By Bowen Kerins

A wide-ranging team worked together to develop the Illustrative Mathematics Grades 6–8 Math curriculum. Many of the authors were and are experienced teachers of Grades 6–8, while others are experienced high school teachers.

My own experience is as a high school teacher, then a high school curriculum writer. One of the ways the IM team’s experiences led to a higher-quality product was the discussion around language and terms used throughout the three grades.

I remember vividly one discussion while building the Grade 6, Unit 2 materials introducing ratios and proportional reasoning. A writer on that unit was discussing the different representations that would appear in the unit, including diagrams, double number lines, and ratio tables. I, honestly, had never heard of a ratio table. When I was informed that it was a table of equivalent ratios, I suggested we should use that term instead.

It’s subtle, but there are good reasons for doing this.

• First, the term “ratio table’ disappears—by that, I mean it’s used in a particular grade or grades and then not used in later grades’ work. When this happens with vocabulary, it suggests that the vocabulary is not really useful and is a candidate for removal.
• Second, the term “ratio table” hides its meaning: that all the rows in the table consist of equivalent ratios. A student might think “ratio table” is a table of any ratios, not just equivalent ones. Additionally, students have just learned the phrase “equivalent ratio” within the same unit, so burying that phrase and concept could lead to substantial difficulties.
• Third, the term “ratio table” is never used in the Common Core State Standards, and is only used in the ratio and proportion progression document after it was defined as a table of equivalent ratios.
• Lastly, reducing the overall vocabulary load keeps the focus on key concepts, and is especially helpful for students who are below grade level or are English language learners.

So, the lesson that introduces this concept refers to a “table”; the next lesson calls them “tables of equivalent ratios”.

You’ll find the same level of care throughout the curriculum, and it leads to a relatively clean and short glossary for each grade.

The same thing is also happening behind the scenes: as a team of writers, we try to speak with the same voice and make the same decisions. For example, here are some decisions the team made:

• Never write 2 cm : 3 cm, because this is not a ratio. Instead, write “the ratio of length to width is 2 : 3.” To indicate scale of a map say, “We are using a scale of 1 inch to 10 feet,” or “On this map 1 in represents 10 ft.”
• Never use the term improper fraction. This term does not appear in the Common Core, and only appears in the progressions documents in a passing mention about converting a mixed number. This term leads students to think 3/2 cannot be a correct answer.
• Satisfy and satisfied, as in “satisfy an equation,” are never used in student-facing text. Use instead: “make the equation true.” The term “satisfy” hides its meaning, and students need to comfortably learn what it means to make an equation true.

These decisions were compiled into multiple style documents, including a 54-page overall writer’s style guide with sections devoted to careful mathematical language. Hopefully this gives you a sense of the attention to detail in the work, as well as some insight into the way a large team can work together to produce something with a singular voice.

What questions do you have? What terms do you like to use when working with students that you might not use when working with teachers? Conversely, what terms do you avoid when working with students?

##### Bowen Kerins

Bowen has been teaching and working in curriculum design since 1997. He began his career as a high school mathematics teacher in Newton, MA and was then a lead author on the high school CME Project Algebra 1, Geometry, Algebra 2, Precalculus, Linear Algebra, along with a three-course Integrated series, while working as a Senior Curriculum Designer at Education Development Center. Bowen has written and delivered the course for teachers at the Park City Mathematics Institute since 2001, and volumes of these courses have been published by AMS. Bowen has led professional development live and online for teachers and coaches nationwide, and presents at NCTM and NCSM annually. Bowen has a B.S. in mathematics from Stanford University and a M.A.T. in mathematics education from Boston University. In his spare time, Bowen plays pinball and is a mathematical consultant for TV game shows.