The effects of early schema acquisition on mathematical problem solving

Abstract

The current paper explores the effects of providing people with schema training at the outset of learning (compared to at later stages) on mathematical word problems modeled after problems from the Graduate Record Examination. Additionally, the ratio of worked examples to problem-solving practice was manipulated. Participants were randomly assigned to one of four conditions and were tested on near and far transfer problems. Participants provided schema training at the outset of learning outperformed those provided schema training after problem-solving practice, particularly on far transfer problems. Likewise, participants provided with a higher ratio of worked examples to problem-solving practice demonstrated better performance during testing than those provided with more problem-solving practice and fewer worked examples. These findings extend the worked example effect to far transfer problems and suggest that providing schema training prior to learning solution strategies may be effective in improving mathematical problem solving.

Appendices

Appendix A

Sample of work-ratio test problems

Near transfer

Cassie is able to write 20 book pages in 6 h; while Amber can write 20 book pages in 7 h. How long would it take both Cassie and Amber working together to write 20 book pages?

• a. $\frac{42}{13 }$of an hour

• b. $\frac{50}{10}$of an hour

• c. $\frac{65}{9}$of an hour

• d. $\frac{72}{8}$of an hour

• e. $\frac{77}{12}$of an hour

Far transfer

A pool filter is able to filter out $\frac{5}{6}$ of the pool’s water in 2 h. How long, in minutes will it take the pool filter to filter all of the water in the pool?

• a) 125 min

• b) 133 min

• c) 138 min

• d) 144 min

• e) 152 min

Sample of distance-ratio test problems

Near transfer

Brian runs from Academy High School to the local library at an average speed of 15 mph. Tina walks from Academy High school to the local library at an average speed of 3 mph. It takes Brian 3 min to arrive at the library. How long does it take Tina to arrive at the library?

• a) 10 min

• b) 11 min

• c) 13 min

• d) 14 min

• e) 15 min

Far transfer

During the first 2 h of a marathon, Madelyn is running at 9 mph. She runs at 6 mph for the remaining 24 miles of the race. What was Madelyn’s average speed for the entire marathon?

• a) 6.5 mph

• b) 7 mph

• c) 7.5 mph

• d) 8 mph

• e) 8.5 mph

Sample of percent test problems

Near transfer

At noon, the stock for Arlene’s company decreased from $40 to $25 a share. At the close of the market, the stock had increased to $45 a share, by what percent (from $25) did the stock increase?

• a) 75%

• b) 80%

• c) 84%

• d) 88%

• e) 90%

Far transfer

Earlier in the month, a leather jacket was on sale at a department store’s outlet center and was priced at $70. Today the leather jacket was moved from the outlet center to the actual department store, where the price on the leather jacket increased by 80%. What price was the leather jacket increased to?

• a) $105

• b) $117

• c) $126

• d) $131

• e) $137

Appendix B

Example of schema training materials

Sample description of a problem type

Work rate problems

Work rate problems typically ask us to figure out how much work will get done in a specified amount of time if we combine the work rates at which two or more rates are operating at. For example:

Bob takes 5 h to make an X number of car parts, while it takes Sam 3 h to make the same number of car parts. How long would it take both Bob and Sam to complete this job if they worked together?

Sample categorization question

Please select the correct problem-type below:

James and Brandon are both race car drivers. James drives at an average of 135 mph, while Brandon drives at average of 180 mph. It takes James 4 min to complete a lap around their current race course, how long does it take Brandon to complete a lap in the same lap course?

• Distance rate problem

• Work rate problem

• Percent problem

• None of the above

Example problem including problem states and goals

Work rate problems

Bob takes 6 h to make an X number of car parts, while it takes Sam 3 h to make the same number of car parts. How long would it take both Bob and Sam to complete this job an X number of parts if they worked together?

Current State (known information): it takes Bob 6 h to produce an X number of car parts and Sam 3 h to produce an X number of car parts.

Problem Goal (what we’re asked for/unknown information): how long it would take Bob and Sam working together to produce an X number of identical car parts.

Example problem state and goal exercise

James and Brandon are both race car drivers. James drives at an average of 135 mph, while Brandon drives at an average of 180 mph. It takes James 4 min to complete a lap around their current race course, how long does it take Brandon to complete a lap in the same lap course?

Problem type:

Current state:

Goals:

Appendix C

Sample worked example

Work rate problems

Bob takes 6 h to make an X number of car parts, while it takes Sam 3 h to make the same number of car parts. How long would it take both Bob and Sam to complete this job if they worked together?

Solution

It takes Bob 6 h to complete an X number of car parts. Thus, Bob can do $\frac{1}{6}$ of the job in 1 h → $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ = $\frac{6}{6}$ = Xcar parts. While Sam can do $\frac{1}{3}$ of the job in 1 h.

→ $\frac{1}{3}$ + $\frac{1}{3}$ + $\frac{1}{3}$ = $\frac{3}{3}$ = X car parts. Next, allow x to stand for the total number of hours it would take Bob and Sam to work together to complete X car parts. Bob and Sam can complete $\frac{1}{x}$ of the car parts in 1 h. Note: it is necessary to take the proportion of car parts that can be completed in 1 h and then add them together. Equation: →

$$\frac{1}{\text{time it takes }A\;\text{to complete job}}+\frac{1}{\text{time it takes }B\;\text{to complete job}}=\frac{1}{x}$$

Note

x represents the amount of time it takes both A and B to complete the job working together.

$\frac{1}{6}+\frac{1}{3}= \frac{1}{x}$→ $\frac{1}{6}+\frac{2}{6}=\frac{1}{x}$→ $\frac{3}{6}=\frac{1}{x}$ → x = $\frac{6}{3}$ = 2. Thus, working together, it takes Bob and Sam 2 h to complete X parts.