Building Bridges After Taking The Scenic Route

Math is hard. You have to take these symbols and string them together with other symbols in a way that makes sense. Lucky for us, mathematicians have developed tried and true rules and patterns to find and follow. Having a solid understanding of numbers and the operations that shape them is the springboard for every other branch on the tree of mathematics like geometry, algebra, or statistics. In my middle school classroom, a majority of the year is spent on working with numbers and operations. We see how one concept transforms into another, eventually evolving into more complex ideas and applications at the year’s end. We begin with converting fractions and decimals into one another, then show how to perform our operations in both of those forms, which eventually leads to strategies that will be used to solve equations in the second half of the year. Many of my students seem to have forgotten about how interconnected math can be. I like to show them how we can trace everything back to different ideas of addition using my Mathematical Knowledge for Teach, more specifically my Specialized Content Knowledge (Hill, 2009), to help create connections to the rest of their understanding of the other three operations. Addition is typically the first operation we learn, which is then followed by subtraction, presented in the form of the opposite of addition or undoing addition. We then get to multiplication, which is repeated addition, and division, which is repeated subtraction. Those four operations are everywhere in any kind of mathematics. They can get so bogged down by learning algorithms that they set aside their basic understanding that got them to this point. Having a strong foundation when it comes to numbers and operations paves the way for success in mathematics.

The impact of lacking a strong foundation in numbers and operations was showcased with Erlwanger’s (1971) student, Benny. Benny was curious about how numbers and operations connected, but his lack of proficiency in numbers and operations led him to find patterns that do not always hold up to mathematical rules. Benny’s main problems came when his definition of a number shifted from whole numbers to rational numbers. Once Benny started seeing the inclusion of rational numbers in his problems, the rules he once knew could not be applied so easily and he was forced to come up with new rules to try and make sense of them. Unfortunately, many of these rules did not hold up for all numbers.

Like Benny, it can be hard for our students to just believe something without knowing why you should, especially with such a rule-driven subject like mathematics. With a myriad of ways to compute operations with different types of numbers, it can be easy to fall into the trap of “just do this” instead of explaining the reasoning behind why you choose the steps you take in problem solving. I can definitely be a victim of this sometimes in my teaching, as I try to justify that the “why” will only lead to more confusion instead of building bridges between concepts. It may not be easy to build a bridge, but it makes it much easier to rationalize strategies once you know the reasoning behind them. Number talks are a great way to help students build bridges of relational understanding. Students are able to showcase their understanding - showing their classmates the bridges that they’ve built - with the teacher there to cement those ideas into place (Kilpatrick et al., 2001).

References:

Erlwanger, S. H. (1971). Benny's conception of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1(2), 7-26.

Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001). Chapter 9: Teaching for mathematical proficiency. Adding it up: Helping children learn mathematics. National Academy Press.

Wix. Math Class [Photograph]. Wix. https://static.wixstatic.com/media/11062b_87a54e59c5114ba0a6a6a0cc1f499ad1~mv2.jpg/v1/fill/w_879,h_587,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/11062b_87a54e59c5114ba0a6a6a0cc1f499ad1~mv2.jpg

Wix. Missing Piece [Photograph]. Wix. https://static.wixstatic.com/media/09e9e6bcb6d68de5b1bb798a05af0b59.jpg/v1/fill/w_879,h_660,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/09e9e6bcb6d68de5b1bb798a05af0b59.jpg

Wix. Student Learning Mathematics [Photograph]. Wix. https://static.wixstatic.com/media/11062b_164f6e4e601647758be55ca317ab4584~mv2.jpeg/v1/fill/w_879,h_587,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/11062b_164f6e4e601647758be55ca317ab4584~mv2.jpeg