“The answer goes here.” That’s how many students interpret the equal sign.
The meaning of equality is actually a first grade standard. And students who are fluent in the language of number sentences understand that equality means balance.
But students who lack this understanding face challenges year after year. They struggle with variables, the meaning of operations, distributive and associative properties, one- and two-step equations, even functions.
And those are just a few of the core standards that build on a solid understanding of equality.
While the concept of equality is a foundation for later standards, it is rarely assessed after the first grade. Students who misunderstand equality may make errors that appear sloppy or hasty. Others may arrive at correct answers, but use inaccurate notation.
If you’re a math educator, there are almost certainly students in your class who don’t really understand the equal sign. It’s hard to assess, as misconceptions look different for each student. Consider the following:
X+7 = 10 = 10-7 = 3
In this example, the student isn’t balancing the equation. Most likely, they are solving in their head and writing out their work as an afterthought. While the answer is correct, this strategy will only work for the simplest of number sentences.
Number sentence proofs will help your students avoid this type of error. And the best part is that you don’t need to add another topic to your curriculum. You can easily turn most standards into a number sentence proof activity.
Why Fluency with Number Sentences Matters
The term number sentence is often associated with elementary school. But it’s usefulness extends beyond elementary, because it encompasses both equations and inequalities. A number sentence is a mathematical statement made up of two expressions and a relational symbol (=, >, <, etc).
An equation is a number sentence whose relational symbol is the equal sign. An inequality is a number sentence whose relational symbol is anything else. Emphasizing the connection between equations and inequalities supports sense making. It also helps students see the similarities in working with equations and inequalities.
By proving and disproving number sentences, students deepen their understanding of equality as balance. This understanding is at the heart of algebraic thinking, and is useful well before students take an algebra course.
1.Arithmetic is not Conceptual
When I was in school, elementary and middle school math was all about arithmetic. The thinking was that students needed to learn how to calculate before they could do the more “advanced” work of reasoning.
We memorized ”math facts” and performed stacked algorithms. As we got older, the numbers got larger (hundreds, thousands, and so on) and smaller (fractions, decimals). But the concepts stayed pretty much the same.
Memorizing times tables and being able to carry, borrow, and perform long division has its benefits. But many students appear proficient in certain topics when they are really just repeating procedures. A hundred years ago, being good at arithmetic could land you a good-paying job. Today, not so much. Using math in your career requires understanding the math and applying it to the real world. Simply learning to mimic a calculator is no longer enough.
2. High Math Achievers Can Struggle in Algebra
When students focus on math facts and algorithms, many end up woefully unprepared for higher math. One out of every five students in the US drop out of high school. The primary reason? An inability to pass Algebra.
The odd thing is that Algebra doesn’t have to be so hard. Schools actually make Algebra harder by emphasizing computation in the early grades, and skimping on reasoning. Many students grow up believing that being good at math is about quickly performing calculations.
Some students are naturally curious. They explore the concepts behind the computation without being explicitly taught. These students transition smoothly to algebra. But there is a large cohort of students who do well in math until 7th or 8th grade. Once they encounter variables and functions, they hit a wall.
3. Number Sentences Help Students See Connections
Contemporary math standards seek to address the difficulty that many students face when transitioning into Algebra. This involves weaving algebraic thinking into elementary and middle school math.
When students perform operations with algorithms, there is little flexibility. They repeat a process, often without realizing why the process works. For example, a student may add 27 and 89 without realizing that when they combine 9 and 7, they create a new 10 and 6 ones. Or that they are combining 8 tens and 2 tens to make a hundred.
But using number sentences, students can break the numbers out to actually see the value of each digit. They can compose and decompose numbers by place value or use other strategies, building their reasoning and mental math skills.
This is just one example of how number sentences can help students develop numeracy and mathematical reasoning – in this case around base-10 understanding. Similar examples abound for fractions, decimals, negatives, percentages, functions, and so on.
Mathematical Reasoning, an Ongoing Challenge for Educators
Math standards have adapted to increase the emphasis on mathematical reasoning. And number sentences can provide a significant support in making the shift.
But standards don’t necessarily translate into effective teaching. Part of the problem is that many schools still rely on textbooks to bring about change, despite the evidence that the choice of textbook has little to no impact on student achievement.
Teaching conceptual math requires high quality professional development and more effective resources. Educators accustomed to teaching arithmetic cannot suddenly shift to teaching algebraic thinking. Many need to learn the algebraic content for their grade – the topics that were never taught when we were in school.
Educators also need to go beyond “I do, We do, You do.” While modeling and repetition may work for teaching computation and other low-DOK tasks, they do not promote deep learning. To develop mathematical reasoning, educators need to offer student-centered and inquiry-based learning experiences.
Engage Your Students with Number Sentence Proofs
A great way to build excitement and support deep learning is with number sentence proofs. Almost any math standard can be taught through the use of number sentences. The first step is to simply unstack the algorithms.
Number sentence proofs help students go beyond disconnected “math facts” to see connections between numbers. For example, teaching that “34 x Y = 68” helps reinforce the concept of equality. It also shows the relationship between multiplication and division. Teaching “4 x 7 = (2 X 7) + (2 X 7)” not only helps students master their times tables, it also reinforces visual modeling and the distributive property.
As students develop fluency with number sentences, the skill becomes a ‘vehicle’ for teaching a range of standards. Number sentences can support multi-digit addition, equivalent fractions, ratio and proportion, and practically any standard you can think of.
But first, students must learn the conventions. They should start by doing proofs on concepts they’ve already mastered. Then use their number sentence understanding to introduce new content.
Teach equality as balance
This may be the most crucial foundation for fluency with number sentences. Show students that an equal sign is like a scale. I actually hold out my hands and pretend things are being added or removed. To stay balanced, whatever happens to one side needs to happen to the other.
This relates to the idea of inverse operations. Take x+5=8. To find x, I subtract 5 from both sides. This demonstrates that addition and subtraction are opposites. It can also be extended to show that division is the opposite of multiplication.
Read Left to Right, and Simplify Downward
Learning the conventions for working with number sentences will allow students to simplify complex, multi-step equations. One such convention is reading number sentences from left to right. Another is writing each step beneath the one before it.
These conventions reinforce the meaning and purpose of mathematical operations. Start with a value. Do something to that value (add, subtract, multiply, or divide). Find the result of the operation.
At first, we want students to become comfortable with straightforward number sentences. One operation, with a result to the right of the equal sign. Then, we can gradually introduce number sentences with multiple operations, variables, grouping symbols, and so forth.
While it’s generally important to support student creativity and exploration, that doesn’t apply to conventions. Students should be taught to adhere strictly to the “left to right” and “simplify downward” approaches when working with number sentences.
It doesn’t matter if a student’s preference is to put their steps in a jumble around the page. Students need to learn that math is a language. If someone can’t easily understand their steps, they’re not communicating effectively.
Working downward (one step per line) is also an important habit that will allow students to organize their thoughts when completing more complex, multi-step problems.
One Equal Sign per Line
Many students who have not developed fluency with number sentences will work across instead of simplifying downward.
You may see students using multiple equal signs per line. This habit even appears in high level math. But while a college professor can bend the rules, students who are still learning to work with equations shouldn’t get fancy.
Keeping one equal sign per line reinforces the idea of balance. After all, we are teaching students to think of a number sentence as a scale.
Try to imagine a scale with three or four fulcra (points of balance) – how exactly would that work?
Keep the Number Sentence Intact
Another habit some students develop is splitting up number sentences instead of working downward. For example:
x – 14 = 17 + 20
17 + 20 = 37
x-14 = 37
37+14 = 51
In the first step, the student takes the expression to the right of the equal sign and create a new equation. In the following step, they combine the simplified expression (37) with the original expression (x-14) from the left side.
The final “answer” is almost correct (we lost our variable along the way), and demonstrates a modest level of proficiency. But the approach shows that the student isn’t able to simplify downward, keeping the equation intact.
You may allow students to work this way at first. But if the habit isn’t addressed quickly (days or weeks), it will prevent the student from moving on to more complex equations.
Additionally, many students who break equations up like this will get confused trying to “recompose” into a single equation, and make sloppy mistakes.
Closed Number Sentence Proofs
When introducing number sentence proofs, start with “closed” number sentences. These are number sentences that have no variables. Students are given a complete mathematical statement and must decide if it is true or false. For example, “⅔ – ⅓ = 1.”
This is similar to “error analysis,” which can be an effective way to promote critical thinking. It’s also a mathematical practice standard (CCSS.MATH.PRACTICE.MP3), and a challenge that appears often on standardized tests.
Students first prove or disprove the number sentence by calculating. A proof by calculation consists of simplifying both sides into something that is obviously true.
For example, an equation that simplifies to ‘1=1’ would be proven true. An equation that simplifies to ‘2=1’ would be proven false.
Then, students complete a proof by reasoning. This is where students explain why the equation is true or false. It shows that they understand the concepts behind the calculation and that they can generalize about the properties of numbers and operations.
A student may prove 3 x (10 + 13) = 30 + 39 by calculating. But if they can’t explain why it works, they won’t recognize other instances of the distributive property.
Sometimes students understand a concept but struggle to express their understanding in writing. For this reason, it can be helpful for students to use visual models as a proof by reasoning. If they can draw the scenario, it’s a strong indication that they understand the concept.
Open Number Sentence Proofs
Once students are proficient working with closed number sentences, we can introduce open number sentences – number sentences with variables.
Because they have variables, many open number sentences will not be “always true” or “always false.” Some will, but we now need a third category for number sentences that are “sometimes true.”
Proofs allow students to think more flexibly about variables. Many students are confused by variables, because they are so abstract. They focus on ‘finding x,’ which reinforces the idea that math is about answer-getting. Many learn to perform math with variables, but still don’t understand the point of them.
But when students complete open number sentence proofs, they gain a more authentic understanding of how mathematicians use variables. Mathematicians didn’t invent variables so that students could ‘solve for x.’
Variables were designed so that we could design, study, and manipulate mathematical scenarios when we don’t have all the information. Completing proofs brings students much closer to the original purpose, making their learning more authentic.
Supporting Fluency with Number Sentences
It actually took me a while to make number sentences a regular part of my classroom practice. The main reason was that I couldn’t find enough resources to teach them effectively.
Over time, I developed lesson plans, sample problems and graphic organizers that made it easy to teach number sentence proofs. The lesson plans and sample problems made sure that activity was tied to the standards I was teaching. And the organizers made it easy for students to structure their thinking.