Communities of Inquiry

In Alan B. Schoenfeld's article, In Fostering Communities of Inquiry: Must it Matter that the Teacher Knows "The Answer"?, he discusses his personal experiences both as a professor of university courses on problem solving and as a participant in a research group comprised of many different levels of experience and education, from masters students to professors.

During the discussion he uses several examples of how, in the problem solving class, usually the students expect him to have "the answer" as he is the teacher. He admits that he knows exactly what will happen in the class; he can predict most of the discussions and questions from students before they are shared.

This is in contrast to his research group, where the progression is more fluid. The participants are welcome to bring data, video, particular challenges, to the group for discussion. The group is a community and no one has "the answer". In the group, there are several understandings: everyone is there seeking knowledge, the authority is the accepted standard of the explanations and there is trust: people are free to share and have ideas compete without personal biases.

Schoenfeld goes on to explain that throughout the semester in his problem solving course, he encourages the students to judge the solutions for themselves, not to look to him for validation of their ideas, but to prove it with mathematics. Indeed, he points out that in a mathematics classroom, the authority is really the mathematics itself, not the teacher.

Overall, Schoenfeld feels that in both the research group and the classroom, the community needs to understand:

a) they are all seeking a particular kind of knowledge and answers are not known in advance

b) the authority is not the teacher, it is the explanations and what is right

c) there is a feeling of trust.

The part of this article that struck me most was the constant and purposeful use of the word community when referring to both a classroom and a research group. This is the core of the article and how he has created a community of learning in both situations. A happy, fruitful community can be a happy place of learning, of discussion, of debate and disagreement; exactly as classrooms should be. I admire his willingness to be fallible. Perhaps, as he is considered an expert in his subject, it is not such a risk for him, but it shows his commitment to acting in the best interests of his students and also his colleagues.

I agree with Schoenfeld that there are 3 elements are essential in a community. I have noticed that in my classroom, it is particularly difficult to convince students that they do not always need to get approval from me, as their teacher, that I am not the expert. The most helpful way I have found is by pointing out my errors and making a list of my mistakes. Students seem genuinely surprised that a teacher not only makes mistakes, but celebrates them. They are highlighted as a learning experience. It has made students more willing to participate, take risks and try out new ideas.

Do you think you have a community of inquiry in your practice? Do you have all the tenets Shoenfeld mentions? Do you think there are times in teaching when the teacher should be the expert?

Topic for class paper

I am interested in researching gender in the mathematics classroom.

I am interested in this because although there have been recent efforts to reverse this pattern. as often there are more men in STEM than women. I have heard many girls in my classes saying that boys are naturally better at mathematics. Why is this? Where does this opinion stem from? How do I effectively work against it in my classroom? Is this a reflection of my school's cultural make up?

Culturally Responsive Teaching

Culturally Responsive Teaching

In Averill et al.'s article, Culturally Responsive Teaching of Mathematics: Three Models from Linked Studies, they examine 3 models for developing and analyzing culturally responsive teaching in mathematics teacher education. The studies took place in New Zealand, where there is a government treaty that declares the relationship between Europeans and indigenous Maori must be honoured. The studies frequently use the term "biculturalism" in reference to this relationship and the treaty, referring to the Maori and non-Maori islanders..

The Components Model was one in which the lecturers in teacher education included as many bicultural activities, aspects and perspectives as they could in their teaching of the preservice teachers. In the questionnaire the preservice teachers completed, students did recall a large range of perspectives being used, but they also did not recall perspectives that could also be linked to pedagogies common to many cultures. The lecturers felt that by not drawing students attention to the bicultural aspects of the course; some of the specific strategies were lost on the students.

The Holistic Model was one in which the lecturers chose a traditional cultural practice (woven panels called tukutuku) through which cultural components were linked. The lecturers used tukutuku to explore problem solving, algebra, number and measurement. Students also had to make their own model tukutuku to

A tukutuku

symbolize and reflect upon their growth as teachers as well as to represent the course content. Students felt that they would not have thought of this type of activity as being related to mathematics before the course and that the tukutuku made them more aware of opportunities to use cultural activities in their classrooms.

The Principles Model investigated how 3 beginning teachers describe their teaching practice in relation to the requirements of the Treaty of Waitangi's standards of bicultural education. The teachers reported that they were using culturally responsive activities in class and had seen positive results from both Maori and non-Maori students. They supported preservice mathematics courses with a strong emphasis on culture. They all mentioned that their experiences were influenced by the school culture and included challenges such as: time pressure, limited flexibility, lack of resources and students' unfamiliarity with the Maori language. They felt as new teachers, they had to "fit in" at their new school and that there was a lack of support for "new" ideas in mathematics education.

Overall, the researchers concluded that the models could be used independently or in combination. Averill et al. felt the following conditions are needed for effective culturally responsive teaching in mathematics:

1. deep mathematical understanding

2. effective and open relationships

3. cultural knowledge

4. opportunities for flexibility of approach and for implementing change

5. many accessible and non-threatening mathematics learning contexts

6. involvement of a responsive learning community

7. working within a cross-cultural teaching partnership.

What struck me most in this article was how, perhaps due to the treaty being law, bringing the culture of the indigenous people into classrooms is normalized in New Zealand. It made me realize how far behind Canada is in this process, with reconciliation still not complete. Will it ever be? Currently, teachers are lacking resources to bring indigenous culture into the classroom, but hopefully this will improve with time.

In my reading, I thought that the holistic method is likely the one that would be easiest to implement, as it has the most potential in the short term: teachers could be shown and taught indigenous skills or projects that could be brought into the classroom. In addition, if these ideas were shared with preservice teachers and there were indigenous teachers or support at the school level, it could be a positive step forward.

I was saddened that the Components Model was not successful overall, as I would think that this method would be ideal for having culturally responsive mathematics classrooms. When you make your classroom a place where many perspectives are examined, through discussion, activities and projects, is that not a place where great learning can occur? When multiple perspectives are always considered? Perhaps this would not work in New Zealand, as the treaty seems quite specific about Maori culture, but it may work in other countries and create more global classrooms.

Questions:

Which of the models do you think shows the most promise? The authors listed 7 conditions needed for culturally responsive classrooms to be effective. Do you agree with them? Do you think there is anything missing from their list?

Mathematics in the Streets and in Schools

This article summarizes a research study done by T. Carraher, D. Carraher and A. Schliemann that focused on children with only a few years of formal education and the computations they could perform mentally.
The study was conducted in a city in north-east Brazil, Recife, that is home to a large number of migrant workers. Many families' incomes are influenced by what income the mother and children bring in, thus it is common for children to be involved in street vending from an early age to help increase the family's low income. The work of a street vendor req
uires many mental math skills and involves all four operations. Pencils and papers were not used.
The study was conducted with 4 boys and 1 girl, ranging in age from 9-15. Their experience with formal education ranged from 1 year to 8 years.
The results showed that when the children were in the market, performing mental math in a contextual situation, they were able to answer over 98% of the questions correctly. In a classroom, when given word problems with some context, they answered 73.7% correctly and without a context at all, 36.8% correctly. Their findings on using real-life and word problems in mathematics "may provide the daily human sense which will guide children to find a correct solution intuitively without requiring an extra step [algebra]."
Overall, the researchers suggest that their study should inform teachers to consider introducing mathematical systems in contexts that are connected to daily life.

This article and its findings were fascinating. The research seems to support the idea that teachers need to connect mathematics to real-world situations in order to help many students "buy in". Indeed, in our current culture, children are surrounded by stimulation and are pulled in so many directions, that effective teaching really needs to be useful, but, perhaps more importantly, also connected to the world they are living in. In many of the mathematics curricular material used in Canada, there is starting to be an acknowledgement of these needs, as problem-based and inquiry-based learning are becoming more common in the mathematics classroom. Both can be challenging to implement, but at the very least can lead to further reflection on one's teaching.
I also agreed with the researchers that connecting math to useful activities will also help retention of skills. We should not always be introducing skills independently, without context, and with certain rigid rules that should be followed to successfully master the skills. In my teaching practice, I have seen many students who have been taught skills and rules, but when faced with a problem that is varied from ones they have seen previously, they are unable or struggle to solve it; they are unsuccessful at analyzing the question and applying mathematic skills. In particular, I find many students memorize formulas and are unable to explain why they work or are correct.

How do you connect mathematics to real-world applications in your teaching? Or how might you? Do you see this as a necessary step in teaching mathematics skills? Does context matter?

Reasonable ineffectiveness in math education

Summary:        

  In this article, Jeremy Kilpatrick discusses some researchers' view that research has a hierarchical structure, with pure, or basic, research at the top, followed by applied research. He offers an alternate view that both types are complementary to one another. Kilpatrick calls this view of the "lens" model, meaning "a study may be basic or applied depending upon the lens you use in reading a report of it." He goes on to discuss the reasons that research in mathematics education is ineffective: the lack of funding (in the US), the lack of a true identity as a community, and more research being conducted for dissertations rather than by true researchers publishing in journals.
            Kilpatrick also found, from looking at 38 U.S. journal articles, that none attempted to link their research to a theory in mathematics education. He gives a few examples of people whose theories had a large impact upon education, namely Thorndike, Piaget and Polya.  He encourages teachers to be involved in research, to not only collaborate on the collection of data, but also the analysis of the results and the writing of reports and articles.  His conclusion is threefold: a strong sense of community is needed in mathematics education, there needs to be theory grounding the research and finally, acceptance both of the limits of research as well as its complexity.

Stops:

I agree with Kilpatrick that all research can be read differently, depending on the "lens" of the reader. The same concept is true in many disciplines. Literature analysis can vary greatly, depending on the experience and background of the reader. It seems natural that whether the research is seen as applied or basic would vary, depending on the reader. As a teacher, I see the same word problem interpreted in numerous ways on a daily basis. It follows that research in mathematics education would be similar.

I wonder if his analysis of the lack of theory in mathematics education research would hold true in other countries. He readily admits in the article that he purposely chose articles from U.S. researchers. Certainly, the U.S, likely has one of the largest number of math researchers in the world, but perhaps researchers in Europe or Asia would have a stronger connection to theory.

I also wonder why it is that teachers do not often participate in conducting research. I have certainly come across the stereotype of mathematics educators being thought of a cold, logical, intellectual people with poor social skills. In my experience, this is not usually true. Teachers in elementary school are often intimidated by mathematics, some due to their own math anxiety. This alone could certainly discourage engaging in discussion and reflection on mathematics; participating in or conducting research would not be desired by such teachers.

Question: Do you agree with Kilpatrick that research cannot be classified simply as basic or applied, but rather is dependent on the lens through which it is viewed?
Do you have any ideas as to why it might be that many teachers to not research? Or do you feel this has changed since the article was written in 1981?

Hello and Welcome!

Hello everyone! I'm looking forward to getting to know you throughout the course and our discussions.