One of the primary reasons student have trouble with problem solving is lack of single procedure that works all the time. Each problem is slightly different. If you misunderstand either the problem or the underlying situation you may make mistakes or incorrect assumptions. One of our main goals here is to make you a better problem solvers, hence, we would like to introduce a framework developed by Polya for thinking about problem solving in this newsletter. In 1945 George Polya published the book "How To Solve It" which quickly became his most prized publication.
At a high level, we can divide problem solving in 4 phases.
- Preparation - Understand the problem: This seems so obvious that it is often not even mentioned, yet student's are often stymied in their efforts to solve problems simply because they don’t understand it fully, or even in part. Trick is to ask right set of questions to understand the problem e.g. Do you understand all the words used in stating the problem? Can you restate the problem in your own words? Can you think of a picture or diagram that might help you understand the problem?
- Thinking Time - Devise a plan: Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy.
- Insight - Carry out the plan: This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals.
- Verification - Look back: Polya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn’t. Doing this will enable you to predict what strategy to use to solve future problems.
For each of these phases, we can have very specific strategies to improve problem solving skills. You can learn more about detailed strategies here.
Learn problem solving by proofs
Elementary School Brain Teaser
"Numbers are placed into the cells of a rectangular table in such a way that the sum of the numbers in each column and in each row is 19. Prove that the table must be a square."
Pockets, Wallets, & Money
Middle School Brain Teaser
"A number of wallets hold 1000 dollars in all, and the wallets were placed in a number of pockets. If no wallet contains another wallet, and the total number of wallets is greater that the number of dollars in any pocket, is the number of pockets greater than the number of dollars in some wallet?" Provide a detailed solution.
You can email your solutions to us. We will be glad to publish if you exhibit good problem solving skills and creativity.
Hard to make 7 friends?
Elementary School Brain Teaser Solution From The Previous Newsletter
“CKSTEM has 25 students in a class, and each student is friends with exactly seven classmates.” Why can’t that statement be true?
Can you prove it using triangle inequalities?
Middle School Brain Teaser Solution From The Previous Newsletter
A straight bar of length 2 m is cut into five pieces with each piece at least 17 cm long. Prove that there are three of these pieces that can be put together to form a triangle.
CKSTEM SUMMER CONTEST REFERRAL PROGRAM
You can help your friends by asking them to join CKSTEM contests. If they enter your name while signing up then you will be eligible for a lucky draw. You will have one entry for each referral. You will not only help your friends to be a part of the most amazing contests, you may also win some cool prizes.
ABOUT THE COMPETITIVE KIDS STEM
CKSTEM is a volunteer-run, nonprofit organization engaged in enhancing kids' education and enabling them to achieve excellence. We focus on teaching problem solving and enabling critical thinking by 'connecting the dots' as well as developing mental dexterity for the students.
We believe that in order to grow, students need to develop beyond their comfort zone, so we ensure they are consistently and appropriately challenged. We not only coach them to understand and problem solve but also train them to compete, nurturing their competitive spirit and mental fitness, similar to competitive sports.