Mathematics is more than just a collection of facts, procedures, and theories. It is a “way of thinking” that harnesses our innate ability to create abstract representations of patterns, and, by applying reason, discover relationships – for example between a circle’s diameter and circumference – that might otherwise be hidden (Jones, 2012, p. 4; Nelson, 2002). The field of mathematics has become dramatically more complex and important in society over the past 100 years, and a “deep” understanding of mathematics – one that allows mathematics to be applied to all aspects of life – is increasingly necessary to be successful in “education, work, and other areas of adult responsibility” (National Research Council, 2012, p. 1; Toulmin, 2007).
Unfortunately, increasing inequality, in combination with high stakes standardized testing, is preventing many students – especially those who are poor, nonwhite, or learning English – from developing an ability that they could use to escape marginalization (Jones, 2012 p.116-118). Because of a vicious feedback loop of poor test performance reinforcing lowered expectations among struggling students, entire groups of students often go through school believing that they lack mathematical-ability, and that mathematics is not relevant to their life-plans (Gutierrez, 2013). In response to this injustice, I seek to develop a deep understanding of mathematics among all students, regardless of what they already know, what social group they belong to, or how much support they have available outside of the classroom.[1]
My effort to put intention into practice is guided by the work of cognitive and developmental psychologists –like Vgotsky, Krashen, Piaget, and Dienes – who make the case that deep understanding results from students connecting new information to what they already know.[2] According to this school of thought, to be able to apply math to other aspects of life, students must first actively develop their own mental representations, or schema, of mathematical concepts, then connect these schema to their knowledge of other subjects, including their unique life-experiences (Jones, 2012, p. 34-36). By first learning how to use mathematical schema in familiar situations, students then gain the flexibility and judgement needed to use the schema in unfamiliar situations (National Research Council, 2012, p. 5-6).
My instructional method puts cognitive learning theory into practice by challenging students to actively work – alone, with technology, and with others – to solve problems that are both personally relevant and tantalizingly within, but just out of their reach to solve. Unlike “traditional” classrooms, that mainly teach students how to execute mathematical procedures to arrive at a correct result, my classroom activities help students discover successful processes that they can use to solve problems. In this environment, “incorrect” answers can yield valuable lessons, and many approaches can be “correct” (Jones, 2012, Ch. 6). To be successful, this method of teaching requires a deep understanding of each student’s background – academic, psychological, and sociocultural – to develop activities that are both developmentally appropriate and personally relevant. The method also requires a classroom culture where students from all backgrounds feel secure and welcome enough to actively participate in (Harrera, 2016). In other words, my teaching practices are guided by cognitive psychology and follow a problem-driven approach because this approach is naturally aligned with my goal of developing a deep understanding of mathematics among each and every student.
[1] The “equity principle” produced by the National Council for Teachers of Mathematics (2000), cited in Jones, 2012, p. 16).
[2] Behaviorism, with its emphasis on developing a consistent response to a stimulus, has limited value in teaching students how to transfer mathematical knowledge to new situations. Therefore, its role in motivating my practice is limited to situations in which students need to develop procedural fluency through repeated practice.
References
Gutierrez, R. (2013). Why (urban) mathematics teachers need political knowledge. Journal of Urban Mathematics Education.
Herrera, S. G. (2016). Biography-driven culturally responsive teaching. Teachers College Press.
Jones, J. C. (2012) Visualizing elementary and middle school mathematics methods. Hoboken, NJ: Wiley.
National Research Council. (2012). Education for life and work: Developing transferable knowledge and skills in the 21st century. National Academies Press.
Nelson, E. (2002). Mathematics and Faith. URL: http://www.math.princeton.edu/~ nelson/papers.html.
Toulmin, C. N., & Groome, M. (2007). Building a science, technology, engineering, and math agenda. National Governors Association.