Research on Strategies for Improving Numeracy Instruction

According to Hattie (2009) direct teaching can play a major positive role in effecting student learning. Direct teaching is where ‘the teacher decides the learning intentions and success criteria, makes them transparent to the students, demonstrates by modelling, evaluates if they understand what they had been told by checking for understanding, and re-telling them what they had been told by tying it all together with closure’.

Effective numeracy planning relies greatly on direct teaching. Carpenter (1989) proposes that conceiving teaching as problem solving is important for understanding teachers’ planning and classroom instruction. The focus here is for teachers to use knowledge from cognitive science to make instructional decisions. He refers to this approach as Cognitive guided instruction. It is based on the premise that the teaching learning process is too complex to specify in advance, and as a consequence teaching is essentially problem solving.

Cognitive guided instruction was initially developed by researchers at University of Wisconsin in 1980’s. Fennema, Franke, Levi, Jacobs and Empson (1996) found that learning to understand students mathematical thinking could lead to fundamental changes in teachers beliefs and practices and that these changes, in turn, reflected in students’ learning. The studies provided sites for examining development of students’ thinking in situations where their intuitive strategies for problem solving were a focus for teacher reflection and discussion. This led to new perspectives on student thinking and on the instructional contexts that supported the development which led to the revision in the approach to teacher development.

Carpenter et al (1999) refer to Cognitive guided instruction professional development as a program which engages teachers in learning about the development of students’ mathematical thinking with particular content domains. To understand student thinking teachers create their own ways of organising and framing knowledge as well as thinking carefully about relationships between this knowledge and their teaching.

Essentially Cognitive guided instruction pedagogy focuses on what students know rather than on what teachers do. Cognitive guided instruction is consistent with independent learning and constructivist pedagogy.

Dr Sam Jebeile (2008) conducted a study with commerce students where Cognitively guided instruction was used as a vehicle to improve student outcomes. The students were performing poorly in many of the numeracy based skills which were embedded in the commerce curriculum.

Students completed a post test, with the results showing substantial improvements in student learning and final assessment results.

Cognitively guided instruction in action

by Dr Sam Jebeile, Macquarie University

Step 1: Identify areas
Teachers identified areas for improvement in a specific topic. They created sample questions which were backward mapped from topics, tasks and assessments to identify the concepts which were potential problems.
Step 2: Pre-test
Students completed the questions as a pre-test. After marking, teachers interviewed students and asked them why they selected the strategies which led to the correct or incorrect answers.
Step 3: Collate findings
Themes emerged and the misconceptions were collated and presented to teachers in workshops. Once teachers understood where students were going wrong in their thinking, they took steps to address and highlight these areas in their teaching.
Step 4: Intent
Teaching and learning intentions were listed for each lesson which explicitly identified areas of common misconceptions, including the level of prior knowledge students bring to the classroom.
Step 5: Modify
Lessons were modified and adjusted to explicitly teach skills required to build new knowledge.
Step 6: Observe
Teachers observed each other’s lessons to ensure the misunderstandings were appropriately addressed, lessons were amended and activities were created to focus on the explicit teaching of the numeracy concepts.
Step 7: Post-test
Students completed a post-test with the results showing substantial improvements in student learning and final exam scores.

cognitively guided instruction diagram

The 21st century numeracy model researched by Professor Merrilyn Goos represents the multi-faceted nature of numeracy that acknowledges the rapidly evolving areas of knowledge, work and technology. The model was created for teachers as a planning and reflection tool. It contains five main elements.

Mathematical knowledge
Mathematical concepts and skills; problem solving strategies; estimation capacities.
Capacity to use mathematical knowledge in a range of contexts, both within schools and beyond school settings.
Confidence and willingness to use mathematical approaches to engage with life-related tasks; preparedness to make flexible and adaptive use of mathematical knowledge.
Use of material (models, measuring instruments), representational (symbol systems, graphs, maps, diagrams, drawings, tables, ready reckoners) and digital (computers, software, calculators, internet) tools to mediate and shape thinking.
Critical orientation
Use of mathematical information to make decisions and judgments; reason and support arguments.

21st Century Numeracy

by Dr Merrilyn Goos, University of Queensland

21st century numeracy

In Professor Goos’ study, teachers were asked to use the 21st century numeracy model (see below) to support improvement in numeracy teaching. In the research a physical education teacher used the model to adapt and modify an existing lesson.

In the adapted lesson you can see the range of mathematical knowledge dealt with such as number, ratio, converting units of measurement and representing and displaying data. Students related physical education concepts to their personal life, making for a rich and meaningful platform for deepening students’ understanding.

Students used physical tools such as pedometers and measuring tapes and digital tools such as Excel spreadsheets and data displays. The teacher explicitly addressed numeracy dispositions by encouraging students to estimate, compare, reason and make sense of the magnitude of the distance. Students used mathematical tools and technology to collate and display data, and were able to explain results which appeared on graphs when comparing charts. Some students chose inappropriate graphs to display data which led them to identify better ways for data representation.

  • Students wear a pedometer
  • Each student records the number of steps they took over a week
  • Data is collected in a table Teacher demonstrates how to convert steps to kilometres on the board
  • Teacher instructs students on how to construct a bar graph showing the kilometres walked per day
  • Students complete the activity in their books.
  • Teacher takes students outside where they estimate a distance of 100 metres
  • Teacher measures out 100 metres Students compare their estimate with the actual distance
  • Students walk the distance and count the number of steps
  • Teacher questions students about how to convert the steps to kilometres using a ratio method, using her own data as an example;

119 paces = 100 metres
119 x 10 paces = 100 x10 m
1190 paces = 1km

  • Excel spreadsheet is used as a tool to collate the data each day for a week
  • Excel chart wizard is used by students to create graphs
  • Teacher and student discussions about data and results produced take place, the results now make sense of the data in a meaningful context
  • After modelling the procedure students complete their own conversions and compare their results
  • Students create data displays using Excel and also Excel formulas to calculate the total number of kilometres for each student for the week.