Junior Mathematical Processes, Problem Solving & Big Ideas by Strand

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1 Junior Mathematical Processes, Problem Solving & Big Ideas by Strand Adapted from: The Ontario Curriculum Grades 1 8 Revised Mathematics 2005, Number Sense and Numeration Grades 4-6, Measurement Grades 4-6, Geometry and Spatial Sense Grades 4 6, Patterning and Algebra Grades 4-6, Data Management and Probability Grades 4-6 Niagara Catholic District School Board Program Department June 2009

2 The Mathematical Processes The mathematical processes are the processes through which students acquire and apply mathematical knowledge and skills. These processes are interconnected. Students should be actively engaged in applying the below seven processes throughout their mathematical program in all five strands, rather than in connection with particular strands. The mathematical processes that support effective learning in mathematics are as follows: problem solving reasoning and proving reflecting selecting tools and computational strategies representing communicating Teaching through Problem Solving Problem solving is central to learning mathematics. By learning to solve problems and by learning through problem solving, students are given numerous opportunities to connect mathematical ideas and to develop conceptual understanding. Problem solving forms the basis of effective mathematics programs and should be the mainstay of mathematical instruction. Problem solving and communicating have strong links to all the other processes. A problem solving approach encourages students to reason their way to a solution or a new understanding. As students engage in reasoning, teachers further encourage them to make conjectures and justify solutions, orally and in writing. The communication and reflection that occur during and after the process of problem solving help students not only to articulate and refine their thinking but also to see the range of strategies that can be used to arrive at a solution. By seeing how others solve a problem, students can begin to reflect on their own thinking, and the thinking of others, and to consciously adjust their own strategies in order to make their solutions as efficient and accurate as possible. Teaching through problem solving as described in the Ontario Curriculum and the Guide to Effective Instruction is not a weekly event (e.g., problem of the week or problem solving lesson of the week), it is a mode of effective instruction to develop conceptual understanding, reasoning and proving, in fact all of the process expectations. The 3 part lesson is a structure for daily implementation with a 50 min to 60 min numeracy block, as all 3 expert panel reports suggest that mathematics be taught in 60 minute blocks. (Kathy Kubota-Zarivnij LNS 2009) The Ontario Curriculum Grades 1 8 Revised Mathematics 2005

3 Suggested Problem Solving Delivery Models Math Congress The congress allows the sharing of selected student responses, analysis of the mathematics used in the solution, and prompts all students to learn from one another. The purpose is to debrief the strategies used by students, unearth multiple representations of mathematical thinking, and assist in the development of deep understanding of concepts. By having students defend and explain their thinking, teachers give their students an opportunity to see and hear different perspectives. This gives students a chance to examine the math very closely it is like laying the concepts and the misconceptions out on the table. Students have a chance to learn from the discourse and clarify their thinking about the math idea. The connection to prior math knowledge and the development of new knowledge is made explicit. Enduring learning is the result. Gallery Walk Ask all participants to post their solutions on the wall. There should be at least two from each table. Conduct a gallery walk by asking all learners to travel around the room and read the solutions from other tables. Learners should each carry several sticky notes and a pen or pencil. During the gallery walk, encourage participants to write any questions for clarification they want about a solution and then stick it on the solution on the chart paper. They can write, I do not understand. or Please explain this step. and so on. Following the gallery walk, ask a group to stand beside their solution. Ask them to explain their thinking and to address and respond to the sticky notes posted on their solution. If time allows, repeat by having other groups share. All this interaction between a person s solution and those of others motivates participants to think more deeply about the mathematics in their solutions, adapt their thoughts, and learn more about the math related to the problem. This occurs no matter what understanding they had when they first encountered the problem. Deep understanding of mathematical ideas, language, and conventions is developed in an iterative way a little today and more tomorrow and more next week. The learning continues to build on what they already know. A typical congress involves a small number of student samples chosen by the teacher and explained by the student. Post the following questions to participants: Which four samples would you choose to use for a math congress? What concept or property would the samples represent? How would you facilitate the discussion in your classroom? During consolidation, lead a discussion about how they could use a gallery walk to identify the range of student understandings of decimals. Some of the solutions could, no doubt, be used to activate analysis of the multiplication of decimals algorithm. This process is also effective when used for the purpose of improving communication. The sticky notes work to prompt editing that improves the communication for other audiences. If you are using this module in a job-embedded format, encourage participants to try the gallery walk strategy in their own classrooms before the next study session. Ask participants

4 to bring along samples of student work from congresses they have run. Be sure to start the next session by revisiting this commitment and making time for debriefing of the process. Ontario Bansho In order to make public the mathematical thinking students use to solve a problem, we need a way of organizing the work so everybody can see the range of student thinking. This allows students to see their own thinking in the context of the similar thinking of others in the class. The matching and comparing process promotes learning as students try to understand other solutions and learn from one another. Japanese educators use a process they call bansho to organize student work and to lead a conversation that offers everybody a chance to learn more about the math used in developing solutions to a problem. A bansho process will be employed to sort and classify participants solutions. The bansho process uses a visual display of all students solutions, organized from least to most mathematically rich. This is a process of assessment for learning and allows students and teachers to view the full range of mathematical thinking their classmates and students used to solve the problem. Students have the opportunity to see and to hear many approaches to solving the problem and they are able to consider strategies that connect the next step in their conceptual understanding of the mathematics. Bansho is NOT about assessment of learning, so there should be no attempt to classify solutions as level 1, level 2, level 3, or level 4. Begin by having representatives from each table post their chart paper solutions on the wall. Facilitate the group in organizing the bansho by using the following prompts: Which solutions show students representing the mathematics using concrete materials or pictures? Which solutions show students working with operations other than division? Which solutions show students using an algorithm (effectively or otherwise) that would have you think they have a deep understanding of the solution? Can they say how and why the algorithm works? How could we organize the solutions to represent a continuum of growth of mathematical ideas? Use the group s thinking and responses to organize the bansho by posting the charts from left to right, grouping solutions that show the same mathematics. Later, during discussion, the mathematics will be named and subsets labelled to describe specific procedures (e.g., counting, array model, repeated subtraction). Ask learners to share how the representations would benefit different students in a junior classroom.

5 Problem Solving Model Understand the Problem (the exploratory stage) reread and restate the problem identify the information given and the information that needs to be determined Communication: talk about the problem to understand it better Make a Plan relate the problem to similar problems solved in the past consider possible strategies select a strategy or a combination of strategies Communication: discuss ideas with others to clarify which strategy or strategies would work best Carry Out the Plan execute the chosen strategy do the necessary calculations monitor success revise or apply different strategies as necessary Communication: draw pictures; use manipulatives to represent interim results used words and symbols to represent the steps in carrying out the plan or doing the calculations share results of computer or calculator operations Look Back at the Solution check the reasonableness of the answer review the method used: Did it make sense? Is there a better way to approach the problem? consider extensions or variations Communication: describe how the solution was reached, using the most suitable format, and explain the solution The Ontario Curriculum Grades 1 8 Revised Mathematics 2005

6 Sample Template Three Part Lesson Plan for Problem Solving (Bansho, Congress) Teacher: School Grade(s) Lesson Title - Learning Goals (Curriculum Expectations) Materials Lesson Components Before (5-10 min) <Activating mathematical knowledge and experience> Task/Problem: Anticipated Responses During (15-20 min) <Understand the problem what information will WE use to solve the problem; record in a list> Problem: Anticipated Responses After Consolidation (20 min)< Choose solutions according to criteria related to learning goal> Mathematical Sorting Criteria Mathematical Annotations Questions to Pose After (Highlights and Summary) (5 min) <Key Mathematical Ideas and Strategies; Mathematical Relationship between the solutions; Generalization Key Ideas Strategy Generalization After (Practice) (5-10min) <students practising analyzing and describing the mathematical thinking of others> Independent Anticipated Responses The Literacy Numeracy Secretariat June 2009

7 Sample Template Three Part Lesson Plan for Problem Solving (Bansho, Congress) Teacher: School Grade(s) Lesson Title - Learning Goals (Curriculum Expectations) Materials Lesson Components Before (5-10 min) <Activating mathematical knowledge and experience> Task/Problem: Anticipated Responses Activating students mathematical knowledge and experience that is directly related to the mathematics in the lesson problem Use a smaller problem similar to previous lesson problem Use student work responses for class analysis and discussion to highlight key ideas and/or strategies During (15-20 min) <Understand the problem what information will WE use to solve the problem; record in a list> Problem: Anticipated Responses Understand the problem. Make a plan. Carry out the plan Understand the problem Ask, What information from the problem will we use to make a plan to solve it? Explain. Teacher records below the problem the information the students identify in a list Students solve the problem in pairs or in small groups After Consolidation (20 min)< Choose solutions according to criteria related to learning goal> Mathematical Sorting Criteria Teacher selects 2 or more solutions for class discussion and decides which solution to share first, second, third Teacher organizes solutions to show math elaboration from one solution to the next, towards the lesson learning goal Students (authors) explain and discuss their solutions with their friends Teacher mathematically annotates solutions to make mathematical ideas, strategies, and tools explicit to students for learning Mathematical Annotations Questions to Pose After (Highlights and Summary) (5 min) <Key Mathematical Ideas and Strategies; Mathematical Relationship between the solutions; Generalization Key Ideas Teacher revisits in the different solutions the key ideas, strategies, and models of representation that are related to the lesson learning goal Teacher records key ideas, strategies, and models of representation separately, so the students see the explicit focus of learning from the lesson Strategy Generalization After (Practice) (5-10min) <students practising analyzing and describing the mathematical thinking of others.> Independent Teacher chooses 2 or 3 problems, similar to the lesson problem for Anticipated Responses students to solve individually (or in pairs as a scaffold) Problems are different by number (choice, size), problem contexts, or variation of problem needing to be solved Students are asked to use a strategy different from the one they used in the lesson to solve the practice problems The Literacy Numeracy Secretariat June 2009

8 Teacher Support during Problem Solving Before (Getting Started) During this phase the teacher may: Engage students in problem solving situation (personalize the problem by using student s names, or by putting oneself into the situation); Discuss the situation; Ensure that students understand the problem; Ask students to restate the problem in their own words; Ask students what it is they need to find out; Allow students to ask questions (e.g., Do you need any other information? ); Encourage students to make connections with their prior knowledge (e.g., with similar problems or a connected concept); Model think-alouds (this can also be done during and after the problem-solving activity); Provide any requested materials and have manipulatives available. During (Working on It) During this phase the teacher may: encourage brainstorming; use probing questions; guide the experience (give hints, not solutions); clarify mathematical misconceptions; redirect the group through questioning, when necessary (students can usually recognize their own error if the teacher uses strategic questioning); answer student questions but avoid providing a solution to the problem; observe and assess (to determine next instructional steps); reconvene the whole group if significant questions arise; if a collaborating group experiences difficulty, join the group as a participating member; encourage students to clarify ideas and to pose questions to other students; give groups a five- or ten-minute warning before bringing them back to the wholegroup discussion that takes place in the Reflecting and Connecting phase. After (Reflecting and Connecting) During this phase the teacher may: bring students back together to share and analyse solutions; be open to a variety of solutions strategies; ensure that the actual mathematical concepts are drawn out of the problem; highlight the big ideas and key concepts; expect students to defend their procedures and justify their answers; share only strategies that students can explain; foster autonomy by allowing students to evaluate the solutions and strategies; use a variety of concrete, pictorial, and numerical representations to demonstrate a problem solution; clarify misunderstandings; relate the strategies and solutions to similar types of problems to help students generalize the concepts; encourage students to consider what made a problem hard or easy (e.g., too many details, math vocabulary), and think about ways to achieve the appropriate degree of difficulty the next time; always summarize the discussion for everyone, and emphasize the key points or concepts. A Guide to Effective Instruction in Mathematics, K-6 Volume Two - Problem Solving & Communication

9 Lesson/Unit of Study Title Grade ASSESSMENT FOR LEARNING OBSERVATION & INTERVIEW Mathematics Lesson Task/Problem Learning Goal/Curriculum Expectations Date Students Math Thinking Mathematical Sorting Criteria Math Annotations (concepts, strategies, actions, models) Highlights/Summary Focus:

10 Sample Daily Lesson Planning for Mathematics Strand: Number Sense and Numeration Grade: 5 Key Concept(s)/Big Ideas: Quantity, Operational Sense Curriculum Expectations: Students will: represent, compare, and order fractional amounts with like denominators, including proper and improper fractions and mixed numbers, using a variety of tools and suing standard fractional notation; read and write money amounts to $1000; solve problems that arise from real-life situations and that relate to the magnitude of whole numbers up to ; add and subtract decimal numbers to hundredths, including money amounts, using concrete materials, estimation, and algorithms. Materials: fraction circles play money chart paper Getting Started Instructional Grouping: Whole class Pose a problem: Steven s class is raising money for a charity organization by selling small pizzas at lunch hour. Each pizza is cut into fourths, and each slice sells for $1.25. The class raised $32.50 on the first day of sales. How many pizzas did they sell? Explain that students will work in pairs. Encourage them to use manipulatives and/or diagrams to help them solve the problem. Ask students to record their strategies and solutions on chart paper and to demonstrate clearly how they solved the problem. Working On It Instructional Grouping: Pairs As students work on problem, pose questions that encourage reflection and explanation of their strategies and solutions: What strategy are you using to solve this problem? How are you using manipulatives and/or diagrams to help you find a solution? Is your answer reasonable? How do you know? How can you show your work so that others will understand what you are thinking? Make notes of strategies students are using (i.e., Students can solve the problem by: counting the number of times they added $1.25 until they get to $32.50, then counting the number of times they added $1.25. Knowing the number of slices, students can then determine the number of pizzas). Reflecting and Connecting Instructional Grouping: Whole Class Reconvene the class. Ask a few pairs to represent their problem-solving strategies and solutions to the class. Ask the students to explain their work, ask questions that encourage them to explain the reasoning behind their strategies. I.e. How did you find the number of pizzas that the class sold? Why did you use this strategy? How do you know that your solution makes sense? Following the presentations, encourage students to consider the effectiveness and efficiency of the various strategies that have been presented i.e. In your opinion, which strategy worked well? How would you explain the strategy to someone who had never used it? Assessment Throughout the lesson, observe students to assess how well they: represent fractional amounts, using concrete materials, diagrams, and symbolic notation; read and write money amounts select and apply appropriate problem-solving strategies; add and subtract money amounts. Use assessment information gathered in this lesson to determine subsequent learning activities. Home Connection Ask students to solve this problem at home and share with a family member how the problem was solved. Two pizzas are the same size and have the same toppings. The first pizza is cut into fourths, and each slice costs $1.95. The other pizza is divided into sixths, and each slice costs $1.25. Which pizza costs more? A Guide to Effective Instruction in Mathematics, K-6 Volume One Foundations of Mathematical Instruction

11 Sample Daily Lesson Planning Template for Mathematics Strand: Grade: Key Concept(s)/Big Ideas: Curriculum Expectations: Materials: Getting Started Instructional Grouping: Working On It Instructional Grouping: Reflecting and Connecting Instructional Grouping: Assessment Home Connection A Guide to Effective Instruction in Mathematics, K-6 Volume One Foundations of Mathematical Instruction

12 Sample Three-Part Math Lesson Template Date: Big Idea/Key Concept(s): Specific Expectation(s): Assessment (What are you looking for? How will you assess it?) As Learning/For Learning/Of Learning: Getting Started: Working On It: Reflect and Connect:

13 Big Ideas in Number Sense and Numeration Grades 4 6 Big Idea Key Points Quantity Having a sense of quantity involves understanding the howmuchness of whole numbers, decimal numbers, fractions, and percents. Experiences with numbers in meaningful contexts help to develop a sense of quantity. An understanding of quantity helps students estimate and reason with numbers. Quantity is important in understanding the effects of operations on numbers. Operational Sense Operational sense depends on an understanding of addition, subtraction, multiplication, and division, the properties of these operations, and the relationships among them. Efficiency in using the operations and in performing computations depends on an understanding of partwhole relationships. Students demonstrated operational sense when they can work flexibly with a variety of computational strategies, including those of their own devising. Solving problems and using models are key instructional components that allow students to develop conceptual and procedural understanding of the operations. Relationships An understanding of whole numbers and decimal numbers depends on recognition of relationships in our base ten number system. Numbers can be compared and ordered by relating them to one another and to benchmark numbers. An understanding of the relationships among the operation of addition, subtraction, multiplication, and division helps students to develop flexibly computational strategies. Fractions, decimal numbers, and percents are all representations of fractional relationships. Representation Symbols and placement are used to indicate quantity and relationships. Mathematical symbols and language, used in different ways, communicate mathematical ideas in various contexts and for various purposes. Proportional Reasoning Proportional reasoning involves recognizing multiplicative comparisons between ratios. Proportional relationships can be expressed using fractions, ratios, and percents. Students begin to develop the ability to reason proportionally through informal activities.

14 Big Ideas in Measurement Grades 4 6 Big Idea Attributes, units, and measurement sense Measurement relationships Student Consolidation of Skills Throughout the Junior Grades Choose units appropriately (by type and magnitude) to measure attributes of objects; Use measurement instruments effectively; Use meaningful measurement benchmarks to make sense of and visualize the magnitude of measurement units; and Make reasonable measurement estimates and justify their reasoning Know and apply measurement formulas; Generalize from investigations in order to develop measurement formulas; Demonstrate relationships among measurement formulas (e.g., the area formulas for squares, rectangles, parallelograms, and triangles); Recognize the role of variables in measurement formulas (as in A = l x w and P = 2 x l + 2 x w) Recognise that formulas can be expressed in more than one way (e.g., P = 2 x l + 2 x w or P = l +l + w + w or P = 2 x (l + w).

15 Big Ideas in Geometry and Spatial Sense Grades 4 6 Big Idea Properties of twodimensional shapes and threedimensional figures Geometric relationships Location and Movement Key Points Two dimensional shapes and three-dimensional figures have properties that allow them to be identified, sorted, and classified. Angles are measure of turn, and can be classified by degree of rotation. An understanding of polygons and their properties allows students to explore and investigate concepts in geometry and measurement. An understanding of polyhedra and their properties helps develop an understanding of the solid world we live in, and helps make connections between two- and threedimensional geometry. Plane shapes and solid figures can be composed from or decomposed into other twodimensional shapes and three-dimensional figures. Relationships exist between plane and solid geometry (e.g., the faces of a polyhedron are polygons; views of a solid figure can be represented in a two-dimensional drawing). Congruence is a special geometric relationship between two shapes or figures that have exactly the same size and shape. A coordinate grid system can be used to describe the position of a plane shape or solid object. Different transformations can be used to describe the movement of a shape.

16 Big Ideas in Patterning and Algebra Grades 4 6 Big Idea Patterns and Relationships Variables, Expressions, and Equations Student Consolidation of Skills Throughout the Junior Grades Generate patterns; Predict terms in a pattern; Determine any term given the term number; Describe a pattern rule for a growing or shrinking pattern; Describe pattern rules in words; Describe patterns using tables of values, ordered pairs, and graphs; Distinguish between a term in a growing pattern and its term number. Can see a variable as an unknown; Recognise and use variables in various forms; Recognise variables in formulas; Can work with missing number equations where two variables covary.

17 Big Ideas in Data Management and Probability Grades 4 6 Big Idea Collection and Organization of Data Student Consolidation of Skills Throughout the Junior Grades Dealing with discrete and continuous data; Use of a variety of data collection techniques (surveys, experiments, observations, measurements); Use of a variety of data representation tools (charts, tables, graphs, spreadsheets, statistical software); Description and explanation of their data collection and organization methods. Data Relationships Deal with primary and secondary data; Read and interpret data and draw conclusions; Use of descriptive statistics (range, mean, median, mode) to describe the shape of the data; Comparison of two related sets of data using a variety of strategies (tall, stem-and-leaf plot, double bar graph, broken-line graph, measures of central tendency); Make inferences and convincing arguments Probability Determine and representing all possible outcomes (i.e., the sample space ) of a probability experiment; Description of the probability of an event using fractions, area models, and decimals between 0 and 1; Posing and solving probability questions.