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Mathematics Teaching 184 - September 2003
Mathematics Teaching is the journal of the Association of Teachers of Mathematics. It is a professional journal sent to all members of the Association. It is not a refereed journal. Submissions are reviewed by the editorial team. Many articles have additional information or associated files placed on the journal website. To make your views known go to the ATM forum add your views, ideas and comments.
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MT184 Contents
Questioning styles - Nikki Martin
I have recently completed a project involving the observation of twelve teachers in different schools, teaching pupils at Key Stage 3 of differing ages and abilities, noting both questioning techniques used and pupils responses. The teachers involved were volunteers, chosen to give a range of experience, styles and school, and included a head of department, teachers on a QTS scheme and teachers whose initial qualification was not in mathematics. The questions in all parts of the lesson were recorded, although plenary sessions were rare. My initial intention was to distinguish between open and closed questions and note the responses of the pupils.
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How should we teach mathematics to 3 and 4-year-olds? - Sue Gifford
Not so long ago, the word teach was anathema to those coming from the early childhood education tradition: for instance, Bruce and Bartholomew (1993) stated that it 'implies adults transmitting, imposing, invading and dominating the child's life'. Teaching young children, in short, was seen as close to abuse. The idea that early years' education should focus on cognitive outcomes like number learning was similarly seen as harmful and threatening to the traditional broad aims of social, emotional and physical development.
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Primary problems solved without sheets - Keith Windsor
Each of these problems encourage children to 'get into' the process of problem solving, and all can be introduced without written instructions A demonstration by the teacher with large hands-on materials can involve all the children effectively and give you the opportunity to highlight key points and deal with possible misunderstandings. All the children should begin with the same task, with support if necessary, using desktop apparatus to help thinking where appropriate.
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Handing control to 5-year-olds - Cynthia Collins
I collected six 5-year-olds and showed them a fresh box of felt-tips and a pile of card squares (about15cms wide). "WE need some new number-cards, 1 to 12. Would you do this for us? Nice, big numerals, but only one on each card please". I counted out twelve card squares, slid in an extra two for luck and left them to it.
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Relational, instrumental and creative understanding - Melanie Reason
A key incident in my learning of mathematics is the time I first remember being aware of myself deliberately doing so. Two years after having left school, I was in a position of having to relearn mathematics after a lengthy time of being unwell. I set about relearning A-level mathematics from a book, meeting definitions afresh and travelling back through texts in order to reconstruct knowledge. I have a picture of the process as walking through a maze until all the paths joined together...
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Putting place value in its place - Ian Thompson
A: What happens when you multiply a number by ten?
B: You add a zero.
A: No, we say that 'the digits move one place to the left.
The context for this exchange could be any one of several scenarios: a primary or secondary classroom, a teacher training establishment, an LEA CPD course venue or a national numeracy strategy consultants' training session. In each context the 'teacher' would be keen to avoid the acquisition of any misconceptions on the part of the 'learner' because, as we all know, the 'adding zero' rule breaks down in the case of decimal fractions.
The spirit of the subject - Nicola Bretscher
I believe that proof is at the heart of mathematics. As Alan Slomson put it in MT155, "Mathematics without proof is like brandy without alcohol: the spirit of the subject is missing". Proof is a topical subject since its recent inclusion into the national curriculum. This change motivated the recent special issue on proof in MT177.
This article is an overview of the research on teaching proof and focuses on four areas: what we mean by proof, pupil's common misconceptions, levels of proof and teaching strategies.
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About ACME - Annie Gammon
ACME stands for the Advisory Committee for Mathematics Education. The origins of this new committee lie with the last two chairs of the Joint Mathematical Council. The plethora of groups representing those involved in mathematics education has had a representative umbrella organisation for some time: the Joint Mathematical Council (JMC). This group of twenty or so members from each society meets termly. Over the past decade or so, when the pace of change in education has accelerated, termly meetings proved insufficient to meet all the demands of responding consistently to the DfES. It was proposed that a smaller, funded group be set up. The Gatsby Charitable Foundation agreed to fund ACME initially for three years.
Who needs to know what? - John Mason
In this piece John reviews two mathematical dictionaries. When might you consult a mathematics dictionary? No matter how careful an author is with clarifying and exemplifying technical terms, there is often something that troubles me even if the author assumes that as Paul Erdos used to say, 'every baby in its cot knows...' That is when a mathematical dictionary could prove useful. The books reviewed are Maths Dictionary by Gillian Rich, published by Letts and Collins Dictionary: Mathematics by E. Borowski & J Borwein, published by Harper Collins.
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Just four points and the 'prepared mind' - John Sharp
Two of my favourite quotations are Poul Anderson's 'I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become more complicated', and Louis Pasteur's 'In the field of observation, chance only favours those minds which have been prepared.'
Supersolids - Sandra Pulver
Infinite series prove important in numerous mathematics applications and, consequently appear in a number of mathematics courses including analysis, calculus, and differential equations. High school analysis often addresses geometric series; calculus courses often devote a great deal of coverage time to the study of convergence of infinite series. To stimulate student interest in infinite series, introduce some geometric examples in which some paradoxes arise...
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Loop cards - Adrian Pinel
Loop cards is the name I coined 25 years ago to refer to combinations of several short sets of 6 to 9 cards. In each set the answer to each card is found on the next card until a closed loop or circuit is made by all the cards. These are not just simple 'question and answer' sets. In loop cards the value on the top left hand corner of each card is used with the operator in the centre of the card to provide the value on the next card. These (original) loop cards are only available through Adrian and Jeni's website.
A Flash movie of a Loop Card game being played
Adrian and Jeni Pinel's website
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Reflections - Paul Andrews
Some time near the start of this last academic year I asked my group of PGCE mathematics trainees to consider what mathematics would look like without proof. Straightaway Tim replied, "Science". This amused me greatly...
Professional Officer's Update - Barbara Ball
In my last Update in June I suggested that maybe all teachers of mathematics should be obliged to attend professional development events in which they do their own mathematics. Throughout June and much of July I had the pleasure of working with many different teachers, who had chosen to do precisely that...
A note from the Web Editor
The resources aspect of the web site is, as always, the most frequently visited section and we aim to build on this over the autumn months.
Reviews
Follow Me! Mental and Oral Card Games for Practising Numeracy Y4; 1089 and All That; Maths Through The Ages - a gentle history for teachers and others; From Conjecture to Proof; Understanding Mathematics in the Lower Primary Years: A Guide for Teachers of Children 3-8 years; Guide To Mathematical Methods.
