Maths to share - CPD for your school
Mathematics Subject Knowledge - Length
Before the session, ask colleagues to read the article by Douglas H. Clements, Teaching length measurement: Research Challenges. It explores the difficult nature of teaching and learning the concept of length, suggesting the difficulty might be due to the fact that it bridges both spatial awareness and real numbers. It also challenges the conventional progression of skills related to the teaching of length.
Ask teachers to discuss the following question…
Share the outcomes of the discussions. Are there common aspects that are shared by all? Do any of the teachers suggest an order for teaching particular concepts?
Provide the teachers with a copy of Length: Key Concepts. Ask them to consider the order in which the statements might be taught or used confidently by pupils. Are there any that are taught throughout? Which, if any, of the concepts is the hardest to teach? Why?
Allow time for discussion and feedback. What do they think are the common misconceptions with the concept of length? Some points for discussion might include:
- When comparing two objects directly, side by side, where no measuring or counting is required, children can fail to ‘line up’ one end of each object. Pupils’ understanding can be challenged by an adult moving one object as shown below.
- Pupils will be moved on by comparing more than two objects, using each as a baseline for comparison, ordering from greatest to least. It is important for them to focus on the relevant criterion, ie length, as they can be sidetracked or confused by the width.
- If pupils use lots of the same uniform unit, for example, placing centimetre cubes carefully along a given line, they can make mistakes counting each unit. Once they progress to using one unit repeatedly (the skill of iteration) they may have difficulty keeping count. It is important that we recognise constraints set by pupils’ counting abilities when setting tasks relating to length. Clements suggests that even when equipment marked in standard units (eg rulers) is introduced, pupils continue to use non-standard units, eg centimetre cubes, alongside.
- When using standard units children do not always have a ‘feel’ for the size of the standard unit. It is only once this is developed that they can use it as a benchmark to estimate.
- When using equipment calibrated in standard units (eg ruler, tape measure, trundle wheel) children need to be explicitly taught appropriate handling techniques. Often children measure from the end of a ruler or from ‘1’ rather than from zero, or count the markings on a ruler rather than the units themselves (or the spaces).
The use of standard measuring equipment is discussed in detail by Clements (1999), alongside an equally strong argument for the development of ‘mental rulers’. In order to be able to develop a ‘measurement sense’ pupils need a great deal of experience estimating lengths, visualising their own ‘mental ruler’. This can then support them when judging and comparing lengths and measurements, or drawing lines of a given length.
The Early Numeracy Research Project (ENRP) describes the development of the concept of length as involving firstly an awareness of length, followed by comparing lengths, quantifying lengths, and finally measuring lengths. Clements (1999) claims that even pre-schoolers can compare two objects directly, and so we, as teachers, must try to assess exactly what level of understanding the pupils have when they arrive in our classrooms.
A common stumbling block for pupils is the language associated with certain areas of mathematics, and ‘length’ is no different. ENRP refers to the need to explicitly teach children to move from multi-dimensional words to uni-dimensional words. An example for length is the early use of the word ‘big’ and the need to give pupils “a wide variety of uni-dimensional words that describe 'bigness' more precisely - 'tall', 'long', 'high', 'wide', etc.” This mirrors work carried out in literacy, exploring alternatives to words such as ‘nice’.
Ask staff to consider, for each of the following, an estimate, a suggested unit of measurement, and appropriate equipment.
What did they use to make their estimates? How accurate does an estimate need to be, in order to be considered acceptable? How often do they ask pupils to carry out similar activities? It is vital that we allow pupils the opportunity to estimate lengths at all stages of their development, even early on when only working with non-standard units.
As for most areas of mathematics, there are numerous ICT programmes and pieces of animated software available to demonstrate measuring skills at all stages of a pupils’ development. Despite the often eye-catching nature of many of these, they simply cannot replace practical activities where the pupils are required to estimate then measure objects and distances in ‘real-life’.
- The use of a floor turtle followed by an on-screen turtle can really challenge some pupils’ thinking and understanding of the concept of length. Floor turtles can be used as a measuring tool themselves, children trying to find out ‘how many turtles wide the classroom is’ for example. They can use their estimation skills to plan for and program the turtle to move around a particular course. You will find more information regarding the availability of on-screen turtle programmes, including some available as free downloads on the Mathsnet website.
- Primary Games has a very child-friendly site focusing on the teaching of measures. It includes resources for whiteboard/large display as well as related worksheets and follow-up ideas. Various units of measure are included.
- The BBC Skillswise site offers some suggestions of activities and includes useful factsheets, quizzes and games relating to teaching length.
- With the development of web-based mapping and location programmes such as Google Earth, a whole host of activities can be developed making use of the tools available. Adam Boddison led a live video conferencing session though Motivate aimed at upper Key Stage 2 pupils. It taught pupils how to use the line and path tools to measure straight line distances on Google Earth. Work then focused around the US Pentagon, the lengths of international airport runways and supermarket car parks, estimating, measuring and converting units of length. Further Motivate conferences can be found on their website.
- The book Length by Henry Arthur Pluckrose, (part of the Math Counts series) shows some wonderful photographic images aimed at encouraging young pupils to talk about the concept of length.
- The Crocodile’s Coat by Cal Irons (also available as a Big Book) is a wonderful way to introduce children to the concept of repeatedly using a non-standard measure. A crocodile goes to the lizard tailor shop to be measured for a new coat, only to find that all of the clothes they make are measured in ‘lizards’. The crocodile is measured by the lizards lying nose-to-tail along his body. The fabric is then measured in the same way, and a beautiful coat is made!