Math and English: Dueling Standards
The Common Core State Standards Initiative created standards for Math and English Language Arts with History and Science as subsets of ELA Literacy. This is obviously because math and language arts are so different—or perhaps that’s not the reason at all. In fact, national standards were created for the two subjects specifically because math and language are regularly and internationally assessed for accountability purposes. Further comparison of the common core standards reveals an important integration of previously divergent teaching practices.
The first comparison is in the focus on content and foundational skills in math and language arts. Traditionally mathematical instruction has prioritized teaching specific mathematic skills over process and language arts instruction has focused on the reading and writing process over specific content. The Common Core State Standards Initiative, however, has given both equal importance to the what and the how. Therefore, the Standards of Mathematical Practice carries the same importance as all the ELA Standards regarding Writing, Language, Speaking and Listening. Likewise, the three subsets of reading standards emphasize the essential ELA content: Foundational skills, Informational text, and Literature. Reading skills of informational text had previously been under prioritized. In math, foundational skills of computation, statistics and probability would never be overlooked or glossed over in this way.
Secondly, there is a real focus on problem solving that goes far beyond solving word problems. The Common Core State Standards Initiative has identified eight specific standards of mathematical practice. Teaching language arts has always focused on the reading and writing process, but teaching math more effectively requires this same attention to the mathematical process or practice. The first standard of mathematical practice addresses how mathematically proficient students make sense of problems: “by explaining to themselves the meaning of a problem…[making] conjectures about the form and meaning of a solution…and [considering] analogous problems …and special cases.” This is where math intervention can be the most effective not in repetition of memorized math facts but in the mathematical equivalent of reading comprehension.
In teaching long division, equivalent fractions and ratios, I’ve been working with a few students that have needed some concentrated focus on various concepts related to division. One student in particular could set up an algorithm for long division using our divide-multiply-subtract-bring down process. The problem was that the student would get stuck at what number to write when dividing the smaller numbers. When he told me “I just can’t do division,” I asked him what he found difficult about division. I rephrased and asked him what division was expecting something about the opposite of multiplication or repeated subtraction. Getting mostly blank stares, it suddenly occurred to me that he might not understand that division is literally dividing a quantity into groups of a smaller quantity. After focused intervention on the key concept of division, he was able to anchor his procedural knowledge of long division and his knowledge of multiplication facts to a concrete modeling of regrouping. Math intervention focused on the relationship between numerals and quantities like this will allow students to more effectively navigate mathematical concepts and problem solve.
Students might be resistant in middle and high school to use number lines or color chips when learning new computational skills, but it’s important to scaffold student’s learning from concrete to intermediate to the most abstract models of numbers and variables. A student might ask: Why can’t I divide by 0? With experience manipulating color chips, students can see that they can never divide a quantity into groups with 0 quantity. Likewise, they can’t divide a quantity into zero groups. Another student asked why a negative number multiplied by a negative number equals a positive number. I explained that the positive repeated addition of a negative number would remain negative and therefore the negative repeated addition of that same number would have to be its opposite and therefore positive. A few students followed my line of thinking, but another student offered a more common sense explanation: If love is positive and hate is negative, then if you love to love or hate to hate, you’re a lover. If you hate to love or love to hate, you’re a hater. That’s when the light bulb went off for the rest of the students. This exemplifies several standards of mathematical practice as well as writing and language standards:
- Model with mathematics; Look for and make use of structure
- Construct viable arguments and critique the reasoning of others
- Research to build and present knowledge
- Knowledge of language; Vocabulary and acquisition and use
It is always easier to remember a mathematical “rule” that is connected to life experience and real world understanding. It’s even more important that students produce their own explanation than passively consuming the teacher’s reasoning.
Even though math looks mostly like numbers, math is a language that is learned the same way any language is learned—by exposure, immersion and interaction. This language can be used to explicitly teach mathematical relationships. Teachers can help students make connections by explaining meaningful metaphors like factor trees and pie charts and root words like commute, associate, and distribute. When reading the decimal 0.75, point out the difference between “zero point seven five” and “seventy five-hundredths.” The former dictates how to write the number and the latter reads the number with its name. Sometimes I check in for an appointment and tell the receptionist that the appointment is for “Melanie, M-E-L-A-N-I-E” but only if they need to know how my name is spelled. This kind of language use in math isn’t obvious but essential to understanding that the numbers aren’t random. Students are always asking,
“Where are you getting all these numbers?” The answer is in the language: 0.75 is 75/100 because 0.75 is actually “seventy-five hundredths” and 75/100 is also “seventy-five hundredths. Mathematically proficient students consistently read, hear, and use this kind of language.