Digging Deeper into … Time (all classes)
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Time is an integral element of our daily lives and therefore an essential, “need-to-know” topic in primary maths, particularly for those children with special educational needs or learning difficulties. However, it is also a very difficult mathematical concept, and one with which many children struggle, for a number of reasons:
- Time is abstract; it cannot be touched, or manipulated, or seen (although we can see its effects when we look in the mirror!).
- The standard units of time do not reflect the familiar structures of our base-ten place value system i.e. 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week etc.
- It can be displayed in analogue and digital forms and, furthermore, digital times can use a 12 or 24 hour system.
- It is not uniform around the globe; each country belongs to a time zone and the time is different in each time zone.
Furthermore, there is a distinct difference between identifying a particular instant/moment in time (i.e. the skill of reading or telling the time) and understanding the concept of duration and the passage of time. In many cases children may know how to tell the time, without having any real understanding of the concept of duration. Remember, that just because a child can read the digits on a digital clock/watch, or even read time from an analogue clock/watch, this does not mean they understand time as a concept.
As explained in all of the previous Digging Deeper posts, Operation Maths is based on a CPA approach. And, while we acknowledge that given the abstract nature of time is can be difficult to represent it concretely, Operation Maths does introduce the empty number line as a way to pictorially represent elapsed time (see below in post).
Telling time requires the children to:
- develop an understanding of the size of the standard units of time eg days, weeks, hours, minutes
- be able to estimate and measure using units of time
- read and tell the time using both analogue and digital displays (12 hr & 24 hr).
The analogue clock has its own features that can further complicate matters; there is a “past” half and a “to” half; when reading times on the hour, the hour is said first (eg 3 o’clock) but when reading all other times the hour is said last (eg half past 3). Other valid questions that a child might have about the conventions of reading time include:
- Why do we only say half past; why not half to?
- Why do we say half past; why not 30 past?
- Why do we say quarter past/to; why not 15 past/to?
More often than not, when teaching analogue time, teachers tend to get the children to focus on the position of the long/minute hand and to use that to tell the time eg “if the minute hand is at 12, it is o’clock”, “if the minute hand points at 6, it is half past” etc. However, this type of explanation can in itself be very confusing, with many children interpreting half past any time as 6 o’clock, quarter past any time as 3 o’clock etc.
In fact, the first thing a child should be able to read is the bigger unit of time i.e. to identify the hour. To do this, they should, logically, look at the hour hand, which although it is shorter than the minute hand, can often be wider on real analogue devices, emphasising its significance. When the hour hand points straight at a number, then it is that time; when it points half way between two numbers, it is half past the previous hour (also the lesser number). In this way, the children will be enabled to tell time in relation to the hour eg “it’s nearly 2”, “it’s just gone past 7”, “it’s around half 5”, etc. Consider also how many children, when drawing hands on a clock to show time, will often have the hour hand pointing straight at the number even if it is half past, quarter past or a quarter to the hour. Focusing initially on the hour hand rather than the minute hand, highlights the fact that the hour hand also travels around the clock as the hour passes, and doesn’t jump from one hour to the next. A real or made clock with only the hour hand can be very useful here to teach this concept. Or use the Operation Maths Clock eManipulative (pictured below), and ask the children to look only at the red hour hand. The Two Clocks problem from NRICH can also be used to reinforce the importance of the hour hand.
Next, draw the children’s attention to the blue minute hand and to the blue minute markings around the edge of the clock. Logically, the minute hand has to be the longer hand because it points, beyond the numbers, at the minute markings which are furthest out from the centre, whereas the hour hand points to the hours (numbers) which are typically closer to the centre, and thus the hour hand is shorter. Emphasise that the minute hand enables us to become more accurate in our measurement of time. “O’clock” is actually an abbreviation of “of the clock”, so then when the minute hand points to the top of the clock , it is o’clock. Avoid saying “when the minute hand points to 12 it is o’clock” as the minute hand is actually pointing to the minute markings around the edge and not to the hours, which instead is the purpose of the hour hand.
At this point, it can be useful to use a cut-out paper circle (see below) to represent the clock and fold it in the centre to show both halves of the clock. Thus we can explain that when the minute hand points at the bottom/base of the clock, it is half past, as the minute hand has now passed through half of the clock. In a similar way, use the paper circle to make quarters and emphasise that when the minute hand has passed a quarter way through the hour, it is a quarter past, and when the minute hand is a quarter away from the next hour, it is quarter to. However, it should be acknowledged that, while it would be mathematically correct to say it is three quarters past the last hour (which prepares them for digital time), the convention used is to describe time in terms of how it relates to the next hour once it has passed the half-way point, at the bottom of the clock. Return to the paper circle at this point and label each half “past” and “to”. “Past” and “to” are also clearly labelled on the Operation Maths Clock eManipulative (see above), along with arrows to indicate the clockwise direction in which the hands travel.
When the children are ready, they should begin to measure time in minutes also (reading time in 5 minute intervals is on the curriculum for third class). Again, focus their attention on the blue minute hand and on the blue minute markings around the edge. In this way, the children may initially begin counting the minutes in ones around the edge before realising that it is more efficient to count in groups of five, which co-incidentally, are also marked by the hour numbers. Creating and using peek-a-boo clocks can help reinforce this idea.
It is essential that when teaching digital time that the connection between analogue and digital is emphasised from the start. The Operation Maths Clock eManipulative is very useful in this regard as both clock types can be shown concurrently (see below). It is also includes the feature to hide/reveal the time in word form, as well as the feature to produce a random time on either clock, which can then be displayed manually on the other clock to match. For ease of use the teacher can also select the options that best suits the class level and ability eg hours only, time to half hours, quarter hours, five minutes, individual minutes.
Equally customisable is the What Time is it Mr. Wolf game from Teachers Pet. A free version can be played online and there is also a paid app available to download from the iTunes store for iPads.
Time as Duration
It is very important that the concept of time as duration is emphasised from the start. Duration of an event requires noting the starting and finishing points of time.
Developing a solid understanding of duration develops from a child’s experience and understanding of:
- Sequencing activities (eg pictures of familiar/daily events, seasons etc)
- The language of time such as before, after, soon, now, earlier, later, bedtime and lunchtime
- The standard cycles of time (seconds, minutes, hours, days, months, seasons etc ) which in turn follows from the sequencing of daily events.
- Measuring duration (using non-standard units initially, and then standard units)
- How long does it take to …..? Estimate & measure
- How many ….can you do in …? Estimate & measure
- What if you do it faster/quicker…? (this in turn develops an understanding of the relationship between speed and time)
- Comparing the duration of two events, (using non-standard units initially, and then standard units) eg what takes longer/shorter? How much longer/shorter is ….. than …..?
Developing an understanding of duration also requires the ability to visualise the passage of time in some way. For this purpose, empty number lines, one of the key problem solving strategies used in Operation Maths, are extremely useful. This can start with drawing an empty number line on the IWB or on the Operation Maths MWBs, to which class/question appropriate details can then be added:
- What hour is after 9 o’clock? 11 o’clock? 12 o’clock?
- Ann’s school starts at 9 o’clock. What time is it 2 hours later? 2 hours earlier? 5 hours later?
- If Ann’s school finishes at 2:30 for how long is she at school?
- Umair’s school starts at 9:30 and finishes at 3:00. For how long is he at school?
- How long is it from the start of the 11 o’clock break to the start of the next break?
In Operation Maths, empty number lines are presented as a viable alternative, to the traditional column method approach, for calculating time. In many other countries, the traditional column method used for calculations involving units, ten, hundreds etc., is not encouraged, or not used at all, for calculations involving hours, minutes etc. However, as the Irish curriculum for Mathematics still specifies the use of subtraction to solve elapsed time problems, Operations Maths has opted to present both ways in the books.
To view an excellent video of students solving elapsed time problems using an empty time line, please click here.
Other tips and suggestions for teaching time
- Refer to, and use, aspects of time as much as possible during the school day, as appropriate to the class level e.g.:
- Assign times for tasks and show interactive count-down timers on the IWB. This loop timer is particularly useful for timing stations in class.
- Reference calendar facts such as the current, previous and next day, date, month, season, etc every morning.
- Have a calendar visible in the classroom, marked with significant dates eg school play, outing, pupil birthdays etc. Ask the children to tell you how long it will be (in hours, days or weeks) until certain events.
- Encourage the children to wear watches themselves, with a preference for analogue, as an awareness of analogue time better develops the children ability to appreciate the passage of time.
- When writing time in digital format always use a colon (as seen on digital displays) as opposed to a dot i.e. 10:30 as opposed to 10.30. Using a dot, which is identical to a decimal point, doesn’t help the child to recognise that the system of measuring time is inherently different from our base-ten place value system.