An Overview of the Keys of Problem 

Solving Professional Learning 

KoPS Instructional Strategies KoPS provides professional development for

math and ESE teachers by: – Using Next Generation Sunshine State

Standards (and CCSS) in problem solving in mathematics

– Infusing researched metacognitive strategies (mnemonic devices, self-monitoring skills, graphic organizers, etc.),

– Including differentiated instruction and technology to meet the needs of all learners.

– Including assessment and evaluation targeted to varying levels/types of instruction.

Rationale for Change

Daily, mathematics is used throughout our lives. The ability to compute, problem solve, and apply concepts and skills in mathematics influences decisions and life choices. From personal, professional, and societal perspectives, the mastery of mathematical skills of number sense and problem solving are necessary. This is especially evident in technology-rich societies, where number sense and problem solving skills have increasing importance, as technology enhances both the opportunities for, yet the demands of, advanced levels of proficiency in mathematics.

~Witzel & Little, 2009

Problem Solving as a Process Standard (NCTM)

Getting Acquainted with a Problem…

•  Where should I start? –  Start from the statement of the problem.

•  What can I do? –  Visualize the problem as a whole as clearly and as vividly as you can.

Do not concern yourself with details for the moment.

•  What can I gain by doing so? –  You should understand the problem, familiarize yourself with it,

impress its purpose on your mind. The attention bestowed on the problem may also stimulate your memory and prepare for the recollection of relevant points.

~Polya, 1945

Area 1: Understand the Problem

•  What are you asked to find out or show? •  Can you draw a picture or diagram to help you understand the problem? •  Can you restate the problem in your own words? •  Can you work out some numerical examples that would help make the

problem more clear?

Critical Questions

•  Do you understand all the words used in stating the problem?

•  What are you asked to find or show? •  Can you restate the problem in your own

words? •  Can you think of a picture or diagram that

would help you understand the problem? •  Is there enough information to enable you to

find a solution?

Area 2: Devise a Plan to Solve the Problem

A partial list of problem solving strategies include: • Guess and check • Make an organized list • Draw a picture or diagram • Look for a pattern • Make a table • Use a variable

• Solve the simpler problem • Experiment • Act it out • Work backwards • Use deduction • Change your point of view

•  Have you seen it before? Or, have you seen the same problem in a slightly different form?

•  Do you know a related problem? Do you know a theorem that could be useful?

•  Look a the unknown, and try to think of a familiar problem having the same or a similar unknown.

•  Could you restate the problem? Go back to definitions

Critical Questions

Area 3: Implementing a Solution Plan

•  Carrying out the plan is usually easier than devising the plan •  Be patient—most problems are not solved quickly nor on the first

attempt •  If a plan does not work immediately, be persistent •  Do not let yourself get discouraged •  If one strategy isn’t working, try a different one

•  Experiment with different plans •  Allow for mistakes •  Work collaboratively •  Check

How do students implement? •  Draw it •  Use calculator •  Use computation skills

Area 4: Reflecting on the Problem: Looking Back

•  Does your answer make sense? Did you answer all of the questions? •  What did you learn by doing this? •  Could you have done this problem another way—maybe even an easier

way? •  ~Reardon Problem Solving Gifts, Inc. (2001)