Problem Posing in Mathematics Education

In this article, Brown and Walter describe five sensitivities that should be embedded to mathematics experiences when teachers introduce problem posing, focusing on pedagogical perspective.

1: An irresistible solving drive:

According to the authors, in some cases, students try to find out answers or solutions that are given without much thought. But it is important for students to think and understand what a problem is, what alternative mathematical entities are, what mathematical view is as problem-solving when they pose their own problems.

2: Problems and their educational potential

The authors question what we can do for a given problem other than coming up with the solutions. They believe students can create the new problems into the situation, and this variation/creativity enriches to understand original problem and enable to pose other possible educational problems aside from problem-solving.

3: The interconnectedness of posing and solving

The authors claim that there might be unexpected logical connections in which problem-posing and problem-solving.

4: Coming up with problems

The authors categorize problem-posing into two types: accepting and challenging. For challenging the given, they suggest “What if not?” strategy to be heuristics problems as following: (p.23)

1. Make a selection
2. Notice the attributes of the object
3. Vary the attributes
4. Ask a question about the new form
5. Analyse the questions

5: The social context of learning

The authors address importance of social context of learning in problem posing environments which are relationship of the individual and the group. They show the readers several ways to create problems for group interaction such as the case of using editorial boards.

 

Response:

  I agree with the authors’ idea of the sensitivities to problem posing in mathematics. As per my teaching experience for immigrant students in Japan, they tend to answer the mathematics problems by utilizing their assumption from parts of the sentence (some words), pictures and figures within the question because they have little Japanese language proficiency to understand the given problems. However, although not always, they surprisingly correctly answered those problems. This fact supports the idea in the first sensitivity. Therefore, problems in textbooks and drills in mathematics have common features across different countries, and I suppose some students who are familiar with it might be able to answer the questions in different languages.

  However, I have never seen the cases that the above immigrant students challenge problem posing. In addition to their limitation of language, there are no contents intend to problem posing in mathematics textbooks and drills and are a lot of the repeated problems instead. I suppose there might be no developments about problem posing in terms of textbooks and drills, thus the mathematics class used those problems might have no chance of heuristic learning for the students including non-immigrant students.

Questions:

Do you think the mathematics textbooks and drills are containing enough contents for the problem posing process? If so, has it been changed over the past years?

My Topic for Assignment

Mathematics learning for English Language Learners (ELLs) focusing on their obstacles of studying in additional language

I am thinking to explore how ELLs are(were) learning mathematics in English or how native teachers teach(taught) mathematics to ELLs. It might be not easy for them to learn mathematics in their additional language, especially for word problems and their participation in the class. Because ELLs consist of variety of people, I will focus on the people with some specific ethnic background such as Asian or Hispanic. I also would like to emphasize how their learning/teaching shifted and developed by the times.

Mathematics as Medicine & Balancing Equations and Culture: Indigenous Educators Reflect on Mathematics Education

<1st article>

Author: Edward Doolittle

Source: Proceedings of the Canadian Mathematics Education Study Group Conference (pp. 17-25).

<2nd article (dialogue)>

Author: Edward Doolittle, Florence Glanfield

Source: For the Learning of Mathematics, 27 (3), 27-30.

The author, Edward Doolittle, has Indigenous ancestors and started to be interested in own root when he was in undergraduate. As he is both mathematician and "Indian" (I deliberately use this word here because he used it in this article), the main goal in his long journey is how to "resolve the apparent incompatibility between being a mathematician and being an Indigenous person." (p.19)

  In his description about interfaces between mathematics and Indigenous thought, he states that mathematics is mandatory to succeed in the labor market. However, he points out people lose something in their culture while they improve their academic attainment in mathematics. Additionally, he claims that ethnomathematics can be "reflective and respectful to Indigenous traditions of thought" (p.20), while he concerns "oversimplification" of ethnomathematics because it does not perfectly reflect Indigenous thoughts. To proceed towards his goal, he raised two suggestions. The first suggestion is to explore the question of how we can "pull mathematics into Indigenous culture." (p.22) The other one is to understand that mathematics is simplifying response whereas Indigenous thought is refining complicated response to complex phenomena. Lastly, he questions that mathematics can "make our lives better as a people, or are its benefits restricted to just a few fortunate individuals." (p.24)

  The dialogue between Edward Doolittle and Florence Glanfield started with the story of "Mathematics as Medicine". In their conversations, they point out many arguments/questions about formal mathematics education and Indigenous culture. Firstly, they discuss that mathematics is required in the societies. Doolittle questions what the adequate amount of mathematics is necessary and has doubt for that the all mathematics school teaches is really necessary. Additionally, they claim formal mathematics education devalued the value of relationship between mathematics and all other things. They say Indigenous education leads people to find or develop own answer to natural phenomena by themselves, but Western does not. Doolittle addresses mass productions can account for this issue. Mass productions force people to have the consistency between input and output, rather than individual control. They suggest to turn into how Indigenous can help Western from how Western can help Indigenous, and they raise the issue "(W)hat way(s) can Western society become 'balance' in valuing multiple strengths and contributions." (p.29) They stress that "knowing who you are and how fit in" is important for balanced development.

Response:

  The critic in the article of Doolittle about oversimplified ethnomathematics by non-indigenous scholar stopped me to think my attitude for research in general. As non-indigenous people cannot become indigenous people, it is impossible for me to have the perfectly same insight as indigenous people. This article is written by Doolittle who has Indigenous root, and it contains the perspective of Indigenous thought. I agree with the idea that "how Indigenous can help Western", not "how Western can help Indigenous". Like international cooperation, it is crucial to think what developing countries need from view of themselves rather than from developed countries. Therefore, to avoid oversimplification of ethnomathematics, the real voices of Indigenous people is obviously necessary to study ethnomathamtics.

  However, I wonder about the idea that Indigenous people lose their culture while they improve their academic performance in mathematics is true. I suppose there are a lot of spaces to learn mathematics other than school, so they can learn cultural contextualized mathematics outside of school while they learn universal mathematics at school. School mathematics may reduce time to learn their culture, but I disagree with that they lose their culture by school mathematics.

Question:

• Canada is known as a multicultural society. How can you provide your students mathematics class which involve well-balanced cultural context? 
• Can mathematics make our lives better as a people, or are its benefits restricted to just a few fortunate individuals?