Developing communities of mathematical inquiry
Case 1, ‘Developing communities of mathematical inquiry’, illustrates how two teachers developed teaching practices that were highly effective for diverse learners. The case focuses on how these teachers accelerated the mathematics achievement of their year 4 to 6 students, most of whom were Māori or Pasifika.
Author(s): Ministry of Education
Date Published: Only released on Education Counts March 2011
Appendix 1. The Mathematics Communication and Participation Framework13
An outline of the communicative and participatory actions teachers facilitate students to engage in as they (the teachers) scaffold the use of reasoned, collective, mathematical discourse.
|Communicative actions||Phase One||Phase Two||Phase Three|
|Making conceptual explanations||Use problem context to make explanation experientially real.||Provide alternative ways to explain solution strategies.||Revise, extend, or elaborate on sections of explanations.|
|Making explanatory justification||Indicate agreement or disagreement with an explanation.||Provide mathematical reasons for agreeing or disagreeing with solution strategy.|
Justify using other explanations.
|Validate reasoning using own means.|
Resolve disagreement by discussing viability of different solution strategies.
|Making generalisations||Look for patterns and connections.|
Compare and contrast own reasoning with that used by others.
|Make comparisons and explain the differences and similarities between solution strategies.|
Explain number properties, relationships.
|Analyse and make comparisons between explanations that are different, efficient, sophisticated.|
Provide further examples of number patterns, number relations, and number properties.
|Using representations and inscriptions||Discuss and use a range of representations or inscriptions to support an explanation.||Describe inscriptions used to explain and justify conceptually as actions on quantities, not manipulation of symbols.||Interpret inscriptions used by others and contrast with own.|
Translate across representations to clarify and justify reasoning.
|Using mathematical language and definitions||Use mathematical words to describe actions.||Use correct mathematical terms.|
Ask questions to clarify terms and actions.
|Use mathematical words to describe actions (strategies).|
Reword or re-explain mathematical terms and solution strategies.
Use other examples to illustrate.
|Participatory actions||Active listening and questioning for more information. |
Collaborative support and responsibility for reasoning of all group members.
Discuss, interpret, and reinterpret problems.
Agree on the construction of one solution strategy that all members can explain.
Indicate need to question during large-group sharing.
Use questions that clarify specific sections of explanations or gain more information about an explanation.
|Prepare a group explanation and justification collaboratively.|
Prepare ways to re-explain or justify the selected group explanation.
Provide support for group members when explaining and justifying to the large group or when responding to questions and challenges.
Use wait time as a think time before answering or asking questions.
Indicate need to question and challenge.
Use questions that challenge an explanation mathematically and draw justification.
Ask clarifying questions if representation and inscriptions or mathematical terms are not clear.
|Indicate need to question during and after explanations.|
Ask a range of questions including those that draw justification and generalised models of problem situations, number patterns, and properties.
Work together collaboratively in small groups, examining and exploring all group members’ reasoning.
Compare and contrast and select most proficient (that all members can understand, explain, and justify).
Appendix 2. Moana’s chart for the ground rules for talk
How do we kōrero in our classroom?
We make sure that we discuss things together as a whānau. We listen carefully and actively to each other.
We ask everyone to take a turn at explaining their thinking first.
We think about what other questions we need to ask to understand what they are explaining.
We ask questions ‘politely’ as someone is explaining their thinking; we do not wait until they have completed their explanation.
We ask for reasons why. We use ‘what’ and ‘why’ questions.
We make sure that we are prepared to change our minds.
We think carefully about what they have explained before we speak or question.
We work as a whānau to reach agreement. We respect other people’s ideas. We don’t just use our own.
We make sure that everyone in the group is asked and supported to talk.
We all take responsibility for the explanation.
We expect challenges and enjoy explaining mathematically why we might agree or disagree.
We think about all the different ways before a decision is made about the group’s strategy solution. We make sure that as we ‘maths argue’ we use “I think … because … but why …” or we use “If you say that, then …”.
Appendix 3. Tumeke School’s expansions of sections of the Mathematics Communication and Participation Framework
This chart illustrates the way Moana further elaborated sections of the Mathematics Communication and Participation Framework. She did this in two steps. She constructed the first step after she began exploring the framework, and she put the second step in place when she observed that her students were managing the first set of expectations and would not lose confidence when expected to engage at a higher level in the mathematical practices.
Making conceptual explanations
Use problem context to make your explanation experientially real.
Think of a strategy solution and then explain it to the group.
Listen carefully and make sense of each explanation step by step.
Make a step-by-step explanation together.
Make sure that everyone understands. Keep checking that they do.
Take turns explaining the solution strategy using a representation.
Use equipment, the story is the problem, a drawing or diagram, or/and numbers to explain another way or backing for the explanation.
Keep asking questions until every section of the explanation is understood.
Be ready to state a lack of understanding and ask for the explanation to be explained in another way.
Ask questions (What did you…?) of sections of the explanation.
Discuss the explanation and explore the bits which are more difficult to understand.
Discuss the questions the listeners might ask about the explanation.
Making explanatory justification
Indicate agreement or disagreement with an explanation and have a mathematical reason for the stance.
Listen to each person in your group and state agreement with their explanation OR state disagreement with their explanation.
Practise talking about the bits you agree with and be ready to say why.
Ask questions of each other about why you agree or disagree with the explanation.
Pick one section of an explanation and provide a mathematical reason for agreeing with it.
Discuss the explanation or a section of the explanation and talk about the bits that the listeners might not agree with and why.
Provide a mathematical reason for disagreeing with the explanation or a section of the explanation.
Think about using material or drawing pictures about the bit of the explanation that there have been a lot of questions about it in the group.
Ask questions of each other (Why did you…? How can you say…?)
Question until you understand and are convinced.
Explain and use different ways to explain until you are ALL convinced.
Appendix 4. Problem example using student misconceptions developed by the Tumeke School study group
Peter and Jack had a disagreement. Peter said that 5/8 of a jelly snake was bigger than ¾ of a jelly snake because the numbers are bigger.
Jack said that it was the other way around, that ¾ of a jelly snake was bigger than 5/8 of a jelly snake because you are talking about fractions of one jelly snake.Who is right? When your group has decided who is correct and why, you need to work out lots of different ways to explain your answer. Remember you have to convince either Peter or Jack, and they both take a lot of convincing! Use pictures as well as numbers in your explanation.