Making Inferences in Mathematical Cognition

“In science, cognition is a group of mental processes that includes attention, memory producing and understanding language, solving problems, and making decisions” (Cognition, 2012).  With this in mind, cognition within mathematics is an important process to help develop the teaching and research of mathematics from the student to teacher level.  Fennema and Sherman (1971/2004) discussed the roles certain factors play on the mathematical achievement of students and in particular how these factors explain differences seen in male and female students.  Zan, Brown, Evans, and Hannula (2006) reviewed literature related to the affects of mathematical anxiety, attitudes toward mathematics, mathematical anxiety, gender factors in mathematics, emotion, value, attitude, and beliefs .  Schoenfeld (1983/2004) discussed the relationship between research based on study participant verbalization and known mathematics, the relationship between what is observed empirically from a problem and how it is solved, and three separate categories of analysis for solving problems:  resources, control, and belief systems.  Thompson (1984/2004) discussed how teachers’ cognitive conceptions of mathematics play integral roles in what is fostered in the classroom and their instructional behavior.

Thompson (1984/2004) described three teachers professed beliefs of mathematics and how these beliefs shaped their instructional practices.  The three teachers, Jeanne, Kay, and Lynn had similar experience, were leaders in the school system, and voluntarily decided to be a part of the study after inclusion in some of Thompson’s pilot studies (Thompson, 1984/2004).  Jeanne viewed mathematics as part of a school curriculum that was separate from real world involvement, thought discourse and student rapport essential, and had the least confidence of the three study participants (Thompson, 1984/2004).  Kay saw mathematics as a tool for the sciences whose creation, validity, and purpose can be found by the student (Thompson, 1984/2004).  Kay showed confidence in her mathematical ability and provided contexts for students to reason, conjecture, and proof (Thompson, 1984/2004).  Kay’s belief about mathematics and mathematics education tended for her to create lesson plans that focused around student questioning and identification of misconceptions (Thompson, 1984/2004).  Lynn’s view as a teacher was managerial in an attempt to lower misbehavior (Thompson, 1984/2004).  Her instruction was instrumental and focused on algorithm fluency with matched her view of mathematics as always having a correct solution, predictable, and absolute(Thompson, 1984/2004).  Lynn devoted little of her time to lesson planning because of her views of mathematics and its absolute structure (Thompson, 1984/2004).  Jeanne was the most inconsistent of her beliefs and instructional practice because of her incorporation of student discussion; however, Kay showed inconsistency with connection of mathematics to the sciences and Lynn’s lack of differentiation and incorporation of real world problems (Thompson, 1984/2004).  These observations of teacher beliefs about mathematics show that there is a direct link to instructional practice.  In essence to change teachers’ and potential teachers’ instructional practice, views, beliefs, and preferences must be sharpened.

Similar to teachers’ views, beliefs, and preferences, students also bring these notions into the classroom.  It was believed in the past that males routinely achieve better in mathematics than females because of ability or social climate.  Fennema and Sherman (2004/1977) use multiple tests related to achievement, spatial visualization, confidence, attitude, etc. to control for comparisons between male and females within mathematics.  When controlling for these factors, it was found that there was no significant difference between male and female abilities within mathematics (Fennema & Sherman, 1977/2004).  Observing the principal components analysis within the study highlights the relationship of factors of ability (factor 1) and motivation (factor 2) (Fennema & Sherman, 1977/2004).  The first factor, in a principal component analysis accounts for the largest amount of variability of the study, but this eigenvalue has not been mentioned and thus the use of dimensionality reduction is impossible to articulate and provide meaning to the study.  Important from this study however is researchers and teachers linkage between mathematical ability and psychological phenomena such as motivation, confidence, attitude, family characteristics, and usefulness of mathematics (Fennema & Sherman, 1977/2004).

Schoenfield (1983/2004) and Zan, Brown, Evans, and Hannula (2006) expand on this psychological relationship to mathematics education.  Zan, Brown, Evans, and Hannula’s (2006) introduction of multiple psychological phenomena related to math education are not to be seen as contradictory but complimentary.  The behaviors that are observed or monitored within math education should provide better theoretical frameworks for the field.  Schoenfield (1983/2004) categorization of a student’s setting; knowledge, belief, and value systems; and degree of awareness can be seen as foundations to understanding what is being observed cognitively in a mathematical research setting.  With this cognitive structure, it is clear that one should be careful to articulate misunderstanding, potential, or ability solely on failure to adequately solve a problem (Schoenfield, 1983/2004).

Human cognition and educational research are becoming very much interrelated.  In order to properly understand qualitative studies and made accurate inferences from quantitative data, it is important to understand the cognitive aspects of human nature.  The four previously discussed articles do much to explain the thinking of students and their relationship to mathematics instruction, learning, and research.  Articles such as these offer constructs for further research that relates to the cognitive aspects of student learning and teacher pedagogy.

 

Cognition.  (n.d.).  In Wikipedia.  Retrieved on October 29, 2012 from http://en.wikipedia.org/wiki/Cognition.

Fennema, E. & Sherman, J.  (2004).  Sex-related differences in mathematics achievement, spatial visualization and affective factors.  In Carpenter, J. A. Dossey & J. L. Koehler (Eds.).  Classics in mathematics education research (pp. 27-39).  Reston, VA:  National Council of Teachers of Mathematics.  (Reprinted from American Educational Research Journal, 14, 51-71, by E. Fennema & J. Sherman, 1977).

Schoenfield, A. (2004). Beyond the purely cognitive:  Belief systems, social cognitions, and metacognitions as driving forces in intellectual performance.  In Carpenter, J. A. Dossey & J. L. Koehler (Eds.).  Classics in mathematics education research (pp. 111-133).  Reston, VA:  National Council of Teachers of Mathematics.  (Reprinted from Cognitive Science, 7, 329-363, by A. H. Schoenfeld, 1983)

Thompson, A. G. (2004).  The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice.  In Carpenter, J. A. Dossey & J. L. Koehler (Eds.).  Classics in mathematics education research (pp. 173-184).  Reston, VA:  National Council of Teachers of Mathematics.  (Reprinted from Educational Studies in Mathematics, 15, 105-127, by A. G. Thompson, 1984).

Zan, R., Brown, L., Evans, J. & Hannula, M.S. (2006). Affect in mathematics education: An introduction.  Educational Studies in Mathematics, 63, 113-121.

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