There’s More to Math A Framework for Learning and Instruction
Math Action Team 2002-2003 Cathy Barr, Monsignor Doyle Julie Clifford, Our Lady of Grace Teresa De Leo, St. Francis (K) Carol Dubeau, Our Lady of Lourdes Audrey Gleeson, St. Anne (K) Paul Henriques, St. Francis (C) Brenda Hunniford, SERT Tierney Hunter, Sir Edgar Bauer Stephen Jones, Holy Rosary Gloria Lasovich, Sir Edgar Bauer Elizabeth Matlock, Mathematics Literacy Consultant Cathy Renda, St. Anne (K) Nancy Snyder, St. Mary’s Eduarda Sousa, St. Peter John Sullivan, Blessed Sacrament Donna Turek, St. Francis (K) Anthony VanLooyen, St. Dominic Savio Joan Young, St. Matthew Maria Zunic, St. Francis (K)
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Background What is Mathematics? “Mathematics is a study of patterns and relationships; a science and a way of thinking; an art, characterized by order and internal consistency; a language, using carefully defined terms and symbols; and a tool.” North Central Regional Educational Laboratory Mathematics allows us to fit models to everyday experiences and situations in order to reason, problemsolve, analyze, investigate, explain, calculate, predict, …
What is Mathematics Literacy? An individual is mathematically literate who can confidently Use mathematics as a way of exploring, modeling, explaining and understanding the world Use mathematics as a way of reasoning and problem-solving Use mathematical procedures appropriately and effectively Mathematics literacy is more than facts and procedures just as language literacy is more than reading and writing. Without the ability to acquire meaning through reading or convey meaning through writing, an individual would not be considered literate. Similarly, the ability to fit a mathematical model to a problem or choose and apply an appropriate procedure, an individual would not be considered mathematically literate. Just as students develop language literacy by using language, students develop mathematical literacy by being immersed in and engaged in doing mathematics, not just by performing procedures, but also by exploring mathematics in contexts and problems that make the procedures useful.
A Contemporary Vision of Mathematics Teaching and Learning A contemporary vision of mathematics teaching and learning emerged in the early 1980’s from an organization called the National Council of Teachers of Mathematics (NCTM). The vision is a departure from the traditional view of mathematics as simply a collection of facts, formulas, procedures and algorithms. It recognizes that for students to be truly “mathematically literate”, they need to possess a strong relational understanding of mathematics. Relational understanding comes from allowing students to develop conceptual knowledge alongside the procedural knowledge that has often been the sole focus of traditional elementary and secondary school mathematics programs. This more contemporary approach to teaching mathematics has been referred to by various names such as constructivist, reform, problems-based, or standards-based. Traditional mathematics programs that focus mainly on proficiency in basic procedural skills do not help students to develop the necessary relational understanding or problem-solving skills needed to transfer procedural knowledge to a new situation or context. Research conducted over the last 20 years confirms that standards-based mathematics programs not only facilitate the development of basic skills as effectively as traditional programs do, but they also allow students to develop transferable problem-
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solving skills that traditional programs do not. Students’ mathematical power increases when students participate in standards-based mathematics programs that allow them to explore and make connections between mathematics concepts through rich problem-solving tasks, interaction and dialogue. The Ontario Curriculum was designed to be taught using a standards-based approach. The Ontario Curriculum
The Ontario Curriculum expectations for mathematics combine curriculum content with process. Each
expectation contains a verb describing what students will do, followed by the curriculum content that students will be working with. Example: Students will present solutions to patterning problems…
Patterning and Algebra, Grade 7 An overall examination of the language within the expectations demonstrates the alignment of the Ontario Curriculum to a standards-based approach to mathematics teaching and learning that emphasizes rich tasks, student interaction and dialogue. Examples: Students will discuss geometric concepts with peers and use mathematical language to explain their understanding of the concepts, Geometry and Spatial Sense, Grade 6
Students will investigate measures of circumference using concrete materials Measurement, Grade 5 Improved students learning in mathematics requires effective implementation of The Ontario Curriculum for mathematics. That implementation requires us to move away from a traditional mathematics delivery model, and move toward a standards-based approach to teaching and learning that is consistent with the design of the curriculum expectations.
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A Framework for Learning and Instruction Some common remarks that teachers make about students and mathematics are: 1. They forget it from year to year. 2. They can’t use their math skills in other subject areas. 3. They can’t problem solve. 4. They only feel confident with a ready-made recipe. 5. They call it one of the ‘hard’ subjects. Research has shown that teaching for understanding helps students to develop transferable mathematical knowledge that addresses the problems listed above. Teaching for understanding involves the development of two kinds of mathematical knowledge: conceptual knowledge and procedural knowledge.
The Learning Piece: Conceptual knowledge vs. procedural knowledge Conceptual knowledge is ideas, relationships, connections, or having a ‘sense’ of something. It is understanding. In contrast, procedural knowledge is like a toolbox. It includes facts, skills, procedures, algorithms, or methods. When we ask students to perform a procedure such as solving an equation, students can often follow an example and get a correct answer without understanding how or why the process works. Questions that call on students’ conceptual knowledge allow us to see whether or not they understand the mathematics that they are learning.
Questions that use procedural knowledge Measure the perimeter of the room If you sleep for 7.5 hours each day, what percentage of the day is spent sleeping? Find the sum of one-third, one-quarter and one-fifth Match the object to its associated volume formula Multiply 24 by 8
Find an equation to solve this problem
Questions that use conceptual knowledge Estimate the perimeter of the room. Justify your estimate. Is it reasonable to state that many people sleep for 30% of the day? Why or why not? Without adding, is the sum of one-quarter, onethird and one-fifth bigger or smaller than one? How do you know? Explain how to determine if you have matched an object to its correct volume formula. In your head, multiply 24 by 8. Explain your method. Try to find another method that works. Find a problem that can be solved using this equation. How can you tell if you are right?
Instructing for the development of conceptual knowledge and procedural knowledge helps more students to understand mathematics. Over the long term, it develops and strengthens a student’s mathematical knowledge base to support more complex ideas as he or she reaches higher grades.
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The Instruction Piece: Rich Tasks and Rich Talk How do we develop conceptual knowledge, and how do we make connections between conceptual and procedural knowledge? Students construct knowledge and connections by engaging in problem solving (rich tasks) and talking about solving problems (rich talk). Problem solving experiences can be intimidating and frustrating for teachers and students when the problems are too difficult, or when the desired learning isn’t clear. Out of care and concern, sometimes students are protected from working on the problems that would allow them to really explore and ‘do’ mathematics. Often we think we are helping students when we give them the answers and the explanations rather than giving them the experiences. By comparison, when we involve children in sports, we don’t limit their experience to basic skills and tell them that the game is still too hard for them to play. We let them develop the skills and the application of the skills at the same time, and we watch them gain confidence in both. That is how it should be to learn mathematics. The key to teaching mathematics using a problem solving approach requires access to supportive resources and professional development that provide appropriate problems (rich tasks) that all students can engage in and discuss (rich talk).
What is a rich task? A rich task is a problem that is not solved by the automatic application of a specific procedure or method. All students can participate in a rich task. It replaces the traditional teacher explanation in a lesson, allowing students to form their own explanations that connect to their unique sets of prior knowledge. For example, consider some traditional lessons on area of a parallelogram. Scenario 1: Topic:
Area of a parallelogram (Grade 7)
Expectation: Students will: evaluate simple algebraic expressions by substituting natural numbers for the variables (PA) Lesson 1. Teacher writes on the board the formula for the area of a parallelogram. 2. Teacher does several examples on how to use the formula. 3. Students complete similar questions on their own, using the examples as models. Some things to note about scenario 1 are these: The students do not know how or why the formula works. The formula needs to be memorized to be used again. Students can complete the exercise and get correct answers without having an understanding of what area is. The lesson only addresses one expectation.
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Scenario 2: Topic: Expectations:
Area of a parallelogram (Grade 7)
Students will:
evaluate simple algebraic expressions by substituting natural numbers for the variables (PA) develop the formula for finding the area of a parallelogram (M) Lesson: 1. Teacher explains to students that the formula for the area of a parallelogram is related to the formula for the area of a rectangle, since the parallelogram can be cut and reassembled into a rectangle. 2. Using the developed formula, teacher then provides examples of how to use the formula to find the areas of parallelograms. 3. Students complete several questions using the examples as models. Some things to note about scenario two are these: The teacher does the math (ie. achieves the expectation), and explains it to the students. There is only one explanation for the development of the formula; there is no opportunity for students to develop alternative explanations. Students can complete the exercise and get correct answers without having understood the explanation, and without having an understanding of what area is. Consider a lesson that uses a rich task as an instructional strategy.
Scenario 3: Topic:
Area of a parallelogram (Grade 7)
Expectations: Students will: develop the formula for finding the area of a parallelogram (M) describe measurement concepts using appropriate measurement vocabulary (M) solve problems involving the congruence of shapes (G) identify two-dimensional shapes that meet certain criteria (G) recognize patterns and use them to make predictions (PA) interpret a variable as a symbol that may be replaced by a given set of numbers (PA) translate simple statements into algebraic expressions or equations (PA) evaluate simple algebraic expressions by substituting natural numbers for the variables (PA) Concepts: geometric vocabulary, properties of 2-D shapes, area, base, height, congruence
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Part 1: Purpose: Students develop and practice geometry vocabulary, and form a concept of base and height of a
parallelogram.
Distribute a parallelogram diagram to each group of 2 students. Without showing the diagram to student B, student A will give instructions to student B that will enable student B to draw a parallelogram congruent to student A’s diagram. Using a different diagram, the students switch roles. After the activity, students will share their drawing instructions in the large group. Vocabulary can be recorded and clarified with diagrams. Teacher should make sure that base and height are clearly established.
Facilitating questions: What are some other ways of saying that? What does that mean? Part 2: Purpose: Students calculate the areas of parallelograms by counting, refine their counting method using
calculations, and arrive as a class at the formula for area of a parallelogram.
Ask students to find the areas of several different parallelograms by measuring them with a grid and counting the squares. Ask them if there is a way to make the counting faster. Some groups will discover the congruent triangles on either end of the parallelogram that, put together, make up full squares. Other groups will notice that the base and height are related to the area of the parallelogram. Students must be able to defend their calculation method by explaining how it works. In the large group, students will share their observations from the activity, build on each other’s answers, and establish the formula for the area of a parallelogram. Students can test the formula on a few parallelograms by calculating the area by multiplying base and height, by calculating the area by counting squares, and then comparing. Part 3: Purpose: Consolidating understanding. Ask students to draw on grid paper a parallelogram that has an area of 15 square units. Compare the diagrams. There are an infinite number of these; students will probably produce parallelograms with a base and a height using 3 and 5 units, but notice that the sides have different slants.
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Ask students to draw on grid paper a parallelogram that has an area of 24 square units. Compare the diagrams. Extension: Ask students to draw a triangle that has an area of 12 square units. Have them explain how they know their answer is correct.
Some things to note about scenario three are these: All students can participate in this activity. The tasks asked of students can be handled in more than one way. The tasks allow students to practice using language. The tasks help students to develop concepts for themselves, and relate procedures to concepts. The extension question sets the stage for developing the formula for the area of a triangle, which is also part of the grade 7 curriculum. The lesson takes longer than one day. The lesson allows students to meet or revisit several expectations. How do we instruct for the development of conceptual knowledge? We need to trust that carefully structured exploration activities and problems assist students to acquire the conceptual knowledge that they need to make sense of mathematics and mathematical procedures. Teaching through rich problem-solving tasks that incorporate student interaction and communication helps students to understand mathematics. What is rich talk? Rich talk is communication or dialogue that helps students to develop or demonstrate mathematical understanding. When built into a rich task, it is the kind of discussion that clarifies understanding, connects and applies knowledge, generates ideas, and evaluates conclusions. Teachers can facilitate rich talk by asking questions and having students respond, however the nature of the questions will determine the ‘richness’ of the talk that will result. For example, “What is the formula for the area of a rectangle?” or “What is the next step in solving this equation?” would prompt students to retrieve memorized information. Information retrieval questions are the most commonly asked questions, but they do not require students to engage in the types of higher order thinking that improve learning. Students do need to be able to recall information, but they also need to be able to respond to questions that call on more sophisticated processes than simple recall. Engaging students in rich talk activates their thinking. Rich talk results when students respond to questions that call on them to process information and then to apply or evaluate it. Processing new information helps students to make sense of it, while application allows them to connect the new information to their own prior knowledge. The need for rich talk does not imply that all learning is done in a group. When teachers use “rich talk questions” in a whole class lesson, teachers can model for students the type of response that is sought.
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As well, students learn to listen to each other, ask for justification and add their own ideas. Facilitating rich talk in a whole class lesson requires that the teacher allow more wait time for students to formulate an appropriate, well-considered response. When students interact with each other through rich talk in pairs or small groups, more students have the opportunity to form the responses that help them to learn. As well, students learn to justify their own arguments, and consider, evaluate and respectfully challenge the arguments of others. The following chart helps to identify the types of questions that facilitate different levels of thinking. Purpose To facilitate recall of information, ask students to …
Verb Complete, calculate, count, define, list, identify, select, recall, describe, match
Example of a question in mathematics What are the characteristics of an isosceles triangle? Solve this equation. What is the definition of a polynomial?
To facilitate processing or making sense of information, ask students to …
Infer, organize, group, explain, categorize, distinguish, experiment or test
Explain why your counting method works. If we change the shape of the rectangle, what happens to the area? Find more shapes that are similar to this one in 2 different ways.
To facilitate application or evaluation of information, ask students to …
Apply, predict, hypothesize, evaluate, extrapolate, generalize, imagine, speculate
Estimate how many marbles can fit into this box. Do you agree with that statement? What does this investigation tell you about how to find the area of any triangle?
Adapted from Teaching Tomorrow’s Thinkers: A Curriculum Framework for Teaching Thinking JK-9, WCDSB, 1992.
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Resources Resources are tools to help teachers and students develop better mathematical understanding. Textbooks, teacher guides, supplementary print materials, technology, people and manipulatives are all examples of resources. A balanced mathematics program makes use of a variety of resources. Type of resource
Examples
Manipulatives
Pattern blocks, geometric solids, clocks
Print materials
Texts, teacher guides, assessment materials (exemplars), literature, resource books
Technology
Computer software (Math Trek), board website, internet, calculators, A/V
People
Teaching partners, consultants, lead students
Student learning is enhanced when students and teachers are trained how to use each type of resource appropriately. Students need exposure to a variety of resources so that they can independently select an appropriate resource to support their learning. Teachers need professional development that introduces them to a variety of resources and demonstrates how the resources can be used in the classroom. Effective teacher instruction is still the best way to improve student learning, and effectively used resources can support that instruction. Resource
Example of Effective Use
Example of Ineffective Use
Manipulatives
Students use manipulatives to test, think, discover
Teacher uses manipulatives, students copy teacher
Calculators
Students use calculators when mechanical operations are not the focus of the task.
Students use calculators when estimation is more appropriate
Textbook
Students use selected text exercises and examples to clarify, consolidate or extend learning
Teacher uses the text as the math program, working from beginning to end
Use of Manipulatives in the Classroom Manipulative use is prevalent in primary grades, but tends to decline through junior and intermediate grades. Manipulatives should be used to support mathematics learning in all grades. When students are explicitly instructed in the appropriate use of manipulatives, and when they are used routinely during learning, classroom management issues are minimized. Manipulatives appeal to a variety of learning styles and provide students with a concrete means of modeling mathematics to develop understanding. They are motivating and engaging for students, and
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when used correctly they enhance student learning at all levels. When students work with manipulatives, they discuss their thinking, facilitating the “rich talk” that is necessary for learning. By observing students and listening to their discussion, teachers can assess student thinking and understanding. Resources to support the mathematics program will be reviewed in the upcoming years. At the present time, the following resources already in schools will be used to support instruction. Current Core Resources (texts) for Mathematics: Grades Kindergarten to Grade 6
Quest 2000 (Addison Wesley)
Grades 7 and 8
Mathpower (McGraw Hill Ryerson)
Grade 9
Mathematics 9 (Nelson)
Additional print resources in schools: The following resources have been purchased for elementary schools and should be available in school libraries. Primary Math Kit
Primary Junior Math Kit
Junior Math Kit
Intermediate Math Kit
NCTM Publications
NCTM Publications (9 issues of each) Elementary and Middle School Mathematics
Daily Problems and Weekly Puzzlers – Grade 1 Daily Problems and Weekly Puzzlers – Grade 2 Daily Problems and Weekly Puzzlers – Grade 3 How to Assess Problem Solving Skills in Math – Grades K – 2 Active Math – Grades JK – 2 Puddle Questions – Grade 3 Primary Problem Solving – Grades K – 3 50 Problem-Solving Lessons – Grades 1 – 6 Math Mysteries – Grades 2 – 5 Real World Math – Grades 1 - 3 Real World Math – Grades 4 – 6 Daily Problems and Weekly Puzzlers – Grade 4 Daily Problems and Weekly Puzzlers – Grade 5 Daily Problems and Weekly Puzzlers – Grade 6 Puddle Questions – Grade 6 Write Starts – 101 Writing Prompts Linking Assessment and Instruction in Mathematics Take a Mathwalk – Grades 6 – 8 Mega Projects – Grade 7 Daily Problems and Weekly Puzzlers – Grade 7 Daily Problems and Weekly Puzzlers – Grade 8 Mathematics Assessment: A Practical Handbook – grades 3 – 5 Mathematics Assessment: A Practical Handbook – grades 6 – 8 Mathematics Assessment: Myths, Models, Good Questions and Practical Suggestions Teaching Children Mathematics - JK – 6 Mathematics Teaching in the Middle School Teaching Developmentally, 4th Edition (John A. Van De Walle)
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Special Education Needs for Standards-Based Mathematics Programs “All children arrive at school with personal differences in experiences and their own conceptual understandings. In order to treat all students equitably, teachers have an obligation to assist all students to succeed in understanding and applying mathematical skills and concepts. Children require early identification and intervention with remediation to ‘level the playing field’ and to nurture their success in mathematics. All children can learn but perhaps not in the same way or at the same rate. Standards-based mathematics programs address these needs by providing rich talk and rich tasks that can immerse students in visual, auditory and kinesthetic learning. It requires a hands-on approach that allows children the opportunity to explore and experience the physical world around them.” (Elementary Mathematics in Canada, Colgan & Pegis, 2003) “Classroom teachers play an extremely important role in the success of exceptional pupils. Teachers everywhere have successfully integrated students who experience varying degrees of challenge into the classroom. By being knowledgeable about a student’s background, abilities, strengths and educational and social needs, a teacher can set the stage for the student’s successes. In many cases, a teacher will be able to draw on the expertise and assistance of school and board staff who can provide support related to special education issues.”
(p. 12 Special Education Companion, 2002, Ministry of Education) Some students need to be recognized as having special needs that are outlined in an Individual Education Plan. The IEP will outline specific teaching strategies for effectively integrating student’s special needs in mathematics learning tasks and assessments. The classroom teacher has an essential role to play in developing classroom lessons that are linked to strategies and learning expectations from the student’s Individual Education Plan. A standards-based mathematics approach will assist teachers in meeting these students’ needs through the use of best practices.
Important Considerations for Planning and Assessment Accommodations: Students may need specific accommodations to assist them in accessing mathematical curriculum materials and concepts at their grade level. “Accommodations refer to the teaching strategies, supports, and or services that are required in order for a student receiving special education programs and services to access the curriculum and demonstrate learning. Accommodations do not alter the provincial learning expectations for the grade.” (p. 14 Special Education Companion 2002)
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Extra time for completion of classroom assignments
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Allowing the student to complete tasks or present information in alternative ways (e.g. through taped answers, demonstrations, dramatizations, role play)
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Providing the student with a variety of learning tools, such as adapted computers for completing writing tasks, and calculators for completing numeric tasks.
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Providing for the use of scribes
Modifications: Students may require modifications of expectations for mathematics at a different grade level based on their individual learning needs and abilities. “Modified expectations refer to the changes that are made to the grade level expectations for a subject or course in the Ontario Curriculum in order to meet the needs of the student. Modified expectations may be drawn from a different grade level above or below the student’s current grade placement. They may also include significant changes (increases or decreases) to the number and or complexity of the grade level learning expectations.” (p. 15 Special Education Companion 2002)
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Expectations from a different grade level e.g. child is in grade 6 but is able to comprehend and demonstrate learning at a grade 2 level in mathematics
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Significant changes to the grade level learning expectations e.g. student is in grade 7 and comprehends only the units of measure specific expectations for the Measurement strand and is not able to complete any of the other two areas in the strand at the grade 7 level.
Alternative Expectations: A small number of students will achieve expectations that are not based on the Ontario Curriculum. They need to have their individual learning expectations considered when planning for their participation in mathematics activities. “Alternative expectations refer to expectations that are related to skill development in areas not represented in Ontario Curriculum policy documents. Examples of such areas include orientation and mobility training, life skills, and anger management. Alternative expectations should represent a specific program that has been designed for delivery to the student.”
(p. 15 Special Education Companion 2002) •
Includes areas such as Life Skills, Social Interaction Skills, Physical Mobility
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Strategies and Accommodations for Mathematics: Conceptual mathematics requires the use of strategies and accommodations for all students to ensure that each individual in a class can actively construct the meaning behind mathematical processes. As a natural result, all students will have an opportunity to jump into a learning activity, regardless of where they perform on the mathematics continuum and thus construct some new understanding. The following list can be defined as “best practices”. There are many more detailed accommodations, listed by exceptionality, in the Ministry of Education’s Special Education Companion. Examples of Best Practice Teaching Strategies for Supporting Students •
Provide pre-teaching of important concepts
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Make explicit links to student’s prior knowledge and experiences
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Use a variety of graphic organizers to provide students with a framework for their thinking
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Provide opportunities for students to display strengths within their individual learning styles e.g. projects and models
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Use teaching strategies and learning accommodations that will meet each student’s individual learning needs. See Special Education Companion for additional examples of accommodations
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Utilize teacher led instruction that involves thoughtful questions and encourage students to participate in meaningful large group discussions
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Provide a range of assessment formats including alternative forms of assessment, depending on the needs of the child.
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Provide opportunities to revisit, and review concepts and skills.
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Provide explicit and direct instruction in how to read mathematics texts.
Some types of accommodations include: •
Graphic organizers
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Concrete manipulative materials to teach math concepts
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Providing definitions of mathematics vocabulary in pre-teaching
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Breaking concepts into sequential steps
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Providing daily allowances for use of mathematical tools such as calculator or math grids, formula sheets
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Promotion of talk about mathematics
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Encouraging students through verbal and visual cues to notice each other’s accomplishments/ideas, strategies
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Encourage students to view mathematics as an integrated subject with connections among its strands.
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In Conclusion: This revised method of teaching mathematics becomes even more important when considering the requirements of students with special needs. “…It is the teacher’s role to establish a mathematical environment to enable students to construct this mathematical knowledge. This environment would provide students with opportunities to hypothesize, test out their thinking, manipulate materials, and communicate their understanding in order to build mathematical knowledge. It is the teacher’s role to facilitate student learning, through setting up problems, monitoring student exploration, and negotiating meaning and understanding with the student. The teacher guides the direction of student inquiry and encourages new patterns in thinking.”
(Mathematics Education: A Summary of Research, Theories and Practice, Homson/Nelson, 2002)
Resources: Special Education Companion 2002 Individual Education Plans Standards for Development, Program Planning and Implementation, Ministry of Education 2000 Special Education Teachers and Special Education Resource Teachers Regular Mathematics Resources within the Mathematics framework
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Professional Development Context and Research Professional development must respond to teacher learning needs. Student learning in mathematics is enhanced when teachers feel comfortable and confident with the curriculum content and delivery approach. The elementary mathematics curriculum is challenging, and the expected delivery approach asks teachers to connect concepts to procedures when they may not have been shown those connections in their own educational experience. Materials and training should support teachers to see necessary mathematical connections, to try alternative teaching strategies that help students to see the connections, and to develop confidence in themselves as mathematics teachers. The research literature about professional development acknowledges what many teachers have already said, that professional development is more effective when groups of teachers are working toward the same goal and can support each other it happens in a classroom system support and training is ongoing allowing teachers time to reflect on and share their experiences.
PD Plan It takes time for alternative teaching strategies to become comfortable and routine, particularly in mathematics that has long been cast as the ‘discipline of right and wrong answers’. If anyone you know confesses that they “can’t do math”, remember that they probably came through a traditional program based on getting right answers by learning procedures by example. Though that program might have served those at the top of the class, it left many students lacking mathematical understanding and lacking confidence in their ability to do math.
In response to the research, the professional development plan that accompanies the implementation of this initiative assumes that in each school, teachers within divisions will work together toward the implementation of the initiative within their school a representative from the division will attend each PD session to receive the PD materials and suggestions for divisional work, and to return with teacher feedback PD to allow for planning and questions will occur prior to teachers using the PD materials in the classroom PD to allow for reflection and feedback will occur after teachers use the PD materials in the classroom PD will continue into subsequent years to allow teachers time to integrate new instructional knowledge
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Principals and Vice-Principals will understand and support the professional development process alternative PD strategies will be used to accommodate small schools with single teacher divisions
PD Materials The more opportunities students get to explore mathematics through rich tasks, the more they practice their own thinking. The more times that the teacher, with the best intentions, prepares the perfect explanation and gives it to students, the more the students practice not thinking. Students own their mathematical knowledge when they can explore it and explain it. All of the sample lessons included here contain the framework components: •
They use open-ended problems (rich tasks) that allow all students an entry point into the lesson
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They provide prompt questions that encourage the desired student dialogue (rich talk)
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They focus on teaching both concepts and procedures.
•
The lessons are intended to be used in two ways:
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In classrooms, as professional development activities
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As models, so that teachers can adapt some of their own lessons to fit the framework
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Assessment, Evaluation, Grading and Reporting The organization of the Ontario Curriculum by strand, accompanied by the Mathematics Achievement Chart that lists four categories of knowledge and skills, has changed the way that teachers must track and report student achievement. The WCDSB Guidelines for Assessment, Evaluation, Grading and Reporting outlines those details for teachers. The mathematics framework acknowledges these reporting requirements and seeks to align instruction with assessment to assist teachers with the task of evaluating students, first by answering some of the common questions surrounding assessment, evaluation, grading and reporting. Why is the mathematics curriculum divided into strands? Strands bring attention to the fact that mathematics is more than number facts, formulas and calculations. Organizing expectations according to strands was intended to balance the content of the mathematics program. Traditional mathematics curricula focused mainly on number sense. Though most of us remember also doing some work with perimeter, area and volume, we were likely tested on our ability to recall the appropriate formula, substitute the numbers in correctly, and calculate the correct answer – in essence, number sense. Many teachers would leave geometry and data management topics until the end of the year to teach only if they had time left over. Why do we have reporting rules that stipulate reporting on 3 strands per term, and each strand twice over the year? Since connections are so important to mathematics learning, strands were never intended to be taught discretely even though the curriculum is organized that way. The curriculum is separated into strands in the curriculum documents only to make it easier to read and manage. The rule requiring teachers to report on 3 strands per term, and each strand twice, was intended to make it necessary for teachers to integrate the strands throughout the year in order to address all of the expectations. For example, patterning can be used to teach numeracy, or measurement can be combined with a data management task. Why are there 4 categories of knowledge and skills for mathematics? We have come to recognize that memorizing isolated facts and procedures does not constitute mathematical learning. Now more than ever, for our students to be successful upon leaving school they require not only knowledge, but also the ability to problem solve, apply knowledge to the solving of those problems, and communicate and justify their conclusions. The ability to problem solve, apply knowledge, and communicate and reason only develops when we ask students to problem solve, apply knowledge, and communicate and reason. All students, regardless of their procedural proficiency, can develop skills in the categories of problem solving, application and communication. Why do we have to assess across the 4 categories of knowledge and skills? The Harvard Balanced Assessment in Mathematics Project in the 1980s identified seven actions of a balanced mathematics program. Our curriculum expectations are written to reflect those actions, using
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verbs like explain, explore, construct, analyze, predict, apply to indicate what students are expected to do when they are learning math. Each of the verbs from the expectations can be sorted into at least one of the four categories in our curriculum achievement chart: (i) (ii) (iii) (iv)
Problem Solving, Understanding of Concepts, Application of Mathematical Procedures, and Communication of Required Knowledge.
In planning instruction, we are required to identify the expectations that will be addressed, teach to the expectations, and then assess what was taught. If we are true to the expectations, including the process verbs, then we are already instructing and assessing across the 4 categories. How does the framework support assessment, evaluation, grading and reporting? Good quality rich tasks often incorporate expectations from more than one strand. The assessment and evaluation tools that will accompany the professional development materials will assist teachers to track student achievement by strand. As well, each classroom task included in the professional development support will be built on the four framework components: 1. Conceptual knowledge 2. Procedural knowledge 3. Rich tasks 4. Rich talk The Achievement Chart from the Ontario Curriculum lists four categories of knowledge and skills: 1. Problem Solving 2. Understanding of Concepts 3. Application of Mathematical Procedures 4. Communication of Required Knowledge We can connect the framework components to the categories of knowledge and skills using the following two analogies: Conceptual Knowledge Procedural Knowledge +
Rich Talk
Understanding and Communication
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Rich Tasks +
Rich Talk
Problem Solving and Application By using the support materials provided to support the implementation of the mathematics framework, teachers will be aligning their assessment activities to the WCDSB Guidelines for Assessment, Evaluation, Grading and Reporting, which are in turn connected to the assessment and evaluation requirements of the Ontario Curriculum.
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