Our Problem Solving Audiobook focuses on Mathematical Problem Solving. We take a look at the theory of problem solving, relating Robert Sternberg's triarchic theory of practical, analytic, and creative intelligences to seven stages of problem solving. We review research from two distinct areas, Thomas Carpenter’s work on Cognitively Guided Instruction and Sriraman’s research on creativity in adult mathematicians. Our audiobook runs about 15 minutes.
We also have two bonus goodies! Our first 4-minute bonus pack looks at the special challenges learning disabled students face in academic problem solving situations and reviews research by Fuchs. The second bonus pack is an interesting review of research on problem solving at the secondary level, with connections to cognitive load theory.
As you listen to these materials, we hope that you will think about our roles as practitioners charged with teaching problem solving. What can we do in our classrooms to thread creative discovery through all stages of mathematical problem solving?
problem solving movie download
In addition you can download a bonus audiobook that covers problem solving as related to learning disabilities and the middle/high school years. This is an optional download.
problem solving bonus audiobook download
Tuesday, May 5, 2009
Bibliography
The 54 entries in our bibliography represent our group’s interests in problem solving psychology and practice. We focused on mathematical problem solving. Our bibliographic Entries span an age-level continuum from early childhood to adulthood. Our interests in the psychology of problem solving spanned creativity in early childhood, cognitively guided instruction and corresponding reduction in extraneous cognitive load in elementary school, as well as adolescent research in collaboration, discussion in problem solving, and back full-circle to creativity, insight in graduate level mathematical coursework, and professional mathematical practice.
Our bibliography includes several seminal books. For an excellent overview of current research on problem solving, start by reading Davidson & Sternberg’s (2003) book, The Psychology of Problem Solving. For excellent material on mathematical heuristics, Polya is the master, and several of his books are listed. For information on cognitively guided instruction, look for articles by Carpenter & Fennema. For information on the relationship to cognitive load theory and problem solving, review the section on middle school and high school mathematics. For a look at the challenges students with learning disabilities and students at risk face, review the section on elementary mathematics and learning disabilities. For newly emerging research in creativity with linkages to mathematical giftedness, an up-and-coming researcher is Siriraman who seems to be on-target with research and practical implications in school settings. Richard Mayer is also looking into creativity.
We hope you enjoy your summer reading on Mathematical Problem Solving.
problem solving bibliography download
Our bibliography includes several seminal books. For an excellent overview of current research on problem solving, start by reading Davidson & Sternberg’s (2003) book, The Psychology of Problem Solving. For excellent material on mathematical heuristics, Polya is the master, and several of his books are listed. For information on cognitively guided instruction, look for articles by Carpenter & Fennema. For information on the relationship to cognitive load theory and problem solving, review the section on middle school and high school mathematics. For a look at the challenges students with learning disabilities and students at risk face, review the section on elementary mathematics and learning disabilities. For newly emerging research in creativity with linkages to mathematical giftedness, an up-and-coming researcher is Siriraman who seems to be on-target with research and practical implications in school settings. Richard Mayer is also looking into creativity.
We hope you enjoy your summer reading on Mathematical Problem Solving.
problem solving bibliography download
Implications
Our group examined problem solving along a continuum that began at the preschool level, and culminated with adult mathematicians. We found that if is useful to look at the problems that students attempt to solve. Some research found that students who were able to solve problems that used what was referred to as inconsistent language actually did better on achievement tests in both reading and mathematics. Other research suggested the use of non-routine problems is helpful. None of this work implied that students did not need skill proficiency, and some research ended with the recommendation that explicit teaching and modeling is useful. The bottom line is that problem solving needs to be an integral and important part of what we teach. Mathematics is not simply a set of skills.
Teachers and preservice teachers need to be apprised of what we know about children’s mathematical understanding. The cognitively guided instruction literature presented good evidence that young children can solve challenging problems by creating their own solutions, and that they are capable of making and testing conjectures. Taking these characteristics of young people into account should indeed influence our teaching. Building on their understanding and encouraging them to think will help them make sense of the mathematics and value their accomplishments. Math is not simply handed down like the family silver.
Fast forwarding into adulthood, we found that many working mathematicians didn’t select the field until well after their high school years. These people valued the creative and collaborative aspects so essential to problem solving that had not been apparent to them in earlier years. Just as cognitively guided instruction showed how far our youngest students can think mathematically, there is evidence that, at the secondary level, the problem solving enterprise should continue to promote mathematical thinking. Polite dissonance/argumentation in collaborative situations has been shown to foster successful work in problem solving. Considering the use a cognitive apprenticeship model where high school students shadow professional mathematicians is another suggestion that might be considered. Teaching at least some secondary mathematics as if it’s a laboratory activity fits well with this effort. In short, what happens in the world of mathematics should not be removed from a secondary classroom. It should be an integral part.
These opportunities need to be extended to all students. What happens in Special Education is a particularly critical issue. As these students are mainstreamed, we need to tap into their creative base. With these learners, skill based issues can be extremely challenging. Schema based instruction has been shown to be effective improving performance in problem solving and computation. These strategies enable students with learning disabilities to access the visual imagery.
Teachers and preservice teachers need to be apprised of what we know about children’s mathematical understanding. The cognitively guided instruction literature presented good evidence that young children can solve challenging problems by creating their own solutions, and that they are capable of making and testing conjectures. Taking these characteristics of young people into account should indeed influence our teaching. Building on their understanding and encouraging them to think will help them make sense of the mathematics and value their accomplishments. Math is not simply handed down like the family silver.
Fast forwarding into adulthood, we found that many working mathematicians didn’t select the field until well after their high school years. These people valued the creative and collaborative aspects so essential to problem solving that had not been apparent to them in earlier years. Just as cognitively guided instruction showed how far our youngest students can think mathematically, there is evidence that, at the secondary level, the problem solving enterprise should continue to promote mathematical thinking. Polite dissonance/argumentation in collaborative situations has been shown to foster successful work in problem solving. Considering the use a cognitive apprenticeship model where high school students shadow professional mathematicians is another suggestion that might be considered. Teaching at least some secondary mathematics as if it’s a laboratory activity fits well with this effort. In short, what happens in the world of mathematics should not be removed from a secondary classroom. It should be an integral part.
These opportunities need to be extended to all students. What happens in Special Education is a particularly critical issue. As these students are mainstreamed, we need to tap into their creative base. With these learners, skill based issues can be extremely challenging. Schema based instruction has been shown to be effective improving performance in problem solving and computation. These strategies enable students with learning disabilities to access the visual imagery.
Monday, May 4, 2009
Chiu (2008)
Chiu, M. (2008). Effects of argumentation on group micro-creativity: Statistical discourse analyses of algebra students’ collaborative problem solving. Contemporary Educational Psychology, 33, 382-402.
Introduction
This article was selected for multiple reasons. The first secondary blog review is about a middle school problem solving experiment. This study examined the work and behavior of high school students. This allowed us to present research about two different chunks of this six-year span. Also, there was a connection to the reviewed work at other levels. Both the elementary and the adult reviews touched upon two characteristics that also connected to this article: collaboration and creativity. Finally, I was intrigued by something the article extolled, dissonance.
Before discussing the actual study, it’s worthwhile to mention how the word creativity was used. The author pointed out that this is not a look at creativity with “big C” meaning that he was not trying to analyze the type of creativity that can affect society. Instead, he referred to the “small c” type of creativity also called micro-creativity. These are moments when group accepts a new idea that is useful in a problem-solving context.
Purpose
The purpose of the study was to take a detailed quantitative look at groups of high school students as they worked in a collaborative setting to solve algebra problems. Arguments and other group processes were examined to identify when and why “watershed breakpoints” occurred. Watershed breakpoints are time markers that separate the creative moments from the periods of low creativity. Although more than one hypotheses were examined in this study, there was an overall pervasive argumentation hypothesis: Polite argumentation will increase a problem solving group’s micro-creativity.
The author presents several reasons justifying the need for his study. He is using a new statistical method to analyze discourse. The method is referred to as dynamic multilevel analysis. He is carefully analyzing distinct time periods to examine the effects of argumentation and to find out when the micro-creativity does and does not occur. Understanding the results, he suggests, might help groups work together more effectively.
Methods
There were 80 ninth-grade participants. All attended an urban California high school that had an “overall [achievement] school score at the 40th percentile.” Although the students probably knew each other, they had not worked together in cooperative groups before the experiment.
Prior to beginning their problem solving work, the students were given a four-question status survey. This became one piece of information used to establish what Chiu called a status hierarchy. Subsequently, the students were given what was referred to as a challenging problem. Their classroom teacher had used the problem to introduce a new unit but neither students nor the teacher tried to solve the problem following the initial exposure. The problem is quoted below:
"You won a cruise from New York to London, but you arrive 5 hours late. So, the ship left without you. To catch the ship, you rent a helicopter. The ship travels at 22 miles an hour. The helicopter moves at 90 miles an hour. How long will it take you to catch the ship?"
Students worked on the problem in groups of four for 30 minutes. They were videotaped and transcriptions of the student dialogues were made. Two trained research assistants who were not privy to any of the hypotheses then coded the transcripts. On rare occasions, when the research assistants could not achieve consensus about coding, the author would make the final decision.
Once coded, the variables were analyzed using a new statistical tool called dynamic multi-level analysis. In 2005, the author was the lead researcher resulting in the publication of a paper about using this new method. T-tests were also used to determine differences between the groups. In all, twenty-five variables were examined. This review focused on a subset of the variables including grade point averages, politeness, rudeness, disagreements, watershed breakpoints, diversity, status differences, and micro-creativity
Results
First, note that this section is a condensation and generalization of a very complex analysis. Second, the author did report some success. He concludes by stating, “This model had a 70% accuracy rate for predicting micro-creativity in any given turn.”
Chiu was not surprised to find that groups with greater mean grade point averages were more successful in the problem-solving task. Other variables that were related to greater moments of micro-creativity included politeness, disagreements and higher social status. Variables that were not related to greater micro-creativity included diversity, status differences and questions. Successful groups were found to have more watershed breakpoints, that is, more transition points that separated periods of high micro-creativity from periods of low micro-creativity.
In the realm of argumentation, disagreements were placed into two distinct categories: rude and polite. The author found that without disagreements, flawed ideas continued to perpetuate more flawed ideas. Rude disagreements connect to micro-creativity in different ways. Although rude disagreements that identified faulty mathematics increased the group’s micro-creativity, they also were related to less micro-creativity on the part of the rude person. Overall, polite disagreement yielded more micro-creativity.
One thing still needs to be said. What about actually solving the problem? The groups with more micro-creative moments actually did have more success solving the problems.
Limitations
The author identifies some limitations. Three classrooms from one school is not necessarily a representative sample. It’s also hard to generalize from on session that used one problem. Other limitations are connected to the previous preparation of the students. While these students were not accustomed to group work, would the results differ for students who were? Another author-identified limitation is related to the number of speaker turns using the new statistical method of dynamic modeling analysis. A speaker turn is tallied each time a person says something to the group. Although the authors suggest their groups yielded enough speaker turns, they do bring out this limitation of the method. Fewer speaker turns could have invalidated the results.
Implications
This article looks at variables that are connected to problem solving success. Paying attention to these variables might have bearing on how groups can function best in the classroom. Teachers might do well to consider mathematics grades, how students well get along, and whether or not students are poised to argue politely when groups are composed. The make-up and the interactions of the group might have an impact on learning.
In addition to group composition, teachers might consider some ways to lead group members into productive modes. If polite argumentation is a good thing, perhaps students should be instructed in the art of respectful arguing. Students might also be given some strategies to deal with rudeness in order to lessen the periods of low creativity.
Although self-regulation was never mentioned in the article, I believe there is a strong implied connection. If group can profit from good processes and desirable individual behaviors, aren’t those prime factors that can enter into student awareness? We have seen that self-regulation can indeed be connected to academic success. Perhaps group success can also be bolstered by self-regulation of the members for the good of the group.
Strength and Weaknesses
The article was remarkably thorough. Chiu truly examined the process minute by minute, and searched for the key moments, that is, the watershed breakthroughs. I was impressed with all the factors the author tried to consider in order to be impartial. The coders were not aware of the hypothesis. Once coded, all the data received numerical values that were subjected to a rigorous statistical analysis. Impressive yes, but in truth, I need to know and understand more about the techniques he used before I make a more definitive appraisal.
I also did find some weaknesses. The article did not say enough about how the variable social status was coded. The four questions on the survey were about choosing group members. The social status variable was connected to social skills, and more than the survey was needed to determine that.
Another weakness was the lack of student work. The author did provide some samples of group dialogues, and that was helpful in understanding his analysis. However, I liked the problem, and I would have found it most revealing to see the student scratch work as they arrived at a solution. How the students solved or attempted to solve the problem was certainly not one of the variables looked at in this study. I wonder if it should have been.
I’m not sure that this is a weakness, but I do have one final concern. Although I never composed groups with the thorough analysis found in this study, I recall a really bad experience when I tried to stratify groups at the grade six level. My students saw through my attempts create a mix when assigned to the groups of four. I overheard children saying things like, “She’s put in this group so we have a smart one.” That caused me to stop the practice. Could group composition based on Chiu’s findings also be transparent?
Introduction
This article was selected for multiple reasons. The first secondary blog review is about a middle school problem solving experiment. This study examined the work and behavior of high school students. This allowed us to present research about two different chunks of this six-year span. Also, there was a connection to the reviewed work at other levels. Both the elementary and the adult reviews touched upon two characteristics that also connected to this article: collaboration and creativity. Finally, I was intrigued by something the article extolled, dissonance.
Before discussing the actual study, it’s worthwhile to mention how the word creativity was used. The author pointed out that this is not a look at creativity with “big C” meaning that he was not trying to analyze the type of creativity that can affect society. Instead, he referred to the “small c” type of creativity also called micro-creativity. These are moments when group accepts a new idea that is useful in a problem-solving context.
Purpose
The purpose of the study was to take a detailed quantitative look at groups of high school students as they worked in a collaborative setting to solve algebra problems. Arguments and other group processes were examined to identify when and why “watershed breakpoints” occurred. Watershed breakpoints are time markers that separate the creative moments from the periods of low creativity. Although more than one hypotheses were examined in this study, there was an overall pervasive argumentation hypothesis: Polite argumentation will increase a problem solving group’s micro-creativity.
The author presents several reasons justifying the need for his study. He is using a new statistical method to analyze discourse. The method is referred to as dynamic multilevel analysis. He is carefully analyzing distinct time periods to examine the effects of argumentation and to find out when the micro-creativity does and does not occur. Understanding the results, he suggests, might help groups work together more effectively.
Methods
There were 80 ninth-grade participants. All attended an urban California high school that had an “overall [achievement] school score at the 40th percentile.” Although the students probably knew each other, they had not worked together in cooperative groups before the experiment.
Prior to beginning their problem solving work, the students were given a four-question status survey. This became one piece of information used to establish what Chiu called a status hierarchy. Subsequently, the students were given what was referred to as a challenging problem. Their classroom teacher had used the problem to introduce a new unit but neither students nor the teacher tried to solve the problem following the initial exposure. The problem is quoted below:
"You won a cruise from New York to London, but you arrive 5 hours late. So, the ship left without you. To catch the ship, you rent a helicopter. The ship travels at 22 miles an hour. The helicopter moves at 90 miles an hour. How long will it take you to catch the ship?"
Students worked on the problem in groups of four for 30 minutes. They were videotaped and transcriptions of the student dialogues were made. Two trained research assistants who were not privy to any of the hypotheses then coded the transcripts. On rare occasions, when the research assistants could not achieve consensus about coding, the author would make the final decision.
Once coded, the variables were analyzed using a new statistical tool called dynamic multi-level analysis. In 2005, the author was the lead researcher resulting in the publication of a paper about using this new method. T-tests were also used to determine differences between the groups. In all, twenty-five variables were examined. This review focused on a subset of the variables including grade point averages, politeness, rudeness, disagreements, watershed breakpoints, diversity, status differences, and micro-creativity
Results
First, note that this section is a condensation and generalization of a very complex analysis. Second, the author did report some success. He concludes by stating, “This model had a 70% accuracy rate for predicting micro-creativity in any given turn.”
Chiu was not surprised to find that groups with greater mean grade point averages were more successful in the problem-solving task. Other variables that were related to greater moments of micro-creativity included politeness, disagreements and higher social status. Variables that were not related to greater micro-creativity included diversity, status differences and questions. Successful groups were found to have more watershed breakpoints, that is, more transition points that separated periods of high micro-creativity from periods of low micro-creativity.
In the realm of argumentation, disagreements were placed into two distinct categories: rude and polite. The author found that without disagreements, flawed ideas continued to perpetuate more flawed ideas. Rude disagreements connect to micro-creativity in different ways. Although rude disagreements that identified faulty mathematics increased the group’s micro-creativity, they also were related to less micro-creativity on the part of the rude person. Overall, polite disagreement yielded more micro-creativity.
One thing still needs to be said. What about actually solving the problem? The groups with more micro-creative moments actually did have more success solving the problems.
Limitations
The author identifies some limitations. Three classrooms from one school is not necessarily a representative sample. It’s also hard to generalize from on session that used one problem. Other limitations are connected to the previous preparation of the students. While these students were not accustomed to group work, would the results differ for students who were? Another author-identified limitation is related to the number of speaker turns using the new statistical method of dynamic modeling analysis. A speaker turn is tallied each time a person says something to the group. Although the authors suggest their groups yielded enough speaker turns, they do bring out this limitation of the method. Fewer speaker turns could have invalidated the results.
Implications
This article looks at variables that are connected to problem solving success. Paying attention to these variables might have bearing on how groups can function best in the classroom. Teachers might do well to consider mathematics grades, how students well get along, and whether or not students are poised to argue politely when groups are composed. The make-up and the interactions of the group might have an impact on learning.
In addition to group composition, teachers might consider some ways to lead group members into productive modes. If polite argumentation is a good thing, perhaps students should be instructed in the art of respectful arguing. Students might also be given some strategies to deal with rudeness in order to lessen the periods of low creativity.
Although self-regulation was never mentioned in the article, I believe there is a strong implied connection. If group can profit from good processes and desirable individual behaviors, aren’t those prime factors that can enter into student awareness? We have seen that self-regulation can indeed be connected to academic success. Perhaps group success can also be bolstered by self-regulation of the members for the good of the group.
Strength and Weaknesses
The article was remarkably thorough. Chiu truly examined the process minute by minute, and searched for the key moments, that is, the watershed breakthroughs. I was impressed with all the factors the author tried to consider in order to be impartial. The coders were not aware of the hypothesis. Once coded, all the data received numerical values that were subjected to a rigorous statistical analysis. Impressive yes, but in truth, I need to know and understand more about the techniques he used before I make a more definitive appraisal.
I also did find some weaknesses. The article did not say enough about how the variable social status was coded. The four questions on the survey were about choosing group members. The social status variable was connected to social skills, and more than the survey was needed to determine that.
Another weakness was the lack of student work. The author did provide some samples of group dialogues, and that was helpful in understanding his analysis. However, I liked the problem, and I would have found it most revealing to see the student scratch work as they arrived at a solution. How the students solved or attempted to solve the problem was certainly not one of the variables looked at in this study. I wonder if it should have been.
I’m not sure that this is a weakness, but I do have one final concern. Although I never composed groups with the thorough analysis found in this study, I recall a really bad experience when I tried to stratify groups at the grade six level. My students saw through my attempts create a mix when assigned to the groups of four. I overheard children saying things like, “She’s put in this group so we have a smart one.” That caused me to stop the practice. Could group composition based on Chiu’s findings also be transparent?
Sunday, May 3, 2009
Griffin & Jitendra (2008)
Griffin, C.C, Jitendra, A.K. (2009). Word problem-solving instruction in inclusive third-grade mathematics classrooms. The Journal of Educational Research, 102(3), 187-201.
I choose to review the Griffen and Jitendra (2009) article, word problem-solving instruction in third grade mathematics classrooms, due my interest in instructional strategies to help special education students remain in their general education classroom. The researchers make note of conducting this study due to the emphasis in standards that place high expectations and strong support for all students, as well as the No Child Left Behind Act of 2001 that places emphasis on measuring educational outcomes for all students.
The authors examined the effectiveness of strategy instruction taught by general educators in mixed ability classrooms. They compared mathematical word problem-solving performance and computational skills of students who receive schema-based instruction (SBI) with students who received general strategy instruction (GSI). Schema-based instruction involves explicit instruction in using schematic diagrams to depict word problems visually before solving them. General strategy instruction involves the use of heuristic and multiple strategies base on Polya’s (1945/1990) seminal principles for problem solving, which is typically found in mathematics textbooks.
The present study focuses on solving addition and subtraction word problems of third grade student in heterogeneous mathematical classrooms. Participants were 60 third grade students randomly assigned to treatment conditions. The participants were pretested and posttest with mathematical problem-solving and computation tests. The tests were repeated throughout an 18-week intervention.
Purpose/Problem
The purpose of this study examined four questions. First, they addressed whether students benefit from either SBI or GSI instruction. Second, they examined whether the strategy effects word persist over time. Third, they questioned whether the change in word problem-solving performance, over time during the intervention phase, was similar for the two conditions. Finally, they assessed the influence of word problem-solving instruction on the development of computational skills.
In the present study Griffin and Jitendra build upon previous research by Jitendra, Griffin & Haria, et. al (2007) that investigated the differential effects of SBI and GSI in promoting mathematical problem solving, computational skills and mathematics achievement of low performing third grade students taught by classroom teachers. This study looked at SBI using explicit instruction and schematic diagrams to depict words problems visually prior to solving them. The results favored SBI over GSI in enhancing students’ mathematical word problem-solving skills at posttest and maintenance. They found SBI with schematic diagrams to be a powerful instructional approach for low-achieving students. The current study looked at the effects of SBI compared to GSI in promoting mathematical problem solving, computational skills and mathematics achievement of third grade students, taught by classroom teachers, in mixed ability classrooms.
Method-Design
The study used a between-subjects design to investigate the effects of word problem-solving strategy instruction. Participants were 60 students (30 boys and 30 girls) from three classrooms attending third grade. The elementary school was located in a college town in the southeastern United States.
Students were rank ordered and then matched on their scores from the Stanford Achievement Test 9 (SAT-9) subtest for mathematical problem solving. Students were matched by pairs and randomly assigned to either intervention (SBI) or comparison condition (GSI). Four instructional groups were formed. Two groups received SBI and two groups participated in the comparison condition receiving GSI. Three general education teachers and one special education teacher participated in the study. They were randomly assigned to the two conditions and provided all instruction in the study.
A word problem-solving unit developed for this study supplanted the classroom mathematics lessons on one day in the five day week cycle. Both conditions included 20 instructional sessions that were implemented for 100 minutes at a time on one day during the week. The instruction was delivered across 18 weeks of the school year. The SBI students received instruction for solving one-step problems that involved two phases, problem-solving schema and problem solution. The GSI students received instruction for solving word problems based on Polya’s (1945,1990) model with four steps: read and understand the problem, plan to solve the problem, solve the problem, and look back and check.
Results
The results answered the four questions the researchers addressed. Did students benefit from either SBI or GSI Instruction? The researchers hypothesized that the SBI instruction would lead to better problem solving performance than GSI because of the schematic imagery. The findings of this study did show that the SBI instruction lead to greater benefits than the GSI condition with regards to word problem-solving in mathematics. The second question examined whether the strategy effects would persist over time. Findings revealed that both conditions maintained the positive learning effect 12 weeks later. The third question asked whether the changes in word problem-solving performance over time during the intervention phase were similar for both conditions. The results of this study show that the SBI approach resulted in a statistically significant effect (effect size = .94). Finally, what is the influence of word problem-solving instruction on the development of computational skills? The researchers hypothesized that students in both conditions would show improvement over the course of the study. The findings of the study supported the hypothesis that students in both conditions would improve their computational skills. From pretest to posttest a positive effect was shown (effect size = .97).
Conclusion
The researchers make note of some limitations to this study and suggest caution when interpreting the findings. One limitation was the lack of a true control, which in turn threatens internal validity. Including a true control condition is essential to conclude an attribute of gains over time to either intervention condition. The 100-minute session per week did not reflect typical classroom practice. The researchers are convinced that this distribution of time could have influenced students’ ability to internalize and retain SBI. Finally, the researchers make note of the importance of reading comprehension as a contributor to students word problem-solving performance and they did not control for the students’ initial reading levels.
Overall the researchers found that SBI and GSI is a recommended strategy instruction, due to the improvement of third grade student’s performance on word and computational problems. If teachers can provide effective strategies and opportunities for problem-solving, this study may help to support children with mixed abilities in general education classroom, which is consistent with the emphasis of standards and NCLB (high expectations and strong support for all students). Another support from this study is in addressing mixed ability students, an effort to help maintain students with learning disabilities in their general education classrooms is made possible.
I choose to review the Griffen and Jitendra (2009) article, word problem-solving instruction in third grade mathematics classrooms, due my interest in instructional strategies to help special education students remain in their general education classroom. The researchers make note of conducting this study due to the emphasis in standards that place high expectations and strong support for all students, as well as the No Child Left Behind Act of 2001 that places emphasis on measuring educational outcomes for all students.
The authors examined the effectiveness of strategy instruction taught by general educators in mixed ability classrooms. They compared mathematical word problem-solving performance and computational skills of students who receive schema-based instruction (SBI) with students who received general strategy instruction (GSI). Schema-based instruction involves explicit instruction in using schematic diagrams to depict word problems visually before solving them. General strategy instruction involves the use of heuristic and multiple strategies base on Polya’s (1945/1990) seminal principles for problem solving, which is typically found in mathematics textbooks.
The present study focuses on solving addition and subtraction word problems of third grade student in heterogeneous mathematical classrooms. Participants were 60 third grade students randomly assigned to treatment conditions. The participants were pretested and posttest with mathematical problem-solving and computation tests. The tests were repeated throughout an 18-week intervention.
Purpose/Problem
The purpose of this study examined four questions. First, they addressed whether students benefit from either SBI or GSI instruction. Second, they examined whether the strategy effects word persist over time. Third, they questioned whether the change in word problem-solving performance, over time during the intervention phase, was similar for the two conditions. Finally, they assessed the influence of word problem-solving instruction on the development of computational skills.
In the present study Griffin and Jitendra build upon previous research by Jitendra, Griffin & Haria, et. al (2007) that investigated the differential effects of SBI and GSI in promoting mathematical problem solving, computational skills and mathematics achievement of low performing third grade students taught by classroom teachers. This study looked at SBI using explicit instruction and schematic diagrams to depict words problems visually prior to solving them. The results favored SBI over GSI in enhancing students’ mathematical word problem-solving skills at posttest and maintenance. They found SBI with schematic diagrams to be a powerful instructional approach for low-achieving students. The current study looked at the effects of SBI compared to GSI in promoting mathematical problem solving, computational skills and mathematics achievement of third grade students, taught by classroom teachers, in mixed ability classrooms.
Method-Design
The study used a between-subjects design to investigate the effects of word problem-solving strategy instruction. Participants were 60 students (30 boys and 30 girls) from three classrooms attending third grade. The elementary school was located in a college town in the southeastern United States.
Students were rank ordered and then matched on their scores from the Stanford Achievement Test 9 (SAT-9) subtest for mathematical problem solving. Students were matched by pairs and randomly assigned to either intervention (SBI) or comparison condition (GSI). Four instructional groups were formed. Two groups received SBI and two groups participated in the comparison condition receiving GSI. Three general education teachers and one special education teacher participated in the study. They were randomly assigned to the two conditions and provided all instruction in the study.
A word problem-solving unit developed for this study supplanted the classroom mathematics lessons on one day in the five day week cycle. Both conditions included 20 instructional sessions that were implemented for 100 minutes at a time on one day during the week. The instruction was delivered across 18 weeks of the school year. The SBI students received instruction for solving one-step problems that involved two phases, problem-solving schema and problem solution. The GSI students received instruction for solving word problems based on Polya’s (1945,1990) model with four steps: read and understand the problem, plan to solve the problem, solve the problem, and look back and check.
Results
The results answered the four questions the researchers addressed. Did students benefit from either SBI or GSI Instruction? The researchers hypothesized that the SBI instruction would lead to better problem solving performance than GSI because of the schematic imagery. The findings of this study did show that the SBI instruction lead to greater benefits than the GSI condition with regards to word problem-solving in mathematics. The second question examined whether the strategy effects would persist over time. Findings revealed that both conditions maintained the positive learning effect 12 weeks later. The third question asked whether the changes in word problem-solving performance over time during the intervention phase were similar for both conditions. The results of this study show that the SBI approach resulted in a statistically significant effect (effect size = .94). Finally, what is the influence of word problem-solving instruction on the development of computational skills? The researchers hypothesized that students in both conditions would show improvement over the course of the study. The findings of the study supported the hypothesis that students in both conditions would improve their computational skills. From pretest to posttest a positive effect was shown (effect size = .97).
Conclusion
The researchers make note of some limitations to this study and suggest caution when interpreting the findings. One limitation was the lack of a true control, which in turn threatens internal validity. Including a true control condition is essential to conclude an attribute of gains over time to either intervention condition. The 100-minute session per week did not reflect typical classroom practice. The researchers are convinced that this distribution of time could have influenced students’ ability to internalize and retain SBI. Finally, the researchers make note of the importance of reading comprehension as a contributor to students word problem-solving performance and they did not control for the students’ initial reading levels.
Overall the researchers found that SBI and GSI is a recommended strategy instruction, due to the improvement of third grade student’s performance on word and computational problems. If teachers can provide effective strategies and opportunities for problem-solving, this study may help to support children with mixed abilities in general education classroom, which is consistent with the emphasis of standards and NCLB (high expectations and strong support for all students). Another support from this study is in addressing mixed ability students, an effort to help maintain students with learning disabilities in their general education classrooms is made possible.
Livne & Milgram (2006)
Livne, N. L. & Milgram, R.M. (2006). Academic versus creative abilities in mathematics: Two components of the same construct? Creativity Research Journal, 18, 199- 212.
Chosen Study
Professional mathematicians show remarkable creativity in problem-solving, quite distinctive from academic procedures we teach in our elementary through undergraduate mathematics classrooms. This article presents an approach for detecting and assessing mathematical creativity in high school students. The selection of this article stemmed from a growing personal interest in an emerging body of research in the role of creativity in mathematical problem solving, and an interest in psychometric multidimensionality modeling procedures.
Problem and Purpose
Livne and Milgram postulate that most K-12 mathematics programs fail to identify students with high abilities in mathematical creativity. Professional mathematicians routinely employ creativity in problem solving, yet creativity is not often addressed or assessed in K-12 students. Students who are truly gifted in creative mathematical problem solving may not be identified or encouraged to pursue mathematical study, especially if their academic abilities in mathematics are not exceptional. One of the barriers to identification of creative ability in mathematics is the lack of a valid and reliable assessment instrument capable of detecting mathematical creativity.
The researchers proposed that creative ability was distinct from academic ability. The purpose of this study was to investigate methods to distinguish between creative and academic ability in mathematics. A second purpose was to explore whether levels of ability within each of the two types of ability could be discerned.
Literature Review
Livne and Milgram begin the article with an extensive literature review establishing the significance and need for the direction of research. Early studies by Hadamard, Halmos, and Muir are cited, showing that achievements of professional mathematicians require creative problem solving more than computational skill. These early studies defined two types of cognitive ability in mathematics – academic ability and creative ability. Academic ability is important for achieving high grades in K-12 coursework, but creative ability is more important for graduate studies and professional mathematics achievement. The literature defines creative ability in mathematics as the ability to discern mathematical patterns and relationships using nonalgorithmic thinking, and the ability to devise multiple original solution strategies to solve problems.
Research in the 1990’s detected the need for establishing operational definitions of creative ability and academic ability in mathematics. Within each of these categories, the literature also suggested the need for defining and assessing levels of ability (ordinary, mildly gifted, moderately gifted, and profound).
Methodology
The researchers developed two new measurement instruments and a new psychometric model to assess and identify creative ability in mathematics in high school students. The proposed mathematical model used structural equation modeling to distinguish whether mathematics ability was composed of four distinct abilities (Academic Ability in Mathematics, Creative Ability in Mathematics, General Creative Ability, and General Intelligence). Within each of the four types of abilities, levels of ability were defined and measured (nongifted, mildly gifted, moderately gifted, profound).
Instrumentation
To assess four types of ability, the researchers utilized six assessment instruments. Two of the instruments, the Multiscale Academic and Creative Abilities in Mathematics (MACAM) and the Tel Aviv Activities and Accomplishments Inventory: Mathematics (TAAI: M) were researcher-developed instruments. The MACAM consisted of eight academic and eight creative questions. Academic questions were computational questions with one solution path and one correct answer. Creative questions were characterized by more than one solution path or correct answer. The TAAI: M was a self-report questionnaire. Participants provided information on leisure time, non-school-based, creative mathematical activities such as solving puzzles or playing games. Scores on both of the researcher-developed instruments were categorized into four ability levels: nongifted, mild, moderate, and profound.
The other assessment measures were from instruments widely used in Israel. The researchers used combinations of scores from the six instruments to measure four types of ability:
Ability Type 1: Academic Ability in Mathematics
1.) Grade in most recent semester of mathematics coursework
2.) Multiscale Academic and Creative Abilities in Mathematics (Academic Questions)
Ability Type 2: Creative Ability in Mathematics
1.) Multiscale Academic and Creative Abilities in Mathematics (Creative Questions)
2.) Tel Aviv Activities and Accomplishments Inventory: Mathematics
Ability Type 3: General Creativity (not confined to mathematics)
1.) Tel Aviv Creativity Test
Ability Type 4: General Intelligence (Verbal and Nonverbal)
1.) Abstract Verbal Thinking Test
2.) Advanced Progressive Matrices Test
Participants were 1,090 10th- and 11th-grade students from 22 public schools in Israel, randomly selected from a list of 571 schools. The participants were divided into four equal groups and all six instruments were administered to each group in three 2-hour sessions, in a counterbalanced design. Scores were analyzed using structural equation modeling as a confirmatory factor analysis procedure.
Results
Analysis 1 - Discriminant Validity Across Ability Types
A complex multidimensional mathematical procedure, structural equation modeling (SEM), was used to investigate discriminant validity for the four types of ability. The mathematical model was statistically significant, indicating sufficient evidence for construct validity for the four types of ability (academic ability in mathematics, creative ability in mathematics, general intelligence, and creative intelligence).
Analysis 2 - Discriminant Validity Across Ability Levels
Scores on the six instruments were divided into four levels of ability (nongifted, mild, moderate, profound). The 4 x 4 model (4 Types of Ability x 4 Levels of Ability) was compared to seven alternative models which replaced the 4 ability factors with various combinations of 1, 2, and 3 factors. SEM findings indicated strong statistical evidence for the existence of four distinct levels of ability, arranged in hierarchical order with nongifted being the most common trait (78% of the sample) and profoundly gifted appearing rarely (1% of the sample).
Analysis 3 - General Creativity Tests Predict Mathematical Creativity
Given the statistical support for the validity of the 4x4 model, the researchers next investigated whether creative mathematical ability could be predicted from scores of general tests of creativity and from intelligence test scores. Scores on the Tel Aviv Creativity Test, a test of general creative ability, produced statistically significant predictions of creative ability in mathematics.
Analysis 4 - General Intelligence Tests Predict Academic Ability
Scores on the general intelligence tests predicted academic ability in mathematics. In contrast, no statistically significant relationship was detected between scores on the general intelligence tests and scores on the mathematical creativity tests.
Discussion
The most important finding of this study was statistical support of the premise that mathematics cognition consists of two distinct abilities – academic and creative. In this research study, scores on instruments measuring academic performance in mathematics did not predict creative ability in mathematics, and measures of mathematical creativity did not predict academic ability in mathematics. Furthermore, although general measures of intelligence were related to academic performance, these measures were distinct from measures of creative mathematical ability. General measures of creativity did not predict academic performance in mathematics but did predict creative ability in mathematics. These findings indicate that instruments measuring general creative ability are more effective in detecting mathematical creativity in high school students than are traditional measures of academic ability or intelligence.
A second important finding of this study was support of using the psychometric technique of structural equation modeling to validate the researcher-developed instruments. Structural equation modeling also supported the 4 x 4 model of ability in mathematics. This 4 x 4 model supports the conceptualization that achievement in mathematics consists of type of ability (academic or creative) and level of ability (nongifted, mild, moderate, profound). The two dimensions (ability and level) interact to create distinct aspects of mathematical ability.
Critique
This study was carefully planned and executed, with complex instrumentation and very complex statistical analysis. One of the major strengths of the study was the innovative use of structural equation modeling to perform a confirmatory factor analysis validating instrumentation and supporting the 4x4 model. Sampling procedures were rigorous, and resulted in a large sample size more than sufficient for the modeling procedures. The article included several tables clarifying use of instrumentation, and summarizing fit indexes and other statistical results. A graphic included in the appendix illustrated the factors and paths of the structural equation modeling procedure, clarifying the design.
One of the major weaknesses of the article was the lack of detail regarding the researcher-developed assessment instruments. Although the researcher used structural equation modeling to confirm validity, the researchers did not report reliability measures such as the KR-20 or Cronbach’s alpha. The TAAI:M inventory, a self-report of the number and difficulty of mathematical leisure time activities, was used as a measure of mathematical creativity. It is unclear from this article whether this is a reliable measure.
Although the researchers developed a complex and statistically significant modeling procedure, there was no reported attempt to cross-validate the model or replicate the procedure using an independent sample. This would be an area for future research.
Theoretical, Methodological, and Practical Conclusions
The theoretical implications of this study support the growing research base that mathematical ability is comprised of two distinct components, academic and creative. This research indicates that measures of intelligence predict academic ability but not creative ability, and instruments measuring general creativity were better predictors of creative ability in math. This suggests important practical implications for identifying creatively-gifted mathematics students, who oftentimes are not identified at the high school level. With their continuing efforts to develop measures capable of detecting mathematical creativity, the researchers contributed a possible methodology for identifying different types of mathematical ability. As these methods are continued to be refined, practitioners can look forward to earlier identification of students with ability for future creative work in mathematics.
Chosen Study
Professional mathematicians show remarkable creativity in problem-solving, quite distinctive from academic procedures we teach in our elementary through undergraduate mathematics classrooms. This article presents an approach for detecting and assessing mathematical creativity in high school students. The selection of this article stemmed from a growing personal interest in an emerging body of research in the role of creativity in mathematical problem solving, and an interest in psychometric multidimensionality modeling procedures.
Problem and Purpose
Livne and Milgram postulate that most K-12 mathematics programs fail to identify students with high abilities in mathematical creativity. Professional mathematicians routinely employ creativity in problem solving, yet creativity is not often addressed or assessed in K-12 students. Students who are truly gifted in creative mathematical problem solving may not be identified or encouraged to pursue mathematical study, especially if their academic abilities in mathematics are not exceptional. One of the barriers to identification of creative ability in mathematics is the lack of a valid and reliable assessment instrument capable of detecting mathematical creativity.
The researchers proposed that creative ability was distinct from academic ability. The purpose of this study was to investigate methods to distinguish between creative and academic ability in mathematics. A second purpose was to explore whether levels of ability within each of the two types of ability could be discerned.
Literature Review
Livne and Milgram begin the article with an extensive literature review establishing the significance and need for the direction of research. Early studies by Hadamard, Halmos, and Muir are cited, showing that achievements of professional mathematicians require creative problem solving more than computational skill. These early studies defined two types of cognitive ability in mathematics – academic ability and creative ability. Academic ability is important for achieving high grades in K-12 coursework, but creative ability is more important for graduate studies and professional mathematics achievement. The literature defines creative ability in mathematics as the ability to discern mathematical patterns and relationships using nonalgorithmic thinking, and the ability to devise multiple original solution strategies to solve problems.
Research in the 1990’s detected the need for establishing operational definitions of creative ability and academic ability in mathematics. Within each of these categories, the literature also suggested the need for defining and assessing levels of ability (ordinary, mildly gifted, moderately gifted, and profound).
Methodology
The researchers developed two new measurement instruments and a new psychometric model to assess and identify creative ability in mathematics in high school students. The proposed mathematical model used structural equation modeling to distinguish whether mathematics ability was composed of four distinct abilities (Academic Ability in Mathematics, Creative Ability in Mathematics, General Creative Ability, and General Intelligence). Within each of the four types of abilities, levels of ability were defined and measured (nongifted, mildly gifted, moderately gifted, profound).
Instrumentation
To assess four types of ability, the researchers utilized six assessment instruments. Two of the instruments, the Multiscale Academic and Creative Abilities in Mathematics (MACAM) and the Tel Aviv Activities and Accomplishments Inventory: Mathematics (TAAI: M) were researcher-developed instruments. The MACAM consisted of eight academic and eight creative questions. Academic questions were computational questions with one solution path and one correct answer. Creative questions were characterized by more than one solution path or correct answer. The TAAI: M was a self-report questionnaire. Participants provided information on leisure time, non-school-based, creative mathematical activities such as solving puzzles or playing games. Scores on both of the researcher-developed instruments were categorized into four ability levels: nongifted, mild, moderate, and profound.
The other assessment measures were from instruments widely used in Israel. The researchers used combinations of scores from the six instruments to measure four types of ability:
Ability Type 1: Academic Ability in Mathematics
1.) Grade in most recent semester of mathematics coursework
2.) Multiscale Academic and Creative Abilities in Mathematics (Academic Questions)
Ability Type 2: Creative Ability in Mathematics
1.) Multiscale Academic and Creative Abilities in Mathematics (Creative Questions)
2.) Tel Aviv Activities and Accomplishments Inventory: Mathematics
Ability Type 3: General Creativity (not confined to mathematics)
1.) Tel Aviv Creativity Test
Ability Type 4: General Intelligence (Verbal and Nonverbal)
1.) Abstract Verbal Thinking Test
2.) Advanced Progressive Matrices Test
Participants were 1,090 10th- and 11th-grade students from 22 public schools in Israel, randomly selected from a list of 571 schools. The participants were divided into four equal groups and all six instruments were administered to each group in three 2-hour sessions, in a counterbalanced design. Scores were analyzed using structural equation modeling as a confirmatory factor analysis procedure.
Results
Analysis 1 - Discriminant Validity Across Ability Types
A complex multidimensional mathematical procedure, structural equation modeling (SEM), was used to investigate discriminant validity for the four types of ability. The mathematical model was statistically significant, indicating sufficient evidence for construct validity for the four types of ability (academic ability in mathematics, creative ability in mathematics, general intelligence, and creative intelligence).
Analysis 2 - Discriminant Validity Across Ability Levels
Scores on the six instruments were divided into four levels of ability (nongifted, mild, moderate, profound). The 4 x 4 model (4 Types of Ability x 4 Levels of Ability) was compared to seven alternative models which replaced the 4 ability factors with various combinations of 1, 2, and 3 factors. SEM findings indicated strong statistical evidence for the existence of four distinct levels of ability, arranged in hierarchical order with nongifted being the most common trait (78% of the sample) and profoundly gifted appearing rarely (1% of the sample).
Analysis 3 - General Creativity Tests Predict Mathematical Creativity
Given the statistical support for the validity of the 4x4 model, the researchers next investigated whether creative mathematical ability could be predicted from scores of general tests of creativity and from intelligence test scores. Scores on the Tel Aviv Creativity Test, a test of general creative ability, produced statistically significant predictions of creative ability in mathematics.
Analysis 4 - General Intelligence Tests Predict Academic Ability
Scores on the general intelligence tests predicted academic ability in mathematics. In contrast, no statistically significant relationship was detected between scores on the general intelligence tests and scores on the mathematical creativity tests.
Discussion
The most important finding of this study was statistical support of the premise that mathematics cognition consists of two distinct abilities – academic and creative. In this research study, scores on instruments measuring academic performance in mathematics did not predict creative ability in mathematics, and measures of mathematical creativity did not predict academic ability in mathematics. Furthermore, although general measures of intelligence were related to academic performance, these measures were distinct from measures of creative mathematical ability. General measures of creativity did not predict academic performance in mathematics but did predict creative ability in mathematics. These findings indicate that instruments measuring general creative ability are more effective in detecting mathematical creativity in high school students than are traditional measures of academic ability or intelligence.
A second important finding of this study was support of using the psychometric technique of structural equation modeling to validate the researcher-developed instruments. Structural equation modeling also supported the 4 x 4 model of ability in mathematics. This 4 x 4 model supports the conceptualization that achievement in mathematics consists of type of ability (academic or creative) and level of ability (nongifted, mild, moderate, profound). The two dimensions (ability and level) interact to create distinct aspects of mathematical ability.
Critique
This study was carefully planned and executed, with complex instrumentation and very complex statistical analysis. One of the major strengths of the study was the innovative use of structural equation modeling to perform a confirmatory factor analysis validating instrumentation and supporting the 4x4 model. Sampling procedures were rigorous, and resulted in a large sample size more than sufficient for the modeling procedures. The article included several tables clarifying use of instrumentation, and summarizing fit indexes and other statistical results. A graphic included in the appendix illustrated the factors and paths of the structural equation modeling procedure, clarifying the design.
One of the major weaknesses of the article was the lack of detail regarding the researcher-developed assessment instruments. Although the researcher used structural equation modeling to confirm validity, the researchers did not report reliability measures such as the KR-20 or Cronbach’s alpha. The TAAI:M inventory, a self-report of the number and difficulty of mathematical leisure time activities, was used as a measure of mathematical creativity. It is unclear from this article whether this is a reliable measure.
Although the researchers developed a complex and statistically significant modeling procedure, there was no reported attempt to cross-validate the model or replicate the procedure using an independent sample. This would be an area for future research.
Theoretical, Methodological, and Practical Conclusions
The theoretical implications of this study support the growing research base that mathematical ability is comprised of two distinct components, academic and creative. This research indicates that measures of intelligence predict academic ability but not creative ability, and instruments measuring general creativity were better predictors of creative ability in math. This suggests important practical implications for identifying creatively-gifted mathematics students, who oftentimes are not identified at the high school level. With their continuing efforts to develop measures capable of detecting mathematical creativity, the researchers contributed a possible methodology for identifying different types of mathematical ability. As these methods are continued to be refined, practitioners can look forward to earlier identification of students with ability for future creative work in mathematics.
Thursday, April 9, 2009
Fuchs et al. (2008)
Fuchs, L. S., Seethaler, P. M., Powell, S. R., Fuchs, D., Hamlett, C. L., & Fletcher, J. M. (2008). Effects of preventative tutoring on the mathematical problem solving of third-grade students with math and reading difficulties. Exceptional Children, 75(2), 155-73.
I reviewed an article on mathematical instructional strategies for students in the third grade. The study looked at the effects of preventive tutoring on mathematical word problem solving of third-grade students with math and reading difficulties. The researchers assessed the effects of an intervention protocol as a tool to conduct secondary preventative tutoring for students in the third grade. A schema-broadening tutoring program was used to teach the students the following: focus on the mathematical structure of three problem types, recognize problems belonging to those three problem type schemas, solve the three word-problem types and finally transfer solution methods to future math problems. Thirty-five students were assigned randomly to either receive secondary preventative tutoring 3 times per week for 30 minuets per session, over 12 sessions or remain in their general education math program.
The researcher suggest a need exist for effective prevention strategies to remediate word-problem deficits early, in the primary grades. This need is based on the 2004 reauthorization of Individuals with Disability Education Act (IDEA) that permits states to switch from traditional intelligence-achievement discrepancy model for identifying learning disabilities to a model of identification known as RtI. Simply put RtI is an intervention that is part of a multitier prevention system that identifies students who are struggling in the “universal” core program. I found this study to be of relevance because it is current (2008) information that could help general educators and special educators as they implement secondary preventative intervention tools to assess students that might be at risk for difficulties in math.
Purpose/Problem
The main purpose of this study was to assess the efficacy of a secondary preventive tutoring protocol addressing math word problems at third grade. In addition the researchers extended their research on secondary tutoring protocols and their research program on schema-broadening instruction in three ways. First, the researchers screened large numbers of students to identify a subset of students with difficulty in math and reading that mirrored many students that are classified as learning disabled. Second, they randomly assigned students who met the specific profile of students that averaged the 10th percentile in math and reading performance to either receive tutoring or remain in their general education math program. This allowed the researchers to estimate the effects of secondary preventative tutoring. Finally, the researchers schema-broadening tutoring protocol addressed math word problem types that the participants would typically see.
The researchers tested out a particular tutoring protocol they called schema-broadening instruction. They built upon a previous study by Jitendra et al. (2007). Jitendra et. al looked at schema based strategy instruction for third grade students experiencing math difficulties. They randomly assigned 88 students, within pairs, to either the schema-based strategy instruction or a metacognitive planning and organizational treatment. The intervention occurred in groups of 15 to 16 students for eight to nine weeks, five days per week, for 25-minute sessions. The schema-based strategy instruction focused on three instructional components: understanding the problem type, recognize basic schema for problem type, and solve the problem type. At the end of the treatment and six weeks later Jitendra et.al found significant effects, in the moderate range favored the schema-base strategy on the authors word-problem posttest.
The Fuchs et. al study differed from the Jitendra et. al study in the following ways. First, the population of students had documented math and reading difficulties. Second, the study provided an intervention that was done one-on-one instead of whole class level. Finally, the researchers protocol relied on schema-broadening instruction in four instructional components: understanding problem type, recognize basic schema for problem type, solve the problem type, and transfer. The Jitendra et al. study focused on the first three instructional components of the Fuchs et. al study with the exception of the far-transfer component.
Method – Design
The study screened 511 third-grade students in 29 classrooms from eight schools in a southeastern urban district, for math and reading difficulties. A sample size of 42 students was obtained by those students who scored below the 26th percentile on the Wide Range Achievement Test (WRAT) subtest for arithmetic and reading. During the study seven students moved leaving a sample size of 35. The researchers determined no significant differences between the students who moved and the sample of remaining 35 students.
A randomized controlled trial was used to estimate the effects of an explicit schema-broadening tutoring protocol, on math problem solving at third grade. This was contrasted with the sole reliance on general education math instruction. Two groups of students made up the study: the students in their general education classroom with instruction on problem-solution instruction (control group), and the students receiving the schema-broadening tutoring. All instruction was given at the same time so students would not miss time from their regular classrooms.
Each session was conducted one-to-one outside the student’s classroom and lasted 20 to 30 minutes. Sessions occurred three times per week for 12 weeks. The first two weeks of intervention taught students foundational skills for successful problem solving. The next three 3-week sessions taught one problem type per 3-week time period, with cumulative review of previously taught problem types built into daily lessons. The final week addressed all three problem types: total problem, combining two or more quantities to make a total, difference problems, compare a bigger and smaller quantity to find the difference, and finally change problems, increasing or decreasing a starting quantity to end up with a new quantity.
Results
The researchers make note that this study, due to the small sample size, is underpowered and they present and discuss effect size for nonsignificant effects to reduce the possibility of rejecting a potentially useful intervention. They caution the readers that effect sizes associated with nonsignificant differences should be interpreted with caution. Most of the foundational skills involving word problems improvement at a statistically significant level did not vary as a function of treatment condition (i.e., control vs. tutoring). Effects sizes favoring the treatment group were large on word problems involving one-digit operands with no irrelevant information or charts, graphs, or pictures (0.69), and on word problems assessing more complex problems with two-digit operands with and without irrelevant information or charts, graphs or pictures (1.80).
Conclusion
In summary the researchers do caution the small sample size and the effect sizes associated with nonsignificant differences, due to inadequate statistical power, may be important to future research. A suggestion from the researchers of a larger experimental field trial with appropriate power may yield statistically significant differences. More importantly the present study, schema-based strategy instruction, can be generalized beyond students with specific math difficulty to students with substantial deficits in math as well as reading. It seems that the kind of treatment used in this study would be applicable to students with disabilities also. Another suggestion from the researchers speaks to the use of a schema-broadening protocol, such as the one used in this study, could be useful to school practitioners as they implement RtI models of learning disabilities identification at third grade.
I reviewed an article on mathematical instructional strategies for students in the third grade. The study looked at the effects of preventive tutoring on mathematical word problem solving of third-grade students with math and reading difficulties. The researchers assessed the effects of an intervention protocol as a tool to conduct secondary preventative tutoring for students in the third grade. A schema-broadening tutoring program was used to teach the students the following: focus on the mathematical structure of three problem types, recognize problems belonging to those three problem type schemas, solve the three word-problem types and finally transfer solution methods to future math problems. Thirty-five students were assigned randomly to either receive secondary preventative tutoring 3 times per week for 30 minuets per session, over 12 sessions or remain in their general education math program.
The researcher suggest a need exist for effective prevention strategies to remediate word-problem deficits early, in the primary grades. This need is based on the 2004 reauthorization of Individuals with Disability Education Act (IDEA) that permits states to switch from traditional intelligence-achievement discrepancy model for identifying learning disabilities to a model of identification known as RtI. Simply put RtI is an intervention that is part of a multitier prevention system that identifies students who are struggling in the “universal” core program. I found this study to be of relevance because it is current (2008) information that could help general educators and special educators as they implement secondary preventative intervention tools to assess students that might be at risk for difficulties in math.
Purpose/Problem
The main purpose of this study was to assess the efficacy of a secondary preventive tutoring protocol addressing math word problems at third grade. In addition the researchers extended their research on secondary tutoring protocols and their research program on schema-broadening instruction in three ways. First, the researchers screened large numbers of students to identify a subset of students with difficulty in math and reading that mirrored many students that are classified as learning disabled. Second, they randomly assigned students who met the specific profile of students that averaged the 10th percentile in math and reading performance to either receive tutoring or remain in their general education math program. This allowed the researchers to estimate the effects of secondary preventative tutoring. Finally, the researchers schema-broadening tutoring protocol addressed math word problem types that the participants would typically see.
The researchers tested out a particular tutoring protocol they called schema-broadening instruction. They built upon a previous study by Jitendra et al. (2007). Jitendra et. al looked at schema based strategy instruction for third grade students experiencing math difficulties. They randomly assigned 88 students, within pairs, to either the schema-based strategy instruction or a metacognitive planning and organizational treatment. The intervention occurred in groups of 15 to 16 students for eight to nine weeks, five days per week, for 25-minute sessions. The schema-based strategy instruction focused on three instructional components: understanding the problem type, recognize basic schema for problem type, and solve the problem type. At the end of the treatment and six weeks later Jitendra et.al found significant effects, in the moderate range favored the schema-base strategy on the authors word-problem posttest.
The Fuchs et. al study differed from the Jitendra et. al study in the following ways. First, the population of students had documented math and reading difficulties. Second, the study provided an intervention that was done one-on-one instead of whole class level. Finally, the researchers protocol relied on schema-broadening instruction in four instructional components: understanding problem type, recognize basic schema for problem type, solve the problem type, and transfer. The Jitendra et al. study focused on the first three instructional components of the Fuchs et. al study with the exception of the far-transfer component.
Method – Design
The study screened 511 third-grade students in 29 classrooms from eight schools in a southeastern urban district, for math and reading difficulties. A sample size of 42 students was obtained by those students who scored below the 26th percentile on the Wide Range Achievement Test (WRAT) subtest for arithmetic and reading. During the study seven students moved leaving a sample size of 35. The researchers determined no significant differences between the students who moved and the sample of remaining 35 students.
A randomized controlled trial was used to estimate the effects of an explicit schema-broadening tutoring protocol, on math problem solving at third grade. This was contrasted with the sole reliance on general education math instruction. Two groups of students made up the study: the students in their general education classroom with instruction on problem-solution instruction (control group), and the students receiving the schema-broadening tutoring. All instruction was given at the same time so students would not miss time from their regular classrooms.
Each session was conducted one-to-one outside the student’s classroom and lasted 20 to 30 minutes. Sessions occurred three times per week for 12 weeks. The first two weeks of intervention taught students foundational skills for successful problem solving. The next three 3-week sessions taught one problem type per 3-week time period, with cumulative review of previously taught problem types built into daily lessons. The final week addressed all three problem types: total problem, combining two or more quantities to make a total, difference problems, compare a bigger and smaller quantity to find the difference, and finally change problems, increasing or decreasing a starting quantity to end up with a new quantity.
Results
The researchers make note that this study, due to the small sample size, is underpowered and they present and discuss effect size for nonsignificant effects to reduce the possibility of rejecting a potentially useful intervention. They caution the readers that effect sizes associated with nonsignificant differences should be interpreted with caution. Most of the foundational skills involving word problems improvement at a statistically significant level did not vary as a function of treatment condition (i.e., control vs. tutoring). Effects sizes favoring the treatment group were large on word problems involving one-digit operands with no irrelevant information or charts, graphs, or pictures (0.69), and on word problems assessing more complex problems with two-digit operands with and without irrelevant information or charts, graphs or pictures (1.80).
Conclusion
In summary the researchers do caution the small sample size and the effect sizes associated with nonsignificant differences, due to inadequate statistical power, may be important to future research. A suggestion from the researchers of a larger experimental field trial with appropriate power may yield statistically significant differences. More importantly the present study, schema-based strategy instruction, can be generalized beyond students with specific math difficulty to students with substantial deficits in math as well as reading. It seems that the kind of treatment used in this study would be applicable to students with disabilities also. Another suggestion from the researchers speaks to the use of a schema-broadening protocol, such as the one used in this study, could be useful to school practitioners as they implement RtI models of learning disabilities identification at third grade.
Pape (2004)
Pape, S. (2004). Middle school children’s problem-solving behavior: A cognitive analysis from a reading comprehension perspective. Journal of Research in Mathematics Education, 35 (3), 187-219
Selection
This article examines problem-solving behaviors of middle school students from the vantage point of reading comprehension. Word problems obviously involve reading comprehension. This was one of the few refereed articles I found that was focused on that aspect of problem solving for secondary students.
This research project had some distinguishing features not found in other similar studies. It used video evidence. This allowed researchers to assess observable behaviors as the problems were being solved. In addition, comprehension categories that had previously been applied to undergraduate students were being used in the context of a middle school. Finally, this was a study that examined problem solving and its connection to academic achievement.
Problem/Purpose
The study was designed to look at the problem solving behaviors of middle school students. The author was seeking to examine the connections between problems solving success and the various ways student comprehend the text. In addition, the study looked at the problem solving behaviors to see if these were related to other variables including individual achievement scores.
Methods
There were 98 grade six and grade seven students from three different public schools who participated in this quantitative study. Two of the schools were in a large Midwestern urban area, and the third came from a Northeastern city. Students were tested on eight different problems in private video taped sessions while a researcher was present. The experimental protocols did not allow the researcher to answer questions about the mathematics content. The researcher did encourage the student to “think aloud.” As a student would finish working on a given problem, the researcher would ask the student to recall the problem.
Students were provided two different types of word problems. One was described as using consistent language and the other used what was described as inconsistent language. One example of a consistent language problem was “Joe runs 6 miles a week. Ken runs 3 times as many miles a week as Joe does. How many miles does Ken run in 4 weeks?” An example using inconsistent language was, “Joe runs 6 miles a week. He runs 1/3 as many miles a week as Ken does. How many miles does Ken run in 4 weeks?” The wording in the first problem is consistent with the arithmetic operation used to solve the problem and thus uses consistent language.
The coding done in previous studies influenced the identified problem solving strategies. A researcher name Hegarty along with others noted that some students took a direct translational approach to reading problems. The students using a direct translational approach focused mainly on the numbers and direct translation of the operational words. Other students used what was described as a meaning-based approach. These students concentrated on relationships and variable names instead of numbers.
Exact transcriptions of the students’ words were made. Other observed behaviors as well as gestures were noted. For example, if a student re-read a sentence, that was noted. This became the raw data for encoding the problem solving behaviors. The author along with two doctoral students encoded the behaviors using what was called “constant comparative methodology.” This involved repetitive viewing and modification of descriptions until agreement on classification was achieved.
Results
ANOVA analyses were used. Both parametric and nonparametric statistical procedures were performed “to provide a more fine grained examination of the relationships.” For the most part, the results were identical. The tests were held to a relatively strict confidence level, .025. Interrater agreement for problem solving behavior, type of error and problem recall was 95%, 99% and 96% respectively. One important aside: I am simplifying a huge amount of complex results. There were some anomalies and exceptions. Be cautious.
The research found that in most cases, the problems that used the consistent language were successfully solved with the greatest frequency. In light of the fact that this was also noted in the literature review, this was not a surprisingly new result. It was also noted that students made more reading errors on the inconsistent language problems.
Pape spent a good deal of time analyzing and fine-tuning both the direct translational approach and the meaning-based approach that characterized the students’ comprehension of the problem and ultimately ended up with five sub-categories that dealt with qualities that included arithmetic proficiency and mathematical reasoning. Within these approaches, he did find there was consistency. Students generally stuck to the same type of approach on most of the problems.
Most of the students, 66%, used a direct translational approach to solving the problem. The students who used the meaning-based approach did score significantly higher on the problem solving tasks. An analysis of the mathematical errors showed that those who used the meaning-based approach made significantly less errors.
Both math and reading achievement scores were available for most of the students. Generally speaking, the subgroups who used the meaning-based approach scored higher on the standardized tests. In math, the mean achievement scores using norm curve equivalents for the direct translational approach students ranged from 44 to 66 and the reading scores ranged from 46 to 61. Norm curve equivalents for the students using the meaning-based approaches ranged from range from 70 to 88 in mathematics and 75 to 80 in reading.
Pape acknowledges that there are limitations. He notes note that talk-aloud analysis might be subject to both validity and reliability concerns. He also points out that his descriptions and distinctions between types of behaviors are subject to errors of judgment on the part of an observer. He also mentioned that more refinements are needed to examine the data that was related to the students’ recall of a given problem.
Implications
Pape notes that students who predominantly used the direct translational approach have less success in the problem solving area. This does have implications for classroom teachers. Consequently, he suggests that instructional practices should encourage students to use a meaning-based approach. In contrast to some socially constructivist approaches, Pape also suggests that explicitly teaching students problem solving strategies might facilitate better problem solving.
Strengths and Weaknesses
Of particular note was the fact that this article connected some seemingly separate areas. Many of the articles/experiments look at a problem-solving task in a controlled situation as this article did. However, this article went beyond the task and made connections to performance in other areas. The fact that reading and math achievement scores were connected to problem solving behaviors is not a trivial matter when curriculum decisions need to be made. Also, the study focused on multiple dimensions. It examined what a student said and did during problem solving situations, and connected that information to successful performance.
Although the article adds to our knowledge, I did find some areas that need consideration. First, I wonder if these results are applicable beyond the setting. The achievement scores for the three schools might suggest that most of these students might be denizens of Lake Woebegone. The percentile rankings for all of the subgroups in reading and mathematics achievement were all above the 50th percentile with one exception, a subgroup of six students who employed the direct translational approach. Are these descriptors useful in settings where the achievement scores are much lower or much higher? In addition, the study did not examine how these students had been taught. Certainly, curriculum and teaching style might also come into play. Finally, the fact that students were observed in very private settings might be an issue. Would these students perform the same in a classroom stetting?
Selection
This article examines problem-solving behaviors of middle school students from the vantage point of reading comprehension. Word problems obviously involve reading comprehension. This was one of the few refereed articles I found that was focused on that aspect of problem solving for secondary students.
This research project had some distinguishing features not found in other similar studies. It used video evidence. This allowed researchers to assess observable behaviors as the problems were being solved. In addition, comprehension categories that had previously been applied to undergraduate students were being used in the context of a middle school. Finally, this was a study that examined problem solving and its connection to academic achievement.
Problem/Purpose
The study was designed to look at the problem solving behaviors of middle school students. The author was seeking to examine the connections between problems solving success and the various ways student comprehend the text. In addition, the study looked at the problem solving behaviors to see if these were related to other variables including individual achievement scores.
Methods
There were 98 grade six and grade seven students from three different public schools who participated in this quantitative study. Two of the schools were in a large Midwestern urban area, and the third came from a Northeastern city. Students were tested on eight different problems in private video taped sessions while a researcher was present. The experimental protocols did not allow the researcher to answer questions about the mathematics content. The researcher did encourage the student to “think aloud.” As a student would finish working on a given problem, the researcher would ask the student to recall the problem.
Students were provided two different types of word problems. One was described as using consistent language and the other used what was described as inconsistent language. One example of a consistent language problem was “Joe runs 6 miles a week. Ken runs 3 times as many miles a week as Joe does. How many miles does Ken run in 4 weeks?” An example using inconsistent language was, “Joe runs 6 miles a week. He runs 1/3 as many miles a week as Ken does. How many miles does Ken run in 4 weeks?” The wording in the first problem is consistent with the arithmetic operation used to solve the problem and thus uses consistent language.
The coding done in previous studies influenced the identified problem solving strategies. A researcher name Hegarty along with others noted that some students took a direct translational approach to reading problems. The students using a direct translational approach focused mainly on the numbers and direct translation of the operational words. Other students used what was described as a meaning-based approach. These students concentrated on relationships and variable names instead of numbers.
Exact transcriptions of the students’ words were made. Other observed behaviors as well as gestures were noted. For example, if a student re-read a sentence, that was noted. This became the raw data for encoding the problem solving behaviors. The author along with two doctoral students encoded the behaviors using what was called “constant comparative methodology.” This involved repetitive viewing and modification of descriptions until agreement on classification was achieved.
Results
ANOVA analyses were used. Both parametric and nonparametric statistical procedures were performed “to provide a more fine grained examination of the relationships.” For the most part, the results were identical. The tests were held to a relatively strict confidence level, .025. Interrater agreement for problem solving behavior, type of error and problem recall was 95%, 99% and 96% respectively. One important aside: I am simplifying a huge amount of complex results. There were some anomalies and exceptions. Be cautious.
The research found that in most cases, the problems that used the consistent language were successfully solved with the greatest frequency. In light of the fact that this was also noted in the literature review, this was not a surprisingly new result. It was also noted that students made more reading errors on the inconsistent language problems.
Pape spent a good deal of time analyzing and fine-tuning both the direct translational approach and the meaning-based approach that characterized the students’ comprehension of the problem and ultimately ended up with five sub-categories that dealt with qualities that included arithmetic proficiency and mathematical reasoning. Within these approaches, he did find there was consistency. Students generally stuck to the same type of approach on most of the problems.
Most of the students, 66%, used a direct translational approach to solving the problem. The students who used the meaning-based approach did score significantly higher on the problem solving tasks. An analysis of the mathematical errors showed that those who used the meaning-based approach made significantly less errors.
Both math and reading achievement scores were available for most of the students. Generally speaking, the subgroups who used the meaning-based approach scored higher on the standardized tests. In math, the mean achievement scores using norm curve equivalents for the direct translational approach students ranged from 44 to 66 and the reading scores ranged from 46 to 61. Norm curve equivalents for the students using the meaning-based approaches ranged from range from 70 to 88 in mathematics and 75 to 80 in reading.
Pape acknowledges that there are limitations. He notes note that talk-aloud analysis might be subject to both validity and reliability concerns. He also points out that his descriptions and distinctions between types of behaviors are subject to errors of judgment on the part of an observer. He also mentioned that more refinements are needed to examine the data that was related to the students’ recall of a given problem.
Implications
Pape notes that students who predominantly used the direct translational approach have less success in the problem solving area. This does have implications for classroom teachers. Consequently, he suggests that instructional practices should encourage students to use a meaning-based approach. In contrast to some socially constructivist approaches, Pape also suggests that explicitly teaching students problem solving strategies might facilitate better problem solving.
Strengths and Weaknesses
Of particular note was the fact that this article connected some seemingly separate areas. Many of the articles/experiments look at a problem-solving task in a controlled situation as this article did. However, this article went beyond the task and made connections to performance in other areas. The fact that reading and math achievement scores were connected to problem solving behaviors is not a trivial matter when curriculum decisions need to be made. Also, the study focused on multiple dimensions. It examined what a student said and did during problem solving situations, and connected that information to successful performance.
Although the article adds to our knowledge, I did find some areas that need consideration. First, I wonder if these results are applicable beyond the setting. The achievement scores for the three schools might suggest that most of these students might be denizens of Lake Woebegone. The percentile rankings for all of the subgroups in reading and mathematics achievement were all above the 50th percentile with one exception, a subgroup of six students who employed the direct translational approach. Are these descriptors useful in settings where the achievement scores are much lower or much higher? In addition, the study did not examine how these students had been taught. Certainly, curriculum and teaching style might also come into play. Finally, the fact that students were observed in very private settings might be an issue. Would these students perform the same in a classroom stetting?
Monday, April 6, 2009
Dow & Mayer (2004)
Dow, G. T. & Mayer, R.E. (2004). Teaching students to solve insight problems: Evidence for domain specificity in creativity training. Creativity Research Journal, 16, 389- 402.
Dow and Mayer’s article reviews current research in creativity, insight, and problem solving. Problem solving can be divided into strategies for solving routine and nonroutine problems. Solutions for routine problems require known strategies, and the problem solver’s task is to select and execute a sequence of steps. In contrast, solutions for nonroutine problems require creativity and insight.
Insight problems are a distinct subcategory of nonroutine problems. Insight problems deliberately lead problem solvers to engage in incorrect solution strategies, by “tricking” problem solvers with extraneous information. To solve insight problems, problem solvers must discard “obvious” solution paths and must creatively invent new procedures.
Dow and Mayer discuss current theory exploring the nature of insight problems and investigate whether insight problems are domain-general or domain-specific. In domain-general theory, all insight problems are grouped into a common cognitive framework and share a common problem solving strategy. In contrast to this prevailing theory, the researchers hypothesized that insight problems are domain-specific. In domain-specificity theory, different problem solving strategies are necessary for three insight subdomains – verbal problems, mathematical problems, and spatial problems.
On page 390, the authors provide an example of a mathematical insight problem:
You have black socks and brown socks in your drawer, in the ratio of 4 black socks for every 5 brown socks. How many socks will you have to take out to insure that you will have a pair of socks of the same color?
Many problem solvers will assume that the sock ratio is important, and will attempt to solve the problem computationally, using the numbers 4 and 5. The correct solution is dependent upon discarding ratio information, and understanding that at most three socks are needed to guarantee a pair of the same color.
In the same way that a mathematical insight problem primes an incorrect solution strategy, verbal insight problems will “trick” a problem solver with words with secondary meanings, and spatial insight problems require a problem solver to recognize and discard self-imposed constraints or imagined rules.
Problem and Purpose
The research study investigated whether insight problems are perceived to be domain-general or domain-specific. In domain-general theory, insight problems comprise a single class of problems with common problem-solving strategies. In domain-specific theory, insight problems are grouped into categories (verbal, mathematical, and spatial), each requiring a unique problem solving strategy. Previous research tended to favor domain-general theory. The researchers wished to investigate whether there was justification for domain-specific theory, hypothesizing that insight problems require distinct domain-specific strategies.
Study 1: Methodology and Results
The first of three reported studies was an exploratory investigation to determine whether students perceive that insight problems are domain-general or domain-specific. Participants were 22 undergraduate students from the University of California, Santa Barbara (UCSB). Participants were given a deck of 67 cards containing a random arrangement of verbal, mathematical and spatial insight problems. Participants were asked to sort cards into categories based upon perceptions of similarities in problem-solving strategies. A hierarchical cluster analysis identified four clusters: verbal, mathematical, spatial, and combined spatial-verbal. The researchers concluded that participants viewed insight problems as domain-specific.
Study 2: Methodology and Results
Given that the results of Study 1 suggested participants perceive domain-specificity in insight problems, the researchers conducted a second study to investigate whether training in one subcategory (verbal, mathematical, or spatial) would transfer to other subcategories. Participants, 63 undergraduate students from UCSB, were divided into four groups receiving training in verbal, mathematical, spatial, and combined verbal-mathematical-spatial insight problem solving.
Training packets contained three insight problems and a three-step procedure for solving the domain-specific problem. For example, the spatial training packet prompted students to identify a self-imposed constraint, remove the constraint, and develop a conclusion. A common spatial insight problem is a 3 x 3 matrix of nine dots, with instructions to draw four continuous straight lines through all nine dots. For many problem solvers, the self-imposed constraint is to assume that it is necessary to stop precisely on a dot. The correct solution is obtained by discarding the constraint, drawing lines beyond the edge of the matrix to arrive at a conclusion connecting all nine dots with four lines.
After completing a training packet for one domain-specific category, participants were tested using a researcher-developed instrument composed of three verbal problems, three mathematical problems, and three spatial problems. Means and standard deviations were reported for verbal, math, and spatial subscores for each training group. Means for verbal subscores ranged from 1.95 to 2.08 for the four types of training. Means for math subscores ranged from 1.74 to 2.07, and spatial means ranged from .62 to 1.50. A one-way ANOVA with training group as the between-subjects factor and domain-specific subscore as the dependent variable resulted in a significant F-test favoring spatial insight problems. When post hoc pairwise comparisons were investigated, participants who received spatial training scored higher on the spatial problem solving subsection than participants who received verbal or mathematics training. Verbal and mathematical insight problems produced nonsignificant results when compared across the four groups.
In the discussion of the results of Study 2, the researchers concluded that the statistically significant results for the spatial scores support the theory of domain-specificity. However, the verbal and mathematics training groups did not achieve higher scores on the domain-specific test questions. The researchers’ thoughtful analysis of these results suggest that poor training materials or prior knowledge of verbal and mathematical insight problems may have contributed to inconclusive results.
Study 3: Methodology and Results
Study 3 improves on the methodology of Study 2 by repeating the same experiment, with the addition of a control group. Mathematical and combined training groups were eliminated, narrowing the focus to verbal and spatial problems.
Participants were 71 undergraduate students from UCSB. Twenty-three participants completed verbal training packets and 24 participants completed spatial training packets. The control group (24 participants) received no training materials. All participants were tested on three verbal and three spatial insight questions. Materials and test questions were identical to study 2, with the exception of the elimination of mathematics sections.
Means for verbal insight subscores ranged from 1.54 for the control group to 1.83 for the spatial training group. Means for spatial insight subscores ranged from .48 for the verbal training group to .96 for the spatial training group. Consistent with the results of Study 2, a one-way ANOVA for spatial subscores resulted in statistically-significant findings for the spatial training group. Pairwise comparisons showed that the spatial training group achieved higher scores than the verbal training group, but showed no significant difference when compared to the control group. Study 3 supported conclusions from study 2, suggesting that training in spatial insight problems did not transfer to other types of insight problems.
Discussion and Critique
Dow and Mayer suggest that the series of studies provides evidence that insight problems are domain-specific. In the first study, participants sorted problems into consistent categories, suggesting perception of subdomains. In the second and third studies, spatial training resulted in higher scores on spatial insight problems. Spatial training did not transfer to other types of insight problems, suggesting domain-specificity.
The series of studies was carefully planned, executed and documented. The first study was a preliminary investigation, the second study a more thorough experimental design, and the third study improved upon a major flaw of the second study by inclusion of a control group. The researchers included all 67 test questions in the appendix and provided a careful description of training materials and methodology, providing detailed information to allow replication by other researchers.
Weaknesses of these studies included uneven quality of training materials, uneven difficulty of the 67 problems included in the appendix, and inadequate assessment of participants’ prior knowledge of specific insight problems. A pretest given to participants before insight training would have assessed prior knowledge, allowing computation of change in scores after training. Also, the study involved only short-term laboratory-setting treatments on college students, perhaps limiting generalizability.
Theoretical, Methodological, and Practical Conclusions
The theoretical implications of this study support domain-specificity theory, indicating that subcategories of insight problems require specific instructional strategies that do not transfer to other problem types. Methodology unique to the study included asking participants to categorize problem types. Indirect practical implications of this study are the researchers’ recommendation that creativity instructional units should focus on specific problem types, without expectation of transfer between domains. This study makes a contribution to the research base by providing evidence of domain-specificity and by defining insight as a component of creativity, recognizing that both insight and creativity are necessary for solution of nonroutine problems.
Personal Observations
In my past employment in a statistical research department, I often observed that professional mathematicians show remarkable creativity in problem solving, quite distinctive from procedures we teach our elementary through undergraduate mathematics students. This article presents alternative training approaches for teaching problem solving, pedagogy that we should incorporate in our classrooms. The selection of this article stemmed from a growing interest in Mayer’s extensive body of research. This article presents clear conceptual background in the areas of creativity and insight in mathematical problem solving.
Dow and Mayer’s article reviews current research in creativity, insight, and problem solving. Problem solving can be divided into strategies for solving routine and nonroutine problems. Solutions for routine problems require known strategies, and the problem solver’s task is to select and execute a sequence of steps. In contrast, solutions for nonroutine problems require creativity and insight.
Insight problems are a distinct subcategory of nonroutine problems. Insight problems deliberately lead problem solvers to engage in incorrect solution strategies, by “tricking” problem solvers with extraneous information. To solve insight problems, problem solvers must discard “obvious” solution paths and must creatively invent new procedures.
Dow and Mayer discuss current theory exploring the nature of insight problems and investigate whether insight problems are domain-general or domain-specific. In domain-general theory, all insight problems are grouped into a common cognitive framework and share a common problem solving strategy. In contrast to this prevailing theory, the researchers hypothesized that insight problems are domain-specific. In domain-specificity theory, different problem solving strategies are necessary for three insight subdomains – verbal problems, mathematical problems, and spatial problems.
On page 390, the authors provide an example of a mathematical insight problem:
You have black socks and brown socks in your drawer, in the ratio of 4 black socks for every 5 brown socks. How many socks will you have to take out to insure that you will have a pair of socks of the same color?
Many problem solvers will assume that the sock ratio is important, and will attempt to solve the problem computationally, using the numbers 4 and 5. The correct solution is dependent upon discarding ratio information, and understanding that at most three socks are needed to guarantee a pair of the same color.
In the same way that a mathematical insight problem primes an incorrect solution strategy, verbal insight problems will “trick” a problem solver with words with secondary meanings, and spatial insight problems require a problem solver to recognize and discard self-imposed constraints or imagined rules.
Problem and Purpose
The research study investigated whether insight problems are perceived to be domain-general or domain-specific. In domain-general theory, insight problems comprise a single class of problems with common problem-solving strategies. In domain-specific theory, insight problems are grouped into categories (verbal, mathematical, and spatial), each requiring a unique problem solving strategy. Previous research tended to favor domain-general theory. The researchers wished to investigate whether there was justification for domain-specific theory, hypothesizing that insight problems require distinct domain-specific strategies.
Study 1: Methodology and Results
The first of three reported studies was an exploratory investigation to determine whether students perceive that insight problems are domain-general or domain-specific. Participants were 22 undergraduate students from the University of California, Santa Barbara (UCSB). Participants were given a deck of 67 cards containing a random arrangement of verbal, mathematical and spatial insight problems. Participants were asked to sort cards into categories based upon perceptions of similarities in problem-solving strategies. A hierarchical cluster analysis identified four clusters: verbal, mathematical, spatial, and combined spatial-verbal. The researchers concluded that participants viewed insight problems as domain-specific.
Study 2: Methodology and Results
Given that the results of Study 1 suggested participants perceive domain-specificity in insight problems, the researchers conducted a second study to investigate whether training in one subcategory (verbal, mathematical, or spatial) would transfer to other subcategories. Participants, 63 undergraduate students from UCSB, were divided into four groups receiving training in verbal, mathematical, spatial, and combined verbal-mathematical-spatial insight problem solving.
Training packets contained three insight problems and a three-step procedure for solving the domain-specific problem. For example, the spatial training packet prompted students to identify a self-imposed constraint, remove the constraint, and develop a conclusion. A common spatial insight problem is a 3 x 3 matrix of nine dots, with instructions to draw four continuous straight lines through all nine dots. For many problem solvers, the self-imposed constraint is to assume that it is necessary to stop precisely on a dot. The correct solution is obtained by discarding the constraint, drawing lines beyond the edge of the matrix to arrive at a conclusion connecting all nine dots with four lines.
After completing a training packet for one domain-specific category, participants were tested using a researcher-developed instrument composed of three verbal problems, three mathematical problems, and three spatial problems. Means and standard deviations were reported for verbal, math, and spatial subscores for each training group. Means for verbal subscores ranged from 1.95 to 2.08 for the four types of training. Means for math subscores ranged from 1.74 to 2.07, and spatial means ranged from .62 to 1.50. A one-way ANOVA with training group as the between-subjects factor and domain-specific subscore as the dependent variable resulted in a significant F-test favoring spatial insight problems. When post hoc pairwise comparisons were investigated, participants who received spatial training scored higher on the spatial problem solving subsection than participants who received verbal or mathematics training. Verbal and mathematical insight problems produced nonsignificant results when compared across the four groups.
In the discussion of the results of Study 2, the researchers concluded that the statistically significant results for the spatial scores support the theory of domain-specificity. However, the verbal and mathematics training groups did not achieve higher scores on the domain-specific test questions. The researchers’ thoughtful analysis of these results suggest that poor training materials or prior knowledge of verbal and mathematical insight problems may have contributed to inconclusive results.
Study 3: Methodology and Results
Study 3 improves on the methodology of Study 2 by repeating the same experiment, with the addition of a control group. Mathematical and combined training groups were eliminated, narrowing the focus to verbal and spatial problems.
Participants were 71 undergraduate students from UCSB. Twenty-three participants completed verbal training packets and 24 participants completed spatial training packets. The control group (24 participants) received no training materials. All participants were tested on three verbal and three spatial insight questions. Materials and test questions were identical to study 2, with the exception of the elimination of mathematics sections.
Means for verbal insight subscores ranged from 1.54 for the control group to 1.83 for the spatial training group. Means for spatial insight subscores ranged from .48 for the verbal training group to .96 for the spatial training group. Consistent with the results of Study 2, a one-way ANOVA for spatial subscores resulted in statistically-significant findings for the spatial training group. Pairwise comparisons showed that the spatial training group achieved higher scores than the verbal training group, but showed no significant difference when compared to the control group. Study 3 supported conclusions from study 2, suggesting that training in spatial insight problems did not transfer to other types of insight problems.
Discussion and Critique
Dow and Mayer suggest that the series of studies provides evidence that insight problems are domain-specific. In the first study, participants sorted problems into consistent categories, suggesting perception of subdomains. In the second and third studies, spatial training resulted in higher scores on spatial insight problems. Spatial training did not transfer to other types of insight problems, suggesting domain-specificity.
The series of studies was carefully planned, executed and documented. The first study was a preliminary investigation, the second study a more thorough experimental design, and the third study improved upon a major flaw of the second study by inclusion of a control group. The researchers included all 67 test questions in the appendix and provided a careful description of training materials and methodology, providing detailed information to allow replication by other researchers.
Weaknesses of these studies included uneven quality of training materials, uneven difficulty of the 67 problems included in the appendix, and inadequate assessment of participants’ prior knowledge of specific insight problems. A pretest given to participants before insight training would have assessed prior knowledge, allowing computation of change in scores after training. Also, the study involved only short-term laboratory-setting treatments on college students, perhaps limiting generalizability.
Theoretical, Methodological, and Practical Conclusions
The theoretical implications of this study support domain-specificity theory, indicating that subcategories of insight problems require specific instructional strategies that do not transfer to other problem types. Methodology unique to the study included asking participants to categorize problem types. Indirect practical implications of this study are the researchers’ recommendation that creativity instructional units should focus on specific problem types, without expectation of transfer between domains. This study makes a contribution to the research base by providing evidence of domain-specificity and by defining insight as a component of creativity, recognizing that both insight and creativity are necessary for solution of nonroutine problems.
Personal Observations
In my past employment in a statistical research department, I often observed that professional mathematicians show remarkable creativity in problem solving, quite distinctive from procedures we teach our elementary through undergraduate mathematics students. This article presents alternative training approaches for teaching problem solving, pedagogy that we should incorporate in our classrooms. The selection of this article stemmed from a growing interest in Mayer’s extensive body of research. This article presents clear conceptual background in the areas of creativity and insight in mathematical problem solving.
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