Student strategies to solve measurement problems Counters, adders and multipliers learn to measure Michael Drake School of Education Policy and Implementation

Learning intentions To develop an appreciation of the difficulties students may have when learning to measure To develop an understanding of the strategies students use to answer measurement problems In the time available I can give you an idea of the issues and introduce you to some things you can do, but that is all – really need another hour… at least!

Get into three groups: Counters Adders Multipliers Handout: scales of measurement

First task Identify the number understanding that students in your group bring to mathematics problems Handout – relevant piece of the number framework to each group

Second task Look at the sheet of measurement problems Try to decide how students with the identified set of skills will approach each of these problems

Third task Now move into groups of three – each group must have someone from the old counter, adder and multiplier groups Compare how each group of students are likely to respond to each question Things to highlight 1) Using the ruler: What is the impact of having counting start at one – but measurement starting at zero? What understanding are students likely to have of the little marks between the centimetres? 2) Weighing the stone: How do you explain the need to zero the scale before weighing? (In counting the biggest number you reach is the answer – demonstrate with unifix cubes or pens…) Which scale should you use? (Isn’t the concept that there are completely different ways of doing the problem – with completely different answers – difficult to get your head around if you don’t really understand how whole numbers work?) 3) Thermometer What experience do students get in class at using non-unit scales? How can a counter work out what the scale is? Students reciting a count (as opposed to skip counting) 4) The broken ruler problem Revisiting the concept that a measurement is an interval – students really need to concept of difference to be able to undertake measurement effectively The number line Didn’t find that contexts added anything to students’ developing understanding of scale How will a counter answer this problem? How will an adder answer this problem? Need for intervals in 20s and 25s to identify what students actually understand (DFES identify the scales marked in ones, tens and tenths lead to higher success rates) 6) Age problem Who was tempted to go back to a counting strategy? (Why?)

MiNZC M2:1 carry out practical measuring tasks, using appropriate metric units for length, mass, and capacity; My answers: Lots of reasons really – firstly the curriculum we have been following Flick up the ruler Started using the ruler too early The ruler has no zero – how can you learn to start from zero if there isn’t one? The cm scale is actually 2 scales nested together. How do you teach using that? (2 options – measure in whole centimetres then change units - cos you use little units for measuring little things, or measure in whole centimetres and swap to points) Second: Differences between measuring length and counting processes [model with unifix cubes or counters on OHP]

The weighing scale Machines like this are very complex instruments – and have two scales as they are sold all around the world… Image downloaded from http://www.scalesgalore.com/salter_250_mechanical_scales.htm

The broken ruler problem Has a number of formats. Commonly used in measurement research – seen in Piaget’s work NEMP 2001 – when measuring the width of a piece of card, only 64% of Y8 students could measure to within ½cm with a broken ruler Handout: some results from students with similar problems Hart (1981) had a similar problem, but the line started at 1 and ended at 7cm. 46% of year 1 (secondary so were 11-12 year olds), 30% of year 2 and 23% of year 3 students answered 7 Questions to discuss: what thinking were students using to get such errors? [Use for broken ruler and weighing scale]

What is going on? Is it good enough that many of these Year 7 and 8 students have little understanding how scales work? What are the implications of such a lack of understanding? What is going on? Scale is much harder to learn than has traditionally been given credit for Handout – student strategies: This is just one aspect of the understanding needed to work effectively with scale Is it good enough? And that many are getting correct answers more by luck than anything else… Implications Ability to learn graphing compromised, and higher maths related to functions may be inaccessible

What can we do? Teachers need to start by finding out what their own students can do Measuring things Hand out activity sheets and talk through it

What can we do? Don’t assume students know and understand about measurement – or can use a ruler Consider using simpler scales with some students or when first introducing measurement tools Point 1: measurement is a practical topic – get students to measure things and compare their measurements – the address what is going wrong! Model this by collating answers on the board – why are there different answers? Is it better to use cm and mm or just cm or just mm? What made measuring this line hard? (and why was it hard?) Point 2: Type ‘ruler into’ google and come up ‘some printable paper rulers’ URL http://www.vendian.org/mncharity/dir3/paper_rulers/

What can we do? NZC has a new approach to the development of measurement concepts We don’t require students to use measuring instruments until they have developed a concept of measurement and scale Handouts: measurement thread: a comparison a revised model for developing measurement processes Highlight the slower introduction of measurement instruments and decimals for measurement – new level 2 Question How would you create a unit for measuring: length, area, Volume, - capacity, - tin cans or cups and spoons weight (mass), - balance scales turn (angle), - temperature time. - hour glass An unusual attribute of an object – eg the bendability of an item GM2-1 Create and use appropriate units and devices to measure…

What can we do? Actively teach how to read any new scale that is being used – rather than just assume the students can use it…