Problem Posing

Source:

"Problem Posing in Mathematics Education." Problem posing: reflections and applications. By Stephen I. Brown and Marion I. Walter. Hillsdale, NJ: Lawrence Erlbaum, 1993. N. pag. Print.

Problem posing in mathematics education counters the traditional way of teaching math where an authority poses a problem and students simply solve it. The benefits of problem posing are that it can interest and motivate students; it can dissipate fear and anxiety caused by imposed curriculum; it can cause students to take more responsibility for their learning; educators can discover that various students have different talents as they pose problems. Problem posing activities should be incorporated into all sections of math rather than isolated; this article poses five themes that should be incorporated into formal and informal mathematical experiences “if mathematics is to be viewed as an act of liberation.”

Sensitivity 1: An irresistible problem solving drive

Students immediately jump to solving a problem even when a problem is not posed; this is problematic because students new mathematics think that the subject is just about solving problems. Instead, students should be aware of the entity they are examining (whether it be a situation, concept, object or problem etc.) and be aware of their reaction to it because doing this allows itself to meaningful enquiry.

Sensitivity 2: Problems and their educational potential

When presented with a problem, there are other meaningful pursuits than simply solving it. For instance, neutralizing a problem is when students create situations and pose problems themselves. When students understand the variations of the problems they pose, they can understand the solutions. They can also ask questions of themselves (ex. how can we group the problems our class generated?).

Sensitivity 3: Interconnectedness of problem posing and problem solving

When we solve problems, we restructure them in order to understand them. Also, we often do not appreciate what we solved unless we have pose a new question. When we ask questions about solutions, we understand our solutions and see connections in unexpected places.

Sensitivity 4: Coming up with problems

There are two ways to come up with problems. One is to accept the given and to ask how it might have been developed and so forth. The second is to challenge the given. Brown and Walter coined the ‘What-If-Not’ strategy. This method involves five steps. The driving force that makes it effective is that this strategy separates varying the attribute of a problem (step 3) and posing a question in a new form (step 4). When these steps of problem posing are separated the variations give rise to ‘possible worlds never imagined’ and problem posing becomes an act of creativity.

Sensitivity 5: Social Context of Learning

Problem Posing counters the notion that mathematics is a solitary activity. Instead it is more true to the way that mathematical ideas actually evolve; having students participate in this way makes them also aware of mathematics as a field.

Stop:

“If learning mathematics is to be viewed as an act of liberation”

I used a few quotes in my summary because there are key quotes that are written quite beautifully in this chapter. This introductory phrase caused me to pause and really ask, what is the purpose of mathematical problem posing? As I read the article the first time I wondered if the authors assumed that as one understands mathematical concepts they are intellectually liberated in some way. I found, rather, that phrases such as liberation and radical reminded me of critical theory and that problem posing can be linked to this; this appealed to me.  Walter and Brown assert that problem posing offers a chance for students to question content, each other and the way we interact; it causes students to question the status quo and what is imposed on them. Not only does it deepen understanding of mathematical concepts it allows for creativity in math and opens up ‘worlds unimagined.’ This phrase reminded me of Maxine Greene, she talks about the liberal arts in this way saying they open up possibilities to the ‘not yet imagined.’   Ultimately, I think providing students a space to see new possibilities in any discipline avails them the opportunity to see new possibilities for themselves. 

The authors conclude that the benefit of problem posing is that it causes students to question content and question how we interact and educate each other. To solve problems in pedagogy is better than searching for the perfect problem-solving curriculum.

Do you (for)see problems in problem posing pedagogy? What problems need to be solved or addressed when considering problem posing as a way of teaching?

EDCP Paper

I am currently looking into the idea of Problem Posing in Mathematics. I specifically want to find out how this can be used in elementary classrooms. I am hoping that by investigating this strategy, I will be better able to form open-ended activities that allow all learners including 'highly able/gifted/precocious' learners opportunities to think deeply about mathematical concepts.

Culturally Responsive Teaching

Culturally Responsive Teaching of Mathematics: Three Models from Linked Studies

Authors: Robin Averill, Dayle Anderson, Herewini Easton, Pania Te Maro, Derek Smith & Anne Hynds

Source: Journal for Research in Mathematics Education, Vol. 40, No. 2 (Mar., 2009), pp.157-186

Be it ever so small, it is as precious as the greenstone.

As though it were a spider web.

This article is guided by the two Maori proverbs above (as translated into English).  The goal of Averill et al. was to model the ways that cultural perspectives of indigenous New Zealand Maori can be “woven” into pre-service teachers’ pedagogy and practice. This study is based on culturally responsive teaching (Gay, 2002); this is the idea that students bring cultural capital into the classroom and that by integrating cultural values and perspectives in the classroom, children’s motivation, mathematical understandings and achievement can improve. In order to reach these ends, raising social consciousness and developing cultural competence of teachers is key.  Averill et al. examined three studies to find out how teachers integrate cultural values and perspectives. Three models are offered as starting points for culturally responsive teaching and research. These are the component model, the holistic model and the principles model.

The component model was developed during a study of 124 pre-service elementary teachers. The key question in this study was to compare lecturer and student perceptions about the bi-cultural content in a mathematics education course.  Researchers used surveys and systemically analyzed course structure and course components .The most notable finding is that while lecturers tried to incorporate Maori content into a seamless and normal way through the use of artifacts and metaphors, pre-service teachers did not identify these specific strategies. The benefit of the component model is that it provides a more planned, coherent and explicit way to identify culturally responsive teaching practices and thus can be used to review and improve existing courses. 

In New Zealand, the Treaty of Waitangi informs education policy which states that indigenous content should be incorporated in public schools (K-12). The holistic model was developed as lecturers created a fourth year university course for pre-service teachers. Lecturers developed the course so that it was based on an indigenous metaphor and linked cultural components to the central metaphor. The metaphor was used as a tool for organizing and representing ideas. Researchers examined the artifacts that pre-service teachers made in order to determine how confident teachers were in their ability to address the implications of The Treaty of Waitangi; to examine how effective the use of the central theme in advancing students confidence and ability to incorporate indigenous components in a course; what specific mathematical links students find in practical activity; and to determine if there was a shift into students’ intentions and attitudes concerning the use of cultural activities. The holistic model can be used to develop courses. In order to do this successfully, course leaders have a confident understanding of cultural aspects in order to show and describe how they are linked to mathematical principles. Also, linking to an expert from the cultural group from which activities are drawn so that their approach aligns with that of the culture (and is thus holistic).

The third study recognized that first year teachers practice can vary considerably from their university learning. In this study researchers conducted semi-structured interviews with three first year teachers in order to examine how the principles of partnership, protection and participation linked to their mathematics teaching. The questions asked centered around teacher expectations, how they demonstrated their understanding of the Treaty of Waitangi and what supports, dilemmas and challenges these teachers faced. All three teachers reported that they felt pressure to conform to the existing values and beliefs in their school communities. School personnel did not talk about the incorporating indigenous teaching and so one idea that teachers struggled with was bi-cultural perspective (indigenous and non-indigenous) versus a multicultural perspective. The results of this study show that teacher education programs should include strategies to address challenges that first year teachers face.

I found this article quite helpful. In my last post, I said how I feel I have haphazardly have used culturally responsive teaching methods; I think what I mean is that I have used cultural components in my teaching but I feel I am on a journey towards having a more holistic approach that incorporates indigenous ways of knowing into units of study.

What I found interesting was that in the introduction, the authors glossed over Demmert and Towner’s finding that stated student achievement is not linked to culturally responsive teaching (p.160). Instead Averill et al. assert that cultural competence and social consciousness are also worthy goals for teachers. To me this seems completely illogical. Firstly, if students are not achieving as a result of culturally responsive teaching we need to look at our assessment measures. Secondly, if students are not achieving we need to critically examine our so-called culturally responsive teaching methods. The point of having cultural competence is to meet the need of the learner.

One aspect I think this article does not address is the implicit way ‘western’ ways of knowing are considered normal and legitimate compared with indigenous (and other cultural) ways of knowing in school. (I think this could be a part of the hidden curriculum) My first teaching job was in a remote Metis community.  One example of how Metis ways of knowing were delegitimized is the way that students did not get any credit for what they knew. As the school as ‘at-risk’ the province mandated that students had to complete ‘culture-free’ math unit tests and submit them to the province’s ministry of education. For instance, in learning shape and polygons, students who could build cabins with their kokums (grandmas) on the weekend but had difficulty spelling a polygon name would technically be incorrect. Although one could argue that we could use cultural capital to teach such concepts, the assessment measures and the types of knowing (valuing writing as opposed to building) need be addressed. The result of these measures is that students feel inferior, like they have failed and that they themselves are de-legitimized. Compounded over years, this has a detrimental effect on student identity and is likely one of the factors that leads most students in this community to drop out in the ninth grade. 

There are aspects of culturally responsive teaching that make me feel uneasy. If 'pure mathematics' is supposedly culture-free will culturally responsive teaching ever be fully accepted?

Adding up the price of fruit-what is the cost? Week 2 Post about Mathematics in the Streets and in Schools

Mathematics in the Streets and in Schools

Carraher, Carraher & Schielmann

British Journal of Developmental Psychology 3(1) 21-29 1985

The article starts with the premise that there may be a difference between solving mathematical problems using methods learned in school and solving them in more familiar contexts out of school. Carraher et. al. tested the notion that a child who might have difficulty with routines learned at school might at the same time be able solve mathematical problems for which these routines were devised in more effective ways in a more familiar context. Five street vendors (between the ages of 9 and 15) from Recife, Brazil completed an informal test in their natural setting and formal test with pencil and paper.

Carraher et. al. findings state:
-participants scored significantly better in natural settings compared to decontextualized math problems in a formal test; 

·      -context-embedded problems were much more easily solved than ones without a context

·      -that in natural settings problem-solvers reasoned using quantities and convenience groups (knowing the price of three items together) rather than school learned routines;

·      -in a natural setting problem-solvers were focused on quantities and in formal testing the participants were focused on manipulating symbols

The authors conclude that their findings conflict with the implicit pedagogical assumption of mathematical educators according to which children should first learn number operations and only later apply them to verbal and real-life problems. In conclusion, Carraher et. al. ask how do we introduce math systems in contexts in order to sustain human daily sense and then depart in order to teach mathematical systems? 

Carraher et al. claim that there is an implicit assumption that math educators should teach math in a decontextualized way and then apply their knowledge. I stopped and wondered if this is true. When I was in elementary school (in 1990-97) we had several worksheets with number operations (12 X 4 etc.) and then word problems on the bottom as extensions. The screenshot on the left is a comic by Charles Schulz makes a joke about the types of word problems often found in math classes; in my teaching, I include it as a part of an introduction handout for 10-12 year olds. 

I realize now, that this joke might be more funny to people my age than my students because the big push when I was in teacher education (for ages 5-12 at least) was starting with a 'student's daily human sense.' Students figure out problems using a method that makes sense to them and then write it down, draw pictures etc. in math journals or on vertical surfaces to share with other mathematical thinkers.

One example from this article is: How much does it cost to buy 10 lemons?

Vendors say: Lemons are 35 cents. Three lemons are 105 so 105...105...105 is 315 and 35 is 350.

If a student were to write this in a math journal, I might say “Oh, wonderful,” and then compare it to another student who uses 35 X 10 and have them explain. I could claim this is a mathematical proof. We came to the same conclusion two different ways. There is a quote I share by Polya, “It is better to solve one problem five different ways to than to solve five different problems.” As Callaher et. al. suggest, I might then introduce an example where it is too cumbersome to do repetitive addition in order to push the multiplication method.

The questions that arise for me relate to motivation and thinking processes: If I were to ask the participants in this study to draw their thinking would they do it or think I am crazy for asking them? After all, they already figured out the problem. What motivation is there to continually ‘explain one’s thinking’? I also wonder, how do the thought processes that are involved in explaining mathematical thinking verbally different than those involved with writing? The novelty of writing on a whiteboard only lasts for so long and drawing out a long answer to a problem is not something one would do in a natural setting. Is drawing on a whiteboard actually sustaining one’s daily mathematical thinking then? Is this teaching method effective and for whom? What implications does this have for people whose mother tongue is not English? The long term implications of these questions will affect a whole generation of students who will make jokes about writing their mathematical thinking on whiteboards.

Please feel free to touch on any of the questions I asked above. Also,

-Based on your math teaching, researching and learning do you think that there is an implicit pedagogical assumption that math educators should teach math in a decontextualized way and then apply their knowledge?

-Carraher et. al. end the article with the question that asks, how do we introduce math systems in contexts in order to sustain human daily sense and then depart in order to teach mathematical systems? Where is the point of departure? 

Ineffectiveness in Research in Math Education

Author: Kilpatrick
Source: For the Learning of Mathematics, Vol. 2, No. 2, (Nov., 1981) pp. 22-29

In the article, Kilpatrick asks the question, 'Why is research in mathematics education so ineffective?' In short, Kilpatrick asserts that the two main reasons are inattention to theory and the disconnect between researchers in math education and practitioners (teachers).  Kilpatrick offers the ‘lens model’ as a way to show how it is both the researcher’s intended purpose and the lens through which the reader interprets research that determines if research is basic or applied.  The ineffectiveness of basic and applied research is found in the connections (or lack thereof) between research and theory and the connection between research and practice. Such ineffectiveness can be seen in the way that ‘new (the article is published in 1981) academic studies are not grounded in common theoretical frameworks and a therefore a community of researchers in mathematics education is not forming. The effect of research on educational practice is mediated by through theory as it is only with a common theory that research can be applied by the reader. Research needs to be more explicit and coherent in its assumptions, frame of reference and point of view so that readers can use a theoretical frame to judge similarities between research and their own situations. Kilpatrick uses a contrast between mathematics and mathematical educational research to help explain the ineffectiveness of research; according to this analogy, researchers need to develop more precise intellectual tools and the practice the ability to adapt tools to various problems and have a realistic expectation of the types of problems that such tools are able to address. Moreover, distinguishing that research in mathematical education is different than mathematics is key as the latter is technical but the former is social.

STOP: Virus

The article includes an example of how teachers mistakenly assume that studying math improves one’s ability to think logically in other domains. Kilpatrick describes how it is the way that teachers understand a theory that has a profound impact on math education; as theories disseminate through books or talks, “Gradually the idea comes into the culture of mathematics education and is picked up by teachers in practice. Sometimes the ideas I banned from colleges of education but lurks in a culture like a virus to strike down the receptive practitioner.”

Kilpatrick uses the term virus to show how an illness can infiltrate a population. This also reminds me of a ‘computer virus.’ I think a part of the trouble is that teachers are often presented viruses in work emails or district workshops.  Similar to a sales pitch, teachers hear, ‘It is important to teach this new concept or work with this type of learner because_______.’  British Columbia adopted a new curriculum this year and so there are a number of terms that teachers are talking about in the lunchroom. One virus or ‘buzz’ word that is ‘coding’ and all of the amazing things that teaching computer coding is supposed to achieve. I have taken a picture of a poster that was given out in a recent school district workshop (the intention is for it to be widely distributed). Linking to Kilpatrick’s example that teachers mistakenly assume teaching math can be generalized to teaching logic in many domains, generalizing problem-solving (which one often assumes is linked to logical thinking) is one of the supposed benefits. Almost insidiously, the limits of such generalizing or transferring potential are left out of the workshop and the poster. In my work, I hear the experiences of children from many schools. Sometimes the curriculum that teachers are presented with at district workshops/emails is taught as little more than a set of instructions to make one thing on a screen move to another location on a screen. I am currently planning a ‘Girls Who Code’ module that starts in February and so this fear of being ‘susceptible to a virus’ is frightening. If I am not careful, I could be teaching children little more than how to follow a recipe. What does successful teaching look like if I cannot trust that the purposes of such learning are embedded in theory? It makes me ask what am I teaching? And what for?

Questions:

1.     In the contexts where you have taught and researched, have you seen a ‘virus’ or idea about education that started in theory but had negative or unexpected implications?

2.     This article was written in 1981. In your research thus far, does your experience reflect Kilpatrick's claim that many math education studies are not imbedded in theory?