Love them or hate them, math standards define how we teach math in today’s schools. While they may be necessary, many educators worry that standards can prevent us from teaching our students in an authentic, meaningful way.

Imagine your 5th grade students don’t understand the concept behind fractions. To top it off, they’re not strong with multiplying whole numbers. But the standards say they should learn to multiply fractions.

So rather than focusing on the concepts they need, we ‘teach the standards.’ And since our students lack the conceptual foundations, they struggle with grade level content. Ultimately, we’re forced to rely on tricks and algorithms to prepare them for the test.

Even when our students are on grade level, standards can prevent us from exploring the richness of mathematics. Schools that adhere *too* strictly to standards aren’t able to explore concepts in depth, or teach real-world applications.

Now imagine the flip side. What would it be like to teach math without any standards? A 1st grader in one school might learn about fractions, while another school might not touch fractions until 5th grade. What happens when students change schools?

Two 6th teachers in the same school might cover different content. How is the poor 7th grade teacher to know where they left off?

Our current obsession with standards is actually a reaction to the chaos of unstandardized schools. Throughout the 1980’s and 1990’s, schools across the country taught different topics in different sequences.

In part, standardization was an equity issue. Students from low-income urban and rural districts received less rigorous instruction and assessments than their wealthy suburban counterparts.

## Unpacking Math Standards

Wherever you stand on standards, it’s hard to deny that math standards are complicated. I’d argue that they’re too complicated.

Here is a 3rd grade math standard, 3.NF.A.1, as found on AchievetheCore.org

When I first encountered this standard, I had to read it several times to even understand what it was talking about.

What does 1/b mean? Or a/b? Do I need to teach my students variables for them to understand fractions? Why is there no mention of numerators or denominators?

This is *the standard* that introduces students to the concept of a fraction. A concept that provides the foundation for many of the most critical standards they will encounter in future math courses. And one that many students struggle with well into high school.

I would propose that we could describe this standard much more simply. In a way that both teachers and students could understand. And which could more easily be translated into a lesson plan:

- A fraction refers to one or more parts of a whole cut into equal pieces
- The ‘denominator’ describes the number of equal parts the whole is cut into
- The ‘numerator’ describes the number of those parts in the fraction
- Fraction notation: a fraction is written by placing the numerator above the denominator, with a line between the two.

## Repacking the Math Standards

I’ve tried to figure out why standards are written in this way.

My best guess is that the authors wanted to create a sort of ‘airtight contract,’ that would leave no room for error, much like how legal documents are written so as to anticipate any possible uncertainty or challenge.

While the standards may be successful in this sense, I think it is a mistake to sacrifice clarity.

Standards are meant to be milestones, or guideposts. Educators (and students) should be able to look at the big ideas and determine the best way to teach them. Maybe with a textbook or an online platform. Maybe by creating their own curriculum from scratch, or through a combination of resources.

I suspect that most standards weren’t written for educators. They were written for textbook publishers. By taking a granular approach, it allows publishers to easily translate standards into lessons.

In a way, this runs counter to the purpose of the common core. The standards were designed to move away from “mile wide, inch deep coverage.” They were designed to help students think deeply and meaningfully about math.

In order to do so, we need to focus on big ideas and connections. And that requires a few changes to how we think about standards.

If we are talking about a math concept that 2nd grade students are to learn, shouldn’t we be able to express it in a way that a 2nd grade student, teacher, and parent can all understand?

We should also think carefully about teaching math as disconnected ideas. Each grade contains dozens of standards. But in reality, the important ideas from each grade level could be summarized in a few sentences.

Phrasing standards in big picture terms serves all stakeholders. Students can understand what will be expected of them. Teachers will be less bound by scripted, one-size-fits-all approaches. And parents won’t feel anxious or confused by what their children are being taught.

## Standard = Concept x Representation

The good news is that math standards can be simplified with a simple formula: Standard = Concept x Representation.

Part of the reason standards seem so complicated is because they combine concepts and representations. While there are only a few of each, the combinations add up quickly.

Think of each concept as an ingredient. And each representation is a type of baked good. Imagine how many types of cookies, muffins, and scones you could make with chocolate chips, walnuts, and raisins.

With just chocolate chips, you would have three options. Add walnuts, and suddenly you have six *more* possibilities. Introducing raisins expands your options to 21!

Now if you were learning to bake, you’d learn how to make cookies, muffins, and scones. And you’d learn about the possible ingredients. You wouldn’t learn how to make chocolate walnut scones one day, and chocolate raisin scones the next.

But that’s exactly what we do with our math standards! Rather than introducing the big ideas and showing students how they’re connected, we cover every last permutation in granular detail.

Here’s an example:

If we can identify the ‘ingredients’ to each grade level standard, the content becomes much more manageable. Teachers and students better understand what they have in store.

And we can teach the content much more efficiently. We would use less class time, and students would come away with a deeper understanding – which would also improve retention from one grade to the next.

## Math Concepts: The ‘Big Three’

Surprisingly, there are only a few core ‘math concepts’ that students need to master at each grade level. In fact, all the math concepts we teach in school can be grouped into three categories:

- Numbers: The idea that numbers can count objects or measure spaces. Over the years the types of numbers become increasingly complex: from counting numbers to Base-10, fractions, negatives, and variables
- Operations: Operations are actually shortcuts to counting. Whether we’re counting up (addition) or counting down (subtraction). Whether we’re repeating addition (multiplication) or repeating subtraction (multiplication). Even exponents and square roots are extensions of multiplication.
- Relationships: This category is a bit more nuanced. While an operation describes an action we perform on a single number, relationships involve comparing or linking two or more quantities. This includes concepts like equality, ratio, and functions.

These three categories are interrelated. For example, a fraction is both a number and an operation. And an equation is a relationship that contains numbers and operations.

There is certainly room for debate about how to categorize the concepts. But regardless of how you group them, it helps to recognize that the goal of each year is to help students deepen their understanding of number, operations, and associations.

## The Five Representations

The other half of our math standards equation is the Five Representations.

These five representations are all ways to represent the same mathematical idea.

Representations are at the heart of conceptual understanding. So a student who truly understands fraction addition should be able to solve using an algorithm. But they should also be able to represent the operation with a visual or physical model. They should be able to explain what is happening, or apply fraction addition to a real world situation.

Traditional math classes focus squarely on symbolic representations. This includes our number symbols, operation symbols, and equal signs.

Some still think of symbolic representations as “real math,” as if the other representations are something other than math.

The common core standards played a key role in spreading awareness that not all math is symbolic. The representations are now woven into our math standards. This enables us to go deeper in our math instruction, but it can also make standards more confusing.

## Dividing Your Math Standards

I think it’s time to make our math standards more accessible. And the simplest way to do that is to divide out the concepts from the representations.

As we introduce each math concept, we can teach students to translate the new concept into each of the five representations. I use graphic organizers to scaffold this way of thinking.

But to do this effectively, we can’t rely on our textbooks. We need to look at our standards and identify which parts are concepts, and which are representations.

### Dividing a 2nd Grade Math Standard

Standard: CCSS.MATH.CONTENT.2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem |

Concepts | Representations |

Base-10 up to 2-Digits Multi-Step Problem-Solving Meanings of Addition & Subtraction Number as Variable | Scale Models Symbols (Expressions, Equations, Variables) Contextual (Word Problems) |

In this standard, we are asking students to demonstrate mastery of adding and subtracting within 100. This requires that they understand addition and subtraction, and to apply base-10 understanding to operate with two-digit numbers. We also introduce variables, asking them to work with “unknowns in all positions.” Finally, they should be able to apply these competencies to multi-step problems. These are the math concepts from this standard.

But the standard also asks students to use multiple representations. So they should demonstrate base-10 understanding with models, with symbols, and in context. The same could be said for the other math competencies.

So, instead of a rambling standard that confuses concepts and representations, I prefer presenting the underlying competencies in a table as concepts and representations.

### Dividing a 7th Grade Math Standard

Standard: CCSS.MATH.CONTENT.7.RP.A.2.A
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. |

Concepts | Representations |

Mutliplicative Comparison Ratio Reasoning Proportional Reasoning Functional Reasoning | Symbolic (“quantities”) Scale Models (“coordinate plane”) Tables (Visual/Symbolic) |

This standard illustrates a granular approach, as we are asking students to test for proportionality. But it doesn’t address the concept of proportionality.

That concept is addressed, but in the ‘parent’ standard, 7.RP.A.2.

That standard contains 4 sub-standards, A through D. Each asks students to find a different way to test for proportionality or apply proportions to real world situations. For example, 7.RP.A.2.D asks students to ‘explain what a point on a graph means in terms of a situation.’

And while these are all worthwhile aspects of proficiency, they leave out many of the ways we could combine proportional reasoning with the five representations.

And presenting each aspect as a separate skill confuses teachers and students. I’ve seen many classes where one day, students will ‘test for equivalent ratios in a table.’ The next, they’ll look for lines on a graph that pass through the origin. But they don’t understand that a coordinate plane graph is just another way to represent that same relationship in the table.

When students learn these skills in isolation, they tend to struggle most with word problems, the “real world” representation.

On the other hand, when students represent proportional relationships using a multiple representations organizer, they can more easily see the big picture.

So while breaking competencies into discrete skills can help us to *assess* student mastery, it is rarely the best way to actual *teach* the concepts. This is why education leaders warn against “teaching to the test.”

## Simplify Your Approach to Math Standards

Organizing your math curriculum around standards and representations has enormous benefits. You’ll cover more content in less time, and your students will develop deeper conceptual understanding.

But it can be challenging to take this approach with a traditional textbook. Their units and pacing guides are designed for the granular approach.

If you’re ready to rethink your approach to content coverage, but don’t want to go it alone, we’re here to help. Our *Learning Journeys* provide 1-on-1 support to teachers and instructional coaches. *Learning Journeys* are customized to your individual needs and goals, providing support in areas like *Lesson & Unit Planning*, *Teaching with Visual Models*, and more.

The program includes all the resources and support you’ll need to create unit plans and pacing guides, implement collaborative and inquiry-based instructional models, and meet the needs of each learner.

To find out if *Learning Journeys* are right for you or your team, schedule a free consultation.

**About the Author**

Jeff Lisciandrello is the founder of Room to Discover and an education consultant specializing in student-centered learning. His 3-Bridges Design for Learning helps schools explore innovative practices within traditional settings. He enjoys helping educators embrace inquiry-based and personalized approaches to instruction. You can connect with him via Twitter @EdTechJeff