Family_D

=WELCOME TO FAMILY D=

**Names:**
Brenda, Szymon, Hartley, Andrea

**Summary of Analysis:**
The common areas for improvement across the grade 3, 6 and 9 EQAO scores were:
 * reading and decoding the mathematical language in the questions asked
 * numeration - understanding basic numeracy concepts
 * conceptual understanding

**Our Question(Smart Goal):**
How can we improve the conceptual understanding of Number Sense (i.e. quantity relationships) in our students from K-12 through problem solving.

Our K-12 Learning Target:
Students will develop an understanding of multiple representations of quantities and effectively describe and utilize the relationships between these representations.

**Strategies for Consideration:**
//__A Guide to Effective Instruction in Mathematics; K-6 Strategies for oral Communication__: Note: Grades 1-3; beginning writing skills do not allow them to demonstrate fully their mathematical knowledge. Thus, oral is important as it helps teachers identify both understandings and misconceptions.// //Grades 1 to 6//; Think-Pair-Share, Show & Tell, Math Reader's Theatre, Math Forum, Cooperative Problem Solving. //Grades 1-3//; Catch the Mistake and Make it Right; //Grades 4-6//- Prove It or Disprove It. Grades 9-12: read problem aloud, observations, real-world projects, develop and defend their own computational methods.

//__Strategies for Written Communication__: Grades 1-8: Mind Mapping, Model Writing, Shared Student Writing,Group Solution Writing, Think-Talk-Write, Thinking Windows, Place Mat, Procedural Writing, Graphic Organizers, Math Word Wall, Math Strategy Wall, Individual and/or Class Journals/Logs, Math Picture Books, Poster Projects, Students' Problem Posing, Math Creative Writing.// Grades 9-12: word walls, real-world problems, develop and defend student's own computational methods, and generalize their findings, portfolios, concept maps, journals, projects.

//__Teaching and Learning Mathematics (Junior grades)__// __//Teaching and Learning Mathematics (Senior grades)//__
 * __Encouraging communication of ideas__ (e.g. explain their ideas clearly, follow others reasoning) deepens students understanding of concepts being taught. (Rich discussion is the heart of mathematics classrooms-students engage in the re-creation of important underlying mathematical ideas). (13/14).
 * In __learning to explain their ideas__ students learn to; 1) write up their ideas in an organized fashion 2) use a variety of forms to explain their reasoning 3) use mathematical terminology and symbolism and thus become more effective in their communication skills and understanding (15).
 * __Math games and puzzles__ which are developmentally appropriate help support mathematical literacy including Number Sense and Numeration (23).
 * __Manipulatives__ in the junior classrooms is a necessary tool that helps support student thinking as well as provide a model for visual learners (26).
 *   Students whom do not have a solid understanding of fundamental mathematical concepts benefit when the mathematical problems presented by the teacher contain a variety of entry points, and are broad enough so that they can be solved through a number of different strategies (37).
 *   “Effective assessment and evaluation practices have the capacity to greatly strengthen a student’s mathematical literacy” (quote, 41). Assessment is a tool which is used to support the learning of each individual student (41).
 * Effecitve teaching and learning begin with the needs of adolescent students and reflect their developmental stages. (25)
 * An effective learning experience is one that connects mathematics with the lives of adolescent students. (25)
 * Students must have a solid conceptual foundation in mathematics in order to apply their knowledge and to continue to learn mathematics. (25)
 * Effective instructionial strategies in mathematics emplhasize the ability to think, to solve problems, and to build on'es own understandings. (25)
 * To improve students' performance, teachers should link instruction more closely with assessment.(25)
 * Teachers of mathematics need professional learning opportunities that strengthen their competence in both mathematics content and the methodology for teaching it. (25)
 * Technology supports leraning and should b eaccessible to all students- especially those who struggle with mathematics. (25)
 * To form an effective learning environment in which the needs of all students are met and their success is promoted, schools and school systems need sound planning and strong leadership. (25)
 * Mathematics learning strategies that benefit all students are a necessity for students at risk, and extra support may also be needed to close the gap.(25)

__//Think Literacy: Mathematics Subject-Specific Examples Grades 7-9//__ Reading Strategies: Writing Strategies: Oral Communications:
 * Previewing a Text - provide prompts for students to explore the particular features of the textbook or resource (p. 2-4)
 * Anticipation Guide - encourage students to make personal connections with a topic so that they can integrate new knowledge with their prior knowledge (p. 10 -14)
 * Finding Signal Words - identify signal words and phrases and their purposes (p. 16-21)
 * Extending Vocabulary - The Frayer Model - create a visual reference for concepts and vocabulary (p.38-42)
 * Most/Least Important Idea(s) and Information - Reading a Problem - decode information presented in a block of text and represent information in an alternate format (p. 44-50)
 * Developing and Organizing Ideas: Webbing, Mapping, and More - identify relationships and make connections among ideas and information (p. 76-80)
 * Think-Pair-Share (p. 96-98)
 * Small Group Discussions: Placemat - provides all students with an opportunity to share ideas and learn from each other in a cooperative small-group discussion (p.102-104)
 * Whole Class Discussions: Four Corners - allows students to make personal decisions on various issues and encourages critical thinking (p. 106-110)

__//Making Math Happen in the Intermediate Years//__ __Transformational Practices (from the PDSB):__ __Strategies:__ __ Problem Solving:  __ //__ Early Math Strategy: __// // __Van De Walle__: // // __A Guide to Effective Instruction in Mathematics- volume 6__: // **1) ****Understanding the problem.** **2) ****Making a Plan** **3) ****Carrying out the plan** **4) ****Communicating/Looking back- occurs throughout the process.** **a) ****Act it out, b) Make a model with concrete materials, c) draw a diagram, d)use the guess and check method, d) work backwards, d) use logical thinking, e) make a table, f) use an organized list, g) solve a similar problem, h) use or find a pattern**
 * Math Conference - a way for students to communicate their mathematical understanding (p. 36-38)
 * Math Journals - encourages students to reflect on the process and content of their learning (p. 50)
 * Manipulatives - allow students to represent mathematical ideas in ways that make them easier to understand and communicate (p. 54-64)
 * Technology - a valuable tool that can enhance students' learning and understanding of mathematical concepts (p. 64-68)
 * According to research "Vocabulary acquisition is crucial to academic development. They need a rich body of word knowledge to succeed in basic skill areas and specialized vocabulary to learn content area." It is imperative that teachers know and use effective strategies for helping students understand both common words used in uncommon ways and specialized vocabulary. (p.28)
 * Developing vocabulary can occur in a print-rich environment, authentic experiences, interactive word walls, Author charts, Resources, Listening centres, Literacy centres, software and websites, through modeling and purposeful talk, and read alouds (pgs25-26)
 * Direct teaching of vocabulary development: a) present words in various ways; b) give a brief explanation or description of the word (print word on blackboard to reinforce oral-visual connection. Repeat word orally several times. c) represent word in non-linguistic ways (e.g. pictures, photographs, sketches) and/or concrete materials for support d) allow students to develop their own explanations of the word (think/pair/share) e) continue to suport student learning in a variety of ways. May include referring to charts, comparing, questioning, and making connections and associations. f) ensure that vocabulary is repeated frequently. (p.27)
 * ** “Problem solving is not only a goal of learning mathematics but also a major means of doing so.” **
 * ** Problem solving should be the mainstay of mathematical teaching. **
 * ** “Problem solving is more than the application of skills. Problem solving in a classroom generally beings with the teacher presenting the problem, students exploring and working on a solution to the problem, and then teachers a consolidating and reflecting.” (p.16) **
 * ** “We use the ideas we already have (blue dots) to construct a new idea (red dot), developing in the process, a network of connections between ideas. The more ideas used and the more connections made, the better we understand.” (John Van De Walle- 3-part Lesson Design) **
 * Mathematical problem solving for today’s classroom has moved beyond the limited view of traditional word problems whose major focus is on the application of arithmetic operations and algorithms. The Ontario Curriculum indicates that the experiences in grades 6-8 should include inquiry and investigations that will provide students with useful problem solving strategies. **
 * George Polya’s: Multi step proves for problem solving and inquiry: **
 * ** “Problem solving strategies are specific methods used for solving problems. These strategies are best explored by students incidentally, within the context of solving daily problems, rather than through direct instruction about the strategies themselves.” (p.38) **
 * Problem Solving Strategies: **
 * Teachers role is to help students articulate the strategies and then to keep an ongoing record of the strategies through the use of a strategy wall. **

__Teaching Mathematics Through Problem Solving: Grades 6-12__ Implications for Teachers
 * knowing how to execute procedures does not ensure that students understand what they are doing
 * to understand, students must get inside these topics; become curious about how everything works; figure out how this topic is the same as, and different from, a topic they already studied; and become confident that they could handle problems about the topic, even new problems they have not seen
 * problem solving leads to understanding - students develop, extend, and enrich their understandings by solving problems
 * posing problems that are just within students' reach, allowing them to struggle to find solutions and then examining the methods they have used
 * all students are to struggle with challenging problems if they are to learn mathematics deeply
 * the key to allowing mathematics to be problematic for students is for the teacher to refrain from stepping in and doing too much of the mathematical work too quickly
 * students should be permitted to choose their own method to solve problems - discussions in class should revolve around sharing, analyzing, and improving methods
 * need to create a learning environment where students are comfortable with taking risks and making mistakes - when mathematics is allowed to be problematic for students, making mistakes becomes a natural part of learning

**Potential Resources:**
1) Teaching and Learning Mathematics; The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario. (Note: 3 part lesson plan pgs 10-15). 2) Elementary and Middle School Mathematics (Fourth Edition), John A. Van De Walle. (Chapter 6;Number concepts and Number Sense). 3) A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 (Volume 2), Ontario Education (Communication, pp 55-84) 4) Vocabulary and Its effects on mathematics instruction http://eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED439017&ERICExtSearch_SearchType_0=no&accno=ED439017 5) Think Literacy Cross-Curricular Approaches Grades 7-12: Subject-Specific Examples: Mathematics, Grades 7-9 (2004) 6) Making Math Happen in the Intermediate Years, Jason Johnston, Troy Parkhouse, et. al. (2003) 7) Education for All: The Report of The Expert Panel on Literacy and Numeracy Instruction for Students with Special Education Needs, Kindergarten to Grade 6 (2005), p. 80 - 90 8) Leading Math Success, Mathematical Literacy Grades 1-23: The Report of the Expert Panel on Student Success in Ontario (2004), p 25-37 9) Transformational Practices (PDSB): Working together to achieve literacy success. (pages 5-25).

1) Teaching Mathematics Through Problem Solving: Grades 6-12
 * Problem Solving**