November 1996

I don't like drill and practice but it works, for some things

This year while teaching a Year 10 maths class I programmed my own Quadratics software in logo for student use.

The impact on the class was immediate and positive. Many students in the class had previously been bogged down in substituting negative numbers into quadratic expressions and getting nowhere fast. Suddenly, for them, things began to fall into place. Freed from the requirements of doing many rapid substitutions and calculations (generate table of values, draw graph, then start looking for patterns) they were suddenly able to see the relationship between the 'a', 'b' and 'c' values and the variation in shape e of the parabolic curve. Rather than having to concentrate on the computation they could begin to concentrate on the patterns. By the 'a', 'b' and 'c' values I mean the values in this equation:-
y = ax2 + bx + c and how changing 'a', 'b' and 'c' will effect the parabolic curve.

I was so encouraged by this turn-around that I began to burn the midnight oil adding extra features to my software. This was an interactive process because I was perceiving students needs in lesson time and changing the software at night to meet those needs.

I hadn't anticipated that so many students in this "extended" class would have major difficulties with "basic" skills that "should" have been mastered in Years 8 and 9. Yet when I presented students with an equation like:-
y = 2x2 - x + 3
and asked them to substitute x = -2 into it, then the success rate was not too high! So, one feature I added to my software was a drill and practice substitution into a quadratic equation. Students were given 'a', 'b' and 'c' values and an x value to substitute and required to calculate the value of the function, or the y value.

For example:
y = ax2 + bx + c
if a = -1 b = 2 c = 3 and x = -1 then what is y ?

I found that the software released me from "lecture mode" and I was able to use much more time meeting some urgent needs of individual students while the others were happily occupied with the program. I could spend substantial slabs of time with a handful of students who really did need quite a lot of help. I could feel the mood changing in the class. Equations and parabolic graphs could be generated in seconds rather than many minutes. The students were able to concentrate on the structure of the parabola and how it was effected by changing a, b and c values without being tormented by their low skill level (in quite a few cases) in calculating the substitutions required to draw the curve. I did receive a lot of spontaneous positive feedback from students about the usefulness of the software.

Another thing I noticed was that the more able students in the class quickly mastered the program. They accepted it as a challenge to be quickly mastered and did just that. Then some of them would boast about it, "too easy sir", comments like that.

So, I began to add more advanced features to my program, to extend the advanced element further, to push out the leading edge. How do you find the axis of symmetry in all cases? How do you find the y value at the turning point? How do you find the x intercepts in certain specialised cases? We have not yet got to the stage of doing the full quadratic formula (that is part of the Year 11 Pure Maths course) but with the aid of my software I was fast approaching that point with the advanced element of the class . The leading edge was being extended, visibly.

So my program was catering for the needs of students across the whole ability range. It could do that because I was writing it and rewriting it on a weekly basis. I see that as a major advantage over a commercial product.

Some students were thrown in their pencil and paper work when the quadratic had a large 'b' value and they had mapped out a table of x values from +3 to -3 and the axis of symmetry might lie on the edge or outside of this domain. Lacking any knowledge of the overall structure of the curve (importance of axis of symmetry and turning point) their performance in mapping the correct graph was poor in quite a few cases.

My understanding and appreciation of this problem and other nuances of quadratics increased dramatically in the course of writing the software. For instance, initially I made the program draw the parabola by starting at one end and drawing to the other end. This created all sorts of problems at the limits because as the equation changed so did the limits. The effect was that some of my curves did not even begin to be drawn, I couldn't keep them on the screen. I eventually solved this frustrating problem by starting to draw the curve at the turning point of the parabola, drawing one side to the outer limits, then jumping back to the turning point and drawing the other side. This problem solving process reinforced in my own mind the central importance of axis of symmetry and turning point in the teaching of quadratics. The mechanical plotting of x values between +3 and -3 often just does not work in the case of quadratics with large 'b' values because the axis of symmetry has moved so far to the right or left.

All the signs of a class being turned around from just battling through to success were there to see. Students became more engaged in the tasks, they asked many more questions than previously, you could visibly see the confidence of many students increase, they became more animated and more positive in their relationship with mathematics and the teacher. Moreover, I felt that I could set more difficult and challenging questions in the program and subsequent tests than I would not otherwise have been able to do.

Looking in my marks book I can see that at least 7 students out of 27 have turned their results around from failing badly to pass marks and in some cases highly successful marks. I'll cite some statistics from my marks book to try to convince, you, the reader (who wasn't in the room to see the change) that a very significant turn around did occur. The Quadratics unit was a 6 week block. I did not use the computer software for the first two and a half weeks because I had not finalised it. In that first two and a half weeks I was mainly using lecture, textbook and homework mode. I also used one interesting activity from MCTP (Algebra Walk, pp. 213-18). In the third week I tested the students only on their ability to substitute values into an equation (two quadratics and one straight line) and plot the graph (first test). The results were poor, average class mark was 56%. I then introduced the Quadratic software and used it extensively for the next 3 weeks. In week 6 I tested the students twice. For test 2 they had to plot a quadratic again and also make predictions from other quadratic formulae about how altering 'a', 'b' and 'c' values would affect the y intercept, axis of symmetry and whether the curve was upright or upside down. This time the average mark for test 2 was 82%, a remarkable improvement over the first test.

For the final test (test 3) I offered students a choice - either do a pencil and paper version or a computer version. Nearly all students opted to practice for the test on the computer and 11 out of 27 choose to do their final test on the computer. One interesting aspect of this was that the computer test was set up for mastery learning. If a student got a question wrong they were invited to try again. They couldn't proceed to the next question until they got the previous one correct. Initially I had programmed it differently, that if a student gave a wrong answer, they got a "no" message and then the problem just disappeared and the next question appeared on the screen. However, when I was doing the test myself, I found this feature incredibly annoying, that when I got the wrong answer, I didn't have the opportunity to try again or to reflect on my mistake in any way. So I changed it. If the technology makes it easy then it seems silly not to use it.

So, conceptually, the final testing process for students who opted for the computer version was very different. They were being continually informed of their progress score as they went along. If they got a wrong answer they were required to persist until they got it right. In their final score this appeared as a larger denominator. If they did the test and didn't like their progress, they had the option of starting over again if time permitted. The program simply generated different questions (of the same type) each time it was run, so it was no difficulty for me to offer multiple chances for retesting.

There was some interesting discussion at the end by students about their reasons for which type of test they chose. Some high ability students said they found practising on the computer very useful but clearly saw it as risky to do their final test on the computer, given their established mastery of the pencil and paper medium. Other high ability students were confident enough to take that risk. Other students said they found it easier to solve the problems on the computer. Some made comments like "its faster". This was interesting because the same problems (actually the computer test had a greater variety of problems) were being set in both mediums but many students clearly felt that it felt very different and expressed preference for one over the other. Another factor was that doing the computer test was more public, less private. The room is set up with the computers around the walls so that all computer screens face towards the centre of the room. This made "collaboration" easier ("cheating") but also made mistakes more public.

A comparison between the final test results was also interesting. I offered 3 tests in total over 6 weeks of instruction (12 * 100 minute lessons), the first two tests were pencil and paper only but in the last test students were offered a choice (either computer or pencil and paper). Mainly due to high absenteeism only 17 out of the 27 class members sat for all 3 tests. Fortunately for the last test (test 3), this group of seventeen split themselves into roughly two equal groups, one group of 8 who chose to do the computer test, the other group of 9 who chose to do the paper and pencil test. For the previous two tests (tests 1 and 2) the percentage results of these two groups was roughly the same (71% versus 68% average). But for the final test (test 3) the group who chose the computer test scored an average of 95% compared with 68% for the pencil and paper group. Quite a difference !

I have explained above that the two tests were not really comparable (even though the questions were of the same type) because the computer based test provided instant feedback and monitored progress. Once again I would argue that it would be ridiculous not to incorporate these features into the computer program since they greatly assist in keeping students focused and motivated. This introduces formative elements into a summative test, which from a learning viewpoint is surely a good thing.

Here is an example of how students who did the pencil and paper test were disadvantaged. One question asked for the 'a', 'b' and 'c' values of this quadratic:
y = x2 - 4

Two of the top students (averages in mid 90's for first two tests) in the class got confused on this question and made this elementary mistake:-
a = 1 (correct)
b = -4 (wrong, the answer is b = 0)
c = 0 (wrong, the answer is c = -4)

Since they made this mistake they also got wrong the y intercept, axis of symmetry and y value at turning point, losing 5 marks in total.

If they had been doing the computer test then they would have received instant feedback on their first error, b = -4, and would have easily corrected it (being in the high ability range), resulting in the loss of only 1 mark.

The program at this stage has these features as displayed in the main menu:-

Final evaluation by students:-

I prepared a final evaluation sheet for students seeking their opinion of how they had learnt about quadratics. Twenty students successfully completed the final evaluation sheet. I asked them to evaluate 8 possible modes of learning according to this scale:-

When I totalled the results the Quadratics software program came out on the top of the list: 10 students wrote that it helped lots, 8 said helped a fair bit, 2 said helped a little bit and none said that it didn't help at all.

"Indicate how much each of the following helped you learn Quadratics using this code. Write a number next to each statement below."


Test 1 (end of week 3):

Average class mark = 56%

Plot these 3 graphs on the same set of axes. Show tables of values:-

Test 2 (week 6):

Average class mark = 82%

  1. y = 2x2 + 4x + 1
    1. Find y when x = 1
    2. What is the y intercept?
    3. Calculate the axis of symmetry (Hint: AS = -b / 2a)
    4. Is the graph upright or upside down?
  2. y = x^2 - 2x - 3
    1. Find y when x = 3
    2. What is the y intercept ?
    3. What is the axis of symmetry?
  3. y = -0.5x^2 + x
    1. Find y when x = -2
    2. What is the y intercept?
    3. Calculate the axis of symmetry.
    4. Is the graph upright or upside down?
  4. y = x^2 - 2x - 3
    1. Calculate a table of values, eg. x = +3 to -3
    2. Draw axes, plot the graph
    3. What is the y intercept ?
    4. Draw in the axis of symmetry.
    5. Work out the x and y values at the turning point.
    6. What are the x intercepts ? (there are two of them).

Test 3 (week 6) pencil and paper version.

Average mark for those who chose this test = 68%
Average mark for those who chose comparable computer test = 95%

  1. y = -2x2 + 2x + 1
    • x = -2
    • Calculate the y value
  2. y = 3x^2 - x - 2
    • x = -1 Calculate the y value.
  3. a = 2, b = 2, c = 0
    • Find the axis of symmetry.
    • a = -2, b = 4, c = 3
      • Find the axis of symmetry
    • y = x^2 - 4
      • Find the a, b and c values
      • Find the y intercept
      • Find the axis of symmetry
      • Find the y value at the turning point
      • Find the x intercepts
    • y = 2x^2 + 4x
      • Find the a, b and c values
      • Find the y intercept
      • Find the axis of symmetry
      • Find the y value at the turning point
      • Find the x intercepts
    • y = (x + 3)(x - 2)
      • Find the x intercepts
      • Then expand the brackets using FOIL and
      • Find the a, b and c values
      • Find the y intercept
      • Find the axis of symmetry
      • Find the y value at the turning point
Bill Kerr, email: billkerr at gmail.com
Woodville High School,
South Australia

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