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Lesson 83

  • Years: 5 - 9
  • Time: 1 - 2 lessons
  • Strands: Statistics, Measurement

At first glance, this lesson doesn't seem dramatically different from traditional practice. However, it does contain several elegant and significant features which contrast somewhat with that traditional approach.

Students in small groups are given a number of graphs representing the average temperatures of various cities over a one-year period. They are challenged as a group to match each graph to a city and to justify their choice. The lesson actively uses a relevant context (regional geography), problem solving and co-operative group work to develop graph reading skills. This approach directly contrasts with a traditional 'text-book' approach.

Extensions using computer support allow the teacher and student to explore a range of similar challenges.



Related Lessons

  • Country Maps

1. Introduction: A puzzle

This lesson is as presented by a particular teacher. The commentary has been added by the observer who was alert to the importance of the teaching craft being displayed. Key features of that craft are collected at the end of the lesson.

To prepare for the lesson, print copies of one of these worksheets:

Answers: Reading from left to right across the rows the answers are Darwin, Perth, Adelaide, Canberra, Hobart, Sydney, Melbourne, Brisbane
Answers: Reading from left to right across the rows the answers are Honiara, Suva, Kuala Lumpur, Jakarta, Auckland, Adelaide.

The teacher handed out the worksheet to students sitting in small groups of 3 or 4.

I have a puzzle for you. These graphs are temperature graphs. There is one for each of the eight capital cities of Australia that are written on the sheet. The top line on each graph is
...(pause for response)...
the average daily maximum temperature for each of the months of the year, and the bottom line is
...(pause for response)...
the average daily minimum temperature for each month of the year.

The major feature of the lesson is the shift from a closed to an open problem solving approach. In the closed approach typical of many texts, if a temperature graph is offered it is usually with no attempt to link the graph to a real place. The skill of reading graphs is then developed through closed questions such as:
  1. What is the maximum temperature in July?
  2. What is the hottest month?
  3. Which month has a minimum of 17 degrees?

In the open approach this skill is still needed and developed, but there are other learning outcomes as well. These include:

  • a genuine link to real cities known by the student (personal knowledge)
  • a much higher level of thinking and reasoning
  • significant opportunities to develop communication skills to justify choices

Lesson Stages

  1. Introduction: A puzzle
  2. Small group problem solving
  3. Group reports
  4. Conclusion: Reflective summary
  5. Extensions: Computer support
The lesson also adds a section at the end titled:

Learning Outcomes

  • Statistics
    • averages
    • graph reading
    • organising and displaying data
    • scales

Software Contribution

A typical text book usually presents only one or two graphs chosen by the author. However, this software allows students and teachers to select their own cities from the data base provided, so challenges can be selected according to interest and ability levels. Each of the first three options produces a random selection of six graphs from the region chosen. The challenge is to match the 6 cities to their graphs. This can be done interactively on screen or the page can be printed, copied as a worksheet. Other options allow students or teachers to add city data from external sources, especially their local town, or choose other cities from a list.

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2. Small group problem solving

Can you guess the challenge I have for you?
Students easily guessed and stated the challenge of matching each city to its graph.
Are you willing to have a big fight (I mean discussion) in each group and see if you can agree on which city goes with which graph?
Can I give you about 6 or 7 minutes? Then we'll see if we can all agree!

I liked the almost casual (but very deliberate) way the teacher encouraged group work. It purposefully allowed for students' existing knowledge to be seen as useful and to encourage the communication and sharing of that knowledge.

There was an assumption in this class that mathematics would be the source of an intellectual challenge worth tackling.

The students then argued (sometimes strenuously) until they had reached agreement about the matching. Much incidental knowledge emerged such as:

Look at this graph - it's always hot - it must be in the tropical area!

3. Group reports

The teacher had to find a moment of intervention, which was when most groups were nearing completion.
Let's see if we can agree on which goes with which. Don't worry if you have not finished yet, or if you want to change your mind as you hear other groups report. That's okay. Which graph 'attracted' you first?

I liked the way the teacher took away the fear of being 'wrong'.

The graphs were argued for one by one. The teacher, rather than give a signal about whether any 'guess' was correct or not, looked to the rest of the class for approval or disagreement.

I'm looking around for nods of approval - or say so if you disagree.
The extreme graphs were easy to get and all groups seemed to agree, but for the last few there was significant discussion as different opinions emerged.

The teacher clearly saw the value not as being right, but rather as using evidence to support and present an argument.

If the discussion did not generate the 'right' answer, the teacher calmly said:

They are rather close aren't they ... but the Bureau of Meteorology thinks that Graph 6 is...
and in so doing brought the discussion to a nice close.

4. Conclusion: Reflective summary

The teacher wanted the class to reflect on their own learning journey and led the class into a discussion with the question:
Well done, we've got all the graphs correct. I'd like to ask you Is this a good way to learn your mathematics? What did you like about the lesson? Or perhaps, what was not so good about it?
With this question, the teacher was acknowledging that there are differences between this lesson and traditional practice and wanted the students' opinions about these. Such opinions can sometimes be very illuminating.

It was much better than the boring text book!, said one student.

Not content with this superficial response, the teacher probed further:

What do you think it was that made it less boring?

I saw a subsequent lesson where the teacher asked students to write down about two things they liked and two things that they didn't like. This journal writing initiated a class discussion.

In addition, students can be invited to reflect on the lesson from the Working Mathematically viewpoint. In doing so they might note, for example, that:
  • the lesson provides a problem to be solved
  • they make hypotheses and check them
  • they consult with colleagues
  • they call on and develop their skill toolbox.

5. Extensions: Computer support

Asia Pacific Option The computer support for the lesson is provided by a data base of various cities, and an interactive page that lets the teacher (or the student) click on a range of cities. Six cities make up a worksheet. For example, selecting Asia Pacific cities produces the on-screen layout shown.

This worksheet can be done interactively on screen or printed as a handout for students.

Each option performs as follows:

  • Asia Pacific:
    The program randomly selects six cities from this region and constructs a worksheet as shown.

  • North America:
    Similarly, selecting this option randomly generates a worksheet of six cities from this region.

  • Any 6 cities:
    If this option is selected, a data base of cities is offered. In this picture, four cities have been selected so far.

    Any 6 Cities

    After six cities have been selected, they appear as an electronic worksheet, as in the other options, that can then be solved on screen or printed and copied.

    The software therefore allows worksheets to be crafted to fit the interests of the group. They can be made simple or quite difficult, depending on the prior knowledge of the group. They can be regional or global, depending on the geography being studied.

  • Edit or add to city data:
    This option allows the student or teacher to use the data base to insert data for any cities for which it is known. Here the new city has been called TEST and the entry of data has begun.

    Own Data

    When all months have been entered and a category selected, the Add Entry button adds TEST CITY to the alphabetical list, and it becomes available to the other options. In this case it was added to the Asia Pacific category.

    Test City

One teacher made a project for the class. Each group chose a city and found their data on the Internet. This was added to the data base and the cities became part of those available for worksheets. For example, in Australia, finding six regional cities in NSW or Victoria or any other State.

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Professional Development Purpose

Lens If this lesson were to be used in a structured professional development workshop or seminar, the object would be to bring out the contrasts with traditional practice. This can be done by scrutinising the lesson through various perspectives or lenses.
For example:
  • Is it good mathematics? (a content lens)
  • Is it good teaching practice? (a pedagogy lens)
  • Do all students learn from such a lesson (an equity lens)?
  • Is this a good lesson for we teachers to use to help us examine our practice? (a professional development lens)

One powerful question I saw asked which I think really focussed the contrasts was: How would a traditional text book treat the same topic?

Answers included:

It would be in Ch. XX Qu. XX
There would be just one graph - not six.
There would be no attempt to make the graph information 'real'.
Under the graph there would be 10 questions. What do you think the 10 questions would be?

The group I was watching laughed as they recounted the closed narrow questions such as:
What is the temperature in July?
Which month has the highest temperature?

Another useful question was:

What about the student who may not be confident about reading scales - someone who does not have a high degree of skill - is such a student excluded?
The group agreed that there are many informal opportunities within the discussion to build up such skills - they don't have to be formalised.


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