Brief Description

The general theme of this unit is to see if one can improve on certain tasks with practice. The comparison uses simple statistical techniques, and discussions help lay an intuitive feel for the sort of conclusion that can be drawn in such circumstances. Since reaction times are often included in science courses at about this age and the ability to estimate lengths and times is an essential part of any scientific training, this unit has major links with the work of the science department.

Design time: 4 hours

Aims and Objectives

On completion of this unit pupils should be able to calcuhte a simple range, find a mode and draw a scattergram. They will have practised tallying, completing tables, drawing bar charts and (in an optional section) calculating a mean. They meet examples of drawing inferences from tables, bar charts and scattergrams, of simple aspects of experimental design, of trends and their interpretation and of the comparison of two sample distributions. They should be more aware of the way variability can make it difficult to see trends.

Prerequisites

Pupils need to be able to measure a line to the nearest ¹/₂cm, to read a non- linear scale to the nearest unit and to complete simple tables by tallying. Some experience in drawing simple bar charts is desirable but not essential. They will need to know the order of the integers.

Equipment and Planning

Rulers and squared paper are required throughout the unit. A scale for taping on to a 30 cm ruler is provided on page R1 (in which case scissors and sellotape are required). Alternatively, the scale can be copied on to pieces of stiff card 30 cm long. Section B5 is optional for pupils who have had previous experience with calculating a mean. Blank tables for the various sections are provided on pages R1 and R2.

The main theme is: 'Do your skills improve with practice?'. This is introduced in general terms in the short Section A. Sections B and C introduce two particular skills reaction times in catching a ruler and estimating lengths. The effect of practice on these skills and on estimating times is considered in Section D.

Detailed Notes

Section A

The problems involved can be set in a wider context with such questions as the following. How accurate were you when you first threw and caught a ball? Are you any better now? Why have you improved? Is everyone as good as everyone else at this? Would they be if they had more practice? How quickly do you react? Can you dodge snowballs? Can you run faster each year? Can you calculate better each year? Can you go on improving for ever? Does practice always help?

The discussion can be led towards: 'Are you quicker at doing things with your right hand or your left hand?' (and hence bring out that any comparison should be between 'writing hands' of different pupils rather than 'right hands'). Which would you expect to show a greater improvement with practice - writing or non-writing hands? (Possibly non-writing, as the writing hand will already be good.) Lead on to the dropping ruler experiment of Section B.

Section B

This first experiment is not described in detail in the pupils' notes because of the need to ensure rapid understanding by the pupils. It is important that both the amount of practice is minimal and that the pupil understands what he has to do. Experience has shown that it is easier for pupils to use a 30 cm ruler for the experiment than a piece of stiff card marked with an acceleration scale. With less able pupils the experiment can be done solely using the centimetre scale of the ruler. There is, however, an important science concept that can be referred to here if the other scale is used, i.e. that it is acceleration not velocity which is constant, and it is the reaction times in which we are interested. Hence we suggest that the scale given on page R1 be stuck on the ruler for this experiment.

As a further link with science it would be useful to discuss the experimental design. Clearly it is important that all pupils do the same amount of practice and use either their writing or their non-writing hand first. The advantage of the latter is that pupils expect to be faster with their writing hand! An alternative is for one of each pair of pupils to use a different hand first.

Instructions
Work in pairs. Rest your arm on a desk with your hand just over the edge. Keep your thumb and first finger about one centimetre apart.

One pupil holds the ruler, as shown in the pupil's notes, with the zero mark level with the upper edge of his partner's thumb.

Without warning the first pupil lets go of the ruler. Without moving his hand downwards, the second pupil tries to catch the ruler as quickly as he can.

Find the reading on the ruler level with the top of thumb. Using the scale on the ruler, find out the reaction time in hundredths of a second. If the ruler is not caught, record a time of 25 hundredths of a second. Repeat a second time with the same hand and use this time for your results. Repeat with the other hand.

This table gives an accurate way of calibrating the ruler (see page R1). T = Time in hundredths of a second D = Distance in mm from zero

The class data are going to be analysed in several ways in subsequent sections. Pupils must therefore keep a record of their results. If you wish them to do B4, then the data should be recorded on the blackboard thus:

Possible questions that can be asked of the data are listed below.

1. Is the reaction time of the writing hand quicker than that for the non-writing hand?
2. How much variability in reaction times is there over the class?
3. (In coeducational schools) Are girls' reactions quicker than boys' reactions?
4. *Is there any relationship between the reaction times of the writing and the non-writing hand?
5. Do the reaction times improve with practice? (This is done in Section D.)

For (i) it is possible that there is a learning effect from one hand to the other, so half the pupils should do the first practice drop with their writing hand and half with their non-writing hand. These pupils would usually be chosen at random. Since the pupils are working in pairs, it is easier to get one to start with his writing hand and the other to start with his non-writing hand.

B1
You will also need to have another blank table on the blackboard or a large piece of paper, as follows:

Each pupil can then come out and make tally marks for his two times.

B2
Blank Tables 1 and 2 are on page R1.

This bar chart is a histogram and should have horizontal axis and vertical bars as shown:

There is no need to start the horizontal axis at zero, but you may like to so as to avoid any possibility of misleading representation. Pupils may need reminding to put a title to their chart.

In mixed schools two separate bar charts may be drawn, one for boys and one for girls. Tracing paper is useful for comparison later.

B3
With the brighter pupils you may like to try a direct comparison of the distribution. This can be done either as a a double bar chart or b by interchanging the axes as a two sided bar chart.

Mixed schools can ask similar comparison questions between boys and girls.

B4
The scattergram can be drawn as in the pupils' notes, in which case there is a possibility of having to put more than one symbol at the same point. (If this happens, they can be clustered or the number written beside the symbol.)

Alternatively, the diagram can be drawn as below and the symbols entered into the appropriate squares. Pupils may need help in interpreting it. Look for things such as most non-writing hands are slower than writing hands or quick rcaction times tend to occur with both hands.

*B5
This is an optional section for those pupils who know, or can easily be told, how to calculate means. The comparison could be of the form 'the mean time for writing hands was ..... hundredths of a second less than the mean time for non-writing hands'.

Section C

Notice that in this section no help is given in the guessing. This contrasts with Section D3 where we hope to see some learning from experience taking place.

C1
The blank Table 3 is on page R2. Lengths are measured in centimetres throughout.

C2
The actual lengths of the lines are (1) 5 cm (2) 6¹/₂cm (3) 4 cm (4) 9 cm and (5) 3¹/₂ cm.

To collect the pupils' data draw a blank table on the blackboard for tally charts to be filled in by the pupils.

C4
The discussion of the class's results could include the following points:

The table gives the distribution of estimates for each line. Look for such things as the bias in estimates, range in estimates, whether vertical lines are consistently underestimated and which lines were overestimated by most people.

The above can be a time-consuming exercise, and it would be valuable to have the bhckboard (or OHP sheet) prepared in advance. Alternatively, it may not be necessary to analyse all five lines in this way.

In C4 we take the absolute values of all the errors, i.e. ignoring the + and - signs, to give some measure of the pupil's accuracy. In this way two errors of, say, +5 and -5, show up as being worse than two errors of +2 and + l.

With brighter pupils you might like to take the + and - signs into account. A mean of 0 here tells you that the estimates were unbiased.

Section D

This section takes up the experiments of Sections B and C to see if pupils improve with practice. If time is short, the class could do one or other of D1, D2 or D3. Alternatively, one-third of the class could do D1, one-third D2 and one-third D3. One difficulty of measuring improvements is that pupils who are good initially do not have room for improvement. Another is that variability of results sometimes makes it hard to spot trends.

D1
In this experiment you hold a watch with a second hand. At the first trial you say 'Go' when the hand touches 0, and 'Stop' at some point which you select between 10 and 20 seconds. After the pupils have written down their estimate, you tell them the true time which elapsed. Move on to subsequent trials, each time telling them the true value after they have made their guess. After trial 10 each pupil calculates his or her 10 errors and plots the results.

Alternatively, to help reduce cheating, the pupils can work in pairs. One pupil says when he thinks a particular time (between 15 and 20 seconds) is up, the other writes down the actual time elapsed.

It is interesting to look for any indication of overcompensation.

D2
To collect the results from the class, have ready two tables like the following (one for writing hands and one for non-writing hands). Column 1 gives the results of each pupil's first attempt; column 2 his second attempt; and so on.

It should be possible to tell from the tally charts whether there has been any improvement, more improvement with non-writing hand, whether the results are inconclusive, etc.

A simpler representation giving just the ranges of reaction times can be drawn as above. This may be good enough to give a general impression.

D3
One difficulty with trying to measure improvement in estimating lengths is that longer lines usually lead to larger errors. The five lines here have all been drawn within a reasonable range to try to reduce this problem. Their lengths are 6¹/₂ cm, 7 cm, 5 cm, 8¹/₂ cm, and 4¹/₂ cm.

D4
This section summarizes the theme of the whole unit: Does practice make perfect? Questions b and c are deliberately open-ended to see how pupils tackle the problems. They may be omitted. Question d can lead to a useful class discussion of all the ideas raised in the unit.

Answers

Most of the answers depend on the pupils' own results.

Test Questions

1. Karen drew this bar chart of class reaction times (writing hand).
   
   Figure T1 - Class Reaction Times
   1. What was the mode (or modal) time?
   2. What were the fastest and the slowest times recorded in her class?
   3. What was the range of times recorded?
   4. How many pupils recorded the fastest time?
   5. How many pupils recorded the slowest time?
   6. How many pupils recorded a time of ¹¹/₁₀₀ second or less?
2. Table T1 shows the reaction times with writing and non-writing hand of 23 pupils in Karen's class.
   

   		Reaction times (hundredths of a second)
   		Writing hand	Non-writing hand
   Boys	1	16	18
   	2	20	21
   	3	18	19
   	4	19	22
   	5	8	9
   	6	22	20
   	7	10	12
   	8	17	15
   	9	19	20
   	10	10	14
   	11	17	13
   Girls	12	20	19
   	13	21	22
   	14	11	14
   	15	23	21
   	16	22	18
   	17	19	21
   	18	21	17
   	19	16	14
   	20	23	24
   	21	20	20
   	22	20	22
   	23	17	12

   Table T1 - Class reaction times (both hands)

   1. Plot these results as a scattergram. Use a '+' for a boy and an 'o' for a girl.
   2. Write two sentences commenting on these results.
3. David guessed the lengths of five lines. After each guess he was told the correct answer before guessing the length of the next line. His results are shown in Table T2.
   

   	True length	David's guess	David's error	Guess too high (+) or too low (-)
   Line 1	5 cm	71/2 cm	21/2 cm	+
   Line 2	6 cm	8 cm
   Line 3	71/2 cm	51/2 cm
   Line 4	3 cm	5 cm
   Line 5	61/2 cm	6 cm

   Table T2 - David s lines

   1. Copy and complete Table T3.
   2. Plot the results on a graph.
   3. Use the figures in column 4 to calculate David's mean error.
   4. David said his guesses improved with practice. Do you agree? Give a reason.

Answers

1	a	20 hundredths of a second
	b	8 hundredths of a second, 23 hundredths of a second
	c	15 hundredths of a second
	d	1
	e	2
	f	4
2	a	Watch for labelling of axes, accuracy of plotted points and title to the graph.
	b	In general: boys' reaction times were quicker than girls' reaction times; times with writing hand were less than times with non- writing hand
3	a	Column 4: 21/2 cm, 2 cm, 2 cm, 1 cm, 1/2 cm Column 5: +, -, +, -
	b	Watch for plotting above and below the x-axis, using the + and - signs as appropriate.
	c	Mean (absolute) error is 8/5 = 13/5 cm (If + and - signs are taken into account, the mean is 3/5 cm).
	d	Yes. His (absolute) error became less over the five lines.

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 1)

Shaking a Six
Being Fair to Ernie
Wheels and Meals
Probability Games
If at first ...
Leisure for Pleasure
Tidy Tables

Units at Other Levels In the Same or Allied Areas of the Curriculum

Level 2

Seeing is Believing
Getting it Right

Level 3

Net Catch
Cutting it Fine

Level 4

Smoking and Health

This unit is particularly relevant to: Science, Mathematics.

Interconnections between Concepts and Techniques Used In these Units

These are detailed in the following table. The code number in the left-hand column refers to the items spelled out in more detail in Chapter 5 of Teaching Statistics 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teaching the unit, but more practice can be obtained before or afterwards by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

Page R1

Time	No. of pupils

Table 1 - Class reaction times, non-writing hand

Table 2 - Class reaction times, writing hand

Figure 7

Page R2

	Line 1	Line 2	Line 3	Line 4	Line 5
My guess						cm
True length						cm
Difference						cm
+ or -

Table 3 - Length of lines.

Table 4 - Estimating time

Table 5 - Reaction times

Table 6 - Line guessing

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