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'Developing the Classroom Culture: Using the Dotty Six Activity as a Springboard for Investigation' printed from http://nrich.maths.org/

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This article supplements the more detailed article Developing a Classroom Culture That Supports a Problem-solving Approach to Mathematics.  This article suggests how to dig deeper into who answers questions in your classroom using the game Dotty Six.

Dotty Six could be a good game to try out in a staff meeting to support the development of classroom culture across the school.

A. Play it together and then look for ways that you could adapt it to suit your class. For example, draw counters in the boxes to aid bonds, use different dice, make the box total different, use fraction dice with ¼, ½ and ¾ and the boxes need to add up to 2.

B. Consider the focus of the learning for the lesson: the lesson objective. It may be that you want to choose a single objective – either a number one or a using and applying one - or a double objective – a number objective and a using and applying objective.

Possible number objectives:

• To recognise the number of dots in each iconic pattern and associate it with its number name and numeral
• To match the iconic representation of the patterns on the die with their symbolic representation using numerals
• To know number bonds to, and within, six
• To develop an understanding of the concept of addition
• To understand the associated language of ‘how many more?’

Possible using and applying objectives:

Here we use the idea of a student being a ‘beginner’ at an element of using and applying and moving on to becoming ‘proficient’. Beyond that we use the term ‘expert’.

Problem-solving

Beginner: engage with practical mathematical activities
Becoming proficient:  adopt a systematic approach

Communicating

Beginner:  respond to questions and ideas from peers and adults
Beginner:  refer to the materials they have used (the die and the grid) and talk about what they have done, what patterns they have noticed
Becoming proficient:  describe the strategies they used

Reasoning

Beginner:    explain numbers and calculations
Becoming proficient:  predict what could happen and give a reason

C. Try out the game in your classroom: you may find some of the guidance below useful (scroll down to 'The lesson' section.

D. Meet back as a staff and share your findings together.

E. Decide on how you plan to develop your classroom cultures in the light of your investigations around Dotty Six.

Other games where you could employ a similar approach and which could be used for a follow up staff meeting are:
Incey Wincey
Shut the Box
Four Go
Countdown

The lesson

Show the students the video clip and ask them to work out the rules of Dotty Six. Collect together a class set of ideas. Refine and adjust them until you are all agreed on the rules.

Let them try out the game themselves and consider whether any of the ideas below are useful to try out. They are based on three elements: asking, listening and responding.

Asking
A useful strategy is to ask questions – open questions - that encourage the students to articulate their thinking. Open questions that could be useful are questions such as:

• How many more dots do you need to fill that rectangle?
• I think you need five more dots to fill that rectangle – am I right?
• How many rectangles have you filled so far?
• If you threw a three, which rectangle would you put the dots in?
• I’ve thrown this ... which rectangle could that go in?
• I’m wondering what to do with this score. Can you help me?
• If I throw a six, how many spaces are left for me to put it in?

It is helpful to use the mathematical vocabulary of 'rectangle' rather than 'box' in the question. When you do this, you are reinforcing key mathematical language for the students that they are in the process of learning how to use for themselves. Whenever we ask a student a question, we need to allow enough wait time for them to respond. Feeling under pressure to answer a question quickly can be very uncomfortable and can prevent us from being able to articulate our thoughts very clearly.

Students learn to join in conversations by hearing what others are saying, listening to how words are being used and ‘playing around’ with those words themselves. This means that some modelling of talk around this game could be useful – between you and your Teaching Assistant, you and a puppet or you and one of the more articulate students in the class.

You may also like to capture some key phrases and words that you hear students using as they talk and put them up on your mathematics 'talk wall' or other display to support the students to use those words. Putting the words inside ready-cut out speech bubbles can be very effective and create an appealing display.

You may also like to stimulate some talk by joining in with a pair/group of students and ‘playing dumb’. For example, you could throw the die and then put more dots in a rectangle on the grid than there should be (you throw a four and put five dots) or you could put more dots in a rectangle than are needed to make the ‘full’ six.

Listening
Listening carefully to what the students actually say is sometimes harder than we realise. We may not hear clearly what they say as we may be expecting them to give us a fixed answer that we have pre-determined – this can be called, ‘guess what is in the teacher’s head!’ We need to be ready to be open to their answers and be curious to understand what they are trying to say.

Sometimes their answer may be part of a sentence and our temptation is to finish the sentence off for them. See what happens if you just repeat back to them what they have said, using the same words they have used, and see if that helps them to finish the sentence.

It may be that their answer is rather jumbled or rambling. Our temptation, in this case, is to rephrase it, reorganise it and repeat it back to the class in what we consider to be its new, improved form. We may hear ourselves saying something like,

‘Thank you, Elspeth. What Elspeth said was ...’

See what happens, if instead, you check with the child, ‘Elspeth’ if you have heard what they said correctly by saying something such as, ‘I think what you said was ...  Am I right?’ When saying what you thought they said try and use the same words that they used.

Responding
After mastering the art of the open, starter question and listening carefully to the students’ responses, we then need to decide how we are going to respond to what they have said. Our aim is to understand as much we can about where they are at with their mathematical thinking and concept development. Often it is helpful to respond to them with another question, phrase or statement that gives us an opportunity to explore their thinking further. This can helps us probe for deeper understanding and evidence of mathematical thinking and reasoning. Here are some examples of possible follow up questions:

Starter Question Follow-up Question/Statement
How many more do you
need to fill that rectangle?
Are you sure?
Convince me.
Show me how you know that.
If you threw a three, which rectangle could you put that in? I am curious to know why you chose that one.
I would choose this one ...  Are we both right?
I've thrown a six, what can I do? What could happen if I threw another six?
How many sixes can I throw and still fit them on the board?

Further ideas for the lesson can be found in the Teachers' Notes that accompany this game.