understand it.