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By NCTM President Linda M. Gojak
NCTM Summing Up, October 3, 2013
One of the most memorable moments I had in teaching mathematics occurred in a fifth-grade class. We began the year using rectangular arrays as a model to develop the concept of prime and composite numbers. We hung student-made posters of the numbers from 1 to 100 with representations of arrays and lists of factors for each number around the room. By the end of that unit all my students had mastered multiplication facts and could factor with facility as we began our work with fractions. The connections among concepts and the use of concrete representations certainly led to deeper understanding. Later that year, students worked with a variety of models to find area and perimeter of rectangles and extended that experience to find the areas of triangles, parallelograms, and trapezoids. Most students were able to generalize a formula, albeit not always the most efficient, for each polygon. One day, a student commented that this was just like what they had studied at the beginning of the year. When I gave a puzzled look, the class pointed to the posters still on the wall from our first unit of study and said, “You know, that factor and multiple stuff.” I had a new appreciation for the power of providing experiences that enable students to make connections among mathematical ideas. My students remembered and understood the mathematics that we had studied months earlier!
Since that experience I have given much thought to the Process Standards in Principles and Standards for School Mathematics, and their impact on teaching. With the current focus on progressions and trajectories of content standards, the potential of the Connection Standard (NCTM, 2000) continues to pique my interest. It’s a powerful standard:
Instructional programs from prekindergarten through grade 12 should enable all students to—
recognize and use connections among mathematical ideas;
understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
recognize and apply mathematics in contexts outside of mathematics.
Too often, rather than making sense of mathematical ideas, students focus on remembering procedures or tricks. For example, how many students learn “flip and multiply” to divide fractions but have no idea why it works? Often those who understand why the procedure works struggle to apply it in problem situations. The procedure alone often leads to misconceptions. Students who work from rote memory often invert the wrong fraction, forget to change operations, or even apply the rule when multiplying two fractions. The meaning of operations doesn’t change from whole numbers to fractions. For example, in the early grades, the understanding that students develop of division of whole numbers often rests on the idea that “9 ÷ 3,” for example, asks how many groups of 3 are in 9. As students move to fractions, it is important to provide them with experiences that connect this whole-number understanding to similar examples with fractions: “9/16 ÷ 3/16,” for example, asks how many groups of 3/16 are in 9/16. In this way, students gain a deeper understanding rather than depending on a memorized procedure and can apply division of fractions to a variety of problem-solving situations and real-world applications.
Many teachers use manipulative materials to introduce a new concept. Manipulatives themselves, however, do not ensure understanding. We must provide experiences that support students’ efforts to make connections between what they are doing with the materials and the mathematical ideas that they represent. This takes time and teacher expertise. Algebra tiles are not an end to teaching basic algebra concepts—when used appropriately, they provide students with opportunities to connect their work to the concepts. And it is these connections that enable students to make sense of the abstract representations.
Although it is important to think about the connections among concepts within the grade level or courses that we teach, it is also important to reflect on the connections across grade levels. This work involves thoughtful discussions with other colleagues about the way that concepts are taught and the potential linkages among those ideas. Many of us learned mathematics as isolated pieces of information. Taking a mathematical concept and considering how it originates, extends, and connects with other concepts across the grades will help teachers to develop a deeper understanding. It is then that we can plan instruction that ensures that our students regularly make connections to help them make sense of the mathematics they are learning.
Learning and Transfer
Processes of learning and the transfer of learning are central to understanding how people develop important competencies. Learning is important because no one is born with the ability to function competently as an adult in society.