Aspects of the Teacher’s Professional Identity
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What was reified/ mobilized
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Evidences
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Self-knowledge
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Selfconfidence. Self-esteem. Self-image. |
Bia:
I’ve got to tell you something: yesterday, working with my students, I think it was the first time I left the room like this, satisfied, you know? [The students] understood, what is really fine is that they understood!! [...] It was amazing!
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Acknowledgement of the fact that the teacher needs to study. |
Ada:
I think teachers don’t have time to study [...] because I’m having difficulty finding time to study [...] We get stuck with [proportional] reasoning. How are we going to develop this type of reasoning if we don’t study?
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Participation in the CoP-PAEM as an emotional support for teachers. |
Bia:
I call these “therapy sessions” moments.
Eva:
This sharing of ideas with colleagues [...] seems to open up our minds, give more security.
Tina:
It’s really a “therapy session”, we get nervous sometimes [...] with things that happen at school and with this conversation, we leave here more relaxed.
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Beliefs and conceptions
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Belief/Concepts change. |
Clea: Something I learned a lot was to look for several ways to solve [a question]. Before I used to see only one way of doing things. [...] Like with the coffee problem, I never imagined I could solve the problem that way. It’s highly stimulating.
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Professional knowledge: mathematics specific knowledge |
Relative Reasoning and measurementRecursive partition principle: to get a precise measurement it is necessary to subdivide the measurement unit into equal and increasingly smaller subunits. By dividing the unit in smaller pieces a fraction will appear. |
Tânia: What did Iara do after 33,3?Bia: As she was looking for precision, she used another decimal place, another measure [...] It was refined. She went on partitioning: a tenth of a kilogram, a hundredth of a kilo.Tina: Iara used decimal register. And Bia used fractions register. [...] This idea of knowing that it is between 1/2 and ¼ is a measurement idea, isn’t it?Bia: Then, when it arrived at 33 and a half with 15 and a half we got 499 reais, I did 33 and 250 , then it went over a bit; then I thought, it must be number between 33 and ½ and 33 and ¼, then I tested again with 33 and 1/3... I arrived at 500 reais.
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Relative thinking and measurementRational numbers density property: allows increasingly closer measurement approximations. There is an infinite number of fractions between any pair of fractions, no matter how close they are. |
Iara
: Yes, then I was subtracting,[...] then I stopped there between 33 and [...] 34 which almost gets there to arrive at 496 and 33 will arrive at 512, then I had to put a number in the middle, then I got 33.5 and started subtracting until I arrived at 33.3 and realized that it had gone over 20 cents, then I started subtracting again. But I was very close. [...] then working with approximate data 16.67 approximately of the 14 reais coffee and 33.33 of the other… since it is a decimal.
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Relative thinking and measurementApproximation principle: how to decide the necessary precision for a determined context. |
Iara: Her (Bia’s) result was more exact because she used the fraction1/3. [...] if it were an exact number then I think that Bia’s result is more correct.
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Relative thinking (multiplicative) Percentage Representation.
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Ada:
If I find for 1 kg [...] how many grams of each type [...] by 50 (total mix).
Cléa:
That’s why she used percentages.
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Quantity and CovariationVerbalize and Symbolize mutable relations: highlight important quantifiable characteristics observing whether the quantities are changing or not under that situation, and if they are, observe changes directions relations. Recognize that variable quantities are not always related. |
Tina: In the table it is possible to see that one increases and the other decreases [...] even without a multiplicative relationship, [...] because it had to arrive at 50, the sum was 50. [...] I think it’s proportional because I already knew that one had more than the other.Iara: On the straight line (representation), on the top she places the prices and below the quantities. There is a covariance relation. If we look at the price, each space represents one real. On the line below, it goes from 50 to 0 and from 0 to 50. From point A to point B, there are six spaces, then, each space represents 1/6 of 50 or 1/3 of 25. So, from the 11 reais position to the 10 reais position there must be a decrease of 1/3 for bean B (1/3 of 25) and add 1/3 of bean A (1/3 of 25).
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Up and down reasoning: the strategy for this reasoning starts by its initial fraction, goes through a unitary fraction, a unit, and, finally, arrives at the desired fraction. |
Luiz:
She (Ada) worked with another unit, another whole. She reduced it to a smaller quantity (1 Kg) to then find what was requested (50 Kg). She used another whole (1 kg).
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Unity flexibilization: To round up before multiplying will increase error by rounding up. |
Ada:
I arrived there between 33 and 34. I got it but there was a difference of 4 cents, which at the end, through multiplication, this difference disappears.
Luiz:
With an approximation, the final amount would give this difference.
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Professional knowledge: Pedagogical mathematics knowledge |
To solve problems without algebraic resources, using only one “reasoning”, mobilizes proportional reasoning. |
Tina: Many times, to place x and y may not make any sense to the student. She needs to know who is x and who is y. That’s why proportional reasoning is important even before using the letter [...] It is important to see in the table what’s fixed and what varies. It’s an important resource.
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The importance to discuss the role of different types of representations and context adequacy. |
Iara: When I teach fractions I compare them with decimals. The fraction is also to indicate an inexact division. So this representation is also important.Tina: Even with 1/3, he will have to know how many grams that represents. There is not a most correct answer. It depends on the context. The scale measures in decimals.
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Students’ difficulties in learning mathematics can be related to teaching that emphasizes resources and formal representations. |
Clea:
Many times the teaching of mathematics does not promote the development of reasoning, prioritizing rules and algorithms. Maybe the difficulties students have lie on this fact.
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A problem can be solved through resources and different strategies. |
Clea:
Something I learned a lot here was to look for several ways to solve [a question]. Because I was a lot like this, I saw only one way of doing thing and had difficulty accepting that students did differently.
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Vulnerability
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Difficulty to consider other strategies and ways of thinking mathematically. |
Laís:
[It’s difficult] to get rid of the devices, of the algebra side. It’s automatic to do it using rules, systems. You “dig out” numbers in the problem and immediately structures [the algorithm]. Many times, you don’t interpret the problem.
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Recognition of one practice that prioritizes rules and algorithms. |
Iara:
Sometimes we use formulas without thinking about what we are doing.
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Confrontation of ideas and opinions. |
Tânia:If you had to validate one or another solution with the students, would make a difference between this answer (Iara’s answer) and this one (Bia’s answer)?Bia: No, the two are correct.Iara: (Bia’s) was more exact because she used the fraction 1/3. [...] if it were an exact number then I think Bia’s is the most correct, not mine.
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Sense of agency and political commitment
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The teacher needs to be responsible for curriculum management: selection and organization of contents taking into consideration time and material conditions. |
Bia:
We have an extremely extensive curricular matrix, filled with contents! [...] We had to rethink this matrix and work well with contents which are more important. There’s a lot to work and we end up working with everything [but superficially]. What’s important for the student?
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