For over twenty years, Americans have been working to improve the mathematics and science education of our students. As home educators, we have a great opportunity and responsibility to help our children become competent mathematicians and scientists. While many home-educated children are competent in math and other sciences, our children can do more than be competent in mathematics. Our children can excel in mathematics.
Many people think of mathematics as sitting down with a pencil and paper and doing lots of calculations. While it is important to develop these skills as young children, math is much more than this. Mathematicians are problem solvers, and the primary tools they use to solve problems are arithmetic, algebra, geometry, and such.
There are many different curricula available to teach these skills, but how do you teach problem solving? Successful problem solvers are not haphazard in their approach to problem solving, but careful and systematic. The following flowchart illustrates this systematic approach.
The processes represented in this figure are a synthesis of John Mason’s (1991) problem solving ideas and of Polya’s (1945) problem solving processes. At each stage of the solution process, there are key phrases for students to remember and key questions for students to answer in written form. The general phases for solving a problem are understanding the problem (ENTER), planning a solution (PLAN), carrying out the plan (ATTACK), and looking back at the solution (REVIEW). At any time, students can consider what to do when they were STUCK and how to proceed once they have an insight into a new approach (AHA) to try.
ENTER is the term used to help students think about understanding the problem; during this phase, students respond in writing to the following four questions. (1) What do I know? (2) What do I want? (3) What can I introduce or assume? (4) What is a reasonable estimate?
For the students who sit down, look at the problem, and then have no idea where to begin, answering these questions gives them a place to be start. Physically writing down the answers helps them process the information and see that they can be successful in this phase. Entering the problem is a crucial part of successful problem solving. It is worth spending several weeks on this. Problems can be solved completely later.
PLAN is the term used to remind students to consider what particular strategies might help them; students are then to make a written note of those strategies. Five specific strategies listed here are (1) looking for a pattern (2) drawing a diagram (3) making a table or list 4) solving a simpler problem, and 5) guess and check strategies. Many others are certainly appropriate.
During the ATTACK phase, students follow their plan, making modifications as needed. So that they can explain their work to others, students should record all of their work and reasoning.
The term, REVIEW, reminds students to CHECK their solution and REFLECT upon their work by considering and responding to the following ideas and questions. (1) Check calculations and reasoning. (2) Is the problem resolved? (3) How accurate was my estimate? (4) Write down key ideas. (5) Justify my solution. The REVIEW phase is designed to help students reflect upon their progress in each of the other three phases. When determining if the problem was resolved and when evaluating the accuracy of their estimates, students refer back to the ENTER phase. When writing down key ideas, students refer back to the PLAN phase. Students refer back to the ATTACK phase when checking calculations and reasoning.
These four phases may be familiar, as similar ideas are often presented in math texts. Usually though problem solving is presented as a straightforward, linear process. In reality, the problem solving process involves give and take as one strategy may be followed, only for the solver to realize another approach may be better. The processes of STUCK and AHA allow for these changes in thinking. If a student is unable to make progress at any point, the following phrases may help him move forward from being STUCK. 1) I do not understand… (2) I do not know what to do about… (3) I do not see how to… (4) I cannot see why… Thinking of a new idea, the student then moves to the AHA phase and completes the phrases, (1) Try…, (2) Maybe…, or (3) But why…
The following solution illustrates the use of the flowchart to solve this particulat problem.
In a group of 37 students 18 neither play an instrument nor like math. Ten play instruments and 13 like math. How many like math and do not play instruments?
What do I know?
I know that there are 37 students. 18 neither played an instrument nor liked math.
Ten play instruments and 13 like math.
What do I want?
I want to know how many like math and do not play instruments.
What can I introduce or assume?
I can assume that all 37 people were categorized as playing an instrument or not and liking math or not.
What is a reasonable estimate?
Since almost one-fourth of the people play instruments, I would guess that one fourth of those who like math play an instrument. One fourth of fourteen is between 3 and 4. So there are between 9 and 10 students who like math and play an instrument.
I will make a table to help me out.
Fill the table with “what you know.”
|37 total students||13 do like math||? do not like math|
|10 play an instrument||?||?|
|? no instrument||?||18|
Make calculations to fill in the rest of the table.
|37 total students||13 do like math||37-13=24 do not like math|
|10 play an instrument||13-9=4||10-4=6|
|37-10=27 no instrument||27-18=9||18|
Answer: There are nine students who like math and do not play an instrument.
If there are 9 students who like math and do not play an instrument, then there must be 13-9=4 students who like math and play an instrument. Then there must be 10-4=6 musicians who do not like math. Then there should be 24-6=18 students who do not like mathematics and who do not play an instrument. This corresponds to the information given.
The answer was very close to my estimate.
I found the number of students who like math and do not play an instrument, as the problem asked.
The key idea was making a table to organize the information. When I filled in the table with the information I knew, then I could fill in the rest of the table with the information I wanted to find.
The sum of the answers in each box of the table should equal the total number of students.
Learning problem solving can be a frustrating experience. Having this flowchart guides the students in this process. Even if a student initially has no idea how to approach a problem, at least he can confidently ENTER the problem. With time and practice, the other phases will follow.
Teaching problem solving can also be a frustrating experience. Keep solving problems. Follow the flowchart. Trust God for wisdom. And wisdom, defined by God, is “first of all pure, peace-loving, considerate, submissive, full of mercy and good fruit, impartial and sincere.” (James 3:17) May God give you and your children wisdom as you solve problems together.
Burks, Linda. (1993) The Use of Writing As a Means of Teaching Eighth Grade Students to Use Executive Processes and Heuristic Strategies to Solve Mathematics Problems. University of Michigan Thesis.
Mason, John; Buron, Leone; and Stacey, Kaye. (1991) Thinking Mathematically. Addison Wesley Publishing Company:New York.
Polya, George. (1962) Mathematical Discovery: On Understanding, Learning, and Teaching Mathematical Problem Solving. John Wiley and Sons: New York, NY.
Polya, George. (1954a) Induction and Analogy in Mathematics. Princeton University Press: Princeton, New Jersey.
Polya, George. (1954b) Patterns of Plausible Inference Princeton University Press: Princeton, New Jersey.
Polya, George. (1945) How to Solve It: A New Aspect of Mathematical Method. . Princeton University Press: Princeton, New Jersey.
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