Mathematical Processes

What does it mean to do mathematics and learn mathematics?  The National Council of Teachers of Mathematics, NCTM, describes “Mathematical processes” as the actions required to think mathematically.  Our Saskatchewan Curriculum recognizes these processes as crucial to the teaching, learning and doing of mathematics.  

 

Communication: Students need opportunities to speak and write about math in order to consolidate their understanding.  “Communication helps students organize and reflect on their own mathematical thinking” (Glanfield, 2007).  How many opportunities do we give students to talk? 

                                                                                               

Connections:  Mathematics is a continuum of learning, one concept built on another.  Everything we understand in mathematics is built on our prior understanding.  There is a rich and important connection between mathematical ideas—connections between symbols and procedures, and connections between math and the real world. If we teach math without exploring and highlighting these connections for students, we teach a set of discreet rules, piecemeal procedures that fosters memorization rather than understanding.  Do we teach math as a series of discreet procedures and skills, or as a continuum of understanding?

 

Representations:  Mathematical diagrams, charts, equations, manipulatives, and patterns do not only display student reasoning, but also foster understanding.  Solutions without representations may not demonstrate understanding. In math, a picture truly is worth a thousand words, as there are many concepts that can only be demonstrated and understood through representation.  Students that construct pictures and diagrams to solve problems add another dimension of understanding to any procedure. Our curriculum promotes representation, as there is a move away from memorization of procedure to understanding of concepts; therefore, our outcomes specify “concrete, pictorial (representational) and symbolic” display of understanding. Do we take time to explore and refine students’ mathematical representations? What can we learn about students’ mathematical understanding by sharing their reasoning?

 

Visualization is the discipline that allows us to conceptualize number, pattern, and shape relationships.  Visualizing mathematical meaning and connections is vital to finding an approach to a situation or problem, yet is often not explicitly taught. Having students communicate their reasoning fosters development of visualizing mathematical concepts.  How can we use classroom dialog to explore and foster students’ visualization?

 

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Reasoning:  Mathematics is the discipline of reasoning.  Mathematics is more than numbers: It is the language of if, then; the language of absolutes and situations, and of evidence and proof.  Math teachers strive to help students develop confidence in their own reasoning abilities.  By communicating, making mathematical connections, and creating representations, we foster students’ ability to reason.  Logical reasoning is integral to mathematics, but applies to logic beyond mathematics. How do we support each student in developing an ability and desire to reason mathematically?

 

Problem solving refers to the ability to reason, visualize, make connections and representations to find a logical path to solutions.  Students that are proficient problem solvers demonstrate an organization of reasoning accompanied by practiced strategies and persistence.  Grit is an important attribute of successful math students and mathematicians.  What can we do to foster grit in our students?

 

Technology is a tool that has greatly enhanced our understanding of mathematics. NCTM encourages use of calculators and software to explore, show relationships, provide pictorial representation, but cautions that calculators How do we use it in the classroom to promote and support learning, rather than to replace learning?