The nature and value of mathematical reasoning
The beauty, simplicity and elegance of information is sharpened, accentuated, and brightened by the lens of mathematical reasoning. Mathematical thinking increases the capacity of an individual to “see” the nature of interdependent relationships.
The skills that are exercised and enhanced in higher math studies include: solving, examining, analyzing, concluding, deducing, connecting, hypothesizing, projecting, and synthesizing. These skills provide the intellectual muscle that facilitate grounded inquiry and provide concrete ways to understand a variety of concepts including convergence, divergence, increasing, decreasing, rate of change, and provide, in general, ways to characterize the “behavior” of a given event or series of events as a function of the qualities of contributing components.
Sound mathematical thinking gives contour, depth and focus to observations. It allows for comparison along different dimensions. It accentuates one’s clarity in describing phenomena. It facilitates interpretation and communication about phenomena. It illuminates and encodes relationships and captures and distills pictures of events. It provides ways to addressing chaotic or disparate phenomena and puts form and reason into those by facilitating the discovery of patterns in the evolution of events. It facilitates mapping of events and facilitates the projection of those events into the future allowing us to see implications of current actions or policies. It allows one to project and make predictions with an expected degree of rigor; it builds power of estimation and provides a mechanism to plan. It facilitates interpolation, the ability to look within data points, and extrapolation, the ability look beyond data points. It allows the capacity to see and understand qualities in relationships between interacting elements and to project based on reasoned and concrete understanding.
The study of “higher math” strengthens the process of logic; higher math connects procedure to understanding and to connecting understandings. It affects positively the ability to see patterns and to convert patterns into mathematical rules which in turn affects positively the ability to make conjectures about those rules. It illuminates measurement and comparison and gives measure to the concepts of “big”, how far, how wide, area, volume, and distance. Number sense, power of estimation and prediction builds one’s ability to hear and apply reasoning to what is heard and to make interpretations relating to a discussion that focuses on quantitative issues.
The dilemma
Many in this country do not value mathematics and/or they do not understand it. Those individuals who do not value mathematics avoid it in high school and college. It results in a gap between the haves, those who have a mathematical orientation and who exercise and develop those skills, and those who have not. The gap occurs along the lines of operational skills certainly but even more in reasoning ability.
Although we in the U.S. have been slow to give high priority to mathematics education, China, India and most other Southeast Asian states as well as many Western European countries have more than enough people who will fill the void left by us and who are more willing to learn those concepts and who do so with a high degree of respect for the processes and the discomforts necessary to acquire those skills. Mathematical sensitivity is central to understanding many elements in the world and universe.
Without a rich experience in secondary mathematics education, a significant proportion of students enter the world of the mathematical proletariat. For this group, whether by virtue of lack of interest, laziness or through solipsistic exercise, mathematics has little place in their lexicon of understandings. These individuals stunt their capacity to experience and to appreciate ideas such as ratio, rate of change, the nature of relationships that constitute different kinds of change, useful formula to understand relationships between variables, the ability to relate to data, to comprehend magnitude, and to understand scientific principles that explain common every day phenomena such as radio waves, the motion of the earth relative to other planets, and rainbows; they are, in short, shut out from almost all thinking about astronomy, physical science, economics, and social science information.
For those individuals who do not pursue secondary mathematics, the equations, formulas, graphs and charts that are used to illuminate complex conditions are “black boxes”; they see the results but they do not see how and why the results manifest as they do. When a formula yields an “answer,” those less informed must accept or reject the result “on faith.” They are not able to enjoin a conversation about why the result occurs as it does or how changes in the composition of the variables may alter the result. Simply put, this population does not “see” or “hear” as well as those who are mathematically literate.
The necessity of having proficiency in these skills is challenged frequently. Those who inquire about many topics and who are always interested in new ways of viewing information, shy away from the same when it is laced with mathematics. Why do I need to know these things? I think that the emergence of that question has to do with our perspective about mathematics from a very fundamental point of view. Most literate intellectuals often dismiss the need to know science-based information precisely because it is more remote. They do not get “it” and the “it” challenges them in ways that most qualitative information does not. Intellectuals who trip all over himself to understand the latest theory on the details of Mozart’s childrearing, reject, out of hand, the need to understand retrograde motion. They tend away from the discomfort of analysis of quantitative information.
The nature of mathematics – a complicated learning process.
Mathematical thinking is complicated. For all of its strengths and advantages, mathematical thinking and the study of mathematics remain remote to many. Mathematical thinking is not all apples, and orienting students to the kind of mathematical reasoning with which they are asked to engage is sometimes understated by teachers of mathematics or, worse yet, ignored. Mathematical tasks require students that utilize very different kinds of thinking. Mathematics is a manner of reasoning as much and more than it is a forum for doing calculations. Mathematical reasoning includes many and different skills that amplify understanding but also can make for students a complex puzzle. Taken together, the repertoire of skills that is necessary to successfully negotiate mathematics courses in high school make the learning process difficult.
Some elements of mathematical thinking involve finding the “right answer.” There is “the doing of math” which is using operations, principals and laws of math to arrive at an answer to a problem. The essential idea is to employ the correct operation to secure the correct answer and in this the operations are rote, mechanical and computational. This is essentially a convergent process; we move from a condition or a set of conditions to a result or solution. The process is, in these circumstances, deductive. The use of memorization and retrieval of facts and laws is critical to have fluency in doing mathematics. Getting from A to F requires going through bcde. Multi-step procedures and multi-step acts of contemplation require patience and discipline. Using facts and laws of process or logic can be tedious and confounding.
Other elements of mathematical thinking involve creating or depicting understandings among various entities. These relationships often have dynamic complexity. In the mathematics of analysis we use visual, theoretical, conceptual, and relational tools in an essentially divergent process to discover or see qualities in particular phenomena; we move, in this case, from a set of conditions about a problem to a depiction of an understanding. This process can be both inductive and deductive. The study of higher math, then, often requires the integration the computational/ operational skills with the conceptual. One learns to do the operations based on an understanding of relationships. In this process, one thinks and acts divergently or convergently at different points.
Conceptualization, the ability to relate social or physical phenomena to symbols and to then utilize symbolic logic enables the manipulation of complex ideas. The essential idea is to represent and manipulate relationships among entities that change with changing conditions such as: time, speed, distance, weight, temperature, population size, grain harvests or profits to name a few; symbols give (or force) form, texture and structure to concepts. Graphs provide a visual construct, which is usually assistive to conceptualization, which depict the behavior of one variable in relation to increases in another. The use of graphs requires that one can interpret inflections and changes in the visual pattern and relate those to changes in phenomena.
Practices and Priorities
Teaching practices in mathematics have changed over time and not necessarily in a productive way. Some methods that had been the bread and butter of sound mathematical training had been under assail for years. Repetition and familiarization in learning math facts became out of favor in the nineteen-sixties and seventies particularly in elementary schools. Rote memorization of a process to perform a mathematical task was equated with the mantra of the time “Death at an Early Age.” “Drill and Kill” another watchword of critics of rote mathematics lasted longer. Today elementary mathematics teachers are, after a fifty year hiatus, speaking positively to the value of “automaticity” with regard to learning facts and executing elementary processes. Automaticity is necessary to clear the brain higher order thinking and the value for automaticity makes far more sense than having students struggle with rediscovering ways to multiply whole numbers. The place of automaticity is as relevant to the execution of simple algebra skills as it is in the execution of simple whole number operations during the early elementary school years.
“Discovery” of mathematical principles is sometimes useful, and many other times it is not. Certainly it is, as argued earlier, beneficial for students to connect thinking about problems and connecting those to operations that produce appropriate results. Sometimes discovery methods in mathematics, particularly when utilized at the elementary level, border on redundancy. Students are sometimes encouraged to “discover” the trivial. It is like having students “discover” the way to spell a common word. What is important to discover and what is important to drill? This is misapplication of the generally beneficial teaching of divergent thinking. Further students who have difficulty conceptualizing an operation, multiplication for example, may find even more complex the examination of multiple methods of attacking a multiplication example, and may have great difficulty keeping straight what to do when as they proceed from problem to problem. Students’ processing is, as a result, slowed. Facts and operations should be memorized and mastered. Elementary school is an impressionable age and what students learn there they carry with them. One of the things that elementary students learn to do as they encounter mathematical dilemmas is they learn to be safe. Generally once an elementary student picks up a technique that works, however inefficient, he or she is reluctant to let it go.
The application of facts and operations to solving particular problems should receive broader examination in complex circumstances. Divergent reflection should be applied those issues whose weight merit creative examination. A discovery approach to learning mathematical principles requires sensitivity and understanding of those principles on the part of the teacher.
Elementary teachers have generally less training in mathematics than other areas of study. Some elementary schools have specialist teachers instruct in mathematics, while other schools have provided in house training in the same. “Our individual experiences with elementary teachers, corroborated by any number of national studies, reveal their limited background in mathematics.” (Greenberg and Walsh) The landscape of teaching practices in mathematics in elementary school and the reservoir of knowledge of that discipline in elementary schools, however, remain generally weak but improving.
Elementary school teachers are, further, naturally protective of their students’ anxieties given that elementary parents have, over the years, become less tolerant of their children experiencing discomfort with lessons. The memorization of repetitive processes and the need to know facts to complete these processes with some efficacy can cause stress to students. Students have come to be viewed, over time, as less able to deal effectively with that stress.
Mathematics, meta-cognition and Gardner’s multiple intelligences.
Gardner’s intelligences suggest that we all have innate cognitive strengths and weaknesses and that all of those capacities benefit from development. If the musical side and the kinesthetic side benefit from development, then the mathematical side does as well. The illumination, years ago, by Howard Gardner of the then “seven intelligences” opened the way to differentiating instruction in ways that are friendly to the acquisition of those various intelligences.
The seven intelligences support meta-cognitive practice. The intelligences bring with them an awareness that is particular to the disciple such as musical thinking, artistic thinking, or scientific thinking. Mathematical thinking necessitates explicit instruction to students about the kinds of processing that is required to engage successfully with a given topic or problem. Understanding this process by the teacher and facilitating understanding of the students is a priority in other counties, some of whom have become great examples. In Japan, teacher training in exposition of mathematical topics and associated thinking skills (metacognition) is practiced daily. In the United States an explosion has occurred in the teaching of thinking skills. 21st century talk has supported meta-cognition but often the mathematical piece is given lesser priority because fewer educators are familiar and comfortable with those skills.
Teaching reading and mathematics – a case in priority
We have given great attention in this country to the concept and practice of fluency in reading while it seems have we underplayed the importance of mathematical fluency. Mathematical reasoning is overshadowed by literacy throughout the grades but especially at the elementary school. Students learn the ABC’s by heart. Students have learned to read for generations (we have almost universal literacy, if only to read the bible.) Students are taught to read by parents reading to their children, by siblings read to younger siblings, by reading teachers, and by teachers of the first three grades in elementary school. The skill of reading is exercised in virtually every arena. The development of associated thinking skills accompanies the process. Whether accomplished through a basal approach, through a phonetic approach, or by any other of several methods, reading becomes an acquired skill by almost all. Reading receives more attention and class time in the early grades of schooling than any other discipline.
The only limit to ones capacity to “read” lies in one’s ability to conceptualize the ideas about which one is reading. A major stumbling block for students in middle school is moving from understanding stories to using reading to understand ideas and using higher order thinking skills to deduce, interpret, induce, synthesize or analyze the information in their texts. All of these skills are, ironically, the very juice of mathematical thinking, the poor sister skill that gets shorter shrift in the early elementary years.
Practice with simple formulas brings students into the use and manipulation of symbols and ideas that then provide the playpen for analysis. The early address of these skills allows the growth of mathematical confidence, familiarity with symbolic expression, and comfort with the inverse nature of multiplication/division and addition/subtraction. Learning one multiplication fact, then, facilitates the easy acquisition of two division facts. These skills are the abc’s of processing and require memorization and habituation just as did the learning of the abc’s for reading/decoding fluency.
If reading is the key to learning then mathematical, scientific, literary, economic, psychological and sociological understanding is the prize.
Recommendations:
1. Students should experience formulas and symbols early in their elementary experience and should become fluent in their manipulation
2. Automaticity should not be viewed as today’s slogan; it is the bread and butter to success with mathematical study as does learning the abc’s affect successful reading.
3. Number facts, operations (ways to do mathematical computation), and processes should become second nature to children’s experience.
4. Discovery methods should be employed as appropriate but abandoned as soon as possible.
5. Increase the amount of time in schools at all levels that is devoted to math teaching and analytical reasoning.
6. Increase greatly the explicitness of instruction that mathematics teachers provide to students about the changes in thinking patterns that are required to successfully ingest explanations or to step through solutions to problems.
7. Increase the level of sensitivity of mathematics teachers to meta-cognition in the teaching of mathematics
8. Adults who do not have a keen sense for mathematical thinking should not teach it.
9. Adults must accept that questioning the value of mathematics education is two steps beyond healthy skepticism.
References
Swartz, Robert J.,Thinking-Based Learning (Cambridge, MA: Christopher Gordon Publishers, 2007), co-authored with Art Costa, Barry Beyer, Rebecca Reagan, and Bena Kallick.
“Thinking Based Learning”, Educational Leadership, February, 2008.
“Teaching Critical Thinking in History and Social Studies”, The Social Studies, October, 2008.
Swartz, Robert J., Thinking-Based Learning (New York: Teacher’s College Press, 2010), Second Printing, Co-authored with Art Costa, Barry Beyer, Rebecca Reagan, and Bena Kallick.
Gardner, Howard (1983; 1993) Frames of Mind: The theory of multiple intelligences, New York: Basic Books. The second edition was published in Britain by Fontana Press. 466 + xxix pages.
Gardner, Howard (1989) To Open Minds: Chinese clues to the dilemma of contemporary education, New York: Basic Books.
Gardner, H. (1991) The Unschooled Mind: How children think and how schools should teach, New York: Basic Books.
Gardner, Howard (1999) Intelligence Reframed. Multiple intelligences for the 21st century, New York: Basic Books. 292 + pages.
Gardner, Howard (1999) The Disciplined Mind: Beyond Facts And Standardized Tests, The K-12 Education That Every Child Deserves, New York: Simon and Schuster (and New York: Penguin Putnam)
Julie Greenberg and Kate Walsh, The Executive Summary of No Common Denominator: The Preparation of Elementary Teachers in Mathematics by America’s Education Schools, available online from www.nctq.org. (National Council on Teacher Quality)
“Cultivating divergent thinking in mathematics through and open ended approach,” Kwon, Park, & Park, Asia Pacific Education Review 2006, Vol. 7, No. 1,51-61.
Understanding and Improving Classroom Mathematics Instruction”, James W. Stigler; Hiebert, James, 1997
Stigler and Hiebert, 2004, Improving Mathematics Teaching, Educational Leadership, February 2004, Vol 61. Number 5, p. 12-17.
“Knowledge Construction and Divergent Thinking in Elememtary and Advanced Mathematics”, Gray, Pinto, Pitta & Tall, Educational Studies in Mathematics, Vol. 38. nos. 1-3 (1999) p. 111-133.