Problem Solving In Middle School Mathematics Using Intel Education Visual Ranking and Seeing Reason Online Tools (see www.intel.com/education)

Resources at end of this page!

Grades: 6-8 Problem Statement or Project Summary Problem-solving is an important skill for everyone to learn.There are many ways to approach problems, and multiple strategies for solving them.The key is to find which strategies are most successful for you, learn how to use them more effectively, and identify ways to improve your problem-solving skills. Final Project/Presentation of Findings The final goal of this project is a Development Plan in which students use what they’ve learned in their Visual Ranking (which problem-solving strategies do they use most and least effectively?) and Seeing Reasons (what other factors influence their ability to solve problems effectively?) activities.In their development plan, students create a 4-week plan, involving a family member and peers, to improve their problem solving skills in specific ways.Students will display their plan and results – and will receive recognition for accomplishing goals (by awards and at Open House).

CA Standards:Mathematical Reasoning (taken from California, Grade 7 but other grades are similar) 1.0Students make decisions about how to approach problems: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. 1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. 1.3 Determine when and how to break a problem into simpler parts. 2.0 Students use strategies, skills, and concepts in finding solutions: 2.1 Use estimation to verify the reasonableness of calculated results. 2.2 Apply strategies and results from simpler problems to more complex problems. 2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques. 2.4 Make and test conjectures by using both inductive and deductive reasoning. 2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.6 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.7 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. 2.8 Make precise calculations and check the validity of the results from the context of the problem. 3.0 Students determine a solution is complete and move beyond a particular problem by generalizing to other situations: 3.1 Evaluate the reasonableness of the solution in the context of the original situation. 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. 3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations.

Learning Objectives

Students will:

Make decisions about how to approach problems.

Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

Use strategies, skills, and concepts in finding solutions.

Articulate reasons for their choices for selecting strategies.

Identify ways to improve their problem-solving skill.

Visual Ranking

Project Prompt: Based on your previous problem-solving experiences, rank the following strategies in order of their value to you in solving mathematics problems.

Sorting List: Write an equation, Solve a simpler problem, Act it out, Simulate or experiment, Make a table, Look for a patter, Draw a diagram, Guess and test, Use logical reasoning, Work backward, Reduce and expand

Seeing Reason

Prompt: What are the causes and effects of being an effective problem solver in mathematics?Please be sure to only identify those causes that you have control over.And also, be sure to make your causes one color, your effects another color, and the factor "Effective Problem Solving" a third color.You can always change colors if you need to.

Procedures 1.Show Keys to Teaching Effective Problem-Solving Presentation (if not already reviewed in previous unit) and engage student teachers in a variety of problem-solving activities. 2.Complete Visual Ranking:Problem Solving Strategies

Visual Ranking Notes a.The items to be ranked are provided.There will NOT be a whole-class discussion of what strategies to rank because we’ve been working these strategies throughout the course. b.Students will be grouped in pairs, or triads, depending on computer access.They will be assigned a team name. c.Ranking will be subjective. d.Two math experts (faculty from the local university) will be asked to rank these strategies as well – so that students can compare to experts. e.Correlation will be explained – at the level provided on the Visual Ranking site. f.Students will be required to provide a justification and example for where they are placing an item. They will be reminded that their justifications should be based on the sorting criteria (what have they found valuable in past problem-solving activities). g.When comparing rankings between teams, students will complete the VR Problem Solving Strategies Worksheet: 1.Identify the team whose list is most different (that is, the lowest correlation) and most similar (the highest correlation) from your own. 2.For the most common correlation, review each others' comments to determine whether they are using the same criteria to make decisions about rankings as you did.Summarize your findings. 3.For the lowest common correlation, identify a strategy that is ranked high by the other team and low by your team. (The idea will be that this will be a goal for them.) 4.What did you learn from completing this activity? h.The teacher should take the time to comment on the work – and then have the students go in and revise.Otherwise, students won’t improve their reasoning skills.

2.Complete Seeing Reason:Effective Problem Solving

Seeing Reason Notes a.After posing the question, have students brainstorm a few factors in the cause/effect relationship to help them see what you are really after.Make sure you identify + and -.Have them identify causes and effects that they will put on their map on their Worksheet.Emphasize that all causes need to be those they have control over. Examples of Causes:skill in computation; ability to identify the pertinent information; understanding of mathematical terms and symbols; missing necessary tools (i.e., calculator, ruler, etc); skill in a variety of strategies; amount of practice; skill in algebra; skill in geometry; consistency in checking your answer, number of school absences Examples of Effects:grade in mathematics, performance on math tests, grade in science, ability to solve problems from real life, ability to make applications to other problems, success in school mathematics classes, performance on SAT and CHSEE, number of repeated courses, amount of summer school required d.After students have finished their first iteration of the map, they should complete the Checklist (self and peer review) and then modify their map again. e.When they are totally finished with the map, have them save to the portfolio and print a copy for submission.

3.When both Visual Ranking and Seeing Reason activities are completed, have students create their Problem-Solving Development Plan, attach all required worksheets, and submit.

Adaptations of Lesson Resource Student: Struggling readers and resource students will be paired carefully with students that will support their participation.Instructional Aides will have time before instruction to review activity so that they can support students. Resource Specialists will be informed of activities and provided all instructional materials and access to Interactive Tools. English Language Learner: Struggling readers and resource students will be paired carefully with students that will support their participation. Gifted Student: In addition to completing their own plan, gifted students will be asked to be involved in the development plans.They will identify 3 strategies in which they are effective and will identify themselves as tutors for those who need assistance in specific strategies.

Student Assessment Students will be assessed throughout the Visual Ranking and Seeing Reason projects. These assessments will be progress-monitoring in that they will inform revisions of the causal map.They will be summative in that they will be given a grade for successful completion of each tool.Finally, they will be assessed on whether they were able to accomplish their plans for improving their problem-solving ability.

Title

Purpose

Pts

Feedback Strategies

Informing Instruction

VR Problem Solving Strategies Worksheet

Identify what they have learned from Visual Ranking

25

Worksheets will be returned to students with teacher comments; students will use suggestions for their Development Plan.

If most (+50%) students have not successfully completed worksheet, instruction will be provided and revisions will be made in class.

Visual Ranking Activity

Assess student participation in activity

50

Teacher comments will be provided electronically during the Visual Ranking activity; teacher will progress-monitor throughout worksheet completion.

Teacher will progress monitor and provide individual, small group, and whole class reinstruction as needed.

Effective Problem Solving Worksheet

Preparation for Seeing Reason Activity

25

Worksheet will be approved prior to work on computers.

If most (+50%) students have not successfully completed worksheet, instruction will be provided and revisions will be made in class.

Checklist for Effective Problem Solving Causal Map

Provide peer and self assessment of causalmap to identify areas of needed improvement

10

Worksheets with peer feedback will be returned to students; students will then self-assess and use that to improve their causal map.

Teacher will progress monitor and provide individual, small group, and whole class reinstruction as needed.

Seeing Reason Activity

Assess student participation in activity

50

Teacher comments will be provided electronically during the Visual Ranking activity.

Teacher will progress monitor and provide individual, small group, and whole class reinstruction as needed.

Problem Solving Development Plan

Reflect on Project learning; identify plan for improving problem solving ability; provide for family involvement

50

Teacher will provide feedback and ask for revision is plan is not of sufficient quality. Students will revisit plan in one month to see if they have accomplished their plan.

Teacher will progress monitor and provide individual, small group, and whole class reinstruction as needed.

Evaluation of Plan (4 weeks later)

Reflect on plan for improving their skills

25

Students will write an evaluation of how successful they were at implementing their pan and improving their problem solving skills.

Plans and evaluations will be posted for open house.Awards will be given.

Resources at end of this page!Problem Solving In Middle School Mathematics Using Intel EducationVisual RankingandSeeing ReasonOnline Tools (see www.intel.com/education)Grades:6-8Problem Statement or Project Summary

Problem-solving is an important skill for everyone to learn. There are many ways to approach problems, and multiple strategies for solving them. The key is to find which strategies are most successful for you, learn how to use them more effectively, and identify ways to improve your problem-solving skills.

Final Project/Presentation of Findings

The final goal of this project is a Development Plan in which students use what they’ve learned in their Visual Ranking (which problem-solving strategies do they use most and least effectively?) and Seeing Reasons (what other factors influence their ability to solve problems effectively?) activities. In their development plan, students create a 4-week plan, involving a family member and peers, to improve their problem solving skills in specific ways. Students will display their plan and results – and will receive recognition for accomplishing goals (by awards and at Open House).

CA Standards:Mathematical Reasoning(taken from California, Grade 7 but other grades are similar)1.0 Students make decisions about how to approach problems:1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.

1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.

1.3 Determine when and how to break a problem into simpler parts.

2.0 Students use strategies, skills, and concepts in finding solutions:2.1 Use estimation to verify the reasonableness of calculated results.

2.2 Apply strategies and results from simpler problems to more complex problems.

2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.

2.4 Make and test conjectures by using both inductive and deductive reasoning.

2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

2.6 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.

2.7 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.

2.8 Make precise calculations and check the validity of the results from the context of the problem.

3.0 Students determine a solution is complete and move beyond a particular problem by generalizing to other situations:3.1 Evaluate the reasonableness of the solution in the context of the original situation.

3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.

3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations.

Learning ObjectivesStudents will:Visual RankingSeeing ReasonProcedures1. Show Keys to Teaching Effective Problem-Solving Presentation (if not already reviewed in previous unit) and engage student teachers in a variety of problem-solving activities.

2. Complete Visual Ranking: Problem Solving Strategies

a. The items to be ranked are provided. There will NOT be a whole-class discussion of what strategies to rank because we’ve been working these strategies throughout the course.

b. Students will be grouped in pairs, or triads, depending on computer access. They will be assigned a team name.

c. Ranking will be subjective.

d. Two math experts (faculty from the local university) will be asked to rank these strategies as well – so that students can compare to experts.

e. Correlation will be explained – at the level provided on the Visual Ranking site.

f. Students will be required to provide a justification and example for where they are placing an item. They will be reminded that their justifications should be based on the sorting criteria (what have they found valuable in past problem-solving activities).

g. When comparing rankings between teams, students will complete the VR Problem Solving Strategies Worksheet:

1. Identify the team whose list is most different (that is, the lowest correlation) and most similar (the highest correlation) from your own.

2. For the most common correlation, review each others' comments to determine whether they are using the same criteria to make decisions about rankings as you did. Summarize your findings.

3. For the lowest common correlation, identify a strategy that is ranked high by the other team and low by your team. (The idea will be that this will be a goal for them.)

4. What did you learn from completing this activity?

h. The teacher should take the time to comment on the work – and then have the students go in and revise. Otherwise, students won’t improve their reasoning skills.

a. After posing the question, have students brainstorm a few factors in the cause/effect relationship to help them see what you are really after. Make sure you identify + and -. Have them identify causes and effects that they will put on their map on their Worksheet. Emphasize that all causes need to be those they have control over.

Examples of Causes: skill in computation; ability to identify the pertinent information; understanding of mathematical terms and symbols; missing necessary tools (i.e., calculator, ruler, etc); skill in a variety of strategies; amount of practice; skill in algebra; skill in geometry; consistency in checking your answer, number of school absences

Examples of Effects: grade in mathematics, performance on math tests, grade in science, ability to solve problems from real life, ability to make applications to other problems, success in school mathematics classes, performance on SAT and CHSEE, number of repeated courses, amount of summer school required

d. After students have finished their first iteration of the map, they should complete the Checklist (self and peer review) and then modify their map again.

e. When they are totally finished with the map, have them save to the portfolio and print a copy for submission.

Adaptations of LessonResource Student:Struggling readers and resource students will be paired carefully with students that will support their participation. Instructional Aides will have time before instruction to review activity so that they can support students. Resource Specialists will be informed of activities and provided all instructional materials and access to Interactive Tools.

English Language Learner:Struggling readers and resource students will be paired carefully with students that will support their participation.

Gifted Student:In addition to completing their own plan, gifted students will be asked to be involved in the development plans. They will identify 3 strategies in which they are effective and will identify themselves as tutors for those who need assistance in specific strategies.

Student AssessmentStudents will be assessed throughout the Visual Ranking and Seeing Reason projects. These assessments will be progress-monitoring in that they will inform revisions of the causal map. They will be summative in that they will be given a grade for successful completion of each tool. Finally, they will be assessed on whether they were able to accomplish their plans for improving their problem-solving ability.

## Title

## Purpose

## Pts

Feedback StrategiesInforming Instruction## VR Problem Solving Strategies Worksheet

## Identify what they have learned from Visual Ranking

## 25

## Visual Ranking Activity

## Assess student participation in activity

## 50

## Effective Problem Solving Worksheet

## Preparation for Seeing Reason Activity

## 25

## Checklist for Effective Problem Solving Causal Map

## Provide peer and self assessment of causal map to identify areas of needed improvement

## 10

## Seeing Reason Activity

## Assess student participation in activity

## 50

## Problem Solving Development Plan

## Reflect on Project learning; identify plan for improving problem solving ability; provide for family involvement

## 50

Students will revisit plan in one month to see if they have accomplished their plan.

## Evaluation of Plan (4 weeks later)

## Reflect on plan for improving their skills

## 25

Instructional ResourcesVisual RankingProblem Solving WorksheetSeeing ReasonCausal Map Checklist for Effective Problem Solving