by NCTM President Henry (Hank) Kepner
NCTM Summing Up, November 2009
Why reasoning? Much of the power of mathematics lies in its coherence, from basic relationships and properties, such as those dealing with whole numbers, to reasoned extensions to relationships, procedures, and applications involving fractions, negative numbers, algebraic systems, and beyond. NCTM’s new publication, Focus in High School Mathematics: Reasoning and Sense Making, released October 6, 2009, defines sense making as “developing understanding of a situation, context, or concept by connecting it with existing knowledge” (p. 4). It uses examples to make the case that the learning of high school mathematics is enhanced when students use sense making and reasoning to connect and justify new observations and rules on the basis of their prior knowledge. However, an emphasis on sense making and reasoning can enhance the learning of mathematics for students at all ages.
In a coherent curriculum presented under a teacher’s direction, students should be expected to engage in sense making and reasoning in analyzing a problem: identifying relevant mathematical concepts, procedures, or representations; applying previously learned concepts; and making preliminary conjectures and deductions. Purposeful questions posed to students on a regular basis are important in promoting reasoning: “How do you know?” “Will that always work?” “What will happen if …?” These questions, framed appropriately, are just as important in first grade as in high school.
Through such questions, we can teach students to expect to provide reasoning that justifies their solutions and to learn from, as well as critique, others’ reasoning in justifying results. As stated in Focus in High School Mathematics, “mathematical reasoning can take many forms, ranging from informal explanation and justification to formal deduction, as well as inductive observation” (p. 4). The communication of mathematics, whether oral or in written words, tables, graphs, or sketches, is based on evidence derived from reasoning about prior mathematical concepts and properties that led to particular conclusions.
Without conceptual understanding and a focus on reasoning, the “learning of new topics becomes harder since there is no network of previously learned concepts and skills to link a new topic to” (Kilpatrick, Swafford, and Findell 2001, p. 123). Without sense making and connections, students are likely to forget procedures as quickly as they learn them. This is one factor that may cause a student to take remedial math courses in postsecondary education after completing high school courses with that same content.
We have a responsibility to help students connect new learning with their existing knowledge. By aiding our students in building on their prior reasoning, we enhance their understanding and retention of new information. For example, instruction about rational numbers (fractions, decimals, and percents) should pose a challenge to students: Given what you know about the properties of whole numbers, how can you make sense of the new procedures and generalizations for rational numbers? A lack of experience with sense making is evident in high school and college students who ask, “Do I need common denominators for the multiplication of fractions or for their addition?
Too often, an easy-to-use rule for this multiplication dominates instruction, with little inquiry into how it makes sense! For many students, one of the strangest multiplication results occurs when they multiply common fractions (positive fractions less than one). Why is the product of 2/3 and 5/8 less than either factor? That never happened in whole-number multiplication, except for a factor of 0.
In geometry, far too many students use a rule as their concept for area—“multiply two numbers!” Reasoning about the area of well-chosen polygons by decomposing regions into known figures is an important approach that should replace the rush to area or volume formulas.
Students must have an expectation—reinforced by multiple experiences—that many phenomena in the physical and social worlds can be seen through a mathematical lens. From this perspective, I look for the day when for every “solve for x” exercise set that students do, they also have an exercise that asks them to create an equation, inequality, or function that models a contextual setting or describes a mathematical problem. Too often our mathematics curriculum goes one way, emphasizing practice in mathematical procedures while minimizing, if not ignoring, experiences in creating equations or functions worthy of “solving or analyzing.”
No matter what the mathematical topic, sense making and reasoning should be major components of all students’ learning. Classroom evidence indicates that students classified as “weak” in mathematics are less frequently engaged in making sense of, or reasoning about, the mathematics that they study than those labeled “strong.” This is an equity issue that we cannot ignore in preparing all students for using mathematics in the world around them and as citizens.
Kilpatrick, Swafford, and Findell, eds. Adding It Up: Helping Children Learn Mathematics.Washington, D.C.: National Academy Press, 2001.