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Working Mathematically


Doug Williams

The aim of a Working Mathematically curriculum is to teach students to work like a mathematician - to generate student interest so they want to solve the problem - and to encourage different strategies for tackling the problem.
As opposed to the traditional 'learn by rote' approach, the Working Mathematically curriculum facilitates investigative learning. The one page document Working Mathematically, details the process. It was composed by questioning mathematicians about their work and is pasted into student journals and enlarged as a class poster. Students learn to ask What might a mathematician do now? and to examine the document for a response.

Shifting from the traditional textbook approach to the Working Mathematically Process requires a balance to be struck between:

  • the process of being a mathematician, and
  • developing the skills necessary to be a successful mathematician.

Working Mathematically and Maths300

Maths300 provides a rich resource of lessons based on this process, all prepared and trialed by classroom teachers. While content skills are important, the Working Mathematically Process emphasises application of skills. As such, it has pioneered a new frontier for excellence in mathematics education.

A guiding principle of Working Mathematically is that we all have the same capacity to work like a mathematician, but not everyone has had the positive experiences and practice necessary for skills to flourish. A focus for the Working Mathematically teacher is to ensure students develop mathematical skills in the context of problem posing and solving.

Afzal Ahmed, one time Professor of Mathematics at Chichester UK once quipped:

If teachers of mathematics had to teach soccer, they would start off with a lesson on kicking the ball, follow it with lessons on trapping the ball and end with a lesson on heading the ball. At no time would they play a game of football.

No doubt his gentle jibe contained a kernel of truth as far as his peers were concerned, but were he to direct it towards Working Mathematically teachers, it would be completely unfounded!

First give me an interesting problem

In the past, mathematics teaching has been solution-focused: children set out to find the correct answer that invariably sat in the back of their textbook. But 'real' mathematicians don't know all the answers - they start with a problem they find interesting, and work through a number of strategies to find a solution.

In Working Mathematically, we teach children to think like mathematicians, but first, we have to get them interested! Each Maths300 lesson begins with a story shell or some other feature, designed to pique students' interest and engage them in the process.

Working Mathematically Curriculum Components

There are no rules about the 'best way' to build a curriculum centred on learning to work like a mathematician. However, in addition to the whole class investigation lessons for which Maths300 is renown, teachers have found that planning around these components enriches the curriculum.
  • Hands-on problem solving play
    In the mid-1970s, Neville de Mestre created the first hands-on problem solving task centre to invite students to work like a mathematician. Mathematicians don't know the answer to a problem when they start it. If they did, it wouldn't be a problem! They have to play around with it; look at it from all sides.

    Each task invites students to play with mathematics in pairs and over the decades many schools have made or purchased their own sets. These schools have the opportunity include this form of mathematics investigation in their curriculum perhaps once per fortnight - perhaps more often than that. The tasks chosen to place before the students as a menu are often selected to integrate with the current strand or topic. The wisdom of practice related to using this resource has been collected at the Mathematics Task Centre.

  • Skill development
    A mathematician needs skills to solve problems. Many teachers find it makes sense to students to place skill practice in the context of Toolbox Lessons which help us better use the Working Mathematically process. Toolbox Lessons are perhaps once or twice a week. Lesson 9, First Principles Percent, is an example of this component.

    A balanced curriculum would contain perhaps 20 of these challenges per year, many of which will take more than one session to investigate. Lesson 25, Sphinx, is an example of this component. In the primary/elementary school it can be included within a unit on perimeter and area. In the secondary school it can be included in a unit on patterns and powers.

When mathematicians become interested in a problem they:
  • Explore the problem to collect and organise data about it.
  • Discuss and record notes and diagrams.
  • Seek and find patterns or connections in the organised data.
  • Make and test hypotheses based on the patterns or connections.
  • Look in their strategy 'toolbox' for problem solving strategies which could help.
  • Look in their skills toolbox for mathematical skills which could help.
  • Check their answer and think about what else they can learn from it.
  • Publish their results.
Questions which help mathematicians learn more are:
  • Can I check this another way?
  • What happens if..?
  • How many solutions are there?
  • How will I know when I have found them all?
When mathematicians have a problem they:
  • Read and understand the problem.
  • Plan a strategy to start the problem.
  • Carry out their plan.
  • Check the result.
A mathematician's strategy toolbox includes:
  • Do I know a similar problem?
  • Guess, check and improve
  • Try a simpler problem
  • Write an equation
  • Make a list or table
  • Work backwards
  • Act it out
  • Draw a picture or graph
  • Make a model
  • Look for a pattern
  • Try all possibilities
  • Seek an exception
  • Break the problem into smaller parts
  • ...

  • Strategy development
    Mathematicians also make use of a strategy toolbox. These strategies are embedded in Maths300 lessons, but may also have a separate focus. Poster Problem Clinics are a useful way to approach this component. A Poster Problem Clinic is perhaps one lesson per fortnight. Lesson 14, The Farmer's Puzzle, is an example of this component.

  • Concept development
    There are relatively few major concepts in mathematics. Examples are place value, fractions and probability. Each took centuries for the human race to develop and apply, yet traditionally, students have been expected to understand such concepts after having 'done' them for a two week slot. Often the concepts are not revisited until the next year. A Working Mathematically Curriculum identifies such concepts and regularly includes development work.

    Typically one or two such concepts are visited and revisited (threaded) over a term, even if only a few minutes per session. An example of this component is the threaded use of Lesson 35, Nine & Over, which addresses Place Value.

The district established four of its five elementary schools with 100 hands-on tasks from the Task Centre Project, one year of Maths300 subscription and one full day of professional development for each school. There are Maths300 lessons to support and extend many of the tasks our schools use. The PD was designed to initiate a local task centre network. Now:
  • Every month the district math team meets and we work on one task.
  • The members then use that task in their classrooms and bring back student work and experiences to the next meeting.
  • It has been very interesting to use the same tasks 0 - 12 and then discuss the different developmental levels of the students.
  • The secondary teachers like doing the tasks in class, and several teachers comment on how much the students enjoy them.
We have also realised that our little ones (Years 0/1) are far more mathematically advanced than we give them credit for!

Sally Collins, District Math Co-ordinator, Englewood School District, Denver, Colorado, USA

Other Key Features

As you create your curriculum, look for opportunities to include:
  • estimation practice
  • use of first hand data
  • outdoor activity
  • open-ended inquiry
  • whole class activity
  • concrete materials
  • an investigative process
  • links to learning theory
  • recording and publishing findings
Uniquely, the Maths300 Search Engine available to members allows you to search for lessons using criteria such as these combined with content and year level.

In-House PD

Many schools have reported success over time using these few steps. More than that, they report both teachers and students enjoying maths more.
  1. Plan a Working Mathematically unit as a team.
  2. Play with it in the classroom to gather data about the students' learning.
  3. Review and refresh the plan based on the team's experiences.
  4. Record the trialed and polished version.
  5. Add a paragraph or two of teacher comment to guide other colleagues.
  6. Print your unit and other relevant ones to build up your syllabus document.
  7. Repeat from Step 1.
Remember, it is our role to collect stories of success, so please tells about your successes with Maths300. Many teachers have already done so. Their work is recorded in Classroom Contributions linked to each lesson and it has often helped us rework Maths300 lesson plans. Our contact link below explains how to contribute.
Please also check our Terms and Conditions when you contribute.


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