Making Marvellous Mistakes

When you were a child in elementary school, how often did you volunteer to give an answer to a problem during math class? Did you avoid answering questions because you were afraid of the embarrassment of making a mistake?

"Typographic Poster Michael Jordan Quote", CalleyFlower, 2013

Science has proven that making mistakes is crucial to achieving success while learning mathematical concepts. Our brains are constantly building new synapses, or connections, between the various pieces of information that we have gathered. In order to create new synapses, we must challenge our ourselves in order to delve into deeper thinking, but unless we are making a mistake within our first line of reasoning, we have not challenged ourselves enough. 

We can imagine our brains like a road map. The individual roads that we know are the synapses in our brain. In order to answer problems, or arrive at our destination, we have to create a route to arrive there. We may consider it a challenge to find a new destination, but if we are able to take the roads that we already know to get there, we aren't learning anything. A true challenge takes us to a brand new road that we have never traveled down before, that we are now able to add onto our road map. 

We, however, liver in a culture were mistakes are feared. In schools especially, students can worry that every mistake they make can result in a drop in their grade. Whenever students see that their marks are deducted because of mistakes that they were not previously aware of, they can begin to develop a fixed mindset about themselves. Personally, I dreaded receiving back my tests after they had been marked, because I was already embarrassed about how mistakes I had made before I was even sure how many or which mistakes I had made. My teachers would comment that they didn't understand why I was making so many mistakes when I seemed to clearly understand the basic concepts when they were taught during lessons. As a result, I believed that I was bad at math, when in reality, I was on the right path but wasn't properly encouraged, or given enough time, to push myself through the challenges and properly learn from my mistakes. After a unit test, we simply moved onto the next unit without addressing the mistakes that we had made on the test. 

Michael Starbird works to promote the idea that "mistakes are happening every day in public." He puts emphasis on normalizing mistakes so that students don't have to feel any shame or embarrassment about them, and instead twists things so that those mistakes instead have the opposite power of building confidence in the students. The key to overturning the negative power of mistakes are the twins questions, "What is wrong?" and "How do we correct it?" 

As a teacher, I will work to normalize mistakes by placing the motto "FAIL FORWARDS" at the top of my math bulletin board, and teaching my students that the basis of failing forwards is reflection. I want to emphasize that mistakes are a vital component of the learning process, so I will give my students opportunities to share a mistake that they made with the class, how they discovered the mistake, and what they did to correct the mistake or solve the problem in a different manner. 

"Math Who", Teachers R Us Homeschool, 2016

"Math Who" is a fantastic game that can be used to develop problem solving and reasoning skills, encourages math inquiry, and demonstrates how useful mistakes can be. In this game, one student chooses a number, and the other students have to ask questions to figure out which number they chose. During the course of this game, students are bound to make many mistakes, and a key to winning the game is realizing where they made the mistake and correcting their path of reasoning in order to discover the correct number.

If you are working with a more advanced class, you could have the students who are guessing choose a number themselves, and by the end of the game, explain why the number they had chosen could not have been the correct answer. This challenge would highlight the skill of recognizing the exact moment where a path of reasoning encounters a mistake that needs to be corrected.

"The Roses of Success", Chitty Chitty Bang Bang, 1968

Math Trickery

Hello there dear reader. I have a request for you, if you wouldn't mind. Would you please pick a number between 10 and 20? Ah, you've picked the number 14, lovely. 

Now would you please pick a number between 20 and 30? 23 is an excellent choice. 

Now I shall perform a card trick for you. Please watch the video below: 

Nicole Horlings, September 18 2017

Now, it isn't a coincidence that I just pulled out two queens. So how did I do this? 

Math: to Hate, or not to Hate?

Math is a popular subject to hate. After all, who likes to be forced to sit in a stark classroom, attempting to answer an extremely long list of questions that all look like gibberish for an hour straight? Or who wants to tell their son or daughter that they have to endure this situation? Regardless of whether this is what the modern math classroom looks like, this is what many people associate with math. Their own personal experiences are combined with the attitudes presented in numerous Hollywood movies, and these together mix to create the negative atmosphere that surrounds math. Before they even begin school, many students are receiving messages that math is difficult, boring, tedious, and pointless. Within this atmosphere, students begin to internalize certain beliefs about math, and about themselves, such as that math is boring and useless, and that they are either good at math or bad at math. Personally, I believed that math followed a strict set of rules, and that there was essentially only one way to solve a mathematical problem. I also failed to see where I would use any of the abstract equations that I was attempting to solve.

I was taught to solve mathematical problems using formulas. These formulas were handed to me as pre-made tools from my teachers, and I had to plug in the correct numbers. This style of teaching did not foster my problem solving skills, not did it inspire any curiosity that would lead me to make new connections between the facts that I had already learned. It was very difficult for me to transfer my knowledge to a similar problem if it did not follow the exact formula that I had been taught, because I didn't have the background understanding of how the formula had been created. Between my lack of interest and struggle to truly understand the concepts being taught, I gave up investing anything more than the minimal effort required for a decent grade.

However, while math is born out of logical reasoning, it is nothing like the rigid structure that I viewed through the narrow lens of formulas, and there are multiple ways to come to the same answer. Many of my teachers taught from the perspective that students must be able to solve questions, but their approach overlooked the importance of us learning the very basic structure of math itself. While they taught us what to do, we didn't understand why we had to do it this way. Teaching through inquiry places the focus on understanding this structure instead, and students naturally find their own ways to the correct answers by exploring mathematical relationships using tools like manipulatives. To quote mathematician Georg Cantor: "In mathematics the art of proposing a question must be held of higher value than solving it." Finding specific answers isn't as important as figuring out a way to find the answers. Students who learn the structure of math through inquiry and exploration are developing problem solving skills that will enable them as they explore increasingly complex concepts.