Some thoughts on language and how we use it

I found a few aspects of the Adler article particularly interesting. The first is the idea that explicitly teaching math vocabulary could be a bad thing. I have never taught it as rigidly as it was explained in this article, but it is a good reminder that we shouldn’t get weighed down by any one aspect. Focusing on any one aspect in math means that you are not giving a balanced approach and are neglecting some areas. It is important that we encourage proper use of mathematical language, but there is a difference between explicit teaching of the language and making the lesson about the language. I think we should teach all learners about how to use the language in order to communicate their ideas clearly, but this is only skill among many

I found this quotation from the Moschkovich article really stuck out to me: “Words have multiple meanings, meanings depend on situations, and learning to use mathematical language requires learning when to use different meanings” (page 91). This reminded me of our discussion last class about the definition and meaning of variables. There were a number of ways that the word could be used that would change how I would explain its use in that context and now I’m wondering what other words I use that have this type of multiple meaning – I am listening to myself more carefully now.

Clarification on Invisibility/Visibility, and other thoughts

I enjoyed Adler’s article, but at times I understood and then felt that I didn’t understand the ideas of invisibility & visibility, and would love to hear from others to see if I am on the right track.

With respect to understanding mathematics and how the acquisition of the mathematical language can be a hindrance to gaining or showing that understanding through the invisibility/visibility lens, my interpretation:

‘talk’ can be a resourceful tool and the mathematical language does need to be seen (has a purpose, ‘visible’) with respect to both vocabulary and notation that we would like our students to learn; however, there are times when the language should ‘disappear’, become ‘invisible’, especially (at the beginning stages?) of student learning – when they are ‘practicing’ what they are learning, showing their understanding, yet they may not be doing so in the conventional way that math teachers, in general, would like to see/prefer/hope for. This would allow the students to gain an understanding of the concepts in their own words before learning to express in a more conventional manner.

I would really like to hear if I am on the right track, please. Thank you.

Similar to Vivian’s concerns, I too wondered if I have been putting too much emphasis on acquisition of math language both vocabulary (do love those ‘let statements) and notation as a way ensuring understanding of math ideas. Then there was another line that struck me as well – Helen’s concern as to whether or not “saying it [was] actually indicative of understanding, of knowing” (Adler, 1999, p.56). The reason why this line struck a chord for me was because I was in fact a student who could do the math without ever understanding what I was actually doing. It was becoming a teacher that helped me to really understand how and why things ‘worked’ in mathematics. Then a few years ago, I took a math course and discovered that as a student I hadn’t changed – I could still be successful without understanding. I have to say that this was somewhat distressing and made me more determined to ensure that my students did not feel the same way. I tried to ensure this understanding through various real world problems and projects, but I too used techniques that were similar to Vivian’s. I felt that good communication, through proper vocabulary and notation, would help students to gain better mathematical understanding. I understand the confusion that Vivian expressed, as I feel it too. I guess the key, if I understood both readings correctly, is to find balance and to allow students to express themselves initially through different mediums and over time, guide to the more conventional math language.

I guess having said this, who is to say that proper math language is truly ever needed? If students ‘get it’, no matter how it is expressed, isn’t that enough? Food for thought…

Mathematics content being lost in language instruction

This week's reading was particularly interesting to me because of my current practice in my math classroom. I've been making an effort to help students better communicate mathematically (translation: getting better marks in the Communication category of their assessments) through three main methods.

1) Creating a word wall - I posted a chart paper at the front of the class and I add new or key vocabulary to it throughout the day's lesson

2) Communication questions - I add test-type questions to a list of "Communication" questions at the end of every lesson (e.g. Explain the similarities and differences between partially factoring and averaging the zeros methods of finding the max/min of a quadratic.)

3) Deconstructing the question stem - I've been making a conscientious effort to go through a word problem line by line, asking students to clarify what the question is asking them to find in mathematical terms

Reading Helen's exchange with her students and her reflections afterward made me think about my own teaching methods in the area of explicit mathematical language teaching. Am I driving my students crazy by telling them over and over again that proper mathematical language needs to be used? Of course. Did I think that I was doing the right thing by my students? I did. Now, however, I'm not so sure...does it matter that the students say "square root" instead of "radical"? Does it matter that they write precise "let statements"? I thought that it did, for proper mathematical form...but now I'm so confused!

I've been emphasizing so much mathematical language acquisition because I too believe that "being explicit about mathematical language benefited all pupils in their mathematics classes, irrespective of their language histories" (p.48). I've been further motivated to be explicit about mathematical language due to a third of my classes having recent Visa students from China; although these students (for the most part) know the math, they don't always know the correct terminology so I felt that the language focus would be most beneficial for them. However, I've also seen the result of having too much focus on the proper language "[obscuring] the mathematics under consideration" (p.62). I've yet to strike the balance and wish that I had some examples of how to properly teach language acquisition in the math classroom while at the same time ensuring that the math content is learned and not having a lesson or teaching style that is too confusing for students to grasp...

Links from last nights discussions

Ourr group discussed many possibilities for meeting the needs of all students and bringing in the community. Here are links for any one interested:

Math Fairs-

A brief intro to Lesson study from professionally speaking. If you want to know more this is the area of my research and love to talk about it.

Job prospects for New Teachers

Technological Studies includes computer science courses

Math and physics used to be the two most hireable of the Intermediate senior qualifications

http://professionallyspeaking.oct.ca/december_2004/reports.asp

http://professionallyspeaking.oct.ca/december_2007/transition_english.asp

Hope these help

Do I teach the way I would like to teach?

The articles by Woodward and Montague as well as Schoenfeld this week have really made me think about my own teaching practice.

I liked the definition of mathematical power where children can “engage in ongoing discourse about mathematics as they solved challenging problems”. I would like to think that I teach my classes using deep conceptual problems and in such a way that students are able to “develop master of skills through problem solving” (Schoenfeld, 23). Yet I fear that in reality, I fall into the category of teachers who provide problems “that could be solved in ‘five minutes or less’” (Woodward and Montague, 91) and who teach in such a way where “students have to master skills before using them for applications and problem solving” (Schoenfeld, 23). I agree that some “memorization and rote learning are unavoidable in education (Woodward and Montague, 92), but it should not be the emphasis. Where is the balance?

One issue that teachers always have is time. I would like to spend time developing concepts with my students (instead of just telling them the facts), but I also want to make sure we get through all of the material they need to learn. I agree that part of the problem is the “’splintered vision’ with curricula that focus on too many superficially taught topics in a given year” (Woodward and Montague, 91) instead of developing a deeper understanding of fewer topics. The other side of it is my own fear that my students will not come to the conclusions I want them to come too. How can I ensure that they have learned in I have not told them what to learn? I think I need to have a little more faith in my students, and recognize that even if they don’t learn everything I want, what they do learn will stay with them a lot longer.

I also agree following the textbook and teaching to the questions at the end of the chapter can be a huge problem. When I look at the textbook I teach with (a technical mathematics book for college students), there are word problems in every chapter, but there is also a chapter called “Simple Equations and Word Problem”. This chapter outlines 4 different types of word problems, making it seem as though there are only 4 types of problems in the world, and if you memorize the steps to solve these problems, you can solve any problem. Clearly this is not the case, but textbooks are set up in such a way that you are not given the opportunity to think about the type of question you are being asked or the choose how you are going to approach it, which is really the most valuable part of solving them problem.

With regards to students with learning disabilities, or students in lower streams, I think that too often we choose to teach them the ‘basics’ because we feel that everyone needs the ‘basics’, and that complex problems are too difficult and over whelming. But really, what we consider to be ‘the basics’ are memorization of algorithms or simple word problems that won’t be useful in their lives, and what we consider to be ‘too complicated’ are the real life problems that they will encounter. Put this way, doesn’t it seem backwards?

Mathematical literacy has become increasingly important in our society. But mathematical literacy does not mean solving equations. These articles have reminded me that, even though I think of myself as a teacher who is more concerned with conceptual understanding then rote learning, I often fall into the traps that most other teachers fall into. I need to make a conscious effort to change my practice so that my students get the most from their experience in my classroom.

THIS WEEK'S READING

I thoroughly enjoyed reading the article of Alan H Schoenfeld this week. It provides an insight into the forces that have driven reform, assessment of these reforms and the issues that need to be addressed if we want these reforms to work best for our students. The impact of reforms can be witnessed in the shape of enhanced skills in mathematics, problem solving and progress made towards achieving equity in maths.

However, another point that i got convinced of after reading this article was that consistency in the implementation of the system and its sustainability can in some cases help more in resolving the issues faced in curriculum, teaching and assessment. The article points out that changes in curricula in China and Japan have been less frequent but these education systems are still ranked as first and eighth respectively in education worldwide. Canada is ranked a respectable 6th. OECD picked out Canada/Canadian educators as one of the four “ strong performers” and “successful reformers”. Point being, constant research and assessment can always help refine the system further but a balance needs to be achieved between reform and consistent application of tried and tested concepts.

I have a few comments about the two articles:

1. I appreciated the specificity contained in the Woodward and Montague article. For example in the Instructions in Computations (pg 95) section they use specific questions to illustrate why learning a complex algorithm and then constantly attempting to master the algorithm is not beneficial to some students. With respect to the point made by the author, during my experience I have found that many learning disabled students that I have encountered will be asked to work on 100’s of complex algorithms, none of which they understand and it appears to be more to keep them busy as opposed to anything else. This is something that they can do and master, but it is essentially a waste of time, because they have no concept of how the algorithm is working. With regards to creating more contextually based problems for students, as advocated in the NCTM document, for some students unless an EA or teacher spends a great amount of time working with the student on a problem, the student will not be able to work through the question alone and will quickly become frustrated. There is a need for a balance between questions that involve practicing skills with questions of a type that are more problem solving based.

2. A line in the Woodward and Montague article that resonated with me involved his reference to Asian countries compared to North American countries:
“More successful approaches, found particularly in Asian countries, tend to focus on fewer topics.” (pg 91)
We try to accomplish too much, and as a result do many things poorly. I believe this is still a legitimate criticism of Ontario’s math curriculum.

3. Finally, the section in the Woodward and Montague article regarding the “emotional” dimensions of problem-solving instruction (pg 97 right side of the page 2nd paragraph) brought back the memories of one student (Rex) that I worked with (real name changed). Rex, had difficulty with math and was a student with special needs. He was working in a grade 11 course with an alternative program. A re-occurring issue with Rex, would arise when Rex had to work through any type of problem type question. Rex had an EA but liked to do things by himself. He was fine with any type of question so long as it was of a fairly basic nature. For any questions that were contextually based such as find the total price (including tax) of a jacket on sale for 30% off, Rex wanted to know a quick calculation to get the answer. This was fine when the questions were of a type similar to the aforementioned but not so fine when Rex was asked to determine the better deal between two sale items. For a question that involved making a choice between two sale items, Rex would quickly become frustrated and either become belligerent and angry or cry. He would not accept any aid from an EA or a teacher but he would not be able to get over the emotional frustration he experienced with the question. The questions/work given to Rex had to be closely monitored because once he started a question he needed to finish it.

4. Teaching as a Profession – With regards to the issue of professional development as discussed in the Schoenfield article, I have found that there is never enough time to get together with colleagues. There used to be professional development days, but they have been taken away or used to make the school year slightly shorter. I believe many issues related to the lack of professional development have arisen because of the inharmonious union/board relationship present within many school boards.

Some thoughts on Schoenfeld's article

Teaching as a profession
I can see how society’s value of teaching can make a significant difference to the type of reforms that are possible. It’s crazy that after a year of teacher’s college we are supposed to be able to teach effectively. I know that my teaching ability is growing exponentially and I haven’t even been teaching for three years yet. I’m also very lucky to be at a school where I have a great number of resources and time for professional development. I also think that once there is more push for people to recognize that teaching is a profession, there will be more pressure on all teachers to act like professionals. I think many of us are doing a great job and are looking to grow and challenge ourselves (like those of us in this master’s program), but there are a few teachers who are getting through on the minimum. If we are expected to grow every year and are recognized for it, there will fewer of this type of teacher.

“economic enfranchisement”
This phrase really stood out to me as a way to explain the importance of learning math. If students don’t have the opportunity to learn math, they are cut off from a number of possible careers and as the article talks about, from the highest paying jobs. Without the appropriate skills in math and technology, students are denied equal opportunity. I like the fact that Schoenfeld makes the connection between the fight for Black rights in the 1960’s and our need to fight for the right of all students to have access to meaningful mathematics education.

Half-life of math students
Another part of this article that shocked me was the quote that referred to the half-life of students in mathematics classes. I had a discussion with a friend over the weekend about two philosophies of math teaching – that all students have the ability to learn and enjoy math or that we should whittle down the number of math students until we get just the really keen ones in senior math. I started my teaching career in the second group and am now firmly in the first. The application of the term half-life makes it all too clear how trying to find the keen students really means looking at the numbers instead of the people.

Mathematics reform and equity

The principles and standards for school mathematics is based on six principles:
-equity
-curriculum
-teaching
-learning
-assessment
-technology
The goal is to make mathematics for all students. In the first article i can see how making math meaningful for all students and catering to individual needs can evade the problem of making accomodations for the learning disabled.
I looked deeper into the principles and standards and found that it lays out five content standards:
-problem solving
-reasoning and proof
-communication
-connections
-representation
These standards refer to the mathematical process through hich student should acquire an dus mathematical knowledge.
I can see how integrating all the five components can ensure catering to the needs of all students irrespective of their requirements.
I also believe that once we are succesful in getting students engaged and motivated, several behavioural issues are already eliminated. Learning needs of students are better met when they are engaged and can associate meaning not necessarily for all strands but even a few could do the trick.
I would definitely argee with the first article in that NCTM standards do have a good vision in mind and if implemented well ensures success for all students.

Looking at NCTM Principles & Standards - both the positive and negative

I may be incorrect, but I felt this week’s reading were an analysis of the NCTM’s documents called Principles & Standards (both the 1989 and 2000). One seemed to be critical of the document with respect to LD students, saying that the document’s ideas/concepts did not take into account the learning/teaching processes necessary to ensure that these students are successful in their mathematical career. Whereas the second took a more positive view with respect to the Standards, stating that if the ideas from this document were fully implemented into curricula then improved results for students would be seen and provided data to support this claim. Now I do confess that these are both very broad inferences, but overall I did feel one reading was negative and the other positive. However, I will be looking at these readings somewhat separately.

Article 1

With respect to the first article about meeting the challenges of help LD students in mathematics, I have to admit I was somewhat shocked to read that, “students with disabilities were neglected throughout the document [Principles & Standards]” (p. 89). I was surprised because I happen to meet a lot of American teachers through workshops and know that Standards is considered a very important part of developing curricula, so see that it does not include an important segment of the student population is unsettling (especially in light of the second article that states full implementation of these ideas shows improvement in students’ results – does this author take into account LD students?).

So I went back to our own math documents to see what our expectations are with respect to LD students in Ontario. I discovered that vocabulary describing these students were slightly different (I say because 9 & 10 was written in 2005, and 11 & 12 was written in 2007). In Grade 9 & 10 the section is called “Planning Mathematics Programs for Exceptional Students” (pp. 24-25) and in the Grade 11 & 12 document, the section is called “Planning Mathematics Programs for Students with Special Education Needs” (pp. 32 – 34). Despite the different names of the section the expectations were similar (more detail was seen in the Grade 11 & 12 document) and they seem to be directed towards accommodations and/or modifications that can be made by teachers in order to ensure that students with difficulties are successful. I took this as a positive sign that the Ministry of Education were taking into account all students; although I will concede there is not a lot of direction to how we can ensure that we are in fact accommodating these students, but it is a start that they are in fact mentioned in a document that we use?

Another thing I found interesting about this article was the fact that special education was based in behavioural theory and that special needs teachers were concerned about the constructionist theory, upon which I believe NCTM Principles & Standards, is based. When I was reading the article the needs of special need students appear to be similar the needs of all our students – making math more meaningful, showing that math is more than skill-based, understanding mathematical conceptually, etc. I found this to be a powerful acknowledgement because I do wonder if expectations are diminished for these students because of their struggles, and time and different teaching strategies would enable these students to be successful as presented in this article.

Last thought (because I am going on yet again) with regards to this article, I thought the following sentence was important “students with learning disabilities are categorically different from those who exhibit developmental delays” (according to Geary, p.94) – this is an important distinction and I wonder if the lines are sometimes blurred? I have worked with both students from both groups and have always struggled to ensure that I am meeting the needs of both. I do worry that I have failed one or both groups.

Article 2

As I have written a lot once more, I will be brief here and respond to others instead. My first impression was that the tone reminded me a lot of the first article we read from EQAO, looking at the Grade 9 Academic & Applied results – very upbeat, and are we not proud of what has been accomplished. This is not to say that the stats are not correct and that there shouldn’t be a celebration of the success seen in Pittsburgh and Michigan, but I don’t know the right way to say this, but it seemed too positive. Maybe someone else will express what I am saying more eloquently. I felt that the some broad statements regarding the success of standard-based reform were a bit too generous, for example: “racial performance gap diminishes substantially” (p. 17), when we are looking at one city (although he did say that there was further success in Michigan). Perhaps I am being harsh?

I did, though, agree wholeheartedly that more professional development and time are needed to ensure that the teachers are prepared to teach students so that they can reach their full potential.

Again, sorry for the length!

Could Wolfram Alpha make Secondary math more accessible

Wolfram Alpha is a free online computation tool that can do all secondary algebra. Here is a link to a video that talks about why one might use it rather than having students master all the calculations: Ted Talk by Wolfram

Once you have seen the video, look at the website. Could you use it to make math instruction more meaningful and accessible?

School and home mathematics

This is a response to Enza’s post, but when I’m at school I can only do new posts.

I also found this article very interesting because I never considered that are different ways of learning math to which students are exposed at home and in different countries. It’s not that I disagree with it in any way, I had just never considered that there were cultural ways of doing math. It is something that I wish had been discussed in my teacher’s college courses.

One of the most interesting aspects of the article for me was the teacher in Brazil’s response to Abreu’s question about whether farming problems were used in the classroom. The teacher’s response was based entirely on her perspective of student interest in future professions rather than referring to what they might already know or experience. I would also guess that it would be pretty intimidating to admit that you wanted to be a farmer in that class. The math in farming was so de-valorized that one student wanted to teach his parents how to do the school math so that they could be better off in life. I can only imagine the effect that this would have on family dynamics when children at being taught this at school.

In my class, I try to focus on different methods to get the same answer and we celebrate new ways to interpret situations and find patterns. I hope that I would be open to a student that had an entirely different way of doing math, but I guess I won’t know until it happens.

The other side of the story

I am not a teacher yet but I am a parent, and when I read the article of Guida De Abreu & Tony Cline I thought I could contribute to this blog as a mother. And since I am from Pakistan I can totally relate to most of the examples and statements of Pakistani kids and teachers quoted in this article.

My seven year old son is attending a private schools here in Toronto. This school is following Cambridge curriculum which is quite rigorous as compared to IB and Ontario's curriculum. I have been teaching my son at home and following the Pakistani curriculum as well. Asian curriculums are generally even more rigorous. However, I chose to teach my son partially because I know that teaching him at home will improve his overall understanding of mathematics. He is able to handle the difference in problem solving techniques. He chooses the method that he is comfortable with and generally the one that is more efficient too. I can see the benefits of this approach because he is much more at ease with what is being taught at school and is being appreciated by his teachers as well.

My experience suggests that we open up our children’s mind by introducing them to the possibility of solving the same problem in more than one ways and that a particular method for problem solving should be chosen because it is more efficient, not because of its association with a certain class. Alternatively, kids can be given the opportunity to choose among the methods that they find more interesting and easy to use.

Hence what we need to think actively about is how to counter the valorization tendencies. One way to achieve that could be by introducing practices that promote the idea that it is very normal to have diversity in problem solving methods. We can encourage kids to share their home learnt methods. In other subjects, parents are usually invited in a class to read a book that interests them or their kids. I don’t see any reason why the same activity can not be introduced for mathematics. If we are not judgmental about other methods they could be an asset in the process of learning.

School for the Real World or the Real World for School?

When I read in the DiME chapter that there was "no statistically significant difference between schooled and nonschooled children for problems of currency arithmetic and ratio comparisons in out-of-school context" but that "children who were sellers did solve more problems drawing on the informal strategies developed in the marketplace than did nonsellers" (413), my immediate response was that the school system seemed to be set up backwards. Schools should be helping students build skills that will help them outside of school. This clearly was not the case in this situation. It's great that students were able to draw on their outside of school experience to help them in the classroom, but it almost looked like success in the classroom was the end goal rather than one step in a long process of learning.

I don't think that every topic covered in a math class necessarily needs an obviously practical application, but I do think that the skills students learn should be transferable to situations outside of the classroom. If the knowledge that students gain stays in the classroom, as it seemed with this group of students, then the learning seems fairly pointless. I think reading this helped me clarify I lot of the problems I have with the way that school privileges some kinds of cultural capital over others. School is supposed to be a place where students learn, but instead it seems to be a gatekeeper to success.

Synthesis of DiME report

Confession time: I found the DiMe chapter really good but really dense!

So I thought I'd create a wordle of the report to see which key words or ideas dominated.  I expected to see identity, culture, equity, race etc.  Instead, I got this:

http://wordle.net

So within the DiME article, there are more references to students and mathematics than any other words.  The fact that these two words are the same size indicates that they were mentioned the same amount of times in the report.

While this was not what I expected, it got me thinking.  Prior to this class and all the readings, I would have said that math class was all about math.  Math remains math, regardless of who was learning it.  But this wordle suggests something else:  mathematics cannot rooted in it's rules and algorithms.  It needs to respond to the students learning it.  Who students are and what they bring into the mathematics classroom is equally important as mathematics itself.

Some initial thoughts on Abreu/Cline and Martin chapters

The chapter by Abreu & Cline had me going down memory lane in a lot of ways. It reminded me especially of the time in Grade 3 when I was learning long division. I came home somewhat confused and asked my mum for help. She couldn’t understand why I was learning division in such an inefficient way; she then showed me how she learned it. It was certainly a much shorter way, but at the time I felt that I had to learn it the way I was taught in school. My mum did help me, but she thought it was silly. As I got older and more confident I used my mother’s way, but interestingly enough I still teach it the way I was taught in school. What does that say about schooling’s influence…?

I was also saddened to read that there still seemed to be a reluctance of students to bring in ‘home’ knowledge of mathematics into today’s classroom. Actually, I think a better word is surprised. I have seen in different schools how teachers do encourage their students to share their methodologies with the class, and I wondered if this was the exception as opposed to the rule? It also made me go through the Ontario math curriculum again because I was sure that it was written that teachers are encouraged to foster different ideas. However, after reading through several different passages, I found that the idea of using diverse methods was implied as opposed to explicitly stated, such as quotations I have listed below:

Page 12 Curriculum Expectations, para 6 (Grade 11/12 Math curriculum)

“Some examples and sample problems may also be used to emphasize the importance of diversity or multiple perspectives.”

Page 35 Antidiscrimination para 2 (Grade 11/12 Math curriculum)

“Learning activities and resources used to implement the curriculum should be inclusive in nature, reflecting the range of experiences of students with varying backgrounds, abilities, interests, and learning styles.”

I then searched through the IB document I need to use because the IB really fosters the idea of internationalism, and again nothing was stated explicitly:

Page 5 Internationalism (IBO Math SL Study guide)

• students are to learn how the attitudes of different societies towards specific areas of mathematics are demonstrated

It was clear that it is up to teachers to be proactive in the idea of showing students that ‘home’ math could definitely have a place in the classroom. I guess a possible concern that I would address if a student did want to use another method would be that they could demonstrate how the method works. For me, math/tricks without understanding, whether they come from the home or the classroom, will not lead to long-term mathematical understanding. There is absolutely value for learning different ways, but I feel that the students need to understand it, not just memorise it.

With respect to Martin’s chapter, I felt this again reiterates the need for teachers to be very aware of how they treat students when they walk into their classroom on the first day; to ensure that preconceived notions do not enter the room and I find this particularly challenging when we have meetings at the beginning of the year that articulates the strengths/weaknesses, the good/bad of students. For me, this can be somewhat detrimental because then you could walk into the classroom with low or high expectations, which is unfair for the students and yourself. I have discovered that students do change tremendously with various teachers, and year to year. I also agree that we need to remember, “that parents and adults are, along with teachers, the most significant influences on the formation of student attitudes, dispositions and beliefs about mathematics” (p. 148). There may be times when we forget how much influence we have especially with students that may be particularly challenging to us as teachers.

Gina brought up the point that she felt that she did poorly in math because she had “no cultural connections with them [teachers]” (pg. 155). I found this point particularly poignant as I have discovered over the many years how detrimental it can be for both student and teacher if this connection is not made. The challenge then becomes how can we ensure this connection is made? Should the connection be based on culture or on emotion? Can it be one or the other, or should it be both? I am fortunate that I have relatively small classes, but for the larger groups, how can we ensure this connection is made? How can we make sure that a student is not lost when there is so much mathematics needs to be covered?

These are some initial thoughts. I will be back.

Quotes from de Abreu & Cline, Martin chapters

This week, I hope the DiME chapter helps you to synthesize some of what we've been reading in the last six weeks, and points out some gaps in the literature. One of the gaps that they highlight is an absence of analysis of race and power in mathematics education. I think the de Abreu/Cline and Martin chapters have some poignant examples of the ways race/power are implicated in mathematics teaching and learning.

Some quotes to think about:

Interviewer: Why do you think then that he was the best at maths?
Child: Because look at him [office administrator], all wearing flashy clothes and like he looks like a rich... and he's got such a good job, and him, he's ... nothing he has to do taxi. ... Like if he's ain't good at something he'd have been like him [taxi driver], like he's probably not good at anything. He's [taxi driver] probably came from Pakistan.
(de Abreu and Cline, p. 125)

Gina: [In high school], the counselors would do rqacial steering, try to steer all the minorities into art and cooking and stuff like that. So if you mentioned math, they were like, "No. No." And they really didn't tell me anything about college preparation and stuff because this was way back in '74. (Martin, p. 156)

Both of these quotes show evidence of master narratives about which groups of people (racial groups, professional groups, ethnic groups, national groups) are capable of high level mathematics. And both of the articles also included counternarratives, about successful children and adults who defied these stereotypes.

I was interested to read the interviews with adults in the Martin chapter. This time through, I noticed that in some of the interviews, there is a potential for these parents reinforcing other harmful master narratives about schooling. Although Keith doesn't quite come out and say it, his comments about how his kids eat and sleep education could be taken as implying that other kids might be unsuccessful because their parents don't care as much about education as Keith and his partner do. (I read the quotes carefully, and Keith definitely does not come out and say this - in fact, he talks about African American children not having the same kinds of opportunities and resources as white children, rather than blaming parents.)

I bring this up only because I want to point out that even people who are challenging some parts of a master narrative might inadvertently support other parts, and the narrative 'those kids just don't care,' or 'those parents just don't care,' is particularly common in education. I think the ideas about social valorization and about mathematical identities help us to think differently about whether and how people 'care.' I hope we get the chance to talk about this further in class.

I don't like statistics but I agree with this man

Chimamanda Adichie: The danger of a single story

Here is the link Leslie was talking about in class today:

The danger of a Single Story by Chimamanda Adichie

Dan Meyer

Diane and I were just talking about Dan Meyer in class. Dan Meyer has some very interesting ideas about math and real world problems.

In case you've never heard of him, here is a link to his TED talk:

Dan Meyer: Math class needs a makeover Video on TED.com

And here his a link to his blog:

http://blog.mrmeyer.com/

Enjoy!

Rohini

Getting Over My Fear of Offending

Reading about Gutstein’s “pedagogy of questioning” I found myself thinking back to Gutiérrez’s notion of equity mathematics in that both aimed to encourage students to become critical thinkers in the way that they questioned the math they learned and how they learned it. In Gutiérrez’s article, she discussed how reform mathematics may encourage such critical thinking but that it did not necessarily lead to students thinking critically about the structures of equity (or lack thereof) that exist in the world they live in (Gutiérrez p.39). In Gutstein’s article, it seems clear that he managed to create a classroom in which students were able to truly reflect on why some groups were approved for mortgages more often than other groups but more importantly that everyone had the opportunity to respond in their own way and to even question the teacher or disagree with him (Gutstein p.62).

The biggest concern that always comes up for me, as a second-year teacher, is to not clash with those I’m working with or working for. As much as this makes me a follower, I’m OK with that to a certain degree (I blame it on my family’s influence…I always had to listen to my parents, and still do to a ridiculous degree as others see it). When Gustein states, “creating a pedagogy of questioning demands that teachers be more open than we are generally used to being. And the more open we are, the more there is the potential to influence students to accept our positions, as well as possibly to cause our positions to clash with those of the students’ families, of the school administration, and of the students themselves” (p.66), I felt an instant shame knowing how much I am afraid to be the teacher that brings up contentious issues in the classroom. I remember when I was student teaching at a vocational school in the Toronto District School Board and my associate teacher was describing her fear of student teachers due to a bad experience with her last one; she told me how he had dared to talk about “common law families” while teaching a lesson on fractions (something to do with common denominators, I think). I remember thinking, “what’s wrong with that? Isn’t that what we’re supposed to do – include different types of families into our classroom teaching rather than always assuming a family has a mom and a dad?” while nodding sympathetically at my associate teacher’s horror story.

I always think about this past scenario when I think about incorporating a social issue into my math instruction. At times I worry that I will never be able to be the type of teacher that continually questions students and encourages students to question her, to create the classroom environment where students become critical thinkers. I suppose this is why I am so attached to teaching the curriculum "as it is" without much thought to how I can frame lessons with a social justice perspective. At other times I placate myself that I'm only one teacher in these students' high school careers...but the problem becomes clear if every other teacher these students have (or even just the majority of the teachers) are also as complacent as me, then when will we ever be able to change math instruction (or, at the very least, think critically about changing it)?

Any suggestions on how I can stop being so afraid?

Social Justice in the Clasrroom: New Understandings and Questions

The three articles this week provided different approaches for teaching for social justice. Each has clarified some issues for me and raised more questions.

I like the specific details that Gutstien provides (especially in Rethinking Mathematics 2006) and I am starting to gaining comfort to raising what could be controversial issues in my classes. The three ideas that stand out to me are (p 55):

- Students need a space to pose their own meaningful questions

- Students aren’t used to posing their own questions; teachers need to seed the process

- Students need the opportunity to name their own realities- I think this means to that students need to identify things in their own environment that hinder their success, so that the students can make informed choices and choose to act in ways that might bring more success.

Gutstein provides strong evidence that his students developed the critical questioning stance that he desires for his students but I wonder about other measures of success. For example his students discussed whether institutional racism results in a lesser rate of blacks and browns getting mortgages, and make the distinction between income and wealth, but does the class build the knowledge of what it takes to build wealth – ‘to play the banking game’- to meet the banks criteria for a mortgage?

One of my strongest impressions from working with Habitat for Humanity revitalising downtown neighbourhoods is seeing $40,000 cars in front of shacks or apartments with holes in the walls. Cars don’t build wealth. Many habitat affiliates now have classes helping families understand how to maintain houses and build a nest egg.

Also how do students do in future math classes- do they have the skills to take gate-keeper courses like Algebra 1?

The Davis and associates article, in contrast, provides evidence of greatly increased participation in college –prep math courses. I find it interesting that Bob Moses developed a program from his studies in philosophy that has many similarities to current research in cognitive psychology ( see for example http://www.nap.edu/openbook.php?record_id=11101&page=1): start where the kids are, give experiences and link those to the mathematical conventions.

I found this article big on the philosophy and big ideas. As a budding researcher, I really liked the idea of examining classrooms using different lenses, but as a teacher I found little concrete that I could use in my classroom- I did find many internet links to the Flagway Game – http://www.typp.org/flagwaycampaign & http://www.youtube.com/watch?v=WD5KB7iK4CA but I still couldn’t use the activity in a classroom.

The Algebra Project chapter also talked about ways they use to engage students – technology, games, and helping students learn to work. This was reassuring because when we have streaming or segregation by neighbourhoods, these are real challenges, which are often glossed over in some articles. I was struck by the idea of presenting being culturally respected and perhaps a better motivator that working silently. I have little exposure to black culture. I wonder if that is a fair conclusion?

I also noted how they work within those presentations to build leadership skills. It reminded me of Lubienski’s assertion that low SES students may get the most out of classes that explicitly work on problem-solving and communication.

What I liked most about Marta Civil’s chapter is the respect for the parents/ communities knowledge. I remember as a student feeling like teachers’ had little respect for the local community and how people lived. However, I need to think about what incorporating family knowledge would look like in a secondary classroom in the community where I live.