Developing Mathematical Inquiry in a Learning Community Professional Development Day 2

Developing Mathematical Inquiry in a Learning Community
Professional Development Day 2

If anything, this programme is about teaching resilience. Never be afraid to make the problems really hard. I’ve been in classes where children do not know the answer but they learning that occurred in finding that answer was HUGE.

Record what has worked well for you and why - it has been cool to see kids who are not confident in maths, or struggle giving it more of a go, and to see kids being challenged and feeling like they are all equal to each other, e.g. mixed ability groups.

Record what you have noticed about how your students feel and act about this way of learning maths - I think at the beginning they were a bit hesitant/ confused. It is a very different way of learning maths to what we normally do, HOWEVER, after establishing the group norms and getting more used to DMIC, the children have enjoyed working in groups and taking charge of their learning. They have also liked being able to work on harder, more challenging problems. I think they feel cleverer - is that a word?

Record what have been your struggles and why -  I have struggled with stepping back and letting the kids figure out problems on their own. It has been a struggle to get the kids engaged and actually wanting to work sometimes. The other tricky part has actually been teaching the children how to talk to one another, and work effectively in a group. It has been a busy term with camp and fiafia and time has been a real constraint.

Ability grouping takes away the opportunity to learn. As a country NZ streams more than any other country in the WORLD!

Use the framework to reflect and record the actions you have taken so your students make mathematical explanations.

Developing Conceptual Explanations:
Bolded areas of framework - this is what teachers should focus on in their first year of DIMC. We don’t say tidy number - we say a tens number.
It is not about the answer - it is the JOURNEY to getting the answer.

Deepen the Mathematical Explanations

When you have the children in their groups - have the students develop 2 or more ways to explain a solution which may include using materials. Bring in the materials at a later stage if needed - don’t tell them

They need to explore 2 ways to answer so they are able to understand the questions and answer problems that arise.

Choosing the types of materials, e.g. if you are working with multiplication counters might not be that helpful.

Compare explanations and develop the norm of what makes an acceptable explanation. Reinforce what makes it mathematical.
Do not accept ‘ because it was easier’ as an answer. Really trying to get the kids to use mathematical language. 15 times 3, e.g. 15 times what three times what, e.g. if it is caterpillars 15 caterpillars, get kids to record this. It might feel like you are slowing down the lesson, but you are in fact going in depth. The community relaises at any point that they may be able to explain what is being explained to them. It is about taking risks, asking questions is a mathematical practice. Ask the child who has asked the question, so does that answer your question. This all needs to be MODELLED, step-by-step. Reinforce acceptability of mulope ways. Support them to make connections to other previous problems. The more challenging you make the problem, the more creative the kids responses will be. Think about your numbers, if you are doing a x number, thinking about using an 28 or 29 so they will have to compensate/ distribution.

If you have a question with a really tricky conext, make the numbers simpler, if you have a simpler question, make the numbers bigger. Number is like spelling, make sure you have the multiplication tables available for them to refer to, eventually they will learn them

Ask the students in small groups to examine their explanations and explore ways to revise, extend and elaborate on sections they think others might not understand.

Have a variety of students from the groups participate in sharing the group explanation. If you have a student who is VERY hesitant to share, for the first few times just make sure they go up with the group and are physically standing there with their group.

It is powerful to sit in a semi-circle, everyone has access to the speakers and can contribute, facing the right way etc.

During large group sharing change the explainer mid explanation. You don’t have to wait for every group to come to a solution.

When questions are asked of the small group, select different members to respond.

Make the use of mathematical practices visible:
Affirm models of students actively engaged and questioning to clarify sections or gain further information
Explicitly note student behaviour which support mathematical practices.

Developing representing as part of exploring and making connection

How can I/we make sense of this for my/ourselves?

Notice kids body language in the class and gestures in the class. Wait for that moment to go over there and say hey what are you doing there…

Communication and justification

How can I explain, show, convince other people?

There will be kids in your class in your small groups that are hiding that  don’t know what is going on.

Representing reasoning
Expect the use of a range of representations, acting it out, drawing a picture or diagram, visuals, making a model, using symbols, verbalising or putting into words, using materials.


A group of mamas are working on a tivaevae pattern.

Representing Reasoning

First position = 12 leaves,    2nd position = 20 leaves

ANSWER = position x 8 + 4

76 x 8 = 372 leaves.

Expect the students to explain and justify using the representation as actions on quantities not manipulation of symbols (use context).

Compare representations to develop connections to big mathematical ideas.


For prize giving Pepe and her family made lolly lei. The used 21 bars for each lei. 9 of the bars were mini crunchie and the rest were mini moro bars. If they want to make 10 lolly lei how many of each mini bar do they need?

9 x 10 = 90 crunchie bars, 120 mini moro bars to make 10 lei for prize giving.

Stopping at each stage to say 9 what … Make sure you are always bringing the context back into the question. Look at the 2 ways, what is the same what is different.

Set some new goals using the Communication and Participation Framework

Remember this is a journey and developing mathematical practices in a community of inquiry takes time because it is ambitious.

First week/s back at school: I really want to focus on the children being able to JUSTIFY why they chose the strategy they chose… and be able to come up with more than just one.. They seem to be only able to find one way of doing things.

How could you use these photos for problems?

What maths can you see in this siapo?

First day - what maths can you see in this image

Day two - started looking at geometry

DMIC maths doesn’t always need to be work problems.

We talked about this in the first PD session now let’s really dig into it. Brainstorm some home contexts you could use to set  your problems.

  • Cooking, how many sandwiches can you make with a loaf bread
  • Supermarket shopping
  • NRL/ sport stuff
  • Fishing
  • Unveilings - sharing cost of the head stone
  • Feeding the family
  • Church activities
  • Traditions
  • Dairy
  • Bakery - e.g. chicken sticks
  • Walking to school
  • Chores (time)
  • Playing in sports teams (rippa, netball etc, set up 5 m from the try line)
  • Music
  • Buying  fine mats
  • Activities that happen at school (fia fia)
  • Playstation, fortnite

Using the Talk Moves

These are a TEACHER tool, for the teacher to orchestrate the discussion. The one talk move that needs to be handled carefully is the REPEAT talk move. IF you hear sometime who has says something awesome, ask someone who you think was listening to repeat, ask someone who is confidence to repeat. If you want to invite a child who is shy, the repeat move is really valuable.

Helping students clarify their thinking and the thinking of others

Wait time - ask the question and then pause, you can give groups ‘think time’. Teachers often find it hard during the wait time not to talk - I am definitely guilty of this!! Is there a point where you do jump in.. we will come back to you.

Turn and talk (circulate and listen) - turn and talk to your buddy - what did they just explain to use .. you as the teacher listen. Sometimes others can explain in clear terms what others are struggling to explain.

Say more (who can say more; tell us more about what you are thinking; would you give an example)

So you are saying … ? (revoicing)

The talk moves are a POWER tool.

TALK MOVES THAT HELP STUDENTS ORIENTATE TO THE THINKING OF OTHERS - these will help children to communicate and participate - these draw everyone into the conversation

  • Who can say that again?
  • Who can put that into their own words?
  • Who can restate what (name) said?
  • Can anyone repeated what they heard (name) say?
  • Tell us what your partner said?

Helping students to deepen their own reasoning - this will be the group that says they are finished
  • Why do you think that?
  • What is your evidence?
  • How did you get that answer?
  • What convinced you this was the right answer?
  • Why do you think that strategy would work?
  • Can you prove that? What makes you think that?
Not just saying they disagree with the whole thing .. WHAT part do they disagree with?

These scripts can help you to push the conversation deeper and keep children engaged in the lesson. Tell the kids, I am using the teacher talk move so you are going to hear me say these things, you can even write it up. The teacher can revoice as well as the students. It is revoicing or repeating the IMPORTANT stuff. During the launch, repeating and revocing what it is WE ARE TRYING TO FIND OUT. You can see there are different talk moves you use in different parts of the lessons.

Instructional Activities (Quick Images) 10 mins, share 2 or 3 max

Instructional Activities (I.As) can be:
  • Used and adapted to use at multiple levels
  • Usefully employed to illustrate aspects of ambitious pedagogy

Steps in A Lesson:

  • Anticipate - predict and record ways students could solve a problem (include erroneous thinking)
  • Set norms and launch, always good to set the norms before you even launch, this week the norm we are working on is… once they going away ….. Don’t give pens when they initially go away. Publically make a deal of only giving the group a pen when they have all had a go at discussing what they might do to solve the problems.
  • Monitor  - make notes of who is doing what, this is a good opportunity to make a note of a particular good is doing some well, put their initials beside their work. Close listening and noticing, questioning to make thinking visible and to allow students to refine or revise their thinking. Pressing students to consider all aspects of the task. Monitoring for engagement.
  • Select
  • Sequence - share in sequence
  • Connect - what is the big idea here?

Looking at some problems and what types of problems they are.
Modelling (children visualising what is happening in the problem) (representing the actions and relations in problems)

In the eyes of children not all addition and subtractions problems are alike. How would they model these?
Mary had 12 biscuits. She ate 5 of them. How many biscuits does Mary have left?

Mary has $5 to buy biscuits. How many more dollars does she has to earn to buy $12 worth of biscuits? (change unknown) 5 + __ = 12

Mary has $5. Tom has $12. How many more dollars does Tom have than Mary? Comparing.

How many more … that really throws kids.
Subtraction can be solved through addition.

Direct Modelling

The first action is removing 5 from 12

The second is additive. They start with a set and add to it

Direct Modelling and actions and relations

How would these be modelled
John had 12 lollies. He put 3 lollies on each cupcake. How many cupcakes was he able to put lollies on?

Doing what comes naturally

Actions and relationships in a problem influence the strategies that the children use.
Direct modelling gives ways to counting strategies.

Addition and Subtraction: Problem Types
  • Join
  • Separate
  • Part-part-whole
  • Compare
Part-part-whole Problems
Involves static relationship, there is no direct implied action and no change over time.

8 boys and 7 girls were playing soccer. How many children were playing soccer?

16 children were playing soccer. 8 boys and the rest were girls. How many girls were playing soccer?


Involves relationships between quantities

Difference unknown
Mark has 8 mince. Joy has 12 minc. Joy has how many more mice than Mark

Compared Set
Mark has 8 mice. Joy has 4 more mice than Mark. How many mince does Joy have?

Referent Known

Joy has 12 mice. She has 4 more mice than Mark. How many mice does Mark have?

Recorded representations

Can be used to model problems using tally marks or pictures

The kids might have drawn 5 things here, then 4 here, written 9, you might need to go in and add the numbers or the sign, e.g. a plus sign. Like when we help them with their writing. You want the record to help you understand what the kid did.

Counting Strategies

Counting on
Count on from the larger number
Counting on to
Counting down

Number Fact Strategies
  • Children know some number facts and drive others from these
  • Need to be able to decompose and recompose numbers to 10
  • Recognise numbers that add to 10 (or 9....)
Mark has 4 bags of apples. There are 6 apples in each bag. How many apples does mark have? x

Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Partition division

Mark has 24 apples. He put them into bags containing 6 apples each. How many bags did Mark use? Measurement division

If apples cost 7 cents each, how much did Jill have to pay for 5 apples? multiplication

Jill paid 35 cents for 5 apples, what was the cost of 1 apple? partition division

Where to next?

Record what you have realised today that you were not aware of this morning: I was not aware of the technical terms of all the different types of math questions and I guess I didn't really understand the importance of the big ideas behind the questions I get the children to do in class.

Record what your next steps/goals are, why, and how you will achieve them?
I really want to get the group norms in my class down pat so that they are work better in their groups, the next step would be to get the kids more confidently in why they chose one strategy over another. 

Use the Communication and Participation Framework and plan the next actions you will put in place in your classroom: Again, I think the main aspect of the framework I want to put into place in my classroom is:

Model providing a mathematical explanation. Use the context of the problem not just the numbers.
Have the students develop two or more ways to explain a strategy solution which may include using materials


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