**There is no shortage of people telling teachers how to improve education. Sometimes it seems that all the educational experts are either cutting hair or driving taxis - or perhaps in public office. So it is useful to find clearly written advice based upon educational research and free from economic motivations. The International Academy of Education (IAE) publishes a series of pamphlets that distills educational research into useful summaries of current teaching techniques that have been found to promote student learning. ”Improving student achievement in mathematics” by Douglas Grouws and Kristin Cebulla provides a succinct summary of how effective educators can approach their teaching.**

Their research suggests that teachers can improve mathematical learning via;

1.Opportunity to learn

The extent of the students’ opportunity to learn mathematics content bears directly and decisively on student mathematics achievement. “As might be expected, there is also a positive relationship between total time allocated to mathematics and general mathematics achievement.” No surprise there I suspect - but this aspect is worth contemplating; “Short class periods in mathematics, instituted for whatever practical or philosophical reason, should be seriously questioned. Of special concern are the 30-35 minute class periods for mathematics being implemented in some middle schools.”

2. Focus on meaning

Focusing instruction on the meaningful development of important mathematical ideas increases the level of student learning. Teachers should stress “...the mathematical meanings of ideas, including how the idea, concept or skill is connected in multiple ways to other mathematical ideas in a logically consistent and sensible manner.”

3. Learning new concepts and skills while solving problems

Students can learn both concepts and skills by solving problems. This clearly addresses the “chicken and egg” issue of some teachers - it is NOT necessary to teach specific computation techniques BEFORE addressing real life applications. “There is evidence that students can learn new skills and concepts while they are working out solutions to problems.”

4. Opportunities for both invention and practice

Giving students both an opportunity to discover and invent new knowledge and an opportunity to practise what they have learned improves student achievement. The research finds that in the USA over 90% of class time is spent on practicing routine procedures. In Japan about 45% of instructional time is spent practising routine procedures, 15% applying procedures in new situations and 45% inventing new procedures or analysing new situations. Like to predict which system is ranked higher in international comparisons?

5. Openness to student solution methods and student interaction.

Teaching that incorporates students’ intuitive solution methods can increase student learning, especially when combined with opportunities for student interaction and discussion. Student interaction - sharing their solutions and the how they approached maths tasks - makes for enhanced student learning. The notion of a good classroom is a quiet classroom with children working in isolation is simply not supported by the research. Students learn better when they interact - which, if the social-constructionist theory of learning is applied, is what we would expect.

6. Small group learning

Using small groups of students to work on activities, problems and assignments can increase student mathematics achievement. Again, co-operative methods of teaching featuring both group goals and individual accountability are associated with enhanced student learning. Teachers would be advised to select mathematical tasks that lend themselves to group exploration rather than simply getting students to “work together” on standard tasks.

7. Whole class discussion

Whole class discussion following individual and group work improves student achievement. The adult in the room need not be the only teacher in the class.

8. Number sense

Teaching mathematics with a focus on number sense encourages students to become problem solvers in a wide variety of situations and to view mathematics as a discipline in which thinking is important. Number sense - that feeling accomplished people get when they get an answer that “doesn’t look right” - is an important part of developing mathematical skills...and it requires that students are actually thinking about what they are doing, why they are doing it, and estimating / predicting internally what sort of result would be reasonable.

9. Concrete materials

Long-term use of concrete materials is positively related to increases in student mathematics achievement and improved attitudes towards mathematics. So, a warning sign of a less than effective teacher may be the pile of worksheets students are expected to complete. Combine this with a lack of manipulatives or concrete support materials and students have a problem - and it isn’t the mathematics.

10. Students’ use of calculators

Using calculators in the learning of mathematics can result in increased achievement and improved student attitudes. Study after study support this notion. The use of calculators enhance mathematics learning. Why? It lets the students think about the mathematics, not the calculation.

Grouws and Cebulla add a caveat to their list of behaviours - the quality of the implementation of the teaching practices listed above greatly impact upon student learning; for example, it is not only whether manipulatives are used but how they are used that determines effectiveness.

Much of this list will not be new to those with an interest in mathematics education. However, it seems to my casual eye that mathematics classrooms are often still the domain of worksheets with a focus on procedural competence rather than conceptual understanding, places of compliance rather than engagement. Reading and discussing research findings such as this may help us improve the quality of our mathematics teaching. Reading the original pamphlet would be worthwhile for all teachers with an interest in the area.

Their research suggests that teachers can improve mathematical learning via;

1.Opportunity to learn

The extent of the students’ opportunity to learn mathematics content bears directly and decisively on student mathematics achievement. “As might be expected, there is also a positive relationship between total time allocated to mathematics and general mathematics achievement.” No surprise there I suspect - but this aspect is worth contemplating; “Short class periods in mathematics, instituted for whatever practical or philosophical reason, should be seriously questioned. Of special concern are the 30-35 minute class periods for mathematics being implemented in some middle schools.”

2. Focus on meaning

Focusing instruction on the meaningful development of important mathematical ideas increases the level of student learning. Teachers should stress “...the mathematical meanings of ideas, including how the idea, concept or skill is connected in multiple ways to other mathematical ideas in a logically consistent and sensible manner.”

3. Learning new concepts and skills while solving problems

Students can learn both concepts and skills by solving problems. This clearly addresses the “chicken and egg” issue of some teachers - it is NOT necessary to teach specific computation techniques BEFORE addressing real life applications. “There is evidence that students can learn new skills and concepts while they are working out solutions to problems.”

4. Opportunities for both invention and practice

Giving students both an opportunity to discover and invent new knowledge and an opportunity to practise what they have learned improves student achievement. The research finds that in the USA over 90% of class time is spent on practicing routine procedures. In Japan about 45% of instructional time is spent practising routine procedures, 15% applying procedures in new situations and 45% inventing new procedures or analysing new situations. Like to predict which system is ranked higher in international comparisons?

5. Openness to student solution methods and student interaction.

Teaching that incorporates students’ intuitive solution methods can increase student learning, especially when combined with opportunities for student interaction and discussion. Student interaction - sharing their solutions and the how they approached maths tasks - makes for enhanced student learning. The notion of a good classroom is a quiet classroom with children working in isolation is simply not supported by the research. Students learn better when they interact - which, if the social-constructionist theory of learning is applied, is what we would expect.

6. Small group learning

Using small groups of students to work on activities, problems and assignments can increase student mathematics achievement. Again, co-operative methods of teaching featuring both group goals and individual accountability are associated with enhanced student learning. Teachers would be advised to select mathematical tasks that lend themselves to group exploration rather than simply getting students to “work together” on standard tasks.

7. Whole class discussion

Whole class discussion following individual and group work improves student achievement. The adult in the room need not be the only teacher in the class.

8. Number sense

Teaching mathematics with a focus on number sense encourages students to become problem solvers in a wide variety of situations and to view mathematics as a discipline in which thinking is important. Number sense - that feeling accomplished people get when they get an answer that “doesn’t look right” - is an important part of developing mathematical skills...and it requires that students are actually thinking about what they are doing, why they are doing it, and estimating / predicting internally what sort of result would be reasonable.

9. Concrete materials

Long-term use of concrete materials is positively related to increases in student mathematics achievement and improved attitudes towards mathematics. So, a warning sign of a less than effective teacher may be the pile of worksheets students are expected to complete. Combine this with a lack of manipulatives or concrete support materials and students have a problem - and it isn’t the mathematics.

10. Students’ use of calculators

Using calculators in the learning of mathematics can result in increased achievement and improved student attitudes. Study after study support this notion. The use of calculators enhance mathematics learning. Why? It lets the students think about the mathematics, not the calculation.

Grouws and Cebulla add a caveat to their list of behaviours - the quality of the implementation of the teaching practices listed above greatly impact upon student learning; for example, it is not only whether manipulatives are used but how they are used that determines effectiveness.

Much of this list will not be new to those with an interest in mathematics education. However, it seems to my casual eye that mathematics classrooms are often still the domain of worksheets with a focus on procedural competence rather than conceptual understanding, places of compliance rather than engagement. Reading and discussing research findings such as this may help us improve the quality of our mathematics teaching. Reading the original pamphlet would be worthwhile for all teachers with an interest in the area.

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**An earlier post - “How to be a great mathematics teacher - what the research says” was mined from another pamphlet in the IAE’s “Educational Practice Series”. The post summarises the content of that pamphlet and provides links to the original source material - which is well worth reading.**

Those with an interest in improving mathematical pedagogy might like to read a related post dealing with the work of Alistair McIntosh - Improving numeracy with the 7Cs.

Those with a general interest in mathematics might enjoy the maths page on this site which collects a range of posts dealing with mathematics.

Those with an interest in improving mathematical pedagogy might like to read a related post dealing with the work of Alistair McIntosh - Improving numeracy with the 7Cs.

Those with a general interest in mathematics might enjoy the maths page on this site which collects a range of posts dealing with mathematics.

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**
Image: via google images: http://www.kingslangley.ps.education.nsw.gov.au/images/Math-Symbols.jpg **

**Credits: All links go to original sources.**

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