“The answer goes here.” To students not fluent in the language of number sentences, this is the meaning of the equal sign.
If you’re a math teacher, you have students who don’t really understand the equal sign. It can be hard to assess, as misconceptions look different for each student. And we rarely ask students to explain concepts on tests. During class discussions, it’s easy for one student’s brilliant answer to overshadow the silent majority. (See Jennifer Gonzalez’s great article on ‘fisheye.’ )
Students who misunderstand equality may make errors that appear sloppy or hasty. Others may arrive at correct answers, but use inaccurate notation. For example:
X+7 = 10 = 10-7 = 3
While it’s beneficial for students to solve problems creatively, there are limits. The method above only works for very simple problems. The student is likely solving in their head and writing out their work as an afterthought.
The Importance of Number Sentence Fluency
To appreciate the importance of number sentences, let’s start with some vocabulary. Number sentences are mathematical statements. They include two expressions and a relational symbol (=, >, <, etc). An equation is a number sentence whose relationship symbol is the equal sign. An inequality is a number sentence whose relationship symbol is anything else.
To work with number sentences, students must recognize that equality means balance. This understanding is at the heart of algebraic thinking. But it’s useful well before students take algebra. Contemporary math standards recognize the need to introduce algebraic reasoning as soon as students begin developing number sense.
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 math 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. Any math-related profession requires that you understand the math and can apply it to the real world. Simply learning to mimic a calculator is no longer enough.
When High Achievers Struggle in Algebra
When students focus on math facts and algorithms, they end up woefully unprepared for higher math. 1 out of 5 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 and skimping on reasoning. Many students grow up believing that being good at math is about performing calculations quickly.
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.
Number Sentences Help Students See Connections
Contemporary math standards shift the progression, weaving algebraic thinking into earlier math. But the standards haven’t yet translated into effective teaching of concepts. Part of the problem is that too many schools rely on textbooks to bring about change. Recent evidence suggests the choice of textbook has little to no impact on student achievement.
Teaching conceptual math requires high quality professional development and better resources. Teachers used to teaching arithmetic cannot suddenly shift to teaching algebraic thinking. Teachers must first learn the algebraic content for their grade. They also need to go beyond “I do, we do, you do.” While modeling and repetition may work for teaching computation, mathematical reasoning requires student-centered and inquiry-based teaching strategies.
A great place to start is by introducing number sentences. Almost any math standard can be taught through the use of number sentences. The first step is to simply unstack the algorithms.
By using number sentences, students and teachers can 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 builds foundations for visual modeling and the distributive property.
How to Teach Number Sentences Effectively
As students develop fluency with number sentences, this becomes a ‘vehicle’ for teaching a range of standards. Number sentences can support everything from multi-digit addition, to equivalent fractions, and even ratio and proportion. First, students must learn the conventions. They should start with concepts they’ve already mastered before using number sentences to introduce new content.
Teach equality as balance
This is perhaps the most central 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 is closely related 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.
Teach the Conventions
There are some important conventions for working with number sentences. When students learn these, they can simplify complex, multi-step equations.
One mistake I see many students make is using multiple equal signs per line. Truth be told, this habit even shows up in high level math. While a college professor can bend the rules, when students are still learning to work with equations, it’s best to not get fancy.
Keeping one equal sign per line reinforces the idea of balance. Try to imagine a scale with three or four points of balance – how exactly would that work?
Students also need to be taught to solve equations in steps. Each step goes beneath the one before it. As much as I support creativity and exploration, this is one area where a strict approach is warranted. I don’t much care if a student’s preference is to put their steps in a jumble around the page. Math is a language – if I can’t easily understand your steps, you’re not communicating effectively.
Some students will break equations into parts. For example:
x – 14 = 17 + 20
17 + 20 = 37
x-14 = 37
37+14 = 51
The final answer is mostly correct. But the solution shows that the student isn’t able to simplify downward, keeping the equation intact. It demonstrates a modest level of proficiency. But if not addressed, this habit will prevent the student from solving more complex equations.
Introduce number sentence proofs
My favorite activity for developing fluency with number sentences is the number sentence proof. Students start with a “closed” number sentence (no variables) and must decide if it is true or false. For example, “⅔ – ⅓ = 1.” Students prove or disprove using calculation, but also explain the reason the equation is true or false.
Next, students can work with “open” number sentences (with variables). With open sentences, students choose among always true, sometimes true, and never true. “X+1 = X” is never true. “3 + x = 0” may seem false to some, but it is true when x = -3.
Build Your Students’ Number Sentence Fluency
It actually took me a while to make number sentences a regular part of my classroom practice. This was mainly because I couldn’t find enough resources for teaching with number sentences.
That’s why I designed activities and organizers to scaffold students through number sentence proofs. If you’d like to try them out, download our Hands-On Fractions Quickstart Guide. It’s free, and it includes a lesson plan and hand-outs to teach fraction operations with number sentences. There’s even a lesson on teaching fractions with visual models.
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