Doubling Down on Number Lines: A New MATHia Feature

Hi, my name is Dr. Bob, and I am a Senior Cognitive Scientist here at Carnegie Learning. One of my primary responsibilities is to design features in MATHia workspaces, such as just-in-time hints (JITs), skills, and showing or hiding information on the screen. I apply the most up-to-date research on cognitive science to help create cutting-edge educational technology so students can learn at their best. It's a cool job because I get to collaborate closely with subject-matter experts, AI programmers, and UX designers. I’ve been with the company for more than 12 years, and each year, we create something that is so extremely cool that I need to shout it from the rooftops. This year, that workspace is a new tool for solving equations: a double number line. 

The Problem with the Equal Sign

What’s a double number line, and why do we need it?

Let’s start with the problem we set out to solve when we were looking for ways to make our award-winning MATHia software even more effective. We knew from prior research that elementary students struggle with the concept of mathematical equivalence (McNeil & Alibali, 2005). In other words, the equal sign doesn’t mean “equal to” in many learners’ minds.

If you talk to students, and ask them a series of probing questions, you will discover that their mental model of the equal sign means, “Give me the answer.” Let’s say you give them the following equation:

4 + 3 + 5 = __ + 7

They might tell you the answer is “19.” In other words, they just perform all of the operations that they see and then give you the answer. Part of the reason for their flawed mental model is that they typically are not confronted with these types of equations. Generally, they see problems in the form of 4 + 3 + 5 = __ . If that’s all you see, then it completely makes sense that you would think of the equal sign as, “Give me the answer”!

An Interactive Solution 

One way to combat this misconception is to provide students with a representation that enforces mathematical equivalence. The double number line does just that.

The double number line tool works by moving one point on one line, which automatically moves the corresponding point on the other line. This built-in constraint visualizes the concept of mathematical equivalence to students, so they can’t automatically jump to a misinterpretation of the symbol “=”. 

Let’s look at an example. Suppose we gave a sixth grade student this equation:

3 = z - 9

Symbolically, there isn’t any constraint on what z can be. However, if you place that same equation on the double number line, then both sides of the equation have to move in lockstep. If I add nine to the top number line, then I have to add nine to the bottom, like in the video below:

Likewise, if you want to solve the equation, then you must shift the equation to the left by nine units. Again, the double number line requires that you move both the top and bottom expressions by an equivalent amount. That’s extremely powerful! 

Shaping the Future of Research-Based Learning

Since its inception, the field of cognitive science has been investigating the impact of representations on problem-solving and learning. Research from the learning sciences also suggests that equation solving is an especially difficult skill to acquire, mainly because it is so abstract. However, the double number line is a wonderfully concrete representation, and it helps prepare students for more complex equation-solving. I’m excited for your students to use it, if your school is already implementing MATHia. If you have not tried our powerful MATHia software yet, the double number line, as well as many other innovative features and tools, will be waiting for you when you do!