Q: I am training as a maths teacher on a PGCE (Secondary) course and I am writing an assignment on the use of calculators in secondary education. I would be grateful if you could send me your views on the (mis)use of calculators.
A: Calculators came into use in schools when I was doing O-levels, but we weren't allowed them in exams. The most obvious "use" is that of a tool enabling students to find answers more quickly. Part of my maths kit was a book of log tables for long multiplication, division, powers, roots and trigonometric calculations.
I showed a set of these to an A-level student recently and he couldn't believe we had to read such cumbersome tables ( I didn't like to tell him about having to turn the handle the number of times we wanted to create a multiplication on our calculating machine). Success in calculations was based partly on accuracy in reading the tables. The earlier "reverse" calculators created their own inaccuracies - for instance, to find the value of sin30 LESS THAN , the confusing sequence was to type 30 then "sin".
When I began secondary teaching an important use of the calculator was to provide "instant corrective feedback" for pupils on completion of paper calculations in homework. An incorrect answer could simply be caused by miscopying a number. This encouraged them to go back and check their work.
Learning statistics at school was boring for me, calculations were lengthy and typing in the wrong number meant having to go back and begin again. The introduction of statistics functions made data entry much easier and the statistic easier to compute. This made data exploration much easier and allowed for investigation of effects of altering the data. The calculator allows for the investigation of patterns in numbers, thus making and testing conjectures easier.
I contacted Stephen Kean, an education consultant for Casio Electronics and these are two of his helpful suggestions:
Orders of operations
S-VPAM (super visually perfect algebra method) means the calculators use the correct priority of operations, (BIDMAS, BODMAS). I tend to demonstrate this by calculating 1 + 2 x 3 to generate the answer 7. What I then do is scroll to the expression and insert brackets to calculate (1+2)x3. This means expressions can be entered directly into the calculator in the same format as they are written on a page. Users do not need to adapt expressions to the way the calculator operates.
The "Four 4's" problem is a good way to explore order of operations and bracketing.
Use any of the four operations +, - , x and V with the digit 4 to create as many answers 1 to 20, e.g. (4 + 4) V (4 + 4) = 1(4 V 4) + (4 V 4) = 2(4 + 4 + 4) V 4 = 3
The red-letter keys can be assigned any value, e.g. 8.A (key presses are 8 SHIFT RCL (-)).
Then ask the question:"What is 2A?" Use your calculator to find out (the calculator will give an error if A2 is keyed in, reinforcing correct algebraic notation).
Then "What is 5A-7?" Though these can be worked out easily on paper, the use of the calculator emphasises the quite abstract concept of substitution.
Multiple letters can be used, e.g. let the red-letter keys have the following values: A.3; B.2.
Use the calculator to work out values for (A+B) (A-B), A2 and B2. The scroll keys can then be used to edit the values for A and B, and these can be larger or decimalfractional values. The values can then be re-calculated by pressing the "=" key.
Some letter games can be played. For instance, place the following on the board: AB+C=10 and A+BC=11.
Ask students: "What could the values of A, B and C be?" They can use their calculator to experiment. A lot can be learnt by playing with the values.
Students like to create their own.
The main misuse of a calculator is using it for simple calculations such as multiplying by 10 as this can interrupt the flow when solving a problem. A project team based at University of Leicester, "Developing effective use of two-line scientific calculators at key stage 3" has created a booklet of ideas with Casio. The team would welcome feedback on their resources or ideas for further development. email: email@example.com
We would like to thank Casio electronics for their contribution of FX83 calculators to support the interactive show, "Algebra without Aggravation", and other workshops that TES Teacher Mathagony Aunt will be running.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses. Email your questions to Mathagony Aunt at firstname.lastname@example.org or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX