Classroom observations prior to the professional development provided evidence that Mrs Stuart had some understanding of the links between arithmetic and algebra. Instantiations of the following types of algebra were evident during the five observed lessons: commutative property, functions, properties of zero and one, associative property, equivalence, inverse relationships, and odd and even numbers. However, Mrs Stuart did not facilitate the students to examine explicitly the properties of operations and numbers. For example, in a lesson involving the commutative property students were asked to share responses related to the following task (see Figure 4):
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Figure 4. Multiplication problem. From MEP practice book Y2b (p. 107), by S. Hajdu, 1999, Budapest: Muszaki Publishing House.
The student constructed equations (e.g., 2 x 4 = 8 and 4 x 2 = 8) implicitly drew on the commutative property; however, there was no further examination of this. In a later lesson, Mrs Stuart began by asking the students what they noticed about the two alternative solutions that she had recorded on the whiteboard and then offered a brief explanation of the commutative property herself:
Mrs Stuart: What do you notice, Otto? Otto: It’s the other way around?
Mrs Stuart: What do you mean by it’s the other way around? Otto: It’s, it’s the same but it’s just changed around
Mrs Stuart: And that’s one of the really important things in multiplication, isn’t it? It doesn’t matter if we do two times five or five times two. (Week Two, Term Three, 2008/2009)
The tasks used by Mrs Stuart with her students were taken directly from the MEP curriculum. Aside from one instance in the fifth lesson, the tasks were not modified.
Mrs Stuart spent a significant portion (between 16% and 27%) of the lessons introducing or orienting the students to the task. This involved students being carefully guided through the steps necessary to complete the task with a focus on a fast pace.
For almost half of the whole class discussions prior to task completion, Mrs Stuart used questioning characterised as leading or funnelling students towards correct responses or teacher chosen solution strategies.
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An example of Mrs Stuart’s task implementation prior to the professional development is provided in the following vignette where Mrs Stuart is introducing her class to a functional reasoning task (see Figure 5):
Figure 5. Functional reasoning task. From MEP practice book Y2b (p. 107), by S. Hajdu, 1999, Budapest: Muszaki Publishing House.
Implementing a functional reasoning task
Mrs Stuart begins to introduce the students to the task by asking them the numbers of legs for each animal. She then asks them whether they think they could complete the table and most of the class put their hand up. She continues to carefully guide them through the completion of the first boxes on the table by acting each scenario out:
Mrs Stuart We’ll do one or two together and then we’ll see altogether, so first column…how many of each animal are we talking about? Gabriel? Gabriel Zero
Mrs Stuart Zero, so here we go. Watch I’ve bought them in especially for you today. Ready. Here you are, zero chickens (Mrs Stuart lifts imaginary chickens in her hands). How many legs on my zero amount of chickens Lorenzo?
Lorenzo Two.
Mrs Stuart How many legs can you see on this chicken? Lorenzo Two.
Mrs Stuart Really? Class Zero.
Mrs Stuart How many chickens are here? Lorenzo Zero.
Mrs Stuart So how many legs are here? Lorenzo Zero
Mrs Stuart Fill that in in the table…Right and here is my (lifts imaginary cat) how many cats am I holding?
She continues to lead them through filling in the table until the first three columns are completed. Then she asks if they think they can complete the table in a minute. After they have worked on the task for one minute she stops them.
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Mrs Stuart Okay, who managed to do some of the table? If you haven’t done all of it, we’ll fill it in as we go. So what do you notice about the middle row? What are those numbers called?
Aaron Even.
Mrs Stuart They’re even numbers, good boy. What else are they, Elijah? Elijah They’re multiples of two.
Mrs Stuart They’re multiples of two. So, are you ready?…Off we go… Elijah Zero, two, four, six, eight, ten, 12, 14, 16, 18, 20.
Mrs Stuart Excellent. Well done. Now if you didn’t do them all you can pop those in now. Can you see they’re multiples of two?
Week Two, Term Three, 2008/2009
At this stage the questioning did not focus student attention on the relationships within tasks, and frequently guided them towards using computation to complete the table rather than to reason algebraically. Whole class discussions after the students had worked independently on a task were used to check that they had correct answers or to direct them to write the correct answer in their work book.
Mrs Stuart regularly used representational forms in her teaching. She drew on representations suggested in the MEP material including tables and concrete materials. Often she directly modelled how to use a representation to solve a problem. She also frequently asked students to use equations to show how they had solved a problem. However, the use of representation was typically limited to a single representational form.
Paired work was a feature of Mrs Stuart’s lessons. After a task had been introduced, the students were requested to work with a partner. While some pairs worked cooperatively on the task, others simply sat next to each other rather than working together. Rather than complete tasks collaboratively, the partnerships appeared to be more used as a support mechanism when the students were stuck. As noted by Patrick, from the focus group discussion, having a partner helped him: because if you don’t know what to do, you have someone right there and the other
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person might know. The expectation that students would record their work individually in their books matched the lack of expectation for students to use collaborative paired work to develop shared understanding of a joint solution strategy.
The teacher domination of discourse patterns during the whole class discussion limited opportunities for student engagement in mathematical reasoning. Frequently students gave answers with no mathematical reasoning or responded with an answer phrased as a question, for example: is it nine? There were no complete correct mathematical explanations provided by the students, however Mrs Stuart provided mathematical explanations at least twice and as many as seven times in a lesson. Often if a student provided a response with no reasoning, she would revoice it and provide a mathematical explanation herself.
The students perceived Mrs Stuart’s role as giving them mathematical knowledge. For example, in a focus group interview when asked what their teacher did to help them, they stated:
Geoff: She explains it for you
Researcher: How does she explain it for you?
Geoff: Because in your book it says fill in the missing numbers and then if you want to do it with your partner she lets you and then you put your hand up if you don’t and she explains it for you.
When asked what it takes to be successful in the mathematics class, they responded: Lorenzo To listen
Researcher: To listen. Why does that help you? Lorenzo: Because you’ll learn every single maths. Researcher: By listening?
Lorenzo: Yeah
Geoff: If you listen then she sometimes gives you sneaky information, if you listen carefully.
These responses illustrate that the students privileged listening to the teacher over talking as a way of participating in the mathematics classroom.
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6.2.2 Analysis of opportunities for algebraic reasoning, classroom and mathematical