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Names: Diego, Estie, Paul, and Saajida

Summary of Analysis:
 * Students had problems with number sense all across the continuum, especially word problems involving number operations.
 * Students also had issues with measurement across the board. We felt that they may not have a complete understanding of the differences between area, perimeter, and volume.
 * We chose to base our question around number sense, since if students have a degree of competence in this area, it will carry over into other areas of the curriculum.

Our Question:  How can we improve at-risk students' strategic competence with number sense in the area of problem solving through the use of operations applied to real-world contexts ? Our revised Question: How can we improve our at-risk students conceptual understanding of simple operations through problem solving? Our Target:

Students across the K-12 curriculum will demonstrate proficiency with number operations and an ability to choose and justify methods of solution.

Strategies for Consideration: Potential Resources Fosnot, Catherine Twomey, Young Mathematicians at Work
 * In the classroom, ensure that problem-solving instruction is //systematic and clear// (see //How to Teach Through Problem Solving: A Concrete Example// Page 79 Education for All, Special Education Needs).
 * School boards should ensure that teachers have designated time for collaboration, easy access to resources and information, and opportunities to participate in mentorship (see page 143 Education for All, Special Education Needs).
 * Young children connect and build on intuitive mathematical understanding. This is done through the use of mathematical experiences that allow students to explore mathematics and to communicate their explorations in a meaningful dialogue with teacher and peers (page 10 Early Math Strategies, Ontario Ministry of Education, 2003).
 * Children learn to value mathematics through activities that help them build connections between mathematical concepts and other areas of their lives. They need authentic math content and real depth to classroom activities that help them to move forward (page 14 Early Math Strategies).
 * Students learn math as a result of exploring problems that provide rich challenges, relevance and engagement. Problem solving helps to support core processes, such as the use of representation, communication and connection between and among mathematical ideas...They acquire their understanding of math and develop problem-solving skills as a result of solving problems, rather than be taught something directly (page 16, Early Math Strategies).
 * When instruction focused on problem solving, children not only became better problem solvers, but also had better mastery of computations than did children whose instruction focused solely on drill and the practice of basic facts. Children need time to practise and consolidate skills, balanced with time to put those skills to use in a problem-solving context. (page 31, Early Math Strategies).
 * Mathematics teachers must support language learning. They must provide the support necessary for English language learners to develop the cognitive academic language proficiency to deal successfully with the mainstream Mathematics curriculum. See //Adding English// (Chapter 12) for specific strategies.
 * Making mathematics meanigful is a key priority for engaging adolesecent learners. Reasearch suggest applying math in ways that are linked to students' experiences, curiosity, imagination, and aspirations Learning must begin with where students are and what they know. //Leading Math Success// (page 28-29 Making Connections)
 * Solving problems in the mathematics classroom involves risk taking. Students do not want to be embarrassed in front of their peers, and will not take risks unless they feel valued and supported. The classroom therefore must be a safe place. //Leading Math Success// (page 44)
 * A good summary to move towards //Leading Math Success// (page 54)

Ministry of Education, //Education For All//, //The Report of the Expert Panel on Literacy and Numeracy Instruction for Students with Special Education needs, Kindergarten to Grade 6,// Queen's Printer for Ontario, 2005
 * Chapter 7: Effective Instructional Approaches and Teaching Strategies for Numeracy / The Importance of Teaching Through Problem Solving (page 77)
 * Chapter 11: Professional Development / Professional Development Models and Formats (page 143)

//Teaching and Learning Mathematics: The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario. (pages 16, 22, 59)//

The Ontario Curriculum, Grades K-12 Ministry of Education, //Early Math Strategy: The Report of the Expert Panel on Early Math in Ontario,// 2003. Fair, Jan and Mary Melvin, //Kids are Consumers Too,// Addison-Wesley, 1986.

Think Literacy - Mathematics Approaches: http://www.oame.on.ca/main/index1.php?lang=en&code=ThinkLit

Elizabeth Coelho, //Adding English,A guide to teaching in multilingual classrooms,// Pippin Publishing, 2004
 * Chapter 12: Integrating Language and Content Instruction

Ministry of Eduaction //Leading Math Success: Mthematical Literacy Grades 7-12

Think Literacy: Cross-Curricular Approaches Grade 7-12, Subject Specific Examples Mathematics Grades 10-12// Ontario Ministry of Education

Assessment, Evaluation, and Reporting Handbook: A guide for Secondary School Teachers and Administrators. (Peel District School Board)

Transformational Practices Grades K to 12: Working Together to achieve literary success. (Peel District School Board)

The following sites have some useful lesson plans that follow the problem-solving approach. They are not necessarily set up as 3-part lessons, but can easily be adapted.

Math Solutions On-Line - Marilyn Burns http://www.mathsolutions.com/

OAME - Ontario Association for Mathematics Education http://www.oame.on.ca/

An incredible library of virtual manipulatives for all grades K-12 http://nlvm.usu.edu/en/nav/vlibrary.html

Some possible questions for mentor/coach:

Coach: As you reflect upon the independent task given to the children, what are some ways that you think the children were able to make connections to the modelled problem? Mentor: How do you think the independent task helps the children to see the connection between multiplication and division? Use of co-operative strategies: Coach: What are some strategies the children might use to be able to feel more confident or comfortable about sharing their ideas? Mentor: How do you think (more) talk-time might have been helpful to make students feel more comfortable about sharing their ideas? Coach: manipulatives Mentor: Do you think that using manipulatives may have been beneficial to those students who need to see a model or manipulate items to visualize their ideas? Coach: Reflecting upon the independent work, what prompts did you find helpful for the students, particularly if a student was struggling? Was it difficult not to give away too much information and allow them to make discoveries on their own? Coach: (based on Key Messages document) Learning Environment: How are you determining that this type of lesson format (problem solving) builds student confidence in their mathematical abilities? Curriculum program: How does this lesson support the students’ learning of the concept of multiplication being related to division? What previous assessment have you done to determine prior knowledge? How does this inform your planning? Classroom instruction: What direction might you want to go to continue to move students along the continuum? What strategies might you use for at-risk students? Those who have achieved mastery to extend their knowledge base?

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COLOR: #f024d7; FONT-FAMILY: 'Courier New', Courier, monospace">OUR PD PLAN

We have determined that the teachers in our family of schools are at stae 4 of the CBAM scale: "How is my use affecting learners? How can I refien it to have more impact?" Based on our collective lesson analyses, we have also decided to tweak our family's question. It is now "How can we improve out at-risk students' conceptual understanding of simple operations through problem solving?"

Saajida Here are some possible PD models that I think might work with our family of schools over 3 - 5 years:
 * A book study revolving around the benefits of teaching through problem solving, and perhaps a learning session around the stages of development/acquisition of number operation development (PRIME)
 * A demonstration (video, presentation, expert speaker) of modelling a lesson(s) through problem solving.
 * A period of team-teaching or coaching/mentoring in which teachers who are teaching the same course have an opportunity to work together to try some new strategies/skills that they have learnt. This would also involve release time to discuss some of the strategies that the teachers plan to use, and perhaps observations of each other's classes.
 * Data analysis to determine what worked well and what didn't during the co-teaching or coaching/mentoring stage (CASL)
 * Perhaps a lesson study to consiladate some best practices and provide a model for future lessons leading to student improvement.

Some points for the checklist: Estie Based on information on the Assessment for Learning area of our studies (I have the York Region Document), as well as Early Math Strategies from the Ministry, there is lots of research that I will focus on this weekend regarding the importance of ongoing, meaningful feedback.
 * The plan will encompass all teachers K-12, but will focus on one or two grade levels specifically.
 * Mathematical content: simple operations (number sense)
 * pedagogy? (working on this one)
 * CBAM level 4

Because this has to be interactive, we could possibly use some Tribes or Instructional Intelligence strategies to demonstrate some inclusive activities, while perhaps modelling a book study, CASL, modelled problem-solving approach. We can combine more than one strategy, i.e., starting out with a book study, then followed by some peer coaching, or CASL followed by teacher mentoring. I will bring the TRIBES book and Beyond Monet on Monday.

In terms of a 3-5 year plan, I suppose we should start small, perhaps two teachers working collaboratively, as coaches, moving onto a Department, to a school, and then to encompass our Family of schools by Year 5.

This would require release time for regular workshops for teachers to continue to have training. As well, there would have to be a team of "Math Experts" who continually plan and deliver workshops. Perhaps we could say one of our goals might be to get Lesson Study groups working at each school, perhaps planning one problem-solving lesson per strand, per year, so that at the end of 5 years, we could have a comprehensive repertoire of diagnostic(?) tools or models for teachers to use and refer to in their practice.

By the way, I had to re-register onto the Wiki site. For some reason, my name was deleted and I could not access the info. Did this happen to anyone else?

Diego Since we took the time to look at our famimly of school data, I would suggest taking the time at the begining to explain the 'Why' the target is what is it to our family. Possibly something similiar to what we did in class except more in depth. In terms of the logistics and $$...this will determine the audience[the size and who] to which we are preaching to (hopefully converting). I think its important to decide this before picking strategies/timelines. I think a pair from each school math dept. might work. and initial book study would be good to start in sharing the knowledge and theory for our "new skills/practices" and a video or hopefully a guest speaker eventually to see it's implementation in the classroom. Im not sure about what the coaching aspect would look like, but i would recommend a wiki type online forum, so that there would be support throughout the entire time (plus it's low cost).

Paul
 * The plan will focus on k-12
 * In terms of mathematical content, we need to be specific. Our target is conceptual understanding of simple operations through problem solving, but we can be specific in terms of which operation(s) / whole numbers / fractions / decimals
 * I think it would be great to focus on addition of fractions for example. This area of content creates problems for students across the continuum.
 * Pedagogy: Since we all seem to be thinking about CASL, certainly part of our focus will be on **assessment**. Given our target ("through problem solving") and some of the misconceptions that I identified in my lesson analysis, I would like to see us focus on how to **develop conceptual understanding through problem solving** (Hiebert and Wearne handout).
 * CBAM: After having analysed my lesson, I would say the teacher that wrote the lesson is past the "self-oriented" type of questioning, but still in the task-oriented type of questioning. i.e. How is my use affecting learners? How can I refine it to have more impact? (4. Consequence)
 * Because we are focusing both on assessment and on effective lesson design in terms of developing conceptual understanding through problemsolving, I think we should start with CASL and then move on to Lesson Study. (order makes sense according to backwards design) These two models are consistent with the CBAM question (How can I refine it to have more impact?).

I've emailed everybody our planning notes for our PD model. We've already discussed and agreed on what each of us will be doing tonight. Until tomorrow morning...