Reasoning and Problem Solving/Putting the Theories to Work: The Case of Mathematics

Reasoning and Problem Solving/Putting the Theories to Work: The Case of Mathematics
Ben Sinclair
Chapter 4 – Information Processing, Memory, and Problem Solving


Reasoning and Problem Solving
Reasoning is thinking directed toward reaching a goal, while problem solving is a means of reaching that goal. Problem solving is the ability to reason. Reasoning builds on generalization and inference to process new information into useable data that can be used for problem solving. In the educational setting, strategies taught in textbooks are usually contrived for the solutions presented in contrast to real world problems that may be more complex and require advanced thinking solutions (Bergin & Bergin, 2012).
Children learn in a predictable manner and begin at birth using generalization to solve problems. Toddlers begin to compare and contrast concepts in order to generate abstract rules. By pre-school age most children have developed enough reasoning ability to use evidence to draw conclusions based on previous experience. This reasoning can be cultivated through training, but at this stage is not flawless. By middle childhood most students have a well developed internal speech for problem solving. Children at this age begin to try new strategies when one that had previously worked has failed.
By adolescence, children think much like adults and can draw conclusions based on numerous variables. Diversity amongst individuals when it comes to reasoning ability is based on many things. Genetics, education and experience, and modeling are key concepts in reasoning ability and problem solving. Student working memory and processing speed are likely beyond the educator’s control, but teacher feedback allows students to revise their strategies in order to find out what works (Malamed, 2013). Questions such as, “How do you know that?” or “Why is that answer right?” allow the student to verbalize their reasoning and allow the teacher to help the student codify their thinking.

Putting the Theories to Work: The Case of Mathematics
While “number sense” is likely innate, mathematical competence in areas such as percentages, algebra or fractions must be taught, and learned. Informal math concepts prevalent in infants and toddlers are acquired without formal training. The math skill possessed by a child at entrance into kindergarten is a likely predictor of academic success, even more so than reading skill. Children of lower economic status may not have developed the elementary math skills needed as a foundation for success (Purcell-Gates, 1995). The ability to memorize math concepts such as the multiplication tables allows the student to quickly move from rudimentary calculations to more advanced math.
The textbook describes four perspectives to explain math ability. Behaviorism believes in direct instruction involving drill and practice in order to learn basic lower level skills and then build on these concepts. According to Piaget, children assimilate what they know into new strategies based on previous experience. Diagrams, graphs and asking the student to explain their answers are frequent strategies of instruction. Vygotsky believed that knowledge is transmitted through social and cultural means and then transferred to other areas. Cooperative learning were children talk about problem solving and opportunities to observe math concepts in the community are concepts for learning.
The information processing model relates that working memory and long-term memory combine for reasoning ability. Executive function keeps the student moving through right steps for solving a problem. Testing is thought to facilitate learning because of forced recall. Spaced practice allows the child to move concepts from short-term to long-term memory and back again for working out the problem. Children likely use learned strategies to become more efficient at solving problems. The information processing model allows for concepts from other theories to be seen as valid on their own without discounting the whole theory (Page & Page, 2011).

Discussion Questions
1. How do reasoning and problem solving contribute to each other?
2. Why is it important for students to verbalize their responses?
3. Why would math competence be a better indicator of future academic success than reading ability?
4. Have you considered using frequent quizzes with explanations required that build on past learning and cumulate in a larger comprehensive test?
5. Why is no single learning theory completely accurate?

Bergin, C., & Bergin, D. (2012). Child and adolescent development in your classroom. Belmont:  Wadsworth.

Malamed, C. (2013). 20 facts you should know about working memory. Retrieved from

Page, R., & Page, T. (2011). Promoting health and emotional well-being in your classroom. (5th  ed.). Sudbury: Jones and Bartlett Publishers.

Purcell-Gates, V. (1995). Other people’s words: The cycle of low literacy. Cambridge: Harvard  University Press.

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3 Responses to Reasoning and Problem Solving/Putting the Theories to Work: The Case of Mathematics

  1. anonymous50 says:

    Why is it important for students to verbalize their responses?

    Students’ verbalizing of responses facilitates the opportunity for feedback from the teacher and allows him/her to better assess student understanding and progress ‘on the fly’. This then enables the teacher to scaffold instruction for the class as a whole or differentiate instruction for individuals, both immediately and in setting long-term goals. Verbalizing responses also allows for the social aspect of learning in which students can question their own thinking, explore the thinking of others, and discover answers and concepts beyond the scope of what is being ‘fed’ to them.

  2. Mary Decker says:

    Why is no single learning theory completely accurate?

    The human brain is a very complex organ. Despite our advances in technology and abundant studies, we are far from determining the exact processes involved in learning new information. As is discussed in the text, proper studies isolate variables to determine causes. However, when studying learning theories, this is practically impossible. There are too many factors which influence how a particular child can/will learn in a particular time period or subject. Variables such as what the child ate that day, issues in the home, self-esteem, prior knowledge, mood, level of attention, perceived importance of material, and even level of likability of the teacher will all slightly affect whether or not new material or concepts are learned.
    Likely, instead of there being one true learning theory, there are parts of each that are valid. Some of the theories build on each other or are complementary. There is also the possibility that different theories apply to different children. If we differentiate our instruction for children, perhaps we should also differentiate the ways in which we perceive how they learn! I guess that is what makes psychology so intriguing!

  3. Andrea says:

    Have you considered using frequent quizzes with explanations required that build on past learning and cumulate in a larger comprehensive test?

    Several years ago my colleagues and I looked at our state MAP results to target key concepts that students were consistently scoring low. We discussed numerous strategies that we could use in the classroom to help students retain information being taught. The upper elementary school (5th & 6th) in the district had an extensive reteaching school wide program. Students were given a pretest, 2 weeks later a formative assessment and the next day or so they were sorted into enrichment or reteach groups. Around two to three days after the formative assessment, student would take the summative assessment. If student didn’t perform up to expectations that were set, then they went to a specific class at the end of the day for several weeks for reteaching and taking summative assessments until students met expectations. They did this for all subjects. We tried this method and found it was very time consuming and that it was difficult to justify stopping class for two to three days to reteach a very small number of students (typically between than 8% -10 % out of 450 students scored lower than a C on summative assessments).

    We decided that we didn’t want to be reactive, but proactive in our teaching. We came to the conclusion to try and get the students to learn the information before the summative assessment. The way we did that was through frequent quizzing and spiraling information throughout the year. We analyze quiz data and determine the areas that student continue to struggle in order to reteach those concepts again BEFORE the summative assessment. Spiraling information helps keep concepts “fresh” in their minds and students were able to make connections between various science topics. Information that was typically spiraled were concepts students continue to struggle with and/or concepts that mentioned repeatedly in the state grade level expectations (ex. Photosynthesis). We have been following this strategy for the last four years and have found great success. Students are able to retain information and are able to recall the information as they progress in school. Currently, on an average 3% – 5% of 450 students score lower than 70% (C-) on summative assessments. Our school has time built into the day(2 days a week) for students to get help in concepts that they struggle in and we have an after school tutoring program to meet the needs of academically struggling students.

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