By: After years of working in math and computer science education, you tend to discover principles of learning that are distilled from thousands of lessons with different learners. A core idea that keeps coming to mind is that beginners need structure and experts need freedom. It seems like common sense, but I think applying this principle can be tricky in practice, especially when there are multiple concerns like student enjoyment and varied learning styles.
As a math teacher, I spent a lot of time thinking about how I should best introduce topics to students. Do I give my students highly structured problems to tackle step-by-step or do I give them something more open-ended and let them discover a solution? I found that both approaches worked well, but it depended on the prior knowledge of my students. For example, if I was teaching my students about linear equations, I could begin with different introductory lessons:
As a math teacher, I spent a lot of time thinking about how I should best introduce topics to students. Do I give my students highly structured problems to tackle step-by-step or do I give them something more open-ended and let them discover a solution? I found that both approaches worked well, but it depended on the prior knowledge of my students. For example, if I was teaching my students about linear equations, I could begin with different introductory lessons:
- Break students up into groups and give each group an action figure and a bag of rubber bands. Give them graph paper and a prompt to tie rubber bands to the action figure and drop it down a stairwell. Have them record the number of rubber bands on the x-axis vs the distance fallen on the y-axis. Ask them to figure out a relationship between the number of rubber bands attached to the action figure and how far action figure drops.
- Explain concepts of slope and y-intercept, giving students techniques to calculate both values. Then have students calculate the slope and y-intercept of various lines and graph lines based on given slope and y-intercept
I've started my units with both kinds of lessons. Almost every student enjoyed the first type of lesson more than the second. But students who were already familiar with the topic of linear equations were able to make the mathematical connections to slope and y-intercept quickly, while students who were new to linear equations usually ended up lost when it came to the actual math. The reason was the huge amount of cognitive load I was imposing on the novices. I was expecting them to perform a hands-on-activity, intuitively recognize the patterns of linear equations, and then create an equation describing that relationship. It was too much to ask for and nothing ended up sticking. In contrast, the more experienced students got a ton out of the exercise, since they were able to draw on their prior knowledge of linear equations and just needed to connect that existing knowledge to this novel, real-world example. It actually broadened their understanding of the topic.
The second lesson may not have been very exciting for my novice students. But by the end of the lesson, they all felt much more confident about their understanding of the topic. Many actually preferred a more structured introductory approach when I polled them at the end of the unit. The more experienced students were not engaged at all, though. Their cognitive load was way too low. Since they already had the basic skills and understood the main ideas of linear equations, the structured problems just felt like busy work, and they became bored and disruptive.
My teaching became much better when I was able accommodate both groups simultaneously with problems that were appropriate to their prior experience.
The second lesson may not have been very exciting for my novice students. But by the end of the lesson, they all felt much more confident about their understanding of the topic. Many actually preferred a more structured introductory approach when I polled them at the end of the unit. The more experienced students were not engaged at all, though. Their cognitive load was way too low. Since they already had the basic skills and understood the main ideas of linear equations, the structured problems just felt like busy work, and they became bored and disruptive.
My teaching became much better when I was able accommodate both groups simultaneously with problems that were appropriate to their prior experience.
We face similar challenges at Digital Adventures. Our mixed-grade classes contain students with a range ability and experience levels. In addition, our project-based learning environment doesn't allow for rote drills. Our solution is to provide student groups with rich projects that allow for both structured practice as well as open-ended and creative problem solving. Depending on a particular student's experience level, we provide her with a different set of challenges.
For example, one of our video game design projects, Spaceship Captains, leads novices through several structured problems that involve the programming concept of loops. Novices receive successive small problems that build on each other, like 'use a forever loop to get the spaceship to turn right'. By the end of the lesson, students who have never seen a loop before will have practiced using loops in multiple contexts and in a non-repetitive environment. On the other hand, more experienced students might get a more comprehensive problem statement, like 'keep score based on the number of asteroids you shoot,' or 'figure out a way to add a second player to the game'. Instead of practicing how to create a loop, experienced students work on the higher level problem of when loops should be used. It takes a lot more work to design each project, but the payoff in having a multi-level classroom where all students are engaged is absolutely worth it.
For example, one of our video game design projects, Spaceship Captains, leads novices through several structured problems that involve the programming concept of loops. Novices receive successive small problems that build on each other, like 'use a forever loop to get the spaceship to turn right'. By the end of the lesson, students who have never seen a loop before will have practiced using loops in multiple contexts and in a non-repetitive environment. On the other hand, more experienced students might get a more comprehensive problem statement, like 'keep score based on the number of asteroids you shoot,' or 'figure out a way to add a second player to the game'. Instead of practicing how to create a loop, experienced students work on the higher level problem of when loops should be used. It takes a lot more work to design each project, but the payoff in having a multi-level classroom where all students are engaged is absolutely worth it.