15
Time
This chapter addresses two aspects of work with time: telling the time and simple problems involving time. We feel that the topic is often underrated in terms of its difficulty. This is probably because time is pervasive in everyday life and we take the skill of ‘telling’ the time for granted. Copeland (1984) observed that at age ten some pupils are still not ready for a true understanding of this concept. If the pupil is dyslexic or dyscalculic, even at the age of ten he may not be ready for mastery of time. In fact, being unable to tell the time is a classic weakness for many dyslexics. The advent of the digital watch and smart phone has enabled more pupils to ‘tell’ the time, but this does not mean they have any understanding of what they ‘tell’.
Telling the time or, preferably, understanding the time is an important life skill. Understanding the 24‐hour clock is an essential skill when travelling, but for many learners dealing with time in a 24‐hour context is much more challenging than using a.m. and p.m. and a 12‐hour clock. Time also shifts the number bases we use to 12 and 60. It also has a time line, as contrasted to a number line, that is a circle. There are other challenging differences, for example the number of words related to time (Haylock and Cockburn, 1997).
Our comments about the paucity of research into dyscalculia and dyslexia and mathematics pale into insignificance when we look for research into time.
What are the Potential Problems with Time?
Time is complicated by the large number of apparent inconsistencies that learners have to address (Chinn, 2001). Time involves new numerical ideas, for example using number bases of 12, 24 and 60. The language of time can be misleading, for example we say, ‘Five past one’ and write 1:05 or even more challengingly, ‘Ten to nine’ and write 8:50. The language of time itself has to be carefully explained and the language we use to explain time has to be carefully chosen to be as unambiguous as possible. For example, a classic mistake that American pupils make is to write ‘Quarter past four’ as 4:25, using the familiar money/dollar interpretation of ‘quarter’ as 25 cents. However, an analysis of the data used to standardise a 15‐minute test (Chinn, 2017a) revealed this as a common error in the UK, too. Another potential ambiguity is with 24‐hour time, where 08:00 is pronounced as ‘O eight hundred’, which does not reinforce the concept of 60 minutes in an hour.
There are directional complications, for example, we count on in minutes after the hour until we reach thirty minutes past the hour and then we countdown the minutes to the next hour, for example, ‘twenty minutes to six’ (Figure 15.2). Fractions are used, but only half and quarters (Figure 15.1). The numbers on a clock face only refer to hours. The user has to work out the minutes. A time may be written in a way that looks like a decimal, but 8.30 is in fact half past eight and 8.50 is not half past eight.
Figure 15.1 The clock face and quarters.
Figure 15.2 The clock face and halves: counting ‘past’ and ‘to’.
After working over the years with dyslexic pupils, we got used to being greeted with, ‘Good afternoon’ at breakfast time.
Setting the scene: the overview
The adage, ‘working from what the pupil knows to what he can know’ applies, of course, to time. Although digital watches, digital clocks and mobile phones are now common, the analogue clock face is still a familiar sight. The advantage of the analogue clock face is that it provides a context for time. It gives 12 a prominent place and 12 is an important part of many calculations involving time. It gives a visual image of time past and time to go.
So, a clock face is a good visual aid. A cheap cardboard play clock has the disadvantage of not having synchronised hand movements. It is possible to buy geared demonstration clocks where the hour hand moves as the minute hand is moved (see Appendix 2, Resources). Watching a working clock gives some idea of the relative values/speeds of second, minute and hour hands.
The clock face allows teachers to explain the key facts: that there are 12 hours, used twice in a day for a.m and p.m. (ante meridian/before midday and post meridian/after midday), that there are 60 minutes in an hour and 60 seconds in a minute, and that each hour mark also represents a five‐minute interval for the minute hand and a five‐second interval for the second hand.
Starting with the assumption that the pupil has some awareness of time, a few questions will determine how large or how little that knowledge base is. Questions that use the pupil’s experiences should be used, such as what time school finishes, what time lunch break starts, or what time a favourite programme begins on television. As pupils give each time, the teacher writes them on the board and shows them on a clock. Work of this kind focuses on showing the pupil the use of hours and minutes to identify the time, without the pupil having to find and read them from a clock.
Work can be focused on terms such as ‘o’clock’, ‘quarter past’, ‘half past’ and ‘quarter to’ in order to fix some key reference times and introduce the concepts of a mix of hours and minutes and of using ‘to’ and ‘past’. So, half past can also be expressed as ‘30 minutes past’. There is this flexibility in the language used for time and it needs to be introduced to the learner. For example, the relationship between morning and a.m. and between afternoon and evening and p.m. should be taught, highlighting the significant times, 12 noon and 12 midnight. If learners can grasp these key reference times they will be on the first step towards accuracy, but will also have an acceptable level of accuracy for many everyday needs. This acceptable level in everyday life often means saying, for example, ‘almost quarter past’ rather than ‘thirteen minutes past’, which can sound pedantic.
Other key ideas that need to be introduced in an overview are the circular nature of the clock, that is, for example, everyday has a 1:00 a.m. and a 1:00 p.m., and time moves in cycles. Also, there is our convention of counting the minutes after an hour only until we reach half past the hour and then counting down the minutes to the next hour (29, 28, 27, 26), so we start to name a particular hour 30 minutes before we reach it and cease using it 30 minutes after we pass it. There is an old conundrum, ‘How far can you walk into a wood? Halfway, and then you are walking out.’
Reading the Time
Times may sometimes be written as though they were a decimal number, for example, 6.21. This can create confusion with pupils writing ‘one and a half hours’ as 1.30 hours. This confusion between the 60‐minute nature of time and decimal notation may also encourage pupils to enter the time into a calculator as a decimal. We would recommend using the colon, as in 5:43 for example, to avoid this potential confusion in early experiences of learning about time.
Digital time is easy to read, but may not give the meaning of reading the analogue time. Older learners may be more comfortable with ‘Five to eight’ rather than ‘7:55’, possibly because it seems more relevant to everyday experiences, possibly because it sounds less pedantic, precise and formal or possibly because it gives them a better sense of their location in time.
Quarter past, half past and quarter to
These key reference times, phrased in the way we use them in the UK, should be relatively easy to master. Pupils may refer back to previous base ten experiences of quarters and half. First we divide a circle to show the hours and then mark the 60‐minute intervals. Pupils can then practise estimation skills by judging the time as being closer to one of these, for example 6:40 could be expressed as ‘almost quarter to seven’. The convention of counting on the minutes up to half past an hour and counting down the minutes to go to the next hour after half past an hour can be reviewed by focusing on quarters. As ever, the structure of any topic should incorporate as many reviews as possible and it is better if these are from slightly different perspectives each time. This strategy can be practised with a clock face. The pupils are shown a time, say 4:11 and need to say the nearest quarter, half or o’clock (‘quarter past four’).
There is some rationalisation in the use of ‘past’ and ‘to’ (Figure 15.2) in that we only refer to the nearest hour, so the nearest hour at 38 minutes past an hour is the next hour. Half past, that is 30 minutes past is the changeover point. There is a similarity here to rounding up and rounding down (Figure 15.3).
Figure 15.3 Four o’clock, half past four, five o’clock.
Minutes past and minutes to
This topic extends the work done on quarters and half. The first 30 minutes after an hour are normally referred to as ‘past the hour’. The next 29 minutes are normally used to count down to the next hour. The quarters can be used for mid‐point check values. So, the further the minute hand goes past the hour, the greater the number of minutes … counting up. The closer the hand goes to the (next) hour, the fewer the number of minutes … counting down (Figure 15.4a and 15.4b).
Figure 15.4 (a) Hours and minutes. Figure 15 (b) ‘Minutes to’.
The 24‐hour clock
While pupils come across times such as 8:30 a.m. and 10:15 p.m. on a daily basis, they will be less familiar with the 24‐hour clock. They may know, and should revisit this anyway, that there are 24 hours in a day, that analogue clocks almost always show only 12 hours and that digital clocks show either 12 hours or 24 hours when programmed to do so.
This topic could be introduced through the use of a train timetable or flight times. There are some quite simple timetables, only listing three stations, for example Taunton, Reading and London. The train timings can be demonstrated by moving the hands of a clock face, counting past 12 to 13, 14, 15 and so on, pointing to the p.m. time. Some idea of the lengths of these sections of the journey and of the whole journey may help in understanding the clock times.
The p.m. time and the 24‐hour time could be written side by side in a simple chart.
The pattern should be clear from this chart, but the additions of time with the 12‐hour clock and the 24‐hour clock produces some strange looking results when viewed with base ten experiences in mind:
Explaining this inconsistency in the rules of addition (because we are using base 12) as applied to the 24‐hour clock may help the pupil’s understanding of its relationship to the 12‐hour clock. This is another example of inhibition, the student has to inhibit their base ten interpretation, and another example of the tendency of first learning experiences, base ten, to be dominant.
A similar chart could be set up by the pupil for a typical day in his life, starting with waking up time, through school time, to dinner time and evening time. The conversion from p.m. to 24‐hour time requires the pupil to add 12 to the p.m. time. The conversion from 24‐hour clock to p.m. time requires the pupil to subtract 12 from the former time. Thus, this is another example of reversible operations.
The classic error is likely to occur when 20:00 is converted to 10:00 p.m. instead of 8:00 p.m. This example may need extra practice or can even be used as a key reference time.
Time Problems
Finishing‐time problems
Some typical questions:
Problem A:
If I start a journey at 9 a.m. and travel for 10 hours, when do I arrive at my destination? or, the more difficult:
Problem B:
If I start a journey at 8:45 p.m. and travel for 2 hours 37 minutes, when do I arrive at my destination?
Several alternative methods are available for these problems, which are essentially addition, but not with base ten. Once again, the use of alternative methods addresses the individual needs of learners and also provides a means for checking an answer.
Using the clock face as a number line and bridging the 12 boundary Problem A (Figure 15.6):
Use 12:00 noon as the key intermediate stage, so:
Figure 15.6 Using the clock as a circular number line.
Using a linear time line
Problem B:
The numbers on a clock face are rolled out to make a ‘time line’ and the journey is represented in stages.
- Stage 1. Move 2 hours down the time line, 8:45 to 10:45 p.m.
- Stage 2. Move 30 minutes down the time line, 10:45 to 11:15 p.m.
- Stage 3. Move the remaining 7 minutes, 11:15 to 11:22 p.m.
The method encourages the pupil to work with easy chunks of time. The principle of ‘easy’ chunks was used for both long multiplication and division (see Chapters 9 and 10). It may be necessary to discuss and identify what are ‘easy’ chunks. These are likely to be hours and half hours, and in some ‘moves’ there may be a back move to compensate for an over addition, for example while adding 55 minutes it may be effective to move down the line by 60 minutes and then back by 5 minutes.
This is another example of the use of the same strategies being used throughout the arithmetic curriculum, for example when nine was added by adding ten and subtracting one.
Conversion to the 24‐hour clock
If time can be converted to the 24‐hour clock format, then the travelling time is simply added to the starting time. This time can then be converted back into the 12‐hour format if required.
- Problem A:
- When 9:00 a.m. is converted as 09:00, then the travelling time is added on:
and 19:00 is converted back to 7:00 p.m. (subtract 12)
- Problem B:
- When 8:45 p.m. is converted as 20:45 (add 12), then the travelling time is added on:
and 23:22 is converted back to 11:22 p.m. (subtract 12).
With additions of this kind, it must be remembered that there are 60 minutes in an hour, so we are working with base 60 at the boundary between minutes and hours. It may help the student if the minutes are added as a separate sum and then converted from minutes to hours and minutes.
Elapsed time problems
These are questions such as the following:
Problem C:
A woman works from 10:00 a.m. until 3:00 p.m. How long does she work?
Problem D:
A journey begins at 7:35 a.m. and ends at 1:27 p.m. How long is the journey?
Problems C and D can be solved by using modified versions of the methods used to solve problems A and B.
Problem C:
This could be solved by using the clock face as a number line and bridging the 12 boundary (see Figure 15.7).
Or Problem C can be solved by bridging 12 noon without the use of the drawing of a clock:
- Time worked up to 12:00 noon, 12:00 − 10:00 = 2 hours
- Time worked after 12:00 noon, 3:00 − 0:00 = 3 hours
- Total time worked = 2 + 3 = 5 hours
The pupil has to understand that 12:00 noon also acts as zero for p.m. and a.m. time. 12:00 noon and 12:00 midnight are where 12‐hour day time returns to 0:00. It is a key fact in the use of a circular time line that is used for the analogue clock face.
Problem D:
Using a linear time line:
Without the time line:
- Time travelled from 7:35 to 8:00 = 25 minutes
- Time travelled from 8:00 to 12:00 = 4 hours
- Time travelled from 12:00 to 1:00 = 1 hour
- Time travelled from 1:00 to 1:27 = 27 minutes
- Total time travelled = 5 hours 52 minutes
Conversion to the 24‐hour clock
This transforms both the problems into time subtractions, where again the pupil must remember that he is using a 60‐minute number base for one hour.
Problem C:
Convert the finishing time
Subtract the starting time
Time worked (elapsed) 5:00 hours
Problem D:
Convert the finishing time <1:27, 13:27, 7:35>
Subtract the starting time
At this stage, there are more options, for example estimating that the answer is approximately 6 hours (slightly less) and a refinement of this estimate could be used to arrive at an accurate answer (comparing 27 minutes with 35 minutes, the adjustment is to take off 8 minutes from 6:00 hours).
Alternatively, we could take an hour from the 13 and change it to 60 minutes, using a decomposition/renaming method, but trading for 60 rather than the 10 used in a base ten number calculation
Summary
The language and vocabulary used for time is full of inconsistencies that will confuse many learners, so the language and vocabulary used to explain this concept must be exceptionally clear and cognisant of the potential problems. Once again, the principles of starting with ‘easy’ examples that can be referred to as exemplars of methods and for estimations may be followed. The bridging strategy and the traditional subtraction algorithm of decomposition/renaming are also used, where 60 and 12 are used instead of 10. The clock face and the time line can be used to provide visual images for the calculation procedures. It is worth reiterating: ‘Telling the time’ is a task where difficulty is frequently underestimated (and may well be taught to children when they are too young to cope with the peculiar challenges).