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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 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”….
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!!)
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
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.
Regular (or even semi-regular..or even the occasional) readers will know that professional learning is one of my soap-box issues.
It was therefore very interesting to read the just-published New Zealand’s Education REview Office report on Professional Learning and Development from Derek Wenmoth’s blog. Well worth a look. Well worth considering in the Australian context. Well worth the reflection. Well worth the thinking. Worth it for teachers’ learning and for better learning outcomes for our students.
“…sharing the same fears. Time it was, and what a time it was, it was. A time of innocence, a time of confidences..” (from Simon & Garfunkel’s Bookends)
Yes, it’s reporting time in Victoria. And what a time it is. Although I’m not sure if we all share the same fears! A time of innocence? Maybe. A time of confidences? Sometimes…depends on how honest the comment, I suppose! Certainly a time of angst. A time when some teachers resent the time that must be taken to write them out of their already crowded and stressed existences, a time when old and barely concealed umbrage and bitterness comes to the fore. A time for martyrdom and indignation (‘How could they ask me to do that as well as write these reports?’) Oh well…
However, because I also have reports to write, this will be a brief post.
Over the last few days, I have discovered that Art Garfunkel has degrees in mathematics and art history (I knew there was an affinity there…that explains why I felt the need to perm my hair in the 80s!!) and that John Brack, the iconic Melbourne artist whose extensive exhibition I viewed yesterday, was also good at mathematics and was always looking for patterns in human behaviour and then creating a visualisation of this in ways that could not be expressed with words. I have written before about mathematics being more like a philosophy than a science and more creative than many give it credit for. Mathematics should, perhaps, be more about looking, musing and wondering.
The second item of this eclectic mix is Elizabeth Wynhausen’s essay On Resilience (part of the Melbourne University’s Press series of little books on big themes). In this wonderful reflection on, mainly her mother’s story, she examines the difference between stoicism and resilience and mentions a study that has identified three traits resilient people share: a resolute acceptance of reality, a sense that life is meaningful and an exceptional ability to improvise. Elsewhere she also writes about a playfulness that is evident in resilient people. I have also written about resilience in students of mathematics elsewhere and how important it is for teachers to try and engender such resilience. Too many of our students endure the subject with something akin to stoicism rather than engaging with its challenges and rolling with the punches, playing with ideas and learning something about the discipline, and themselves as learners, as they do so.
Finally, I have been fortunate to have spoken with, and heard Ron Ritchhart, of Project Zero out of Harvard University, present over the past month or so. He is always an interesting speaker and says things about education that make me think and wonder. He is also an ex teacher of mathematics. He has recently started up his own website here and there you will find some of his recent presentations from his Melbourne visit. It is well worth a look.
