Basic Skills and Technology- not a Contradiction, but a Completion |
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by Josef Böhm | ||||
Bundeshandelsakademie
St. Pölten T3 - Austria and DERIVE User Group |
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Abstract | ||||
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At almost all discussions on the use of CAS in teaching mathematics the question appears mathematical basic (manipulating) skills are lost by using computers or not. At the same occasions it is emphasized by the members of the discussion that there is a need in investigating which basic skills are still necessary and in which amount. I start from my point of view that - at this moment - we cannot and should not do without certain basic skills in mathematics. I feel unable to give any forecast for the future. I am sure that we could significantly rise the acceptance of technology in teaching mathematics by sceptic and critical teachers or technology refusing educational systems if we would be able to apply technology not only to present real life problems or new - for some people radical - didactical approaches but also to improve basic calculating and manipulating techniques and mathematical capabilities. We can very consciously practise all those techniques using technology which could be neglected by a (too?) intensive use of the technology. It would be necessary
and helpful to set up a list of such basic skills which teachers would
like to have practised by their students. From my own experience I can
say that it is very useful to provide tools for the pupils - or students
which enable them to help themselves in case of troubles and difficulties.
It is not sufficient to have a textbook with lots of examples followed
by the solutions and the students don´t find the way from the problem
to the solution. At the other hand it is not sufficient to have problems
followed by a step by step solution. Sometimes it could be useful to work
out a student - computer interactive strategy to help overcoming some
deficiencies. |
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Incited by ideas from Jan Vermeylen (Belgium), Heinz Rainer Geyer (Germany) and Johann Wiesenbauer (Austria) I tried to produce training- and practise programs for my students (secondary level II), which I want to present as seed for further developments. The platforms are DERIVE and the hand held revolution, TI-92/89. The only reason for this choi ce is because I mainly use those to CAS. I am sure that you can use each of the available systems. The goal should remain the same: developing softeware products which will meet our - the teachers´ and the students´- very special wishes, ideas and visions. I have tried to work
with such tools since long. In my first years working with computers I
produced programs written in BASIC. It is quite nice to follow the development
of hard- and software on one application: The Rule of Vieta, presenting
random generated examples. |
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Then I changed to DERIVE working with a class in the computer lab: | ||||
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The idea
remained the same, the tool changed and it changed once more to hand held
technology. I rediscovered my old BASIC-program, converted it to the TI´s syntax and my students practised "Vieta" in the lessons, in the breaks, at home, sometimes in the train or sometimes just for fun or – how a girl told us – as a means to find her concentration before learning for other subjects. |
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Practise the Rule of Vieta! | ||||
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Allow me another memory of my BASIC past: a program to revise the students´ knowledge on set theory. I am still using this tool, the students like it and it doesn´t take too long time to achieve quite reasonable results. | ||||
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Some other examples are following: | ||||
Practise factorising (Jan Vermeylen) | ||||
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POL(n) produces n random number generated problems. The students have to factorise using pencil and paper and then check their solution. The file could easily be changed to practise special types of factorizations if necessary. Settings of factorization modes - Rational, Complex, ... – gives a nice variety of points of view. |
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(This example is some years old and could be adapted to other platforms) Then it could look like the next trainings tool (JosefBöhm). My students like it and use it: |
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I will
demonstrate the program including error analysis and Help function. We want
to practise cubics of binomials ![]() |
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The screen shots are self explanatory. | ||||
They practised squaring and cubing binomials, squaring trinomials, multiplying binomials mentally and with "The Quiz" they were presented a random generated sequence of problems. So you can have competitions in the class room and the pupils become very,very busy, even the weaker ones. | ||||
The next example is from H.R.Geyer to help students improving their mentally calculation skills. | ||||
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![]() J. Vermeylen produced a DERIVE file to have lots of "standard problems" dealing with arithmetic series. "verschil" is Dutch, it is the difference d of the series. |
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The next screen shot
is the final result of a very busy discussion in the DERIVE community
to produce a sort of report when solving an equation and applying various
equivalence transformations. So students are able to reproduce, what they have done using paper and pencil and check their results with the computer´s ones. They find their mistakes by themselves. |
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I found it useful and necessary as well that students can find the equation of a linear function, seeing its graph. They also should be able to have an imagination of the graph seeing the linear function very quick: "Match the Line!". Then I added some other tasks, as finding parallel and perpendicular lines by inspecting the grid and applying the various forms of the equation for a line | ||||
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Another course of the Lines-Menu: "Find two points!" You enter the coordinates of two points which should ly on the given line, see two "balls" on the grid and if you are lucky the line passes the two "balls". The linear function appears in various representations (explicit, implicit, ..) |
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Finally I produced a "game". Give some random points on the grid and let the students hit them with as few lines as possible, or do that with parabolas ...... | ||||
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Another tool helps my calculus classes to improve their calculus techniques. I transmit the program calc() on their TI-92/89s and then they are able to practise whenever they have some time and what they believe to be necessary for them: | ||||
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I know that there can be lots of improvements. It would be better to explain the way how to do it correctly to have really a learning tool. What do you say to the next program, written by Philippe Fortin from France. He calls his program "stepder" from "Stepwise deriving functions": | ||||
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The last example is part of a whole sequence to teach the pupils how they can help themselves reaching a sound level in working and manipulating with fractions. | ||||
Before working with fractions we had together developed a function kgv to calculate the LCM of two general expressions, to check the common denominator and then we used a selfmade function ew(expr) to find the expanding factor for the denominator expr with a common denominator gn. | ||||
Step by step the students can perform their calculation and check it using the calculator. | ||||
From the TI 92 Worksheet 3 | ||||
We have predefined a function ew(expr) which returns the expanding factor for a fraction with denominator expr and the given common denominator gn | ||||
Train your skills in calculating and manipulating with fractions | ||||
Example 2: ![]() |
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Calculate without the TI. Write down your result | ||||
Here it is. | ||||
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Can you find a reason why this calculation could cause a very special mistake? | |||
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Compare
the common denominator and the factors used to bring all the numerators
over a common denominator.
Explain the last part in the edit line: -4 * gn: How
to work with an expanding factor |
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The remaining
work is very easy:
Explain the relation between left and right hand side in the last row of the history area: |
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The underlying basic idea sounds very strange and controversary: | ||||
In addition to all the wellknown reasons to include modern technologies into mathematics teaching, which I like and support in a very high degree, I also use the technology in many cases to improve manipulatings skills which seem to be no longer of any importance by using these technologies. If
you would like to try some of the tools presented in this paper, then
please contact me. I would appreciate any suggestions and ideas dealing
with this topic. The TI-92/89 programs algebra(), lines() and calc() are
distributed together with the bk teachware books (references). |
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qu= presents a square a random generated square, ku= a cube, tr= a square of a trinomial, pr= a product of two binomials of form (a var1 + b var2) (c var1 + d var2) and the student is asked to perform the multiplication mentally, di= gives a product of form (a var1 + b var2) (a var1 - b var2) and finally te= brings a randomly chosen task on the screen. qub(n)= presents n problems of form qu, etc. You can see that the learning and practising tool will not only provide the correct result with res=, but and that is important gives an advice where to look for the mistake. |
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The underlying programming structure can be found in [9]. | ||||
I used a similar idea to practise factorising general expressions and/or polynomials | ||||
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DERIVE "offers" the task and the student who should perform the manipulation using paper and pencil and then check his/her answer. See one example for factoring out ("Herausheben") and a sequence of three problems to apply factoring out together with factoring a difference of squares. | ||||
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The acceptance and the enthusiasmus of the students were the best confirmation for my work. | ||||
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One can
factorize polynomials on different levels of mathematical knowledge:
ch= gives rational linear factors only, chw= returns also the irrational way and if appropriate chk= shows the complex factorization. |
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References: [1] Geyer, H. R
,Vermeylen, J.,Wiesenbauer, J., Welke St. (1998). From Nested Ifs to
a MACRO for |