Who am I?
I am a teacher at St Hilda's Collegiate School in Dunedin New Zealand. I am HoD Mathematics at St Hilda's and also the Course Facilitator for the University of Otago EDU336 paper (training physics teachers). I am the author of several textbooks for delivering Physics in the New Zealand Curriculum, published by Pearson Education. The main reason for starting this wiki is my involvement in the Dunedin based 'Algebra Project' looking at methods of enhancing and measuring the attainment of algebra concepts and processes. Within this project are the following teachers and researchers who are also keeping a wiki to store our journal of professional reading, teaching ideas and feedback from the Algebra Project.
--Newton 08:10, 9 February 2010 (UTC)
The impact of the Algebra Project on my teaching to date At the start of this TRLI research project, we were asked to diary our thinking, reponses to papers we read and the impact on our teaching. I fully intended to do that in almost real time, but the realities of time constraints during the year have meant that I have neglected this task. So this section, is the link from the last (Sfard) to now (early Term 4, 2010). Early in the year, in the throes of the background academic papers that we were wading through, I was somewhat worried about how we would tie down the abstract nature of our discussions (minutes from these meetings on our TRLI wiki) to practical applications in the classroom. Over the year it has become seamless, partly due to the instinctive nature in the planning of the year, partly in reaction to the needs and learning styles of my actual current students and partly in trying ideas and foci from our Algebra Project discussions. In a roughly chronological order (any other order would require a vast amount of thought/discussion/dispute), the impact on teaching has been as follows. Planning For the implementation of the revised curriculum (NZ), I had written the year 9 and 10 mathematics programmes as a collection of three week chunks of work that have number and algebra as 'integrating' strands and underpin these skills and concepts with the teaching of context. For example, the starting unit for year 9 is 'angles' which allowed geometry to be a context for algebra. If there are five equal angles around a point, each of x degrees, then x+x+x+x+x=360, etc. The continual use of algebra (and see later entry for why number as well) in context has led to a very strong acceptance and understanding of algebraic convention and processes in relatively junior students. e-learning and unstreamed classes Our school, in 2010, introduced lap tops for all year 9 students and I removed the streaming structure. Both initiatives rely on each other. We have been able to provide a 'differentiated' learning environment, helped by the wonderful teaching resources available electronically. I would have to mention 'Mathletics.com' here as being particularly helpful in managing a mixed ability class effectively. With regard to removing streaming, it has become very apparent in frequently assessing number and algebra competence in a variety of contexts, that the students are truly diverse. There are abstract thinkers and those who can only solve when a meaningful context is given. There are those who can match numerical operations to algebraic, yet there are many whose number knowledge (basic + and x families of facts particularly) is so poor that when faced with even entry level linear equations, they are unable to 'see the wood for the trees'. I have found teaching in small ability groups (a la primary school) to be essential! increasing the focus on = I have always been a big fan of the set of scales as an analogy to the equation. Now, I am an addict. A great deal of teaching and discussion involved arms outstretched like the wingspan of an albatross, ensuring the same thing to both sides. During the sections of pure algebraic manipulation, I also adopted an increased focus on the terms in an algebraic sum. The students needed to make decisions as to whether they wanted to work on single terms ( collecting like terms or simplifying a rational expression for example) or a whole side (to extract 'x'). With the aid of a very kinesthetic learner in my year 13 calculus class, we developed a two hand expression of working on a part (hands close together as if holding a cup) or the whole side (hands wide apart as if picking up a large box). This action helped distinguish between 'changing the look but not the value' work such as expanding brackets, simplifying or modifying fractions, etc and writing and equivalent equation by 'doing the same thing to both sides'. Bringing back formal language and habits Over the past couple of decades, the formal top down, deductive reasoning used in proof work and solving has relaxed a little. The precise use of symbols and terminology has nearly disappeared. Both of these areas of slackness have been to the detriment of teaching algebra. With an effort to be more rigorous in my use of terminology and to encourage deductive reasoning set out vertically, I have found student confusion to have been minimised. Without ambiguity, students can focus on the real issue/problem. For example, it is helpful if they know they are solving for an unknown, or finding a generalised expression. It is helpful if they know the solutions to a problem are elements only of natural numbers, or could be from the set of all real numbers.
On the Dual Nature of Mathematical Conceptions Anna Sfard Well, this paper had me seething from the abstract. I, personally, don't see that algebra may be viewed as either structural or operational, but rather that the structure view is a subset (albeit, usually a solution) of the operational view.On page 2, the basis of the highly successful educative process of student centered learning is dismissed. Further to this, the assignation of an ontological status of mathematical constructs should be well outside the bounds of mathematics education. The epistemological status is, however of interest. Sfard wants to assert that algebra is apriori knowledge, whereas I see it as an elegant combination and apriori and aposteriori. And all this by page 2 .. I guess I'm like the little boy in the story saying that 'the Emporer isn't wearing any clothes'. I rather hoped that the paper would explore the link between abstract and concrete and to what extent mathematics is actually aposteriori. I heartily agree with the statement on page 6 regarding visualisation. The graphical approach in education is essential. As I read on and see that Sfard develops more of a tolerance of the operational/structural duality (the number i being a very good example of such duality), I feel much more settled reading this paper, but still do not see the point of view being argued.I do not see that there is always an order of derivation to be followed for all cases. The number i for example must have been developed from the abstract firstly, but then has shown its concrete application in situations such as quantum mechanics. I also disagree with the wholly 'interiorization' in the process of learning and would like the possiblity of reification coming first as students are often very beleiving of empirical data.
Prerequisite Skills for Learning Algebra Linsell and Allan, 2010 This was an interesting study - showing that student grasp on inverse operations were, on the whole, better than their understanding of equivalence. This is likely to be directly related to strategies shown by teachers. The amount of time and energy devoted to the = sign and its meaning is probably far smaller than that spent on 'undoing to solve'.
Education Week Spotlight on Algebra, November 2009 The recurring theme in this collection of articles, being 'algebra readiness' is one which has been at the back of my mind in teaching junior (years 9 and 10) mathematics over the past few years. The work that we do in number needs to be explained by the user in such a way that s/he is able to seamlessly transfer the number strategies to algebraic processes. I have mainly taught senior (years 12 and 13) classes in the last 12 years and have seen the after effect of number and algebra being treated as separate entities at an early stage. The students are quite ready to put up a mental block as a barrier to learning algebra in the senior years. To counteract this, I have rewritten the St Hilda's junior units of work to sprinkle algebra throughout the course so that it is not a stand alone mountain to scale. This technique seems to be in line with the strategies for algebra readiness used by the examples given on pages 3, 12 and 13. The article on Computer-Aided Instruction (Sean Cavanagh)is of personal interest as this year, as well as the Algebra Project, the new units of work in the junior school where algebra is taught throughout the topics as an extension of the calculative work and implementing the revised national curriculum, the Mathematics Department has removed streaming of classes (reasons given below) and the school has introduced e-learning with all year 9 students having their own laptops (Apple). What seemed daunting a couple of weeks ago, as the new term began, is now exciting and successful, mostly due to the wonderful extension to the classroom offered by the two interactive maths tutorial programmes that we are using as part of our range of activities in this 'differentiated learning environment'. The programmes are ten ticks and mathletics. The reasons for removing streaming are that we are a small school (three classes per year level)with a high level of numeracy and literacy in the cohort. The cut off mark for a top set would mean that half the year group would be in one class and only 6 students in the lower set. Because of such numbers, we noted a significant negative effect on the morale and lack of progress in the lower set class. To a lesser extent, this was also noted in the middle set, more able students feeling put off by not having made it into the top set. Further to our immediate observation, many research papers showed that mixed ability grouping and raising the expectations of all leads to enhanced learning. So far into this term, that is proving to be the case.
Early Algebraic Thinking: Links to Numeracy, Linsell et al It is interesting to see that the development of algebraic strategies has been broken into sequential steps. I know that we have, in the past, endeavored to teach in such a way, but am not sure that this is quite such a one way street. I do think that there are 'eddy currents' in the learning process as the student grasping algebraic strategies looks back to how those strategies worked in number. I seriously question guess and check as a valid, separate stage.