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In our jobs at New Classrooms, we are constantly thinking about how to create and refine personalized paths to guide students through the mathematical landscape. 

This post originally appeared on the New Classrooms blog.

In our jobs at New Classrooms, we are constantly thinking about how to create and refine personalized paths to guide students through the mathematical landscape. This requires us to consider myriad factors—what, where, when, how, and with whom— that shape a student’s learning experience. Often, this means confronting long-held misconceptions about how students learn and ensuring that our program—Teach to One: Math—doesn’t reinforce these myths.

As former teachers, both of us have experienced the frustration of operating under these myths and have seen the differences it made for our students when we were able to get out from under them. Last month, we facilitated a discussion about this topic here at New Classrooms, and it was interesting to hear about these myths from many different perspectives—from teachers and college professors to parents, former students, and working professionals. To continue the conversation, this blog will debunk some of these myths by highlighting best practices and research on cognition and student learning.

Myth #1: Students are born with a fixed amount of Intelligence

There is a common belief that intelligence is something people are born with, which is often held even more firmly when it comes to math. We all have friends and family who say that they just aren’t “math people.” Research shows that intelligence is not fixed at birth. In fact, people can actively increase their cognitive abilities at any time in their lives, regardless of what their starting point is. There are even research-based strategies to help! (And yes, this applies for all subjects—even math.)

The most important thing is having a growth mindset, which is the idea that thoughtful effort, rather than innate ability, is the key to learning. Studies have shown that when students simply believe in the idea that their intelligence can grow through effort they’re more likely to succeed.

One pitfall of this myth is that parents and teachers often praise children for being “smart,” according to Carol Dweck, a leading researcher who pioneered the idea of a growth mindset. Her research has found that students who received this type of praise were more likely to feel inadequate when they fail and shied away from difficult challenges for fear of failure.

Myth #2: Students should be praised just for their effort

Merely praising children for their effort isn’t enough. It can even backfire, especially with older students.

Helping students build a growth mindset means regularly emphasizing to them that learning is about the process, not the outcome. (Dweck recommends using the word “yet” with students to remind them that learning is a process.) Alfie Kohn even argues that any praise can be harmful for children because, he argues, it hinders their ability to assess their own accomplishments and could cause them to lose interest in activities when they are not praised for them. He advocates for more value-neutral, observational feedback.

Myth #3: Students perform better when instruction matches their preferred “learning style”

The concept that people have different learning styles is a popular myth that consistently gets debunked when held up to rigorous study. There is a tendency for people to believe their preferred way of learning should also be their only or primary way. In fact, researchers have found that people learn best when they learn in a variety of styles: visual, auditory, reading and kinesthetic.

That is not to say that a student can’t or won’t get “more” out of one learning style over another. For instance, a student might consistently retain more information when using visualization tools like, say, a concept circle or an area model than through a lecture-based modality. But that student will learn more deeply when he or she engages in a mix of different approaches as opposed to just one.

Myth #4: There is a “best way” in which to learn math

There are always multiple approaches to solving math problems, and research shows that critiquing and making connections between those approaches is a powerful learning experience. For example, if one student represents a math problem visually, and another does so algebraically, discussing the similarities between the approaches can enhance their ability to use multiple representations and deepen their mathematical understanding.

On the other hand, if a problem is presented in a way that suggests there is only one approach, or that prescribes only one particular method, the message is that math is all about repeating procedures and not about sensemaking.

Myth #5: Students are learning if they’re getting answers right

Many students are adept at memorization tricks and procedures for problems that help them answer correctly. But that doesn’t mean they have a true conceptual understanding of the problem. Furthermore, when students are getting all the answers correct, they may not be learning much at all. In fact, Jason Moser’s neurological research into what happens when people make mistakes found that getting the wrong answer increases brain activity and can lead to learning growth.

Myth #6: Students will learn a concept as long as it is covered by instruction

This myth assumes that all students have the same background knowledge, the same skill gaps, and learn at the same pace.  These assumptions, plus the pressures of teaching a year’s worth of grade-level math standards to 30 or more students, can often lead to a situation where we are trying to move through a breadth of mathematical concepts and skills without ever pausing to focus on depth.

In general, students need time to play with and absorb ideas and make connections. One study, for instance, found that students performed better in college if they had extra time to investigate a topic deeply in high school than peers who studied more topics over the same amount of time. The goal of learning, after all, is not just to cover a concept, but to uncover it.

Myth #7: The “fastest” students are the ones with the most mathematical ability

Students who complete problems at a fast clip aren’t necessarily demonstrating that they’ve figured anything out. As with the case in Myth #5, it usually means that these students have memorized a trick that can be applied to quickly arrive at the right answer.

Authentic learning in mathematics happens when students are presented with a novel problem, given an opportunity to consider the mathematical principles that can be applied, and then required to apply them. This usually involves some trial and error, which can take different amounts of time depending on the student, but it is no less valid a process if it takes longer. In fact, Jo Boaler, a professor of mathematics education at Stanford University, has found that timed testing can lead to math anxiety for children of all ages, starting as young as five.

Myth #8: Students should master mathematical procedures first, then try and understand them

The key shifts to Common Core-aligned instruction call on teachers to build both procedural fluency and conceptual understanding. Many teachers reason that concepts are easier to understand when students are already fluent with the mathematical procedures involved, but for many students, teaching procedures first is not the best approach.

Research shows that many instructional practices, which start by using manipulative, visual models or familiar real-world contexts to help make those procedures more meaningful improves outcomes for students. When students are able to connect the process of a mathematical procedure to an idea that is visible, tangible or that they already understand, they are able to synthesize the procedure more easily and are more likely to retain it.

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Gabe DeAngelis & Brad Cameron

New Classrooms

Gabe DeAngelis and Brad Cameron are from the Instructional Content and Progressions team at New Classrooms.