Using videos in the classroom

When Lippi mentioned using videos to teach concepts and students learning on their own at home, I remembered The Khan Academy's website:

www.khanacademy.org

Their philosophy is that students can use the videos to learn at their own pace and then spend classroom time in one-on-one interaction with the teacher (i.e. what we would give now as homework questions are now done in class).

I understand people's fear of a) students not having the access to technology to view the videos, and b) students not being motivated (or not having the time, in many cases) to watch these videos on their own. The more I think about it though, it seems like if we are advocating for more inquiry-based tasks and that students can learn the mathematics by working on the activities, it would be better to spend what class time we have to work on group-worthy tasks and having that one-on-one time with students, helping them with questions individually...even if they didn't all have a chance to view the videos the night before, working through the class activities can often result in the students learning the material anyway.

Reform: When and how?

It was very refreshing to read this article since it gives a new perspective to researchers. Even though i am pro multicultural education, differentiated instruction and inclusion i somehow felt that different aspects were being left out in reform articles that focused on various aspects of education.
The framework provided by William Tate seems to cover the three most important aspect of education : time, quality and design. Upon reading each aspect I could identify how models outlines by Talbert and McLaughlin and Spillane are integrated in some ways as well. I feel that sometimes its the external factors that inhibit various reforms. Tate's framework is open-ended yet structured well enough to explain for various external factors that could determine teacher and student success/responses in a classroom with respect to reform.
Its amazing how we get bogged down with the process without understanding the factors that might limit us in implementing reform successfully. Kitchen (2003) emphasizes the importance of being able to understand and analyze specific contexts in order to be able to be successful in implementing reform that supports equity. The article shows that by understanding and investigating the problems that affect “time and quality” we would automatically reform the design as a result of this understanding. Design , which the third component of the framework relies on the first two and it would be crucial for us factor in all the possible limitations and the reasons for it , which obviously would be context specific and then develop a system that better supports the teacher as well as the student in that particular context to overcome these limitations.

Digging Deeper

This article was an interesting read to me because I read it from the perspective of a professional development provider, not as a classroom teacher.  So, while Rousseau and Powell's questions were specific to their study,  they are also my questions.  I've been grappling with them the past three years as a math coach.  It is the main reason I enrolled in the M.Ed. program.

Why did the professional development program not appear to result in more substantial and consistent change in her teaching practice?  Why did Teacher A seemingly make more progress along the continuum of traditional-to-reform? (p. 29)

While I agree that using Tate's framework offers "attention to factors associated with equity", I would have liked Rousseau and Powell to dig deeper into the individual characteristics of these teachers.  Again, I agree that attributing Teacher B's narrow implementation of reform mathematics to "personal traits or lack of effort" is inappropriate.  But, I think a discussion of each teacher's self-efficacy would have added depth to our understanding of what happened at the classroom level of implementation.  My request comes from two specific quotes found on page 27:

"..Teacher A appeared to feel less pressure to focus on test preparation.  He expressed confidence that his students would pass the test, even without explicit efforts on his part to prepare them"

"Teacher B repeatedly expressed not only her concern that her students pass the test but also her view that teacher for understanding and preparing students to be successful on the test were, to a large extent, conflicting goals"


Bandura (1997) describes self-efficacy as ‘beliefs in one’s capabilities to organize and execute the course of action required to produce given attainments”. Teachers with a strong sense of individual efficacy believe that they can and will make a difference in student learning. They believe that all students can and will learn (Goddard, Hoy & Woolfolk Hoy, 2000).

Teacher A appears to have strong self-efficacy, despite not having a math degree, rich tasks, or teaching mathematics for the second year.  This made me wonder:  What was he like as a learner of mathematics?  What is his vision of successful implementation of reform mathematics?  

Teacher B appears to have lower self-efficacy because she did not believe that teaching for understanding would have prepared her students for the state test.  Surely, her participation in the professional development would have exposed her to that data that indicates that "reform students do as well on skills as students on an understanding of concepts and problem solving" (Schoenfeld, 2002)?  

Therefore, I had the same questions for her. Gabriele & Joram (2007) suggests that one reason why the transition from traditional to reform-based mathematics teaching may be particularly difficult for teachers is that typical sources of evidence that teachers use to judge their teaching success, which in turn support their sense of teacher self-efficacy, are no longer operative. Success is defined differently when teaching reform-based mathematics, therefore self-efficacy is threatened. Was this the case for Teacher B?

While I agree that understanding the "barriers to change experienced by teachers in different contexts is to better support the implementation of reform", I think we also need to understanding what experiences and beliefs teachers bring to these contexts.

References
Bandura, A. (1997) Self –Efficacy: The exercise of control. New York: W.H. Freeman

Gabriele, A. J., Joram, E. (2007) Teachers' Reflections on Their Reform-Based Teaching in
Mathematics: Implications for the Development of Teacher Self-Efficacy. Action in Teacher Education,
29 (3), 60-74.

Goddard, R., Hoy, W.K., Woolfolk Hoy, A. (2000) Collective Teacher Efficacy: It’s meaning, measure, and impact on student achievement. American Educational Research Journal, 37(2), 479-507.

Schoenfeld, A. (2002) Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity. Educational Researcher, 31(1), p. 13-25

It is unfair to paint everyone with the same brush

“The context, therefore is important” (Rousseau & Powell, p. 29)

This article and this line in particular really hit me in a unique way because recently I ran a workshop at a school that is struggling to find the right path that will enable their students to improve their test scores and for the school to stay open. It really struck me talking with this group of dedicated teachers how lucky I have it to work in a school that where I have classes of sizes twelve to twenty, students have laptops and resources are fairly easily attained. Here was a group of teachers that were learning about a new programme that the administration have decided to try and implement to help ‘turn things around’. There were times where I felt awkward because I would talk of expectations, resources and ideas that would help ensure the implementation of a successful programme, and I believe they were thinking ‘when and how am I going to do this?’ It was so easy for me to talk about how wonderful this programme was, but I didn’t really understand what their day-to-day teaching life was like. I didn’t fully comprehend how adding these new expectations could potentially overburden an already heavy and highly pressurised workload. In other words, I wasn’t fully aware of the context that these teachers were working in, but how could I? I was brought in to talk about the programme, but I learned very quickly that these teachers were unaware of the potentially sweeping changes that were coming their way. What this experience taught me for future workshops – never make assumptions and always make sure to learn about what constraints these teachers may face in their schools as you may just never know that what may be a straightforward implementation of a programme to you, may be an overwhelming prospect for others.

Now this may seem like a story that digresses from this article, but I think there is a tie-in and that is when judgements are being made about teachers and the lack of apparent success according to outsiders, one needs to better understand the context by which these teachers are working under. Although NCTM talks about equity as “ALL students should have access to the type of high quality mathematics curriculum and instruction” (p. 20) in their documentation, which is certainly a key goal, it is critical to understand what might be stopping teachers from ensuring this from happening in the classroom. Outsiders need to take a closer look at what is happening in the classroom and in the schools and understand the concerns as expressed here (large class sizes, student absenteeism and state exams) by Teacher B that is stopping the implementation of reform mathematics. It must be disheartening for teachers who are working hard to teach mathematics in the best possible way to be judged by one set of exams. As these authors have illustrated, one should not make sweeping judgements without fully understanding the context – it is unfair to everyone involved.

Article Rant…Significance of Context: …Equity and Reform

“…those of us in mathematics education have yet to take seriously, from an equity perspective, the significant differences between teaching contexts with respect to mathematics reform.” (Rousseau & Powell, 2005, p. 29)

My efforts at a mathematical reform pedagogy kept hitting roadblocks and this article nicely articulates all of those blocks (and I have a couple others I could have added). Yet, looking at those mentioned is at least a hopeful starting point for change towards equity. Far too often much of this type of research has been dismissed as an American issue that is not as relevant here in Ontario. I and many other educators that I had previously discussed similar issues with did not see the disparity in educational equity as drastic here, in Toronto as that of the experiences of “those Americans”, but for me that had recently changed.
When I began my own change in pedagogy towards mathematical reform I started to experience many of the same constraints to change as mentioned in the article. I had no time for instruction in the classroom (in addition to barriers that were mentioned in the article, a lot of my time was also spent on teaching social skills and group dynamics), and I had very few available resources. The NCTM lists what should be in every classroom as a basic for student resources/manipulatives, but in my school this does not exist. The less than adequate number of resources for a K-8 school is located in one open math resource room. Thus, it is not easily possible to gather supplies when needed and student use of manipulatives at their free will does not exist (I have spent hundreds of dollars of my personal money trying to remedy this). In addition, planning time is wasted on moving from room to room finding an area to work, trying to gather/beg for resources, doing redundant paper work and filing instead of preparing class activities, planning and reflecting on practice (not to mention the wish of moderated marking).
The disparities became even more apparent when I experienced a concurrent comparison of Toronto (urban) and Peel (suburban) classrooms. I had recently changed the school my daughter attended from TDSB to Peel district and hearing her perspective on the differences in classrooms as well as my own observations of the differences reinforced the inequities. Last year she was in grade 4 while I taught grade 4 at the school in toronto that she had left. This year she is in grade 5 as I teach grade 5 in the same school in toronto. She is still in contact with many of the students in my class (through msn, phone, emails, on line games etc.) and still attends gatherings with them. Therefore, I constantly hear from my students and her (regarding curriculum and class activities) “how come they did not do this, how come she gets to do that, how come they did not learn this, how come she got to go there? Although I am very glad for the opportunities and experiences my daughter gets, I am extremely frustrated at the lack of opportunities my students in an urban setting are afforded, especially when it is known that these students need the opportunities even more (students of colour in lower SES communities).
I am not sure how long it will take before the message of Rousseau and Powell, on these systemic constraints, has a great impact on what is happening in our urban schools, but it really is time to stop blaming the individual teachers for the inequitable progression of these students.
Sorry for the rant, but that is how I felt after reading the article.

The significance of the context

As I was going through this reading, I found that I was continually asking why. Why are students that we already know are disadvantaged getting access to the fewest resources? Why are their teachers getting less time to prepare and less professional development? Why are their class sizes the largest? I realize that the purpose of this article is not to address these questions, but that was my reaction over and over again.
It also made me reflect more critically on my own teaching practice. I wonder where I am on the spectrum of traditional and reform teaching and what I can do to move further toward what I know is the best for students.
When we looked last week as large scale assessment, we briefly discussed teacher pressure to teach to the test. The major difference between Ontario and the Classroom B context is that the EQAO is based on the Ontario curriculum, which means that by teaching the curriculum, teachers are preparing their students for the test and by preparing them for the test, they are teaching them the curriculum. I also agree that there are different pressures in different contexts about how important test results are and that this can change how willing teachers are to take risks with new teaching methods, even if they have research backing.
I also had some personal conflict with the content in this article because I teach at an independent school where my students come with significant economic and social advantages. I also have more preparation time, smaller classes, no standardized tests, access to many resources and weekly teacher development. I realize that in some ways, this is how the world works – those who have money, will pay for the best. At the same time, I wonder what can be done so that all students have teachers with the same support. Rather than stop independent school from providing this, I think that public schools should also have the resources and support that I get (although that might put me out of a job!)

Blogfolio hint

I discovered that if you look at the list of blogs on the right (grouped by month) that the blogs you posted or commented ar in grey instead of brown (with a few exceptions).

It made it much easier to find the posts I forgot to save in a folder.

“What we have described above is a set of predispositions and experiences with which teachers approach the task of constructing their knowledge of students.”

This quote made me think of my own practice with regards to evaluating students, and how the mark is in some ways a construct of my own knowledge of a student in relation to his or her mathematical knowledge. As a result, because I am always limited in the information I am privy to, that evaluation will always be lacking.

With regards to the first study mentioned in the Morgan and Watson article, I think it would be interesting and unnerving to have someone sit in the back of my class and observe the class on an ongoing basis – it would probably feel like teacher’s college all over again. I would imagine, like the teachers in the study, there would be behaviours that I would fail to notice. I would like to believe that I do not mix my assessment with my impressions of a student, but there are undoubtedly times that I likely do. I think a third party would be better able to observe when I do this. Related to this thought is the quote:

“On the contrary, the study suggests that informal assessments, including those that furnish information for summative assessments of performance, are inevitably and unavoidably influenced by a variety of factors that may have little to do with mathematical achievement.”

This illustrates the problem I have with informal assessment – how do I do it without being biased? I know that this is next to impossible and as a result I don’t think it is fair to count towards a grade. If need be, in parent teacher interviews or discussing marks, I would rather give my impression of what I think is going on and make it known that I only have a limited snapshot of the situation pertaining to a student’s work or achievement.

Using Research to Improve Large Scale Assessments

This weeks papers made me realise how easy it is to become complacent with one’s assessment practices. While I have worked very hard to make my math classes more accessible to students I don’t t really challenge my assessment practices, even after being part of a pilot for a reform assessment program (all subjects). I changed my practices, but don’t routinely question my assessment practices. I am fairly comfortable with other teachers watching my teaching, but I would be very uncomfortable with having others examine my assessment practices.

In assessing, I will now be looking for systematic biases (do I assume students with IEPs know less for the same written content?) and want to look at recorded biases against specific groups. Also I recently had a long conversation with a mother of smart boys who get discouraged by unclear evaluations. My own daughter (grade 11 functions) just got discouraged by a rubric marking rather than simply right or wrong. The rubric was bang on, but my daughter had truely never been graded on communication and thinking before.

Unlike some in the class, I didn’t see the articles as complete critiques of the EQAO and teacher assessment practices, rather a call for further improvements. There have been changes to EQAO since this report. I have noticed that recent EQAOs do not have questions broken into parts see question above or http://www.eqao.com/Educators/Secondary/09/BookletsandGuides.aspx?Lang=E&gr=09&yr=10

And the students get one mark that encompasses all categories. So I wonder if EQAO now has a clearer matrix as to which topics are covered to ensure more consistency to the curriculum and year to year.

Our board has been using moderated marking with grade 9 teachers for 3-4 years to help teachers prepare for the test. It has helped change teacher attitudes about what is important to teach. I think our students are seeing more rich problems and understanding what good answers look like (with flexibility). We have now extended it to all college and Workplace math teachers to push their assessments and learning activities from short calculations to richer problems. So even if the current EQAO isn’t promoting investigations it is promoting rich tasks.

I appreciated the Morgan & Watson article because it provided a vision of how assessment of investigations could occur- using moderated marking to develop more consistent assessment practices and then have teachers assess in class investigations with an consistent evaluation rubric and possible invigilation. The current Ontario “summative project” with little guidance and only an exemplar is not pushing more investigations in the classroom. If this is a important goal for the Ontario curriculum, Morgan & Watson provide some guidelines to creating an somewhat equitable assessment of investigations. As Suurtamm Lawson and Koch argue, assessments “defin[e] what is worth knowing, hence what is worth focusing on in classrooms” ( p.41).

Morgan & Watson: Assessment of students'...Issues of Equity

I approached the reading of this article with hopes of discovering information to help improve my assessment practices. I have felt, like others have mentioned, as if I will have had to explain my assessment/teaching practices one day since I have chosen not to “teach to the test”, but allow for creativity and logical understanding. This has left me on my own in my department and open to much comparison this year. Since reading the articles (Suurtamm as well) my conclusion is that I can only continue what I am doing, unless I find colleagues to do moderated marking with, and to keep a continual awareness that there are "possible alternative perceptions and interpretations" (p.102) and that there is not only one correct way of assessing my students’ work. If I am able to get the opportunity to dialogue and incorporate moderated marking with colleagues that would be an improvement, but until then I can only keep in mind, and possibly quote, “reliability is not a concept that can be applied simplistically to assessments of students’ mathematics. …tighter specification of criteria will lead to stereotyped responses from both teachers and students –in opposition to the value ascribed to creativity, openness and authenticity within ‘reform’ discourse” (p.103). With this in mind I can have improved confidence that what I am doing is the best I can do if I am going to continue my attempt at reform pedagogy.

Exercising professional judgment when it comes to assessing

As I was reading Devika's post this quotation stood out for me:

"Ultimately, without clear direction, teachers make their own decisions [...]" (Reys and Reys 1998, p. 237)

This quotation that Devika posted reminded me of a discussion we had a while back in this course about what it means to be a professional in the field of education. I remember Indigo pointing out that unlike doctors who ultimately have the same goal of saving a patient's life, the goals of teachers are often dissimilar because educators have different views on what content should be learned and how learning should take place. Part of this may lead to teachers doing amazing things in the classroom and encouraging students to become critical thinkers, while another part of this lack of consistency may lead to teachers acting on a misguided view of what professional judgment means. However, without a clear direction from the Ministry, teachers are truly left on their own to figure out what they should teach, how they should assess but at the same time be able to take responsibility (hence, the accountability piece) if ever a student, a parent, a colleague, or an administrator takes issue with how the teacher is doing his/her job.

My question is, how does this belief (that a teacher should exercise his/her professional judgment in what specific content/topics are taught and how assessments are carried out) align with practices like EQAO? It seems like a much easier and safer path to follow for teachers to "teach to the test" in the case of something like the EQAO if for no other reason than to not have to justify one's professional reasoning skills when, for example, an administrator or a parent asks why another teacher's class spent weeks preparing for the test while this teacher's class carried on with regular lessons not tailored to EQAO. Are there any ways to get around this issue, I wonder? Even if we had more holistically-assessed inquiry tasks as suggested in the Suurtaam (2008) article, would it just bring up more questions about teachers' professional judgment when it comes to assessing a piece of work without a clearly defined marking scheme?

Connections (in response to Praboda's post)

My comment in response to your post got too long so I decided to make a separate post:


I completely agree with you: we're walking in circles and having the same conversation, over and over again!

You brought up several key points in your post. There are two that specifically reminded me of some previous readings:

1."Although reform-oriented mathematics education seeks to de-emphasize procedural knowledge, we certainly cannot dismiss it entirely."

I agree and so do Hiebert et al. (1997).  They had this to say about teaching skills (procedural understanding) and conceptual understanding:

  

In spite of our belief that understanding and skills can and should develop together, we must make it clear that we assume the primary goal of mathematics instruction is conceptual understanding.  But we must also make it clear that setting conceptual understanding as the primary goal does not mean ignoring computation skills (p. 6).

2."If we are to ‘authentically’ promote the notion of interconnected-ness in mathematics, then can we have a curriculum that communicates a high degree of specificity in terms of individual expectations?"

Alan Schoenfeld (2002) referred to this exact point in terms of the Curriculum and Evaluation Standards for School Mathematics from NCTM in 1989:

..the Standards were long on direction and short on detail - it was a vision statement rather than a blueprint.  In hindsight, that was a good thing, for both political and intellectual reasons (p. 15)

He goes on to argue 'no set of curricular recommendations would have been politically viable" and that we would have missed out on 'significantly enriched curricular discourse" among the mathematical community.

Interestingly, Reys and Reys (1998) disagreed with Schoenfeld years earlier when they wrote this:

Ultimately, without clear direction, teachers make their own decisions on the basis of the mixed messages received from the collective forces of parents. fellow teachers, standardized assessments, curriculum materials, backgrounds that their students bring to the learning environment, and their own beliefs about how children learn (p. 237)

Surprisingly, NCTM did get specific by releasing Curriculum Focal Points:

NCTM's Curriculum Focal Points

Personally, I adopted a philosophy a while back to take my cues from my students.  They told me which parts on the curriculum to focus on.  My circle walking days are not nearly over though :-)

References:

Hiebert, L., Carpenter, T.P. & al. (1997) Making Sense: Teaching and Learning Mathematics with Understanding.  Portsmouth, NJ: Hieneman. Preface & Chapter 1, p. 1-15

Reys, B.J., & Reys, R.E. (1998) Computation in the Elementary Curriculum: Shifting the Emphasis.  Teaching Children Mathematics, 5(4), p. 236-241

Schoenfeld, A. (2002) Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity.  Educational Researcher, 31(1), p. 13-25

The (Antithesis of Problem Solving in Large-Scale Assessment)^2

I was struck by some of the points mentioned in Suurtamm (et al) article regarding large-scale assessment. The authors did well to illustrate how large-scale assessments, such as the EQAO Grade 9 mathematics assessment, may ironically conflict and contradict the tenants of reform-oriented curriculum they were originally designed to support.

Reform-oriented mathematics encourage learners to develop a deeper understanding of mathematics and promote “problem solving approaches to teaching mathematics” rather than memorizing procedural knowledge (p. 32). Using the content strands and the achievement chart (process) categories from Ontario’s Grade 9 math curriculum, the EQAO assessment divides each of the multiple choice items (questions) and short answer items into one content strand and one process category. The extended response tasks, on the other hand, are filtered (or “mapped”) into 1-2 content strands. These tasks, which are designed to promote and demonstrate critical thinking, are broken down into scaffolded questions in order to cover all process categories (knowledge, application, problem solving and communication). Once this is accomplished, all EQAO assessment items are then paired with the specific curriculum expectation(s) from the Ontario Ministry documents.

As the authors have pointed out, the structure for developing the Grade 9 EQAO assessment does not maintain the integrity of reform mathematics. By breaking down extended response tasks into smaller, scaffolded questions, the authors argue that the EQAO assessment “artificially isolates the processes” and once again, “problems are presented as a series of smaller steps”. Ironically, the EQAO assessment seems to value procedural knowledge and certain algorithms of problem solving. Also, the authors also note that assigning EQAO items with process categories can be extremely ambiguous and subjective as these categories are “neither easily separated, distinct, nor distinguishable” (p. 38). This lead to several inconsistencies and once again, EQAO assessment structures imply that these process components can be compartmentalized and segregated into mathematical tasks. Suurtamm et al. also attack the Grade 9 Ontario mathematics curriculum stating that it “gives no clear indication of the important ideas that should be focused on and hence assessed” (p. 41). Ultimately, a curriculum that perpetuates an “insufficient degree of specificity” leaves large-scale assessment incomplete.

After reading this article, I couldn’t help but wonder what exactly is so wrong about problem-solving tasks being broken up and scaffolded into smaller questions. Although reform-oriented mathematics education seeks to de-emphasize procedural knowledge, we certainly cannot dismiss it entirely. (After all, we’re still using “process” categories to measure and map assessment items that seek to shift attention away from procedural tasks – hmm). Considering elements of universal design, I think all students, regardless of mathematics ability, can benefit from scaffolded questions (like those found on the EQAO assessment) during problem solving tasks. Of course, many students may have different methods to solve a task and scaffolded questions may be somewhat limiting, but I don’t think this format would completely discourage a learner’s thought process. If we want to engage students into thinking in “multiple” ways and through different mediums, then perhaps we can consider these scaffolded questions as an example of “a way of thought”.

To add more complexity, I thought there was (dare I say it) a potential paradox with the arguments that the author(s) brought. Suurtamm et al. pointed out the lack of specificity with the curriculum but at the same time, they also value the broad and interconnected strands of mathematical problem solving and inquiry. If we are to ‘authentically’ promote the notion of interconnected-ness in mathematics, then can we have a curriculum that communicates a high degree of specificity in terms of individual expectations? A curriculum that is highly specified may also come across to many educators & stakeholders as being prescriptive and narrowing as well. Also, with the subjectivities that the authors pointed out in their article, I would also wonder who exactly gets to design and choose expectations for a focused curriculum? Again, I feel like we are walking in circles.

Large scale assessments are used to measure student achievement but they continue to provoke intense debate. Educators, teachers and parents are increasingly looking into the purposes of these tests such as student’s learning, teacher’s goals and accountability of education system. The concept of large scale assessment is not new, it has been a norm in many countries for a long time. I have taken these tests myself in my home country but I guess reform mathematics is changing what was being initially expected from these large scale assessments.

It is important to point out that Canadian teachers are responsible for the development and grading of provincial and territorial large-scale criterion- referenced assessments. The national SAIP instruments are also developed and graded by classroom teachers. These activities are coordinated with respective ministries or departments of education. This is in stark contrast to the United States, where large-scale assessments are typically norm-referenced and are run primarily by commercial organizations outside of the education system. This distinction is important and may account for the fact that Canadian testing programs tend to account for greater linkages with classroom practice than their American counterparts (Gambell & Hunter, 2004).

The reliance on criterion- referenced testing also suggests that Canadian testing programs tend to be more aligned with mandated curricula, and as our Ontario curriculum prompts to take an investigative approach to learning mathematics then its not fair with the students and teachers that our assessments do not go far enough in addressing investigative component.

The article of Suurtaam (2008) states that problems in EQAO are the scaffolded version with multiple steps and suggests that problems should be rich and open ended, but I was wondering, are the inherent issues of grading such open ended questions and the excessive time that might be involved in solving such problems, the restraining factors keeping us from achieving this goal?

Large-scale and Equitable?

“Hence, the question is to what extent can large-scale assessment represent reform-oriented curriculum and what are the challenges in trying to design such an assessment?” (Suurtamm, 31)

This is a question that I’ve wondered about before, though not in such eloquent language. My wondering sound more like: why are we doing massive EQAO tests that don’t use the same methods that we know from research are great methods for teaching? And what is the value of a test that requires so much teaching time to prepare students for its particular methods? I can see that there are advantages to large-scale testing for certain parties, but I’m not sure what the value to students is. I think this article does a great job of presenting the challenges that face teachers when there are pressures to adopt new teaching methods and at the same time prepare students for a test that does not use any of these methods. Despite the best efforts of the EQAO team to make sure that the test covers a broad range of topics and methods, it has failed to put emphasis on any particular topics as the most important and it fails to recognize new methods of assessment that have been adopted.

While the article by Morgan and Watson, I was very disheartened. Although I agree with almost all of the points that were made, I find it very depressing to realize that my assessment is so arbitrary. I have had many conversations with other teachers who think that my marking is easier because it is either “right or wrong”, which I always disagree with, but this article sheds new light on just how complex marking in math can be. I too had questions about the solution that Steven had. Maybe I would be able to tell more if I had read the entire interviews, but I thought it was disappointing that a few of the teachers didn’t even try to understand the student’s method. I hope that I have not done the same. I can only hope that through the awareness of the problems that exist with providing equitable assessment, that I can make better decisions!

Happy Pi Day!!

Hi everyone,

I just couldn't resist! I hope everyone is having a wonderful 'pi' day!

LM

Resource List

1.  Literacy and Numeracy Secretariat (Ministry of Education)
http://www.edu.gov.on.ca/eng/literacynumeracy/initiative.html

Communication in the Mathematics Classroom

This monograph talks specifically about three consolidation strategies during the last part of the lesson: gallery walk, math congress, bansho.

Problem Based Learning in Mathematics

Specifically addresses this question: What kind of mathematics problems help students develop deep, conceptual understanding? 

ESL in the Math Classroom

How teachers can support ESL learners to succeed in mathematics?

There are tons more!

2. EduGAINS
http://www.edugains.ca/newsite/index.html
There are tons on resources here on Literacy, Numeracy, Differentiated Instruction, Assessment etc.  There are videos as well.

The Math GAINS site is packed with resources and definitely worth sifting through!

Comments on Race, Culture and Mathematics (March 9th)

Our group discussed recomendation #4: Recongnizing race and ignoring race. These are the points we talked about/found interesting:

- Race has no barring on student achievement and should not be discussed in the context of what a student is capable of doing “Knowing the race of a specific child offers no information whatsoever about that child’s current or potential achievement”

- “There’s no shortcut to getting to know the student as an individual. The teacher must interact with the child and his or her family to gain the information needed to tailor the provision of learning opportunities for the student”

- " In those quintessential moments of teaching and learning, race means nothing. In contrast, biography, culture, and relations of power may mean everything."

- Race may be considered in the context to student experiences (how they have been treated by others based on race). Students may have negative self-perceptions about their ability to succeed as a result of those experiences that need to be addressed by the teacher.

Devika, JWallace, MP, Rohini

Why group work?

Hi, everyone!
I've been enjoying the thoughtful and interesting posts on this week's articles, and I've been wondering about a couple of questions. I'd love to hear your opinions.

When teachers talk to me about their use of group work, they usually tell me two things.

1. They think group work is helpful because it lets the 'strong' kids explain things to the 'weak' kids.
2. They think group work is helpful because the teacher can go around and help the 'weak' kids individually, while the 'strong' students help each other.

What do you think about these uses of group work? Helpful, unhelpful?

Group Work vs. Cooperative Learning

I mentioned in my other post that I loved the way that the Boaler article talked about interactions between students and this article also discusses the relationships that students have with each other when working in group settings. This is a very important aspect of group work that I often forget about because we assume that students already have relationships with each other. Although this might be true, students don’t always have the most positive relationships and we need to explicitly teach how to be respectful and how to properly contribute to a group.

We have talked a lot at my school about cooperative learning this year, which emphasizes the importance of students working in groups and each student having a specific role. It prevents students from sitting in the group and not participating. Vygotsky is referenced as promoting the opportunity for students to talk about what they are learning, which means that group discussions are important, but they are only valuable if all members are able to speak and be heard.

Chizhik: Equity and status in group collaboration...

I have always believed in the necessity of discourse and have prescribed to Vygotsky's theory of the connection of thought to language. I have used discourse at every opportunity possible in my pedagogy (mathematics and all other subjects). This article by Chizhik has me now rethinking several areas of my teaching practice. Firstly, the structure and use of tasks within group work. Although, as Chizhik states, single answer tasks have its place, I will still need to ensure I am using more variable answer tasks. Secondly, although I already give roles to students within group work, I was not doing so with the urgency that I now feel regarding the impact of inequitable group collaboration on individual learning. As pointed out in Chizhik's research, I will now also inform groups of the multiple skills required to be successful, that all group members should understand the final answer and encourage them to learn from each other. Collaboration is a problem with students in my school and realizing the impact on individual learning I am hopeful this added focus will allow for a more equitable pedagogy. Lastly, although Chizhik points out that, the "race-specific findings... should not be used to stereotype individual students" and "...the differences...arise from the hegemony with American culture." p.197 I have to say, this research has me rethinking the arguments for Afrocentric schools. Although I am adamantly against segregated schooling, I now have a slight understanding of the argument some may have for the necessity of this grouping for improved learning. I am not completely convinced though, because as Chizhik concludes, if we (as educators) pay attention to the impact of equity and status within group collaboration and the tasks given (single vs. variable) it is possible to better influence the learning of all students.

Key points, questions, and reflections from the Boaler article

In reading the Boaler article, there were several points that resonated with me in her discussion of complex instruction and the work at Railside:

1.  "Put simply, when there are many ways to be successful, many more students are successful" (page 42)

The use of open-ended problems is at the heart of this statement.  I think we all know that we should use more open-ended problems in our classrooms but what are the barriers preventing us from doing so?

2.  "The roles contributed to the complex interconnected system that operated in each classroom, a system in which everyone had something important to do and all students learn to rely on each other" (page 42-43)

This felt like a vision statement to me.  I've played around with roles but have not had this type of success.  I found that some roles, such as resource manager, were not 'meaty' enough.  This role in particular is finished fairly quickly and the child is left watching as the others play out their role.  I would have liked Boaler to address some of the explicit teaching and modelling that the teachers had to do to get students to be an effective 'facilitator' or 'team captain' etc.  These skills do not come naturally to students.

3.  "If student feedback is to address status issues, it must be public, intellectual, specific, and relevant to the group task" (page 43)

Assigning competence is a new term to me but the practice is not.  In reflecting back, I can think of specific students in my elementary classes who benefitted from this.  I wish I did it more often and was more intentional.  But I wonder, how would secondary students react to this?  Would they welcome it or resent the spotlight?

This statement also connects to her later statement about effort over ability. If we give students feedback based on their work, it gives them the message that their effort will bring about success.  Dylan Wiliam calls this descriptive feedback. 

4.  "...it was critical to the success of the students that teachers kept the demand of lessons intellectually high, by providing complex problems, and by following up with high-level questions" (page 44)

The statement contributes to our discussion about the deficit model of thinking about low-achieving students.  It is so easy to dismiss the use of complex problems with a 'my students can't handle those kinds of problems' comment but Boaler clearly articulates that the use these problems contributed to student success.  

5.  "...students learned something extremely important that would serve them and others well in their future interactions in society, which is not captured in conceptions of equity that deal only with test scores or treatment in schools" (page 45)

This (relational equity) is not easily achievable so this makes Railside's success even more amazing!  Teaching is complex and success does not come with quick fixes.  

I really appreciated Boaler's summary that, "at the heart of this system is the work of the teachers and the many different equitable practices in which they engage".  

With that in mind, I wonder about the types of supports that were provided for these teachers.  I'm sure there were obstacles and challenges on the way, so how did the school get the teachers to persevere with this way of teaching?  

Comments on the Adler Article

There were a few quotes in the Adler issue that I found interesting and would like to comment on:
In reference to Helen’s reflection she indicated that she had “gone on too long.” I find that whenever I discuss the vocabulary associated with a concept the pacing is really important. If I stay on a subject too long then I lose the attention of a large portion of the class. For example, when discussing slope with relation to any given linear relation and graph, it’s important for me to recognize when it’s time to move on and come back to the concept at another opportunity. I realize that slope is an important idea but there is always a point when the students tune out.

“some mathematical ideas are difficult for pupils to verbalize precisely and with meaning”
I find that some students hate verbally communicating what they are working on. I was like that. Sometimes the concepts are hard to put into words, and a student will make the argument that in showing the work mathematically they are still ‘communicating’ what they are doing, just not verbally. I still believe that communicating using words is an important activity, but only when appropriate. In addition, I sometimes feel that because much of the math is often out of any context, communicating what is happening is made that much more difficult.

Bilingual mathematics learners

Reading through the article by Moschkovich, i feel that there has been a connection made to the term "deficiency" , "of vocablary" and "multiple meanings". This association has a very negative conotation to it as in classifying students entirely incapable due to such gaps in their language. However, i feel we cannot ignore these gaps as they influence theire capability in communicating in a foreign language.Even though it is sad that assessment is based on students' capability of interpreting mathematical concepts in the foreign language, it is only fair to expect that students' be able to convey their understanding to their peers. This would ensure their success as citizens in the foreign country.
Therefore, i cannot completely dismiss the fact that emphasis should be given on vocabulary building and understanding of multiple meanings.
I do however, agree with the article in that too much emphasis on english vocabulary can deter students from advancing further as they would get de-motivated from constant stress. Their success should be celebrated as well, if they are capable of explaining a concept through diagrams,gestures and stories in simple language it is crucial to make account of this and motivate students further to make such associations.

Example of a word wall

Here's an example of a word wall that has a picture or example attached to it.  It's from a Grade 7 classroom.

I wonder if a tool like this would be helpful for students and make room for the mathematics.

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