Math Problem Solving and the Use of Generative Strategies

A recent Facebook post by the Secret Society of Happy People (September 17, 2014) highlighted a common concern among many educators and students regarding math word problems.  The post read, “Every time I see a math word problem, this is what it looks like:  If I had 10 ice cubes and you had 11 apples, how many pancakes will fit on the roof?

How does one approach a word problem like this?  What strategies are in place to make a meaningful connection with the mathnproblem and reach a solution?

We recognize, of course, that this word problem is not one to be solved, but often students are presented with math word problems that appear insurmountable to them.   How can we, as educators, provide an approach and a strategy to tackle the solution of perplexing word problems?

Well-designed instructional strategies are referred to as generative strategies because they prompt or motivate students to generate or construct meaningful connections with the material around which they are working, as well as facilitate longer retention of what they are supposed to learn (Roberts, 2012).  A generative strategy could provide a means of solving challenging math word problems.

In a recent study (Swanson, Moran, Bocian, Lussler, & Zheng, 2012), students used generative strategies to approach math word problems that required paraphrasing (a skill crossing all content areas) text orally or in writing for various purposes.  A drawing of the main idea to connect it with prior knowledge could be part of the paraphrase.

Focus of the paraphrase:

Restate – paraphrase of the question propositions (statements)
Relevant – paraphrase of the relevant propositions
Complete – paraphrase all propositions:  question, goal, relevant, irrelevant

What might this look like in practice?

Math Word Problem at the Middle School Level:
(
http://www.uwosh.edu/coehs/cmagproject/many_word/index.htm;http://www.uwosh.edu/coehs/cmagproject/many_word/documents/Morris_MSLevel_3.pdf)

John received 3 detentions in the fourth quarter. He received 4 other detentions for 1st and 2nd quarters combined. He received 12 for the whole school year. How many detentions did he receive in the 3rd  quarter?

Restatement of Question: How many detentions did John receive in the 3rd  quarter of the school year?

Goal:  Determine the number of detentions received during 3rd quarter.

Relevant Information: 

4 detentions in the 1st and 2nd quarters combined
3 detentions in the 3rd quarter
12 detentions for the entire school year

Irrelevant Information:  None with this problem.

Complete:  Collectively include restatement, goal, and relevant and irrelevant information items as paraphrased when considering problem solution.

Another Math Word Problem at the Middle School Level:
(http://www.uwosh.edu/coehs/cmagproject/many_word/index.htm;
http://www.uwosh.edu/coehs/cmagproject/many_word/documents/Morris_MSLevel_2.pdf)

Jerritt had 15 beef sticks. Leonard gave him some more. Than Jerritt had 42 beef sticks. How many beef sticks did Leonard give Jerritt?

Restatement of Question:  Need to find the number of beef sticks Leonard gave to Jerritt.

Goal:  Determine the difference between Jerritt’s original number of beef sticks and the number he had after Leonard gave him some more.

Relevant Information:

Jerritt had 15 beef sticks originally
Jerritt had 42 after Leonard gave him some beef sticks

Irrelevant Information:  None in this problem.

Complete:  Collectively include restatement, goal, and relevant and irrelevant information items as paraphrased when considering problem solution.

Study results indicated the following outcomes with regard to the use of generative strategies in solving math word problems.  Improved problem solving accuracy and allowance for those with a math disability to catch up was evidenced with relevant and complete paraphrasing propositions.  Improved problem component identification was evidenced with complete paraphrasing propositions.  Finally, improved operation span performance was evidenced with restate and relevant paraphrasing propositions.

In short, the results support the use of generative strategies for improving the problem-solving performance of children with a math disability (Swanson et al., 2012).  Further, data in another study (Swanson, Moran, Lussier, & Fung, 2014) support the use of generative strategies for improving problem-solving performance in children at risk for math disability.

Resources

Classroom Cognitive and Meta-Cognitive Strategies for Teachers: Research-Based Strategies for Problem Solving in Mathematics K-12 The Paraphrasing Strategy, p. 15

Intervention Central – Math Problem-Solving: Combining Cognitive and Metacognitive Strategies

References

Roberts, L. (2012). Designing the instruction:  Strategies.  Source:  Designing Effective Instruction by Morrison, Ross, Kalman, and Kemp. Slideshare, June 11, 2012. http://www.slideshare.net/leesharoberts/instructional-strategies-13283812

Swanson, H. L., Moran, A. S., Bocian, K., Lussier, C., & Zheng, X. (2012). Generative strategies, working memory, and word problem solving accuracy in children at risk for math disabilities. Learning Disability Quarterly, 36(4),202-214.

Swanson, H. L., Moran, A. S., Lussier, C., & Fung, W. (2014). The effect of explicit and direct generative strategy training and working memory on word problem-solving accuracy in children at risk for math difficulties.  Learning Disability Quarterly, 37(2), 111-123.