Question: Seven apples were on the table. I ate some apples. Then there were four apples. How many apples did I eat?
Student response: 7 + 4 = 11
Question: Renzo has 15 books. He has 7 more books than Morris. How many books does Morris have?
Student response: 15 + 7 = 22
Question: Jill had some pencils. She gave 18 to Kelley and now she has 14 left. How many pencils did Jill have to start with?
Student response: 18 – 14 = 4
Question: Skylar has 10 times as many books as Karen. If Skylar has 620 books, how many books does Karen have?
Student response: 620 x 10 = 6,200
Question: Oliver has 496 pens. Bernard has eighteen times as many pens as Oliver. How many more pens does Bernard have than Oliver?
Student response: 496 x 18 = 8,928
In our quest for the “right answer” are we encouraging students to simply manipulate some numbers and move on?
Each of the students above ignored the first and last steps of problem solving: first, what are we trying to find, and last, is our answer reasonable.
The first grade student did not indicate the solution, but if we assume they meant I ate 11 apples, then how can that be true if there were only seven to begin with?
In the problem with Renzo, again we don’t know which number is the solution. If we assume the student’s answer was that Morris has 22 books, then he has more than Renzo’s 15, which contradicts the second statement that Renzo has more books than Morris. Similarly, how can Jill have 4 pencils to start and then give away 18? And Karen can’t have 6,200 books when we are told that Skylar has more.
In the last question, Bernard does indeed have 8,928 pens, but that was not the question that was posed.
There are many models for the steps of the problem solving process, based on the work of mathematician George Polya. Here is a simple one for students and teachers to work with:
1. What are we trying to find?
2. What do we know?
3. How will we solve?
4. Have we answered the question and is the solution reasonable?
Before trying to solve any problem, students should identify the question and write a complete sentence for the answer, leaving a blank for the solution. Once they have solved, they fill in the blank and revisit the question, checking to see if the answer makes sense and fits with the statements in the problem.
If the above students had taken these first and last steps, they probably would have realized that I couldn’t have eaten 11 apples, Morris can’t have 22 books, Jill couldn’t have had 4 pencils to start with, Karen can’t have 6,200 books, and that they were asked to find how many more pens Bernard had than Oliver.
More about step 3 in a future post.