Preventing Conceptual Gaps: An Argument for Mastery Based Education

Richard Bitgood

Richard Bitgood

I’m just not a math person.” It’s a sentence I have heard countless times from students during my years of teaching math in the classroom and online. It’s not just students – parents, friends, even colleagues have articulated similar phrases solidifying an identity that distances themselves from my beloved subject. Perhaps this willingness to define one’s identity in opposition to an entire field of study is a symptom of math anxiety. Perhaps it is the cause. Either way, it is not much of a leap to see a correlation between math anxiety and a willingness to write off the subject for good. In fact, studies show that the “majority of adolescents report worry and tension in math classes and when doing math.”

One solution that has been tried is creating math courses for “non-math” people. While many US schools offer advanced honors classes for mathematics, states such as Florida have created “liberal arts” mathematics programs. In addition, many Canadian provinces have “academic” and “non-academic” streams of math courses in high school. This was likely done under the noble intention to ensure that all students, regardless of ability, have the opportunity to complete some high school math courses and therefore graduate. However, we could argue that the very existence of these streams reinforces the identity crisis: “Look at all of us together in this easy math class. We must be the kind of people who can’t really do math.” In addition, the student population of non-academic streams of mathematics are disproportionately made up of marginalized groups. This has led to the province of Ontario recently deciding to destream it’s math courses, removing the option for non-academic streams of math and hopefully increasing the future opportunities for those students.

Compounding Conceptual Gaps

In teaching online math I had the privilege of doing thousands of 1-on-1 help sessions with students and started to notice a pattern – over half of my time was spent on helping students fill gaps in their knowledge and understanding of previous concepts. The reason a student couldn’t complete the trig proofs, was because they struggled to simplify basic fractions. Another couldn’t comprehend exponential growth, because they never internalized that exponents are repeated multiplication. Time and time again the problem wasn’t today’s question, but a conceptual gap from three years ago. And speaking of exponential growth, those gaps are compounded year over year: a student who fails to master the concept of negative numbers one year, will fail to master basic algebraic equations the next year, which will make all of math seem impossible the year after.

From this compounding of conceptual gaps we can see the roots of how negative math identity takes root. “I don’t understand this question” turns into, “I don’t get this whole chapter” which when repeated too many times becomes, “I’m not good at math.” That quickly becomes, “I hate math” and, “I’m just not a math person.” This person will probably avoid further education and career opportunities that involve mathematics because of this internalized identity.

While it is intuitive to see how the curriculum builds progressively in mathematics, It’s important to note that conceptual gaps aren’t unique to math. A prerequisite to being able to balance chemical equations is an understanding that molecules are made up of atoms. In history it’s crucial to learn how some events led to others, and every 1st year English professor I have ever met states that some students show a lack of essay-writing ability. It’s hard to write a coherent essay if you can’t construct a coherent paragraph, or sentence for that matter.

Mastery Based Learning

How do we address the problem of conceptual gaps? Mastery based learning is an idea that has been around for a while, but has become much more possible in the context of K-12 online education. The basic premise of mastery based learning is that we don’t allow conceptual gaps to occur in the first place. Sometimes referred to as competency based learning, the principle of mastery learning is to ensure that a student has reached a high level of competence (mastery) on any given topic before moving on to future topics that depend on it. Done! No more gaps. 

Given that mastery based learning seems like a common sense option, why hasn’t all of K-12 adopted it? The main reason we don’t have mastery-based learning in classrooms is because we have something else: time-based learning. We decide that we only have so much time for students to gain mastery of a topic. Once the time is up, their level of mastery is assessed, the students are given a grade that we expect to be distributed on a bell curve, and we move on. In time-based learning, time is held constant, and mastery is variable.

But in online learning, we no longer have the same constraints around time that traditional schooling. We can use the flexibility to create the opportunity for mastery based learning, by making the desired level of mastery the constant, and time the variable. What if the amount of time spent on a topic more resembled the bell curve, and the grades students received were consistently at the mastery level? Online schools can adopt this, and indeed have, preventing conceptual gaps to form in the first place.

Techniques for Mastery

So how do you implement a mastery-based system? The first step is to believe that students are capable of learning every topic in the course you teach, in addition to believing that it is important that they do so. I’ve argued for that above. Secondly, a belief in mastery learning also has to take root in every student you are working with. This is a negotiation with every single student. It takes patience and intentionality to call out that potential.

Finally, it takes a course structure that is designed for this. Courses need to have checkpoints for understanding as well as built in opportunities to redo work if necessary. Having review assignments is an opportunity for teachers to assess if a student is ready to attempt the test. We, at StudyForge, suggest that these assignments don’t receive any marks, but rather teachers give feedback on specific areas of learning that are needed. Teachers assign a “good to go” in which students can move onto the test, or a “needs corrections” in which case students are encouraged to gain mastery before moving on. Another important consideration is to have large test banks where questions can be pulled from, so that students can retake tests if necessary. All of this is based on the negotiated goals set up between the teacher and the student.

Avoiding conceptual gaps creates the opportunity for an upward spiral of confidence to take place. When a student can say, “I figured out this question!” that can build into “I mastered this topic” which over time becomes “I’m good at math.” This person will be empowered with the confidence to pursue further education and career opportunities because of this new internalized identity. Maybe you are a “math person” after all.


Want to learn more about how StudyForge uses these techniques to encourage mastery learning? You can request a preview of any of our courses within your LMS by contacting our Client Success Manager, Toni Lyons at info@studyforge.net

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