Ronna Williams
Case Study (Part II).
I have been holding word problem sessions prior to the scheduled class time. I started
off with three students. One student dropped (B.M.). As the word of the sessions got
out to the class, many other students came for the lessons on word problems. I was
amazed at how many students would arrive early for these lessons.
Students were taught guess-n-check, eliminating possibilities, looking for patterns, and
using a matrix. With all techniques students were taught to use their chosen procedure
to build algebraic equations.
Overall I discovered that students displayed more confidence in their ability to solve
word problems.
Pre-lesson questions
Problem #1 If nine times a certain number is subtracted from 7, the result is 52. Find
the number.
Student (A.E.): This student is low in math abilities, but she keeps on trying. She is trying to teach her
daughter pre-algebra, so that her daughter can pass the GED.
Student built an equation 9(7 – x) = 52; 6 x 9 = 54 – 2 = 52.
Student is not able to build an equation. She was really confused but willing to try.
Student (S.D.): This student is in the top of the class. He seems to have no problems with pre-algebra
work.
His equation: 7 – 9x = 52 was the correct equation.
His final solution was 6.5. This student made an error in algebra. He added 7 to each side of the
equation instead of subtracting.
Student (B.M.): This student is working and struggling. She is an average student in pre-algebra.
Her equation: 9x – 7 = 52. The equation is set up backwards. Then she subtracted 7 from each side of
the equation rather than adding it.
Her solution: 5
Problem #2. Find four consecutive whole numbers such that the sum of the first three
numbers equals the fourth number.
Student (A.E): Her equation: x + x + x = x / x^3 = x
Her error: Equation is set up incorrectly. There was confusion over the definition of consecutive. Then
she added the “x” incorrectly.
Student (S.D.): He chose numbers, in this case 2, 4, 6, and 12. He showed that if you added these
numbers together 2 + 4 + 6 and arrived at 12. His error: not knowing the definition of consecutive
numbers.
Student (B.M): She chose the numbers. First she tried 1, 2, 3,4 that didn’t work. Then she tried 2, 3, 4,
and 5, that didn’t work. Here third attempt produced the correct answer 0, 1, 2, 3.
Follow-up Question results.
Problem #1. Kendall paid $244 for a CD player and six CDs. The CD player cost ten
times as much as one CD. Find the cost of the CD player and the cost of one CD.
Student (A.E.): Student took 244/10 to arrive at $24.40 for the cost of each CD. She forgot to take into
account the cost of the six CD.
Student (S.D.): Student declared the variable. He set up the equation correctly. He arrived at the correct
answer and checked his answer.
Problem #2. If eight times a certain number is subtracted from 27, the result is 3. Find
the number.
Student (A.E.): Student arrived at the correct answer. She did not declare a variable, but the equation
was correct.
Student (S.D.): Student had the correct equation. He didn’t declare the variable. He checked his
answer.
Comparison of Problem-solving Strategies
Student (A.E.): This student is low in math abilities. She puts a lot of effort into her
school work. Many times her errors occur because she doesn’t understand the
instructions.
Initially the student was not able to set up an equation at all. She couldn’t use any
problem-solving strategies. After teaching problem solving strategies she is able to
solve some problems. When she is presented with a word problem is feels like there is
something that she can do to start the problem.
Student (S.D.): This student is in the top of the class. As the semester progressed he
started have a few problems in pre-algebra work, but overall he is able to complete
most assignments.
Initially he was able to set up the correct equation but made errors in algebra. He
previously tried to use the guess-n-check method. He was not able to produce the
correct answer using this method. After teaching problem solving strategies this
student declared the variable. He set up the equation correctly. He arrived at the
correct answer and checked his answer. He is now able to use guess-n-check and
other methods to solve word problems.
My goals in teaching these students.
1st goal: I would like to discuss the definition of consecutive whole numbers. My
student work on vocabulary, but do not really work on many word problems. I think that
trying to apply their vocabulary to a word problem was . . . well a problem.
Result: I spent time in class teaching vocabulary. Students are able to practice
their vocabulary skills with worksheets and quizzes.
2nd goal: To teach these students how to discover what is being asked in the question.
Result: As a group, we spent time going over each question. We discussed how
to find the question that was being asked. We also went over checking that the
question was answered.
3rd goal: Teaching the students how to declare a(n) variable(s) based on what is being
asked in the question. Also teach the students how to develop a plan. This plan would
not only solve the problem, but also help them to estimate the answer. These
techniques could be guess and check; eliminating possibilities; drawing pictures, looking
for a pattern, or making a systematic list.
Result: The students were able to implement problem solving strategies. The students
were able to take these problem solving strategies and declare variables. Teaching
students strategies gave many of them hope. They did not feel overwhelmed by the
task of problem solving.
4th goal: Using the variables and what is being asked, learning how to set up an
equation or expression.
Results: Students who were able to declare variables were able to set up the equation
correctly.
5th goal: Review algebra rules used in isolating variables. Solve the equations.
Results: After teaching algebraic processes one of the students was able to solve the
equations.
6th goal: Deciding if the answer makes sense, was the question answered and does
the answer check mathematically. Also teach the students to state the answer correctly
with units.
Results: Students started checking the answer without prompting from me. The
students learned to apply the units to the answer during regular class instruction.