She is a good teacher. She is good in mathematics. Nothing deeply wrong with her mathematics. She has no problem with classroom management. The class is not crowded. Almost every learner does the homework. Learner takes turn to solve problem on the board without any strict instruction. It is like voluntarily from learners. Learners participate actively such as answering teacher' questions, no only Yes or no, but also give his/her reasons.
The challenge is why this lesson is still problematic. It means there is wider pedagogic cultural problem that frames ......
This is why we need to analyze carefully about the object itself because the object (the intended object of learning) is still problematic for some learners.
What aspects then we need to look at carefully. Things that related to the mathematical object learned by learners.
They are about:
- examples in relation to the object
- representation in relation to the object
- questions in relation to the object
- mediation in relation to the object
- explanation in relation to the object
- what learners are doing or stating or questioning or reasoning in relation to the object. One important thing is that the teacher is not the only one that can provide explanation. The learner can do it and the teacher can re voice/paraphrase or the teacher can use learner's contributions to prompt other learners' thinking.
the - coherence of the lesson in relation to the object can be assessed or established based on those aspects.
Those are part of our analytical framework.
This framework seems are powerful in two ways:
1) As a tool to analyze the lesson/video to be able to talk about teacher mathematical judgement and the changes
2) This can be as a tool to develop or enhance teacher mathematics judgement.
Let's remember the TM1.
When the teacher was asked to reflect on the teaching by CP, teachers talked a lot. They talked about what the like and don't like, talking about the need to mention of the objectives, and the issues that this lesson is only for smart learners or who knows already about this.
Using this framework seems productive to focus teachers' attention to the object, which is about the intended object of learning and how the teacher create learning conditions for learners.
They were challenged to identify what examples, representations, questions and what learners did in relation to the object.
After recording all those, teachers could see and commented the relationship among them or to get them to see the coherence of the lesson.
The use of this framework prompt them to think about the object itself, such as why is that example, what about the order of examples, what possible things to see by the learners from those presentations. This shows the teachers that choosing examples to present require mathematical judgement.
There are 3 levels of judgement:
- how they explain
- what explanation they need to provide
- what examples and representations need to display
- what questions need pose
- what learners contributions are expected?
- what types of learner's engagement could be stimulated or enhanced.
All these need MJ.
We are interested to see school mathematics discourse. We have to have a good operational explanation. For example, how do we see routine?
Routine:
Is it a pattern of interaction between teacher and learners?
Is it in relation to mathematical instruction?
What questions do we need to ask ourselves to do the description of the lesson?
The nature of TMJ is able to see in the selection of examples and explanations in relation to the intended object.
If we want to see improvement, we want to see the examples used, a more greater leverage in relation to the intended object.
The teacher judgement can be detected through his/her way use languages, representations or in terms of generating examples, the way he/she endorse.
What's the operational criteria for what's legitimated?
In the teaching, teacher may state something,establish something, asserting something, explain something.
What we have seen from the videos that teachers work mainly at the level of statement, state what it is, state just what to do, so we miss objectivication, opportunity to .....to the object.
We see in the ICCAM, the results of imitating something rather than ....
We are wanting to see in the examples and representations greater leverage of object, that could be through started examples, opens up, references example.
The choice of examples may be haphazard, picking whatever comes into mind, as opposed to rational or well planned example.
In order to have good judgement, we have to be clearer in being able to identify and describe the intended object. Teachers have to be able to say this is "it" they want learners to know, and be clear about what they want them to do with it.
The major constraint is about the learners, the resources available, text book. The biggest constraint is the cultural script that are operating. That seems so normal.
In our PD, in each session, we need activities to reinforce this ability. For example:
- Here is the problem given.
- Here are some of the learners answered.
Now, what question do you ask the learners?
What explanation do you offer?
What do you want them to understand?
What examples and questions might you give?
What explanation would you give?
What do you want the learner to do?
All the time, what is it? What examples? What representations? How you explain? Why do you want the to do?
We can do this in the range of setting.
In video reflection, the will not necessarily focus on the way we are seeing but we need to bring them closer to our intention.
The reflection is about:
Are the explanations good for that thing?
Are the examples powerful for that thing?
Are the learners doing things that we expected?
From this writing so far, how do we turn these ideas into a way of being able to code our transcript?
What counts as an explanation?
Some of them by stating.
Whatever we capture, we need to be able to say what is it endorsed?
What criteria teacher recognise?
In T2 lesson, what we can say?
There are nice range of examples. She moves quite nicely through things. She gave useful representations to what distributive law look like?
But in fact what counts as explanations or endorsement is about steps that you need to do. They don't need to do about multiplication. How do we get to see that? How do we describe the endorsement?
For her, you know what to do by indicating the arrows. Whatever you find outside, you need to multiply the inside. So the endorsement doesn't reinforce words... But it doesn't establish the product or multiplication. It doesn't point what it is to multiply?
So we are not talking how do we deal with the lesson, but what could we see inside the lesson?
We want to see, what is carry about the intended object. What is there? How do you tell whether the explanations have relationship with the object.
The main issue is:
What is enacted here and how this relate to the object that we understand.
The is intending that learners would be able to do that and think that is sufficient. Good teaching involves learners. But something is missing here. Why her kids are still doing mistakes.
There is something. About disappearing object. What we are trying to do is to bring back the object into focus. When you are think what you are doing, you need to think about what is it.
Let see CP teaching:
What the examples and representations used?
He displayed verbal representation or words. Then writing algebraic equation. The represented in a table with particular cases. First is generating the value that if you multiply, give you 12.
T2, all the way through is symbolic.
Here there is the relation between examples, explanations, learner activity. The relation between this that constitute the coherence.
MJ is about how is the examples relates to the object.
How is the representation relate to the object
What examples are displayed
What explanation? Just describe restate.
When they see it? Do they able to recognise it?
Every time you see negative, it's not polynomial.
Have you got the means to deal with 1/x-2.
It is assertion. It is not rational for this so if they don't oppose with examples, learners cannot recognise it.
So it is not about how many. It is about critical features. They don't offer to discern .... All they being offered ......
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