Stephen ARNOLD
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Technology and General Mathematics:
Mad if you don't!

Stephen Arnold
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Over the past two years I taught a senior General Mathematics class populated by those students who have always found their mathematics less than a satisfying educational experience. These are the students that the Board of Studies would claim should not be doing mathematics at all for their Higher School Certificate. And yet there they are – and few would drop this subject at the end of the Preliminary Year, no matter what I recommended to them or their parents. This situation, I believe, is repeated in every school across the State: students who previously may have been well-suited to the Mathematics in Practice course, now facing a subject that is quite clearly way beyond them.

What are our responsibilities as teachers in this regard? I counselled the students (and their parents), pointing out that they need not be doing mathematics at all, that this course is a very challenging one, even for many students who studied Advanced Mathematics at the School Certificate… even that they should be presenting their best subjects for their HSC – and this is unlikely to be one of those. And yet, for a variety of reasons, they sat there on the other side of those desks and looked to me to prepare them as best I could for the external examination at the end of year 12.

I decided to do something I had not done before – to take full and unashamed advantage of the available technology; in this case, Texas Instruments TI-83 graphic calculators, which these students have had since Year 9, but which had not been utilised to their full potential up to this point. And even though I have been using the technology for a long time now, I was surprised at some of the advantages that this technology offers my students. I have spoken before about technology potentially offering a more level playing field for all students – over the past two years I began to see what that could mean in reality.

Using the tool well

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Did you know that all graphic calculators will substitute accurately into formulas? Simply store the values into the variable names required, and then enter the formula (as it is provided on the Formula Sheet). Think about the number of formulas used in the General Mathematics course, and realise the extent to which this simple ability shifts the balance of power for our students.

Remember, these are students who have tried unsuccessfully for years to master the mysteries of number and letter. If they have not learned by the end of year 10 to successfully manipulate mathematical symbols and values, then there is little point in spending another two years trying to teach them. Just as we do with the standard calculator, we decide that certain computations will always be done with the aid of technology, and move on from there. My students had quite a good understanding of the principles of substitution, and the ideas behind the various formulas, but the calculator takes the guesswork out of the evaluation.

I also advised my students to use the MODE key to advantage. When working with money, set the mode to 2 decimal places and let the calculator do the rounding: correctly! If the question requires 1 decimal place, set the mode accordingly. I call it “taking the guesswork out” but for most of my students, it may mean the difference between getting some marks or none at all.

This use of variables goes further still. Consider graphical and tabular questions associated with compound interest. Again, they set the values for the principal and interest rate, but what if we wish to find the number of years it would take for my $2500 to double at this rate of interest? Enter the formula into the Y = menu, as shown, using x instead of n in the formula (since this is what we wish to find). Now simply view the table of values. Not 186 years, of course, but 186 months. There is still interpretation and understanding required. Technology can never be effectively used without some understanding of the mathematical principles involved – and that should be what we are about in our teaching, not rote learning of meaningless procedures. The advantage of using the symbolic formula in such questions lies in the ease with which students can change the variables and recalculate the result. “What-if” games remain a powerful teaching technique.

Perhaps even more powerful mathematics lies in the in-built Equation Solver found in graphic calculators. I teach my students how to solve linear equations – and they learn to do them well. It is something they become proud of. This allows them to perform simple manipulations. Consider solving the question above: when does P(1 + R)N = 5000. On the TI-83, they must bring everything to one side, leaving P(1 + R)N – 5000 = 0. No problem. Now solve for N.

All this without even touching the in-built Finance package available. And it can be used in so many different parts of the course (I am thinking areas and volume questions, particularly).

Did you also know...?

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The list editor and statistics capabilities of the graphic calculator may very well be its most powerful and useful feature, although perhaps under-utilised in this state in the past. Certainly, the ability to easily produce a range of data plots is of great advantage to students, but did you know that they can also produce frequency distribution tables from raw data?

Consider a set of twenty scores on a test out of 5 marks. Have students enter these directly into L1. Create a simple histogram, adjust the window settings to whole numbers and view the graph in TRACE mode. Students can easily read off the frequency of each score – no tally marks, no errors. One more place to remove both guesswork and likelihood of error.

Now choose the CALC submenu from STAT, and choose 1-variable stats to easily calculate all the statistics needed from this data.

Of course, if the data is already given in a frequency table, this can be entered straight into L1 and L2, and graphs and calculations follow readily. Anyone who claims that using a graphic calculator offers no advantage over a scientific calculator in the field of statistics is kidding themselves.

When we were studying shares, I had my students select three companies from the Stock Exchange listings, purchase enough to make up to $1000, and then follow their ups and downs over a six-week period. For my students, making three sets of numbers add up to $1000 was a major challenge, so we used the list editor. We entered the numbers of each we wished to buy in L1 (the first attempt were just guesses) and the share prices in L2. Then out to the home screen to calculate the sum of L1*L2. Repeat this process until we were close enough to $1000.

This was a surprisingly involving and empowering lesson for my students. The calculator supported them in making meaningful mathematical decisions in order to reach a goal. They got a taste of something positive involving numbers. They then enjoyed tracking their share prices and viewing the graphs of their progress.

The topics involving graphs and related algebra, of course, are also well supported. Certainly, topics such as simultaneous equations would be out of the question for my students, without the assistance of their graphic calculators. And yet, with these tools such difficult topics become achievable. The level playing field at work again.

The Y = editor becomes their friend when doing algebra work rather than statistics. Using the table of values, my students have become quite proficient at linear modelling questions. They learned to recognise the gradient (What are the numbers going up by?), put that next to the x, and then usually use trial and error to find the constant term. This is an active and intelligent use of the technology to support a mathematical process of some value.

Interestingly, my students have also been introduced to one of the most powerful features on the graphic calculators: regression equations. Consider the standard conversion of degrees Fahrenheit and degrees Celsius. Given the two relationships for freezing and boiling points, these can be entered into the list editor, then linear regression chosen to produce both the linear model and, if desired, the graph of the conversion. Powerful mathematics indeed, but a meaningful and simple means to an end for these students.

I am sure that some would argue against this approach, and I do not recommend it for every class and every situation. More able students should be exposed to mathematical procedures for these topics…and yet I begin to sense a lingering doubt about the value of time spent in teaching procedures such as simultaneous equations to students studying a terminating mathematics course. Be that as it may, if I choose not to offer my students such clear advantages for their HSC study, whose interests am I serving?

Tools for teaching

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The various features described above are available on just about any graphic calculator, and will be with the students in every examination and test. Over the past year or so, however, a growing array of additional tools have begun to appear from Texas Instruments. These are called APPS (little applications) and they certainly represent a significant shift in the nature and capabilities of graphic calculators. These are a far cry from the simple programs we may have used in the past – these are fully-fledged teaching and learning aids, not available for examinations, but still important additions to any classroom.

When I taught gradient and linear functions earlier in the year, I was able to make use of a wonderful app called Topics in Algebra Part 1. This is almost a textbook in itself, with a wonderful selection of explanations, worked examples, and even animated displays. This would be great learning material on a computer – on a calculator it is just amazing. I coupled this material with students using a CBR™ motion detector, in order to build up an active and robust understanding of gradient as a measure of velocity in Distance-Time graphs – always great fun and a powerful learning learning experience at any level.

The excellent AreaForm app offers students clear animated definitions, examples, explanations and quizzes on the plane figures required for the course. Like many of these new generation tools, students are actively supported in both the learning and assessment of major topics of the course. ProbSim offers a fast and effective probability simulator for half a dozen different experiments, including tossing coins, drawing cards, spinners and more, all customisable, all visually effective.

Possibly even more exciting than such specific purpose software tools may be the growing range of open-ended apps, which now extend to include a full spreadsheet (CellSheet™), a word processor (NoteFolio™), personal Organizer, dynamic geometry package (CabriJr™) and even a StudyCards™ app which allows teachers to create their own tests and lesson displays involving both text and graphics – a poor man’s Powerpoint®, if you like!

These new generation calculator tools are not only transforming my teaching and learning practices, they are even changing my day-to-day organization. My TI-83 Plus Silver Edition has become my personal organiser (I have my daily timetable, appointments and contact details on hand always), my markbook (I have created a markbook template in CellSheet™ which is able to link with Excel®) and now even my jotter and notepad. I have always used a TI-92 Plus (or the new Voyage™ 200) as my personal organiser. I now use my 83 Plus. With the advent of the new keyboard (and dynamic geometry as well!) I am finding less reason to look elsewhere.

It is my belief that these tools are fast becoming the next generation of personal technology for schools. Laptops for students (and even their teachers) may be more powerful, but they are far too expensive for most schools, too heavy and not robust enough for the day-to-day demands of school life. These calculators are designed specifically for these conditions, they are compact, able to be used in an increasing array of subject areas and far more affordable than any other alternative. The challenge will be for schools and administrators to stop seeing these as “calculators”, as tools for mathematics and science only, and to realise that they truly have become “personal learning tools” which are cross-curricular, appropriate and affordable for students and their teachers.

Use such tools in your teaching in the future? Mad if you don’t!

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