When I work with teachers, I spend a lot of time helping them plan. In a coaching role, we plan lessons together. We plan assessments or instructional strategies. In a workshop setting, we think more like strategic planning. How do topics flow within a unit? What are the pitfalls and challenges of teaching a particular topic?
Every now and again, someone will say something like, “I just need them to know that when you add two numbers you get a bigger number.” Then, my heart stops beating momentarily.
When NOT to Reach Out and Touch Someone
Karen Karp is one of my personal math heroes. So much so that I allowed myself to exert privilege as a CAMT Board member to get a picture with her at CAMT 2016 in San Antonio when she was an opening speaker! Serious fanboy moment.
In the above scenario, I reach deep inside and ask myself for strength from Dr. Karp. Fortunately, she and her colleagues Sarah Bush and Barbara Dougherty have given us a lot of information with their article series, “Rules that Expire” for Grades K-5, 6-8, and high school. These articles appear in NCTM’s professional journals. Having access to those archives justifies your NCTM membership.
A rule that expires is something we tell our students that works right now but at some point becomes no longer true. In each of their articles, they highlight a certain number of rules that teachers tell their students that at some point in their mathematical careers are no longer true. They’re very convenient in the short term and usually have caveats that teachers leave out, like “…for whole numbers only.” What I like most about the Karp, Bush, and Dougherty articles is that not only do they make these rules explicit, but they then tell us when the rules expire so that we may consider alternatives.
Why is this problematic? An excellent question! When those rules expire later on, students are bewildered because their beloved teacher told them it was always true. That bewilderment becomes confusion, and that confusion becomes a distaste for mathematics. As Yoda once said, fear leads to anger. Anger leads to hate. Hate leads to suffering.
How do we cause suffering in future mathematics classrooms? Here are some examples.
Elementary Rules that Expire
- Addition and multiplication make numbers bigger. This rule that we teach in first grade addition and third grade multiplication is only true for particular numbers. As soon as students add 0 or integers, then this rule for addition goes out the window. When students multiply by fractions between 0 and 1, multiply by 1, or multiply a positive number and a negative number, this rule expires for multiplication.
- Improper fractions should always be written as a mixed number. This rule created lots of problems for me when I taught high school algebra and geometry. Writing improper fractions as mixed numbers helps students in elementary grades see the number of wholes and fractional parts that the quantity represents. However, when students reach 6th grade and begin studying rates and ratios, it’s sometimes helpful to have improper fractions so you can see the relationship between the two quantities. In 8th grade and Algebra 1, slope is much easier to understand as an improper fraction than as a mixed number.
Middle School Rules that Expire
- Multiplication is repeated addition. This rule is true but it’s not exclusively true. Multiplication can also be scaling or represented with an array. A focus on this rule without its conceptual underpinnings may also cause students to confuse exponents with addition.
- A variable represents a specific unknown. Yes, but… A variable can represent many things, including a specific unknown. It also represents parameters (y = mx + b), labels of particular dimensions (A = bh), or generalized quantities. Presenting a variable as an unknown with a particular values limits students’ understanding of the power of using variables alongside numerals in mathematics.
High School Rules that Expire
- FOIL. Go ahead and sharpen your sword. FOIL is a useful acronym to structure the double-distribution of multiplying binomials. It is but one way to multiply two binomials. The problem with FOIL is that it expires shortly after teach it when you move to multiplication of binomials and trinomials. That’s when you reach for the distributive property and recycle your FOIL. I would make the claim that if you teach double-distribution the first time (go green and don’t use FOIL) then you’ll save yourself lots of reteach time later when you teach multiplication of other polynomials.
- You cannot find the square root of a negative number. Wanna bet? You may not be able to express the square root of a negative number in 8th grade or Algebra 1 because all you know are real numbers. And it’s certainly true that you cannot find the square root of a negative number using real numbers. Doesn’t the mere presence of the term “real numbers” imply that there must be “unreal numbers?” And, indeed, when students get to Algebra 2 they discover imaginary and complex numbers. And this rule goes in the dustbin of mathematical expediency alongside “adding two numbers always gives you a bigger number.”
These are just some of my favorite rules that expire because they illustrate the caveats that we should really place when we use procedural tricks-of-the-trade in mathematics classrooms. Karp, Bush, and Dougherty identity 12 rules that expire in elementary grades, 13 rules that expire in middle grades, and 13 rules that expire in (or before!) high school. Their Rules that Expire articles make for great conversation in your mathematics department or professional learning community (PLC). Using a protocol to run a text-based discussion on an article like these helps teachers build their mathematical knowledge for teaching while thinking about how they can better serve their students.