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NYC Helpline: How To: Teach Math
Keys to Successful Problem Solving in Mathematics
Luzviminda “Luchie” B. Canlas

 “Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work… A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.” —George Polya, How to Solve It, 2nd ed., Princeton University Press, 1957.

Have you ever wondered why our students are not performing as well as students from other countries when it comes to problem solving? Would you like to know how to teach problem solving in mathematics effectively?

The first thing you need to do is ask yourself this question: What is the main purpose for learning mathematics? Why am I teaching mathematics to my students? If your answer is to prepare students to solve problems in real life; to help students make informed decisions, and to help students make sense of their work so that they can explain their solutions coherently and clearly to others – then, you are on your way to leading them to a successful life.

Teaching students how to explain their solutions and strategies clearly and coherently to others is as important as helping them find the solutions. Solving, alone, is not adequate. Students should be able to explain why their solution is valid. This is where the power and beauty of math lies. Teach students how to organize their thoughts verbally and in writing before presenting to the group. Model constantly.

Students should also be able to apply what they’ve learned to solve problems in new situations. We need to keep on challenging them to give alternate ways to solve a problem. The more ways, strategies, and methods they know the better. Knowing how to evaluate which of these ways is the most efficient is essential, as well. We need to instill in the minds of our students that mathematics is a way of life; it is all around us. Mathematics is incredibly valuable in our daily lives, so we need to give students multiple opportunities to apply what they’ve learned in real life situational problems.

As teachers, we should inspire students to create their own problems. This practice will make math more meaningful for them. This also has a tremendous impact on their self-esteem. Teaching students to create their own problems is one way to empower them and to make them realize that they are primarily in charge of their learning. Constantly ask them thought-provoking questions, and encourage students to ask their own questions. One way to teach students how to ask good questions is by modeling. We also need to allow them to talk with one another. Set a stage where they are constantly giving, receiving, and sharing their ideas. Providing a variety of opportunities and tasks to make them more curious about math will lead to its deep appreciation and understanding.

Setting up a classroom that encourages risk-taking, exploration, investigation, and experimentation will make students more reflective thinkers. They will not be afraid to make mistakes because they know that these will lead them to understand the concepts better and become more careful and fluent in applying math procedures and processes. In the end, their products will reflect mastery and rigor.

Before you can effectively teach students, you need to know your mathematics very well. This is a must. It also helps if you do your very best to apply the National Council of Teachers of Mathematics (NCTM) standards on problem solving, communication, and reasoning and proof when teaching:

Instructional programs from pre-kindergarten through grade 12 should enable all students to

Problem Solving:

    • build new mathematical knowledge through problem solving
    • solve problems that arise in mathematics and in other contexts
    • apply and adapt a variety of appropriate strategies to solve problems
    • monitor and reflect on the process of mathematical problem solving

Communication:

    • organize and consolidate their mathematical thinking through communication
    • communicate their mathematical thinking coherently and clearly to peers, teachers, and others
    • analyze and evaluate the mathematical thinking and strategies of others
    • use the language of mathematics to express mathematical ideas precisely

Reasoning and Proof:

    • recognize reasoning and proof as fundamental aspects of mathematics
    • make and investigate mathematical conjectures
    • develop and evaluate mathematical arguments and proofs
    • select and use various types of reasoning and methods of proof

Note that it is not sufficient for students to know the problem-solving standard. They also need to be able to communicate effectively and to provide sound arguments, reasoning and proofs. Our students are not performing as well as others because we try to teach them numerous math concepts, facts, and skills without teaching them the principles and the “why” of mathematics. We are in this predicament because we focus on quantity instead of quality. We have contributed to limiting their ability to think. We tend to focus more on rote learning and recall and forget that this alone will not equip our youngsters to meet the immense demands of the future. We need to elevate the level of teaching. We focus on content and product and not so much on the process.

We need to change. We need to solve fewer problems daily – instead, focus on high-quality problems. Refrain from boring drills and mechanical computations that do not require our students to engage in higher order thinking. High-quality problems are those that will challenge them to think more deeply, ask questions, apply previous concepts and principles learned, make relevant connections, and encourage communication and reflection. We need to give our students enough time to think the problems through so that they can analyze them on a deeper level, develop and apply strategies/skills, build new concepts/extend learning, and ask better questions.

Moreover, how often do we ask our students to write about their thinking? Make it a habit to ask them to explain their thinking thoroughly on paper. This way they have a written record of their thoughts that they can always refer back to in order to refine their thinking further if needed. This written record is then also available for others students to look at , learn, and compare and contrast.

Always push students to find another way to solve a problem. Test if they can apply their strategies in other situations. Encourage them to engage in mental math arguments so that they can “defend” their thinking. Let them learn that they need to support their statements with facts or evidence. Ask them always, “Does anyone have another solution that is different? Explain.” Design lessons, tasks and activities that promote active discourse and exchange of ideas. Teach problem-solving strategies so that students will gain confidence as they solve problems. Teach them how to apply George Polya’s1 four-step problem solving plan [Understand the problem; Devise a plan; Carry out the plan; Look back.]

George Polya’s Four-Step Plan to Solve Problems

  1. Understanding The Problem
    • You have to read the problem carefully. Can you restate the problem in your own words?
    • Do you understand the vocabulary?
    • What is given? What is not known? What is the problem about? Identify the facts.
    • What are you being asked to find out or do? Identify the question.
    • What is relevant information? Irrelevant information?
    • Draw a figure if this will help you understand the problem.
    • Can you represent the problem using numbers?
    • Can you estimate?

  2. Devising A Plan
    • Think which of the problem solving strategies you know is applicable to solve the problem. Here are some suggestions.
      • Draw a sketch, picture, illustration, or diagram to help you solve the problem.
      • Make an educated guess and check it.
      • Find a pattern by using inductive reasoning.
      • List ideas in a chart or table.
      • Make an organized or systematic list.
      • Solve algebraically.
      • Use smaller numbers to simplify the problem.
      • Act it out.
      • Work backwards.
      • Use deductive reasoning.Use logic.
      • Use manipulatives to represent ideas.
      • Visualize the problem.
      • Eliminate possibilities by using the given information.
      • Apply the trial and error method.
    • Think of what operations or materials you will need and use to solve the problem.
    • Choose the strategy*.

  3. Carrying Out The Plan
    •  Solve the problem. Use computational and operational skills to find out the answer.
    • Try to use another strategy

  4. Looking Back
    • Inspect your solution. Is it close to your estimate?
    • Does your solution answer the question?
    • Does your answer make sense? It is reasonable?
    • Can you check your answer? Can you support your argument?
    • Can you solve the problem in a different way?
    • If you didn’t get it right, why not? Where did you get it wrong?
*Check textbooks and online resources for examples of each of these strategies. Then you can make your own to match the needs of your students.

I hope that these ideas have given you more confidence in teaching problem solving in mathematics. You need to create tasks that are appropriate for your students. Always remember: you don’t need plenty of problems, you just need high-quality ones that will challenge thinking and guide instruction. Hopefully, students will appreciate the beauty and value of mathematics while gaining the confidence and satisfaction of successfully solving interesting and meaningful problems. This will make math fun and alive. We need to be a nation of math lovers not math phobic individuals. We can do this!!!

If you have a question or comment about this article e-mail Luchie.


1George Polya (1887-1985) was a famous mathematician who believed that problem solving should be taught and that anyone could learn it if taught the right way. In his 1945 book, How to Solve It, he gave us four steps or principles to follow to solve problems effectively.

 

 

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