Concurrent Enrollment Math Instructors Fall Seminar Polya’s Problem Solving Strategies Ruth Ann Abbott

Outline for the talk Some thoughts on education today The Rigor/Relevence Framework How Polya’s problem solving techniques fit in. Closing thoughts.

A students guide to problem solving Rule 1: If at all possible, avoid reading the problem. Reading the problem is time consuming and leads to confusion. Rule 2: Extract the numbers from the problem in the order in which they appear. Be on the watch for numbers written in words.

Rule 3: If Rule 2 yields three or more numbers, the best bet for getting the answer is adding them together. Rule 4: If there are only two numbers which are approximately the same size, then subtraction should give the best results. Rule 5: If they are not close to the same size, divide. If this gives a decimal, multiply instead.

Rule 6: If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem. Rule 7: If Rules 1-6 don’t seem to work, make one last desperate attempt. Perform about two pages of random operations using these numbers. You should circle about 5 or 6 answers on each page in case one of them happens to be the answer. You might get some partial credit for trying hard. Rule 8: Never, never spend too much time solving problems. This set of rules will get you though even the longest assignment in no more than 10 minutes. Joe Dodson

An example by Andrew Wiles

The primary aim of education is not to enable students to do well in school, but to help them do well in the lives they lead outside of school.

Job Outlook 2002, National Association of Colleges and Employers (NACE)

B D C A RIGORRIGOR RELEVANCE Rigor/Relevance Framework High Low Relationships Relationships of Little Importance Relationships Essential Relationships Important Relationships Important

RIGORRIGOR RELEVANCE A B D C Increasing Rigor/Relevance High Low

Relevance Rigor Relationship Rigor, Relevance & Relationship Framework (3-D Model) Grand Junction, Colorado

Relevance Rigor Relationship Rigor, Relevance & Relationship Framework (3-D Model) Now, picture the capacity for student learning as the volume we create within our 3-D model….

Relevance (5) Rigor (20) Relationship (5) Rigor, Relevance & Relationship Framework (3-D Model) The capacity for student learning is the volume we created within our 3-D model…. 20 x 5 x 5 = 500 learning units

Relevance Rigor Relationship Rigor, Relevance & Relationship Framework C D A B

Relevance Rigor Relationship Rigor, Relevance & Relationship Framework High Rigor with Low Relevance & Low Relationships We might ask students to learn the rigorous content without the meaning of application or the support of relationship…..

Relevance Relationship Rigor, Relevance & Relationship Framework Rigor High Relevance with Low Rigor & Low Relationship We might push students into the high levels of application without providing them the foundation of core knowledge coupled with the safety to take risks.

Relevance Relationship Rigor, Relevance & Relationship Framework Rigor If students do not experience high expectation for learning the content as well as applications for their learning, we may become more like a friend than a facilitator of their learning. Low Rigor and Low Relevance with High Relationship.

Relevance Rigor Relationship Rigor, Relevance & Relationship Framework To maximize a student’s learning, it is critical to keep the balance between all three dimensions. To optimize the learning for each student, we must adjust the dimensions to best fit the situation at hand.

Dimension R Rigor, Relevance & Relationship Framework IMPORTANT: To maximize a student’s learning, it is critical to keep the balance between all three dimensions. To optimize the learning for each student, we must adjust the dimensions to best fit the situation at hand. Imbalance or incongruence of one dimension could affect the capacity for a student to learn.

high RIGORRIGOR Express probabilities as fractions, percents, or decimals Devise a scale to test consumer products and graph data. Plot the coordinates for quadrilaterals on a grid. Make a scale drawing of the classroom. RELEVANCE low high low

Select best measures of Central tendency to support a point of view. Develop a sampling plan for a public opinion poll. Predict the probability of events using ratios. Measure small metal objects using calipers and micrometers. Relevance low high RIGORRIGOR low high

Instructional Strategies Brainstorming Presentations Cooperative Learning Problem-based Demonstration Project Design Guided practice Research Notes/Graphic Organizers Inquiry Simulation Instructional Technology Socratic Seminar Lecture Teacher Questions Memorization Work-based learning

Now Presenting Polya’s Principles of Problem Solving 1) Understand the Problem 1) Understand the Problem 2)Devise a plan 2)Devise a plan 3)Carry Out the Plan 3)Carry Out the Plan Review/Extend Review/Extend

Understand the Problem Do you understand all the words? What are you asked to find or show? Can you restate the problem in your own words? Can you think of a picture/diagram that might help? Is there enough information?

Devise a Plan Guess and check Make a list Solve an equation Look for a pattern Draw a picture Solve a simpler problem

Carry out the plan Care, patience and persistence. If it continues not to work, choose another plan.

Review What worked? What didn’t? Record strategy with problem category.

Examples

Go to /redesignsite/resources/additional_res ources/guide_to_problem_solving/problem_s olving.pdfhttp:// /redesignsite/resources/additional_res ources/guide_to_problem_solving/problem_s olving.pdf For many other examples and varieties of strategies.

Critical websites for CE Projects mathSite/index.htmhttp://t4.jordan.k12.ut.us/teacher_resourses/ mathSite/index.htm SuccessfulPractices.org Follow links: =>publication excerpts=>instructional strategies For Gold Seal Lessons and How to design such. e/index.htm (Best) e/index.htm

In Conclusion: Rigor, relevance, and relationships must all be met. We are preparing our students for real life, not for a 4.0 Students need the skills of problem solving – Guess and check, Make a list, Look for a pattern, draw a picture, solve a sumpler problem, use a model, equation or work backward.

Invisible difference Passion Commitment

Concurrent Enrollment Math Instructors Fall Seminar Polya’s Problem Solving Strategies Ruth Ann Abbott