Anchor representations are learning destinations – common experience points – that classroom communities can use to relate and build on individual understanding.

I flew back to Seattle on Sunday after visiting family in Indianapolis over the weekend. I have a bad habit of trying to be a “cool” traveler, pretending to fit right in with the business travelers and pretend like it’s no big deal that I’m flying through the air at 30,000 feet. This was working quite well until I saw this:

I did a double take, immediately forgot I was trying to be cool, smiled like a little kid, pulled out my camera, and snapped a picture or two. Can you blame me? As with all sunset pictures, the image simply doesn’t do the scene justice. Even though I’ve flown enough to know pretty well what I’m likely to see looking out of the window, it still endlessly fascinates me to peer down on cities, rivers, and clouds, taking in the sights from a new, more complete perspective.

Changes in perspective like this help me to put the landscape around me into a broader context. It’s the same way with students. Experiences that give students a better vantage point on the conceptual landscape allow them to piece together individual ideas, lessons, skills, and thoughts in a more cohesive way. I think it's our job as teachers to create these experiences for all of our students.

## What Makes a Concept Stick?^{1}

In my teaching experience, I’ve encountered certain units and lessons that seem to be much better than others at helping students truly grasp concepts. I’ve taught algebra students to solve equations using Lab Gear, I’ve introduced the notion of geometric proof with Proof Blocks and I’ve used extensive metaphor to help students understand computer science concepts. In each of these cases, these alternate representations helped students to first understand ideas in a different context. After that, I work diligently to help students connect their understanding back to the “real” (or traditional) representation, whether that is an algebra equation, a geometric proof, or computer code.

I’m immediately reminded of my own experience as a learner in college, using the Fourier Transform to analyze and solve electrical systems. The Fourier Transform creates a completely different representation of difficult math problems by mapping them into a domain where things like integration and differentiation are represented using simple algebraic relationships. Along with this transformation comes a completely new way of thinking about signals and systems. This new perspective is affectionately called the frequency domain or the s-domain. I learned to live and to think in this frequencydomain for some time by learning to use the Fourier Transform. It is not enough to just analyze problems in this new domain, however, and I also learned how to translate my understanding from the frequency domain back into regular descriptions of electrical systems. Certain tasks, like filtering out particular sounds in an audio recording, are almost impossible to do without using the lens of the frequency domain. The work that I did as a student in understanding both the regular representation and the frequency domain representation strengthened my conceptual understanding of how everything fits together. I still geek out sometimes when I realize that something I see around me is an example of frequency domain concepts. One reason why I started this blog was so that I didn't have to lose any more friends after spending an hour telling them how cool the Fourier Transform is...

So, these representations – these *things* that we use as teachers to support learning are very important. I think some representations are stronger than others. I call these *anchor representations*. I want to discuss what these anchor representations are, what makes a “good” representation, and why this matters for student learning.

## What are Anchor Representations?

An anchor representation is a framework for examining, exploring, and understanding a particular target concept or standard. A target concept is a concept that you want students to understand and be able to apply as a result of a lesson, unit, or course.

Anchor representations include the notation, language and word choices that support the use of a particular construct for understanding. They often use metaphor as an explanation model. Good anchor representations are consistent, reusable, and able to be repurposed for future concepts. Anchor representations also have one or more breakdown points. These points define where and when the representations stop becoming useful. I’ll share an example of something I consider to be an anchor representation in my algebra class for teaching students to write linear equations for situations.

## Staircase Tables^{2} - An Anchor Representation in Algebra

In Algebra, I like to put analyzing linear relationships early in the year. Even before we’ve focused explicitly on equations and expressions, we explore linear relationships and their equations through the lens of pattern finding. We start by looking at patterns represented in tables. I work to build a rich set of words, notation, rules, and metaphors around these tables to help build out the anchor representation. I call them “staircase tables” for their structure and for the metaphor that I use in explanations. Here is an example problem that is solved using the staircase table as a representation for understanding. Key terms and vocabulary that support the representation are italicized.

`find the equation of the line that goes between (-5, -3) and (17, 8).`**1. The setup (shown in blue):**

We identify early that in order to write an equation for a linear relationship, we need to know the *starting point* or y-intercept, and the *unit rate* or slope. I teach students to create a table using the information from the problem and I always tell them to “leave a space for 0.” One rule I push fairly strongly is that students should create their tables with the x-values increasing. I tell students that this is not a requirement for a valid table but it helps them keep things more organized.

**2. The unit rate (shown in red):**

Students start by using two points that they know. They draw “matching” *change arrows* between them, showing how much change there is on both the x and the y side between two rows. Students capture both the direction and the sign of the change (“matching” change arrows have to go in the same direction). This helps them correctly calculate the slope of the line by just dividing the change in y by the change in x. Dividing the change in y over the change in x is one of our "rules of the road," things that we just need to know as mathematics students in order to effectively communicate with other mathematical thinkers.

**3. The starting point (shown in black):**

We talk extensively in class about linear relationships as “staircases” with a certain “step size” and direction. Once students have identified the *unit rate* for a problem, I use the analogy of the staircase to help them work to find the *starting point*. Students start by choosing a point where they "know the numbers” (both x and y) and draw a *change arrow* to the *starting point* on the x-side (where the value is 0). If we’re working back to 0, meaning a negative *change value*, this change arrow shows “how many steps back we’re taking,” as opposed to "steps forward."

We learn from investigation that the change in y between two rows is the same as the change in x multiplied by the *unit rate*. In the staircase analogy, this is the idea that if we know the constant size of each step, and we know how many steps to take towards our *starting point*, we can use multiplication in lieu of repeated addition. At the end, students record their answer in a slope-intercept equation. I emphasize the concept of why the *unit rate* needs to go next to x and why the *starting point* needs to be added at the end on its own, leaving more details to later in the year once we’ve focused more on algebraic expressions.

This representation is an alternative to the oft-used method of starting with a canonical forms for linear equations (slope-intercept form, point-slope form, etc.) and plugging in information that is known, then using algebraic manipulation to simplify the equation in a desired form. These methods are not necessarily worse, but they do tend to cloud the underlying concept of a linear relationship: x and y change together through a constant additive pattern.

The staircase table is an example of an anchor representation towards the target concept of modeling situations using linear equations. Focusing exclusively on this representation in the beginning of the year has helped more of my students gain a more complete understanding of the concept. Additionally, students seem to pick up other methods with ease by relating them to and making sense of them in the context of this anchor representation. This is just one example of what I consider an anchor representation. It's a thorough and detailed framework for understanding

## Why Search for Anchor Representations?

If anchor representations are strong enough, they provide a home base for conceptual and procedural growth for all students.

Imagine that as a teacher, you’re a tour guide. Your students are eager tourists (just like every day in class!) in your guided tour of the city and your goal is for them to be able to get around on their own after your tour. I moved to Seattle about 3 years ago and for the first year, I struggled to understand the landscape of the city. The roads are crazy and because of the hills, you have to think about directions in three dimensions, adding height to your understanding of maps. Does Mercer St. actually intersect Highway 99 like it looks on Google Maps? (No.) Can I exit from I-5 Northbound to get right to Safeco field? (Yes, but keep your eyes peeled because the exit isn't marked until you’re on it and if you miss it, you won't have another change to get on the west side of the highway for a while.) I could have used a good tour guide when I first moved here, but I eventually figured out how the neighborhoods and roads of the city piece together.

As teachers, we are the tour guides. We are given the task of coming up with a path to take our students on (our curriculum) that will give them the best overall perspective on their city of math concepts. We want them to be independent thinkers after they leave our tour and to be able to navigate around the city as well as they can. Anchor representations are like hills in Seattle that let you see how the neighborhoods and roads fit together. Anchor representations are learning destinations – common experience points – that classroom communities can use to relate and build on individual understanding. No matter where students go afterwards, they can still relate their understanding to these common experiences from anchor representations.

Guiding students to anchor representations can help them learn more from each other. When students have the common language, experience, and understanding of a particular concept by using an anchor representation, they are better positioned to communicate together about that concept. Also, if individual students contribute different representations, methods, examples, or explanations, the classroom community can tie these back together to the common understanding of the anchor representation. If anchor representations are strong enough, they provide a home base for conceptual and procedural growth for all students.

## Anchors Away!

Anchor representations can be hard work. I don't think that finding and integrating these representations is easy or even doable for every unit or topic. Effectively using an anchor representation requires deep conceptual understanding from teachers and also requires strong familiarity with and a committment to the details of a particular anchor representation. Despite these pre-conditions, I believe strongly that as teachers, we should be striving to find anchor representations and use them whenever possible.

Determining what makes a “good” anchor representation is important, but I don’t think it’s ultimately about arguing about the best anchor representation for each concept. It’s much more important for teachers to identify, understand, and use *an* anchor representation that they feel is strong and that will help their students. Taking students up to the observation deck in the Columbia Tower might give them the highest vantage point of Seattle, but a visit to Kerry Park might still give them a better understanding of where the Space Needle is relative to downtown. Ultimately, I think that a teacher’s confidence using a particular representation is as important as the strength of the anchor representation in an objective sense.

What are good anchor representations that you use for other math concepts? How do you see curriculum design and materials supporting teachers to identify and use strong anchor representations? What barriers exist to helping other teachers leverage particular anchor representations that have worked well in your classes? These are all things I’m wondering about. Thanks for reading!

**1 **Ironically, I wrote this just before I got home to see the latest Mathematics Teacher magazine devoting an entire issue to this same question.^{↩}
**2 **I owe much of this approach to discussions with an amazing colleague at Garfield HS in Seattle, WA. Thanks!^{↩}