Theme 9: Tools and representations

Effective teachers carefully select tools and representations to provide support for students’ thinking.

The stories below highlight the variety of tools available to teachers, including the number system itself, symbols, diagrams, models, notation, stories, technologies and a range of ‘concrete’ materials. They discuss the importance of tools in assisting their students to make connections between operations, concepts and their symbolic representations.  They show that tools must be carefully selected and that teachers should ensure that connections between concepts and representations are explicit, linking the model to the key focus ideas. This requires sound knowledge by the teacher if misuse of tools is to be avoided and if deeper mathematical understanding is to be developed.

Appropriate tools and skillful teacher guidance should support students in their discussion of their ideas, and in their recording of these. This communication of their ideas is critical for students in their own learning process but it also provides essential insights for teachers into the students’ thinking and into any misconceptions. In scaffolding learning using a variety of tools, including an increasing array of technologies, teachers make learning accessible, support independent enquiry and make links with the students real world.

It is evident in some of the stories below that the use of particular tools led to some real teacher insights.

     

Exemplars:

They could recognise some patterns on a dice but did not understand what they meant.  I therefore worked on moving the students onto patterning and to seeing patterns in a lot of different ways. I used a variety of materials and representations so the students could model, explain and justify their thinking in a multitude of ways.
The selection of appropriate tools and representations to provide support for students’ thinking was essential in assisting them to make connections between mathematical ideas.
‘With the help of an appropriate tool, students can think through a problem or test an idea that their teacher has modelled’ (Anthony, G. & Walshaw, M., 2009, p.23)
Students needed the opportunity to explore a range of materials to make sense of the mathematical ideas. We found that we could not rely on one piece of equipment to model the idea but had to use several different pieces to ensure that learning had been met. For year 3 students we used a range of representations of arrays linked to a story context. Year 5 children physically represented arrays using a range of materials such as sports equipment. This allowed children the opportunity to make connections between the representations and the mathematical concepts and ideas. To support this idea it was essential also for the children to see the connection between the symbols and the representation. It was important to provide children with adequate scaffolding of mathematical concepts in order for them to succeed.
Equipment was a significant component of my trial. Concepts being developed were explored with materials to consolidate understanding. Examples of this included:
Fly flips, beans, unifix cubes – derived multiplication
Pop sticks, place value houses, beans, number lines, arrow cards – place value knowledge
Place value houses – place value knowledge
Hundreds boards – place value knowledge, addition/subtraction facts
Deci-pipes – decimal place value knowledge
To consolidate multiplication facts, children were engaged in a number of ‘games’ such as loopy, family of facts trading game and many more.
Materials were used daily to help the students to express their thinking and to solve the maths problems. Knowledge was built up with the help of tens frames, card games, dice, iceblock sticks, place value flip book, blocks and number cards. Most of these materials were also used to develop strategies, with the additional inclusion of laminated number lines and representations of number lines drawn by students in their books. The materials were essential for the students to visualise and represent what they were doing when solving the problems. By manipulating the materials, the students were able to mathematically prove their solutions and become totally involved with the strategies being taught. In addition, computers were used once a week to practice memory games which the students found to be motivating.
I used materials that linked words, digits and a physical representation. Read, Say, Do times two. For example, READ: the word twenty and digits 20 correctly. SAY: twenty as two lots of ten or twenty singles. DO: make two bundles of ten ice block sticks and twenty single ice block sticks.
  • I have discovered it is important to always use materials when initially introducing a new concept and for as long as the students need it to talk about their learning.
  • When students were able to give me the correct answer I discovered they could not always show me evidence of their findings. As the students used a variety of materials, their knowledge was consolidated.
  • Visual and  concrete materials should be used to consolidate learning. Materials should be available at all stages for the student, for introduction of new learning, to show their thinking, and to demonstrate their learning when appropriate.

 

Having a clear and structured programme with relevant tools (activities) to accompany each learning area, made it successful. The programme was easy for the teacher to follow, broken down into manageable steps. The weekly charts were particularly useful and related easily to accompanying assessment materials and resource (activities) information. To make this easier for teachers the activities could be included on the actual weekly plan.
Plenty of equipment and tools were utilised as well. Having space meant we could spread out. Money, place value blocks, place value houses, decipipes and decimats were used the most with these year 8 students. The decipipes became my favourite item of equipment. I would say, “Picture the decipipes in your marvellous mind and describe what you see for (1.82).” They were fantastic for demonstrating addition and subtraction of decimal numbers and the students enjoyed using them.
One day we were using magnetic counters to model subtraction using compatible numbers (Numeracy Project Book 5: Teaching Addition, Subtraction and Place Value, page 26). The required number of counters to be removed, was physically moved to the side of the board by the children. BUT as we discussed our answers the students began moving the put-aside counters back into the remaining pile. They found it difficult to mentally set aside the removed amount. In subsequent teaching I therefore took off the removed amount of an equation and ensured that it was out of sight of the students. I routinely discussed this with the students encouraging them to mentally remove a number from their mental pictures as we explored subtraction. Discussion with other teachers led us to reflect that newer equipment might be creating differing mental images for some of our students.
The first part of the programme was based around knowledge, particularly place value and basic facts. It included lots of work with tens frames, bundling sticks, place value houses and place value money. Peter Hughes' place value work from his paper “Mathematical Literacy for Lower Achieving Students” was used, as were ideas from John Van de Walle’s Elementary and Middle School Mathematics.
Be very aware of the fact that underachieving students in numeracy may have learning difficulties and experiment with a range of equipment and materials. “Play and learn by doing.”
Allow students to use equipment as they may also have difficulty in holding an image (memory).
Try a process of pairing students. Use concrete objects and discussion, drawing pictures and discussion, think-pair-share discussion, and provide opportunities for solving problems by using objects or drawings plus discussions and questioning.
I found this process longer but it began to enable students to eventually hold the images in their minds. Don’t be in a hurry to get to the abstract number properties. Some students will need 2-3 weeks to make the leap, while others will take 6-7 weeks. Acknowledge that each student learns at their own rate in their own time.
Something that was a major factor in accelerating learning was the variety of material that was available for children to use. We used a range of materials such as number lines, bundles, tens frames, hundreds boards etc. If a student didn’t click using one type of material I tried another. I think an important factor is that the children tend to think that the equipment is for “babies” or that it’s only the teacher who manipulates it. Once they found the material that worked for them I saw them selecting material to use of their own accord while doing independent work. Another spin-off was seeing other students using materials when working independently, after seeing their peers using it.
Many of these resources are used within numeracy teaching, but the teacher has to have a clear understanding of how to use these in a systematic and logical way for children and the hierarchy of understanding required for each of the resources. For example, children can see all the popsticks in a bundle of 10 but cannot see the single dollars in a ten dollar note, so the ten dollar note is a more “sophisticated” representation of ten than the bundle of popsticks . We started with numbers that children had already encountered (teen numbers) and we represented them using a variety of place value resources and practiced the language of teen numbers being (….and ten). The exceptions (11 and 12) were explored and we went into some of the linguistic history of how these numbers got their names.
I initially thought the children would have a solid grasp of this understanding but many of them had not previously made the connection. It was a surprise to them that the link was so obvious.
As the project evolved it was evident that the children had difficulty understanding place value. We spent considerable amounts of time partitioning numbers. We used place value arrow cards, place value blocks, place value houses, flip charts and a place value card game to help learn how numbers are made up. We used mathematical language to describe digits, thousands, hundreds, tens and ones.