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