Malcolm Swan Lecture

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

Sound familiar?

He then contrasted this list with the one for a collaborative learning/intellectual challenging mathematics classroom:

*network of ideas

*social activity/discourse

*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.

Noticing

I’ve been spending some of these holidays looking at some web resources and came across John Mason’s website. It is full of some great ideas and he has generously put his presentation resources on this site for others to download and use. There are PowerPoints on developing and supporting creativity in mathematics, encouraging mathematical thinking and the use of examples in mathematics instruction. I have previously blogged about his great little book, Thinkers. This is a fantastic site and has given me all sorts of ideas for 2010. In these presentations he gives examples of how to use some of the Thinkers ideas in classes.

I am thinking along the lines of making “Noticing” the faculty theme. Teachers noticing when and what students are doing and thinking (and intervening to correct misconceptions – assessment for learning) and students noticing, in terms of looking closely, recognising patterns, seeing when it’s appropriate to generalise and when it is not, what the invariants are in a situation, what the limitations are etc.

This site is well worth many visits.

Keeping NAPLAN in perspective

I was very interested, and pleased, to receive the following email message from Judy Anderson, President of the Australian Association of Mathematics Teachers today regarding the recent release of NAPLAN  results:

TO: All AAMT members and interested others

 

Dear colleague

 

You will be aware from the media and in your school that NAPLAN (National Assessment Program in Literacy and Numeracy) results are now being distributed to parents. The release of the summary report has resulted in a range of comments in the media. Interpreting the data for parents and at the class and school levels is likely to be on the agenda for many members of AAMT. I write to remind you of the AAMT’s position on the matter: these comments refer to the ‘Position Paper on the Practice of Assessing Mathematics Learning’ and may be useful to you in the conversations about the NAPLAN results that are likely to be occurring all over the country. The Position Paper can be downloaded from http://www.aamt.edu.au/Documentation/Statements

 

One of the main issues is the use of NAPLAN data. The Position paper makes some clear statements about this. It states that Education authorities’

assessment should:

 

€ ‘make claims that can be related directly to what is assessed’:

the format of NAPLAN tests limits what they are able to assess.

 

€ ‘prohibit the publication of league tables of schools from their data’:

league tables and similar coarse comparisons of schools’ performances are not helpful to teachers and the teaching and learning of mathematics.

 

€ ‘provide information that maximises opportunities for teachers to capitalise, in their teaching, on the assessment information gathered’:

it is more than just providing the data — education authorities need to support teachers and schools to analyse NAPLAN results to diagnose students’

strengths and weaknesses. (p. 7)

 

There is no doubt that the NAPLAN results can provide diagnostic information about your students so I encourage you to seek out that information and use it to inform your teaching. However, the Position Paper argues for ‘appropriate assessment’ by stating that teachers should:

 

€ ‘assess the full range of learning goals by using a range of strategies’:

NAPLAN tests are only one component of a comprehensive approach to assessment in mathematics. (p. 3)

 

Education authorities and schools need to further support the development of quality assessment in mathematics.

 

The release of the NAPLAN results is a single event in this whole year of teaching mathematics. I hope that you are able to be involved in productive thinking and conversations about the process and the data as part of what we all recognise as the ‘main game’ — high quality teaching, learning and assessment in mathematics for all our young people.

Creating a Culture of Thinking in the Mathematics Classroom

I’ve had the great opportunity to work with Prof Ron Ritchhart (his website has pdf versions of most of his presentations) from Project Zero, Harvard Graduate School of Education, today.

He started with a great activity that could be replicated in department meetings. First, ask participants to write down all the actions that students engage in within mathematics classrooms. Then, people select from this list to create their own private lists of about 5 items for:

• Actions your students spend most of the time doing during mathematics classes
• Actions that are most authentic to the discipline (what is ‘doing maths’ about)
• Actions you remember from a time when you were learning new mathematics (eg at uni, teaching a unit of work with content you are not familiar with) and what you did to develop understanding

If actions arose that were not on the shared list, these could be added.

As a group, participants then created a shared list of the above. This was a great, unthreatening way of eliciting people’s underlying beliefs about what mathematics is about and what it should be about – for teachers and for students.

We then looked at 4 Useful Patterns of Thinking (whilst these are not inclusive of all types of thinking we might want to foster in our students, these are useful for teachers to nurture in the mathematics classroom):

• Speculation: What might be going on here? What are the possibilities..and the limitations? What questions, puzzles or issues are being raised?
• Generalisation: What principles, strategies or formulas can we identify from this situation that will always hold true?
• Analysis: What relationships and connections can we identify here? What are the various parts and facets? What is their role, effect or purpose?
• Proof: How can we convince both ourselves and others that our findings, assumptions and conclusions are valid?

These broad umbrella terms struck a chord with me. These fit in very nicely with what ‘working like a mathematician’  means. Wouldn’t it be great if we could plan our lessons, topics, curricula, assessment around these ‘dimensions’ of thinking? What if we made these explicit to teachers and students about what is expected in a mathematics classroom, that these were the goals we had in mind and aligned everything we did to these?

We were then given a transcript of a lesson that started with the question: What shapes can be made by a straight line and a square?

Such a simple question, such rich opportunities presented, and developed, by the teacher. Students were challenged to think about whether a 30,60,90 triangle could be formed. What congruent shapes could be formed? “How do you know?” Ron referred to this as ‘bumping up the thinking opportunities’…what I call ‘ratcheting up the thinking’. When I plan learning experiences, I use one of Project Zero’s thinking routines: Connect, Extend, Challenge. I connect to previous knowledge, extend this into the next phase of constructing knowledge and understanding about new material then challenge students to transfer this knowledge into an unfamiliar context to check for understanding. Ron mentioned that one of the most popular thinking routines in mathematics is Claim, Support, Question. He described a class where students had 3 dice numbered 0 to 5. The students were told that the aim was to create even 3-digit numbers from the digits thrown on the dice. A point was awarded for every even number created. After doing this activity for a while, the teacher asked for ‘claims’…eg “If all three digits are even there are 6 even 3-digit numbers possible”. So much mathematics, so much thinking..and the students own it…it hasn’t been passed down to them from on high like a lot of mathematics seems to be.

We ended the day by watching a teacherstv video on Revising Polygons.

An invigorating, energising day that extended and teased out my ideas on thinking and learning in mathematics.

Visible Teaching & Visible Learning

Received a copy of John Hattie’s recent book – Visible Learning - last week.

Quoting from Chapter 3: The Argument – What Teachers do Matters:

…what some teachers do matters – especially those who teach in a most and visible manner. When these professionals see learning occurring or not occurring, they intervene in calculated and meaningful ways to alter the direction of learning to attain various shared, specific and challenging goals. In particular, they provide students with multiple opportunities ..for developing learning strategies based on teh surface and deep levels of learning …leading to students building conceptual understanding of this learning which the students and teachers then use in future learning. The act of teaching requires deliberate interventions to ensure that there is cognitive change in the student: thus the key ingredients are awareness of the learning intentions, knowing when a student is successful in attaining those intentions, having sufficient understanding of the student’s understadning..and knowing enough about the content to provide meaningful and challenging experiences in some sort of progressive development. It involves an experienced teacher who knows a range of learning strategies to provide the student when they seem to not understand, to provide direction and re-direction..and thus maximise the power of feedback.

..in the right caring and idea-rich environment, the learner can experiment (be right and wrong) with the content and the thinking about the content, amd make connections across ideas. A safe environment for the learner is an environment where error is welcomed and fostered-because we learn so much from errors and from the feedback that then accrues from going in the wrong direction or not going sufficiently fluently in the right direction. In the same way, teachers themselves need to be in a safe environment to learn about the success or otherwise of their teaching from others.

To facilitate such an environment, to command a range of learning strategies, and to be cognitively aware of the pedagogical means to enable the student to learn requires dedicated, passionate people. ..teachers need to be aware of which of their teaching strategies are working or not, be prepared to understand and adapt to the learner(s) and their situation(s), contexts and prior learning, and need to share the experience of learning in this manner in an open, forthright, and enjoyable way with their students and their colleagues.

We rarely talk about passion in education, as if doing so makes the work of teachers seem less serious, more emotional than cognitive, somewhat biased or of lesser import. Passion reflects the thrills as well as the frustrations of learning- it can be infectious, it can be taught, it can be modelled, and it can be learnt.

Learning is not always pleasurable and easy; it requires over-learning at certain points, spiralling up and down the knowledge continuum and building a working relationship with others in grappling with challenging tasks. This is the power of deliberate practice. The greater the challenge, the higher the probability that one seeks and needs feedback. ..

..the message is not merely to innovate- but to learn from what makes the difference when teachers innovate. When we innovate we are more aware of what is working and what is not, we are looking for contrary evidence, we are keen to discover any intended and unintended consequences, and we have a heightened awareness of the effects of the innovations on outcomes.

Lots of good stuff here. I look forward to dipping into other chapters and reflecting on my own practice.

Highly recommended. You can read more about the book at Bruce Hammonds’ blog, Leading and Learning

Dali Thinking

Went to see the Salvador Dali exhibition at the National Gallery of Victoria earlier this week. Surrealism isn’t really my ‘thing’ in art, I have to say. There were some beautiful drawings, however, some great perspective drawing, in particular, and some of his earlier work was very appealing (I particularly liked Girl’s Back).

 

What fascinated me was the mathematical precision and physical composition of his more surrealist pieces. Despite the seeming lack of realistic structure in his art, there was a strict, even rigid, mathematics behind all of it. Viewers could see the gridlines he used to create a matrix of cells on the page and he filled in each of these cells with some image that was then balanced or counter-balanced with some other image in another cell on the page. There seemed to be almost desperation in his desire to ’speak’ to the viewer and expound his views on the world and its philosophies via these paintings…along the lines of “it’s so obvious to me, here..let me show you…there…do you see now what I see?” In addition, there was a painting of a rhinoceros’ horn and the accompanying descriptor mentioned it as the perfect example of a logarithmic spiral. I will investigate…

He is actually quoted as saying that he felt for a work of art to be considered as such, it must be based on mathematics, composition and physics.

Other quotes I liked were:

• “grandiose, geological delirium”
• “mineral impassiveness”
• “accumulated and chronically unsatisfied tension”
• “verbal colour”
• “photography offered the most secure vehicle for poetry”

Also of interest to me was a quote Dali made after trying university and giving it up in frustration: “Expecting to find limit, rigour, science, I was offered liberty, laziness, approximations”

It made me think…again…do we, in fact, limit our students’ potential by not providing sufficient challenge to extend their boundaries?

Mathematics and the Iranian ‘election’

Discovered this post through Twitter: From the Huffington Post website:

The other story about people sucking at math that’s a bit more surprising has to do with the Iran election. First came the report from British think tank, Chatham House, which showed that Ahmadinejad received 13 million more votes than he and other conservatives got in 2005, an unlikely occurrence considering his waning popularity. They also found that in two provinces, Mazandaran and Yazd, turnout was more than 100 percent.

Then Bernd Beber and Alexandra Scacco, two Ph.D. candidates in political science at Columbia, performed their own mathematical experiment, publishing their results in a Washington Post op/ed. Beber and Scacco looked at “digit frequencies” in the vote counts–when numbers recur at certain rates it suggests human tampering–to come up with a statistical probability that the election was fair.

And, according to their findings, the probability that the election was fair came out to .005 percent.

What does all this mean? The Iranian election riggers–Ahmadinejad & Co.–really really really suck at math.

Barbie turns 50

Barbie has a lot to answer for.

I never had a Barbie – she was too expensive. I had a Cindy instead. At least Cindy never spoke the words “Math is hard” and gave rise to what I fear has been a generation of mathematics-avoiding females.

Imagine what could’ve been if she’d said “Math is fun”….

Developing Mathematics Syllabus Documents

I attended one of the Dean’s Lecture Series seminars, at the University of Melbourne’s Graduate School of Education, last week. It was titled ‘Competencies and the Fighting of Syllabusitis’ and was given by Associate Professor Tomas Hojgaard from Denmark.

He has recently been working with the Mathematics Faculty of the Graduate School of Education. He was also one of the people behind ‘Mathematical Competencies and the Learning of Mathematics’ by Mogens Niss (2002), a paper that has guided the re-writing of syllabus documentation for mathematics in various European countries.

He started with saying “Curriculum only works if it works in the classroom”

He also stated that many curriculum documents he has seen focus on content and pay little attention to the purposes of teaching the ‘what’ that is to be learnt. Guidelines for assessment also seem to refer more to the ‘what’ and ‘how’  (ie the format of the assessments. For example: the number of tasks, the timing of these tasks, the structure of these tasks, consequences for late submission etc) rather than a purposeful focus on what it means to be working mathematically.

The analogy was made that creating a syllabus is like creating a house. If you were to present the builders with a list of the content required to build a house (the number of bricks, amount of cement, number of pieces of wood etc) this would not be of much use to them. It wouldn’t lead to the construction of something with a purposeful structure nor inform one on how the various elements link together to create the house.

His contention is that the, sadly, majority of syllabus documents that are merely a listing of content should be replaced with a series of competencies. By his definition, a competence is something that is being enacted, the use of knowledge to show understanding. His formal definition is “someone’s insightful readiness to act in response to the challenges of a given situation”. ‘Insightful readiness’ is a way of encapsulating the difference between merely doing a skill and being capable of understanding and knowing how to go about approaching a problem set in an unfamiliar context.

He believes that the answer to “What does it mean to be mathematically competent?” is addressed in the following 8 competencies:

• Mathematical thinking competency: carry out and have a critical attitude towards mathematical thinking
• Problem tackling competency: formulate and solve both pure and applied mathematical problems and have a critical attitude twards such activities
• Modelling competency: carry out and have a critical attitude towards all part of a mathematical modelling process
• Reasoning competency: carry out and have a critical attitude towards mathematical reasoning, comprising mathematical proofs
• Representing competency: use and have a critical attitude towards different representations of mathematical objects, phenomena, problems or situations
• Symbol and formulation competency: use and have a critical attitude towards mathematical symbols and formal systems
• Communicating competency: communicate about mathematical matters and have a critical attitude towards such activities
• Aids and tools competency: use relevant aids and tools as part of mathematical activity and have a critical attitude towards the possibilities and limitations of such use

Not only would instruction be geared towards these, but assessment as well. He thinks that focussing on competencies as ‘bands of strength’ would mean a more positive approach to education instead of the can/cannot, either/or dichotomy he believes currently exists. These competencies underly the essence of the discipline of mathematics.

Prof. Hojgaard sees two key foci in an authentic education:

• The need for student directed processes and
• The need for maintaining educational focus (the essence and rigour of the discipline)

To enact this curriculum in the classrooms, teachers need to engage with thinking about the competencies and how they can be achieved in their learning plans…the need to develop teacher-directed autonomy. (As an aside, it is interesting to consider this in the light of the proposed national curriculum. Will the way the curriculum is designed allow for, or indeed compel, teachers to control their learning plans by thinking about how to teach so that a coherent set of linked ideas create a mathematical structure in their students’ minds? Or will it try to take the control away…and thus, in my opinion, treat teachers more as transmitters of knowledge instead of designers of their students’ creation of knowledge?)

So the question then becomes: “How can a syllabus document be structured so that these competencies can be integrated across a content-rich curriculum so that they are a focus and not able to be ignored by teachers?” There needs to be a way that teachers are compelled to address the competencies and not just teach the same way as done before. Teachers have a difficult job – to develop and target a learning focus but, at the same time, encourage their students to engage in autonomous learning. It is imperative, if students are to internalise the learning, that they take more responsibility for the teaching of what it is teachers want them to learn.

Prof. Hojgaard’s suggestion for the design of syllabus documents is as below:

		Number	Algebra	Probability & Statistics	Geometry	Measurement
Mathematical thinking competency		The body of the document would have sample questions, examples, suggested
Problem tackling competency		activities for learning etc.
Modelling competency
Reasoning competency
Representing competency
Symbol and formulation competency
Communicating competency
Aids and tools competency

 

An interesting talk…and it is very interesting to consider this idea in concert with the Understanding by Design curriculum framework that I am currently toying with.

At the conclusion of the talk, I had the opportunity to chat with Tomas for a while and he agreed it was imperative to compel teachers to engage in thinking more about the purposes for specific content and then delivering a learning plan that addresses those purposes. Too many teachers, he fears, see themselves as deliverers of a curriculum and a methodology that is determined elsewhere and by others. (And he laughed when I asked whether Denmark’s educators were as sick and tired of having Finland given as the ideal example of education, as we in Australia were!!)

Something of Interest

This came up in my regular email alert from the online Curriculum Leadership Journal:

An interpretive scheme for analysing the identities that students develop in mathematics classes

Journal for Research in Mathematics Education

Volume 40 Number 1, January 2009; Pages 40–67

Paul Cobb, Melissa Gresalfi, Lynn Liao Hodge

 

The ways that students relate to mathematics can affect their interest and persistence in mathematics learning. Analysis of the identities that students develop in mathematics classrooms can therefore help inform class design and pedagogical approaches. The article reports on a study that has used an identify framework to analyse the classroom-based mathematics identities of middle-school students enrolled in both a regular algebra class and a collaborative, inquiry-based class, designed for this research. Students’ identities were measured in terms of normative identity, indicating how well they met class-wide norms of mathematical competence; and personal identity, the extent to which students identify with these classroom norms, for example whether they are actively engaged, merely cooperate, or openly resist the teacher. In the algebra class, authority was distributed to the teacher, who determined the methods students could use to solve tasks and was the judge of the legitimacy of their responses. Mathematical competence was constituted as the ability to use set processes to reach appropriate solutions. Interviews with students confirmed that they saw that their role was to take notes, ask clarifying questions, and demonstrate knowledge using processes legitimised by the teacher. Students’ responses indicated that they were merely cooperating with the classroom obligations, and were not developing a sense of affiliation with mathematics in this classroom: their sense of obligation remained directed toward the teacher rather than toward themselves. In contrast, authority in the inquiry-based class was distributed, as students and the teacher jointly determined the legitimacy of responses. Students had agency to select their own methods for developing and explaining analyses, and to challenge those of other students. Mathematical competence was demonstrated by students’ ability to justify their solutions. Students believed that their role was to justify and explain their reasoning, ask clarifying questions, and to explain reasons for disagreement with others’ results. Their positive evaluations of their obligations indicated that they were developing a sense of affiliation with mathematics in this classroom. Different approaches to mathematics teaching foster different learning identities, perceptions of competence, and affiliation with mathematics.