Teaching Mathematics? Make it count

This was the title of the annual ACER (Australian Council for Educational Research) Research Conference for 2010 which was held in Melbourne August 15-17.

I had the opportunity to attend 2 days of this conference. You can read more about the keynote speakers, program and download the papers presented here. 

To open the conference, Geoff Masters (CEO of ACER) spoke of the importance of high expectations; our expectations, students’ expectations, other subject teachers’ expectations, systemic expectations. He introduced me to the term ‘soft bigotry’ (which is apparently Julia Gillard’s) which is what results when we have lower expectations for sections of our student community based on their socio-economic status, gender or race. He believes that more emphasis needs to be made on the growth of individuals from different starting points and not always measured against an achievement standard for all at a particular school level. He spoke of limiting beliefs about mathematics that are held by many students and teachers eg. that only a small % of students can excel at mathematics. He mentioned that about 6% of Y8 Australian students were classified as ‘advanced’ in TIMSS, compared to 45% of Chinese-Taipei students and over 40% of Singaporean students. Our students aren’t any less capable than these other countries, so what’s making the difference?

Professor David Clarke then had the first keynote presentation. I liked the phrase ‘conceptions of accomplished practice’. He spoke of how what we consider to be accomplished practice is steeped in the culture to which we belong. As I have written before, it is widely recognised that increased classroom discourse and verbalising of reasoning, in which students use appropriate mathematical language and have this re-inforced through their interactions, positively affects outcomes in mathematics. Interestingly, it is mainly the Western countries that believe that this needs to be student-student discourse. Clarke’s data shows that a highly skilled teacher, who can guide the discussion from him/her to students and back to him/her can have equally effective outcomes. Choral responses (in which the class all responds) can also enhance the learning in positive ways (as long as it’s not just all students saying ‘yes’). In the classes observed in Shanghai, for example, no student is permitted to talk privately to another student. All utterances are ‘public’ (for the class). Only in the classes that were observed in Seoul, where students were not permitted to speak at all, was there a negative effect. During the presentation, it was mentioned that US classes in which there was a lot of collaborative group work, had the greatest effect on students using mathematical language correctly and linking it to previous learning during the post-lesson interviews to see how much students had learnt from the lesson. Clarke’s message was that effective mathematics instruction needs to be prescribed differently for each cultural setting.

Also part of this presentation was a discussion on the importance of language. Not only appropriate mathematical language, but the actual language of the culture. English, for example, is one of a few languages that has separate words for ‘teaching’ and ‘learning’. In other languages, the two are considered to be part of the same and there is only one word to describe these, acting together.  If an activity can be recognised, it can be named. What if the language you’re using has no name for it? Does that activity lose value? What activities constitute accomplished practice?

French speakers have ‘mise en commun’ to describe a particular teaching/learning activity.

Chinese speakers have ‘pudian’ (a deliberate, planned connection between students’ prior learning and what is to be explored in the current lesson). Japanese have ‘matome’ (see David’s paper)

I liked the three questions he encouraged us to ask of our own practice: Am I doing it? What does it look like? How could I do it better?

If we can promote a language of quality (for both technique of teaching/learning and mathematics) then we would be much better teachers.

The first session: Professor John Pegg: Promoting the acquisition of higher-order skills and understandings in primary and secondary mathematics

I have had the privilege of working with, and hearing, John Pegg previously at the one and only Summer School for teachers (see earlier posts). He was arguing for constructive alignment between assessment, pedagogy and syllabus content. I liked the phrase ‘cognitive architecture’. Working Memory(WM) is the ability to hold information in the mind whilst transforming it. We can make WM more efficient (but cannot increase its capacity) by bringing chains from long term memory to bear whilst considering higher-order reasoning. We can assist students to do this more effectively by enabling them to put ‘stuff’ into their long term memories. This, of course, is what the fluency proficiency strand is about in the Australian Curriculum K-10. Human intelligence comes from this stored knowledge, not from WM. Skilled performance consists of building complex schemas from the chains we store in long term memory. This means we need to improve the automaticity of students (another long-term bee in my bonnet) and orchestrate deliberate practice at what he calls the unistructural level of learning. Students need to develop their own connections in their own way to strengthen their neural pathways in ways that their brains can assimilate and connect up the new learning to prior learning. Teachers can also help students by creating an environment that makes relational thinking happen. What opportunities are there for this to happen? These need to be deliberately planned for. Non-routine problem solving is an excellent vehicle for this. ‘Reversible questions (what I call ‘backwards questions’) are also excellent ways to develop relational thinking. For example: what two fractions, with different denominators, will add to give you 2/5? What he doesn’t advocate, however, is ‘problem solving’ as a vehicle for developing concepts and skills.

Second Keynote: Phil Daro : Standards, what’s the difference? A view from inside the development of the Common Core Standards in the occasionally United States

Phil made the point that curriculum standards are meant to be ‘the same for all’. And to reduce unwanted differences. They can function as a platform for managing or mismanaging instruction. He spoke of the motivating and demoralising effects of standards. He also made the point that students who only do mathematics to get answers will have a poor attitude towards doing it to learn mathematics. Answers are part of the process but shouldn’t be the product of school mathematics. The product should be mathematical knowledge. Phil said that educators in the US got a wake-up call after the TIMSS video survey study in which it was obvious that US teachers were focused on answering the question: “How do I teach students to get the answer to this problem?” versus the Japanese teachers who seemed to be more focused on: “How can I use this problem to teach the mathematics of this unit?” He exhorted us to not use problems that can be done using an easier strategy to the one we are trying to introduce/exemplify. It must be an authentic problem to the concept or skill we want to engender. He also affirmed my belief that ‘nifty tricks’ that will help students get the answer are not to be taught as it just re-inforces the belief that it’s all about getting the answer as opposed to teaching mathematics. He mentioned the butterfly trick of adding two fractions together (see below). Nifty…always ‘works’ but what happens when students are asked to add three fractions together…or when they reach algebraic fractions? They have no conceptual understanding on which to build.

Butterfly fractions:  3/4 + 1/5     Write with horizontal vinculums first. Circle the 3 and the 5 – multiply these. Circle the 4 and the 1 – multiply these. Add these numbers together to form the numerator. The denominator is the two denominators multiplied.

Similarly, he believes FOIL should be banished as a technique (will this help students multiply a trinomial bracket by a binomial one?) and to always put in the 1s when cancelling, otherwise students think that 0 is left after cancelling. The message was to avoid short-term goals to do the maths today but short change the thinking for tomorrow. (Nice). He also recommended not giving a grade and feedback at the same time as students will focus on the grade and ignore the feedback.

Third Keynote: Kaye Stacey: Mathematics Teaching and Learning to Reach Beyond the Basics

Kaye has four wants for school mathematics:

  1. Not rules without reasons
  2. Deep and robust understanding
  3. Flexible problem solving
  4. Mathematics as a way of thinking

Working Mathematically has always been a goal for school mathematics, it’s just proved to be an elusive one. She and colleagues have analysed Y8 textbooks and classes, in particular. She has found a plethora of questions asked in texts that are close repetitions of similar problems, low procedural complexity, an absence of mathematical reasoning, a low % of real-life problems and that teachers and their students tend to shy away from making connections; that the focus was on getting an answer. She emphasised that it was important to present the same concept in different contexts to force students to make these links. When doing patterns in junior secondary classes, don’t just find a pattern from a table of values (for example when finding the rule that links the number of visible faces to the number of cubes when cubes are progressively joined in a line) but insist on asking for a reason why the rule ‘works’ in relation to the physicality of the situation.

Session with Merrilyn Goos – Using technology to support effective mathematics teaching and learning. What counts?

Favourite phrase: Educational affordances of technology in mathematics. Merrilyn made the point that the way technology is used in classes is closely aligned with teacher beliefs about mathematics. If you consider maths to be a fixed body of knowledge then technology is that of an efficiency tool. If we consider tools as providing access to new understandings then technology can be a part of a conceptual construction toolkit. I liked her idea of using technology in mathematics as a form of ‘agent provocateur’, to create cognitive dissonance and push thinking. Examples used included “Draw a line root 45 units long” and “When will a population of 50 000 bacteria become extinct if the decay rate if 4% per day?”. The focus should be to focus on mathematical thinking and use technology to explore this thinking as a partnership.

I was also pleased to hear her decry the horrendous statement on technology in the Senior Australian Curriculum draft which is “Technology can allay the tedium of repeated calculations”. That’s it. Its sole purpose. Unbelievable. I highly recommend you have a look at Merrilyn’s paper at the ACER website above.

Whilst exhausting, the conference was definitely worthwhile. It affirmed many aspects of the teaching and learning of mathematics for me personally but it also made me look at many things through new eyes and that’s what education is about, isn’t it?

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About Linda

I have been involved in secondary mathematics education in Victoria, Australia for over 25 years.
This entry was posted in Ideas for teaching & learning, Pedagogy, Systems, Technology, The discipline, The profession, Things that engage, Thinking. Bookmark the permalink.

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