Math = Patterns

Fibonacci Sequence Powerpoint, Nicole Horlings, Oct 23 2017, Screenshot 

How on earth is a comparison between a pine-cone and the Parthenon relevant during math class? Well, to answer that question, I first need to ask you a different question:

How do you define math? 

Is the answer:

a) the study of numbers?
b) a set of rules?
c) a set of patterns?

While none of these answers are incorrect, the answer that you chose for this question is still important. As Dr. Boaler explains in this video, most people see math as numbers. This answer seems pretty obvious, since you can't do math without numbers, right? Well, the mathematicians who see math as a set of patterns probably wouldn't agree with the previous statement. Patterns, such as the spiral of a fern, exist in nature without requiring any numbers. The Fibonacci sequence is a special pattern that is extremely prevalent in nature, as well as in a variety of other locations, such as art and architecture. Thanks to this pattern, students could do an inquiry project based off of the question, "What similarities are there between a pine-cone and the Parthenon?"

But why is the way that you define math important?

Throughout my youth, I was implicitly taught that math is a set of rules about numbers. This made math feel abstract and disconnected from nature, which made it difficult for me to make connections between the concepts that I was learning about and how I could apply them to my life. I believed that I had to follow certain procedures to answer math problems, and felt closed off from exploration and inquiry.

As a teacher, it is important for me to view math from a different mindset. Defining math as a set of patterns opens it up to exploration and inquiry, allows for a variety of entry points for understanding, and directly connects math to nature. Suddenly, there are ways to study math in other subject areas such as music, dance, sports, and biology.

If you're still not convinced, just consider which came first: patterns that exist in nature, or the numbers that we use to describe the patters that we see?

The Myth of Memorization

When I think back to my experience in math classes in elementary school, one of my strongest memories is attempting to get perfect on math drills. These drills focused on memorization and speed, which are two topics that are greatly discussed and debated in today's teaching world. But where do I stand on these topics?

Personally, I am not a very fast mathematical thinker. Math drills always frustrated me as a child, because I knew that I was able to answer all of the correct answers, but I would so often be forced to leave the last four or five questions blank because of the arbitrary time limit set on the math drill. If I hadn't known the answer, I wouldn't have felt so badly about the blank questions, but I absolutely hated it when I wanted to show my knowledge but was blocked from doing so. In order to answer all of the questions, I rushed and became more reckless, then would receive back my drill and discover that I had wrong answers to questions that I did actually know correctly.

Receiving a low grade in a subject not because you answered questions incorrectly, but because you weren't given enough time to complete the work and show your capability is a humiliating experience, and causes students to believe that they aren't capable of doing math even though they are perfectly capable of doing so. I do not think that timed math drills are a beneficial teaching technique, however, I do think that the repetition that certain exercises provide can be useful to develop math fluency.

Being mathematically fluent in basic multiplication enables a student to use this base knowledge for performing mental math, and also for quickly realizing when the solution to a math problem doesn't seem right. Mathematical fluency is closely partnered with mathematical intuition. This fluency is far beyond shallow memorization, but contains a deep understanding of math that, like a muscle, has been strengthened to work more efficiently. What's dangerous is when we confuse memorization and deep understanding, believing that a student knows a concept better than they actually do, since mathematical concepts build on each other and require firm prior understandings. One of the huge problems with memorization without understanding is that the information, such as a specific formula, that we are using is not stored permanently for future use, but is disposed of after we have no immediate need for it. Another issue is the fact that while our brain is holding these temporarily memorized facts, it has less processing power left to use for tasks like problem solving.

For students to develop a lasting understanding of math, we need to encourage their mathematical intuition. We need to guide students towards internalizing the big ideas in math without telling them the answers, but giving them the space to struggle through problems and develop their own solutions. Sebastian Thrun, the former VP of Google, believes that we should be "only willing to manipulate equations [we] understand" and avoid "fixed rule manipulations [that] are devoid of an intuitive interpretation." Formulas in and of themselves are not problematic, but we should only use formulas when we understand why and how they work, and "expose students to as many methods and means of representation in math as possible so they can decide for themselves what suits their brain the most" (Bernadette James). For some students, this deeper understanding will not come through formulas, but through drawing.

To celebrate alternative methods to solving math problems, here is an interesting video that shows how to do multiplication using lines.

The Danger of the One-Way Approach

As I mentioned before in this blog, when I was a student, I was taught math in school with the one-way approach. My teachers were focused on our procedural understanding, and taught us specific ways to find answers using a particular method or formula.

In other words, I was taught that that 4 x 5 = 20. 

I saw math as something that was structured in a very specific way, that had to be memorized, and that was linear. You followed the instructions step by step, and if you couldn't find the answer, you must not be mathematically minded. There wasn't room for creativity or exploration in math. I developed a fixed mindset in math, and began to seriously dislike math.

What if I had been asked 20 = ? 

Suddenly, math becomes creative. The answers are endless!

20 = 4 x 5
20 = 10 + 10
20 = (8 x 3) - 4
20 = -20 + 40
Etc.

This may not seem like a big change, nor may it seem to be teaching any curriculum expectations, but this shift in thinking is important. This shift from robotic plugging in numbers to exploration and curiosity adds intrigue into the math classroom, and frames math as a wide, interconnected web rather than as a single pathway. If math is supposed to be all about problem solving, how can we expect students to think in different ways if they are only ever shown math linearly?

But seeing math as linear wasn't the only way that the one-way approach impacted me. It also affected my ability to do mental math.

For example, when I see a question like 18 x 5, I immediately want to grab a calculator. My immediate explanation is because when I was in elementary school, we didn't learn multiplication facts past x 12, so I don't have 18 x 5 memorized. Looking back, I realize just how much emphasize my teachers put on memorization, how little time we spent on exploring the relationships between numbers, and how we were not taught to do any mental math beyond the math facts that we had memorized. Also, we were always taught to use the fastest, most efficient and direct way possible to get to the correct answer at the center of the problem, usually through formulas, instead of exploring around the perimeter of the problem and discovering our own path to the answer.

So when I was asked to solve 18 x 5 mentally this past week, after I resisted my initial urge to grab a calculator, it took me a moment to realize that I didn't have to directly answer 18 x 5, and that I could travel around the problem to find an easier way to answer it mentally. I decided to round up 18 to 20, since I knew that 20 x 5 = 100. Then I removed 5 twice to find 19 x 5 = 95, and then 18 x 5 = 90.

What fascinated me afterwards was the fact that there were other ways that people took to answer the question, such as breaking the question up into 10 x 5 = 50 and 8 x 5 = 40, then finding that 50 + 40 = 90. I was amazed at how many ways there were to solve the question, and at how they could each be visually represented.

In school, I was presented mostly with closed tasks, where we had to find a single answer to a problem, but as a teacher I will be striving to present my students with rich tasks, where my students will be encouraged to figure out multiple answers and explore math while doing so.

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.

Mathematical Measurements

How confident are you when you make guesses about someone's height?

This was a question that one of my teachers asked our class before, and our class had a range of answers from "hardly" to "pretty confident". This question was a precursor to an experiment where we had several student volunteers come to the front of the classroom, and have everyone guess their height in centimeters, then afterwards compared our guesses to their actual height. The results were interesting, with a range of accurate guesses and wildly wrong guesses. What fascinated me the most was the fact that while making their guesses several of my classmates had wrongly assumed that certain volunteers were taller than others, which completely skewed their guesses as to how tall the volunteers were.

When we did this experiment to test our observational skills, I began to think about how important accurate observational skills are, and how I had totally overlooked them before. For example, eyewitnesses are vitally important in our justice system, and they need to provide the police details about suspects so that the police will be able to take the correct person into custody. If an eyewitness gives the police a very inaccurate estimate about the height of the suspect, suggesting that the suspect was average height when he was really above average height, this misinformation can hamper the search for the suspect, or cause them to take the wrong person into custody.

I do not recall learning how to make accurate estimates about lengths when I was in elementary school. I remember learning about the importance of specific measurements, and measuring something correctly, but not how to judge lengths from a distance. As an adult, I wish that I had learned this skill when I was a child.

In order to teach students how to accurately estimate lengths, it's important to teach them several benchmarks to use to base their estimates off of. In Casey's video, she suggests having the students use their hands as one such benchmark for future estimates. One of the things that I really like about this suggestion is the fact that students always have their hands with them while they usually don't have a ruler nearby. One problematic thing about teaching students to use their hands is the fact that elementary students are still growing, and so their hands are also going to grow larger, which will skew their benchmark unless they consciously continue to remeasure their hands regularly to update their benchmark.

There are several ways to teach students about estimation. I really liked the activity on page 36 of this resource that suggested that teachers have students examine several different packaging options for 3D objects, use estimates to make educated guesses about which packaging uses less material, and then measure to test their predictions. I like how this activity goes beyond a 2D concept of distance or length, and ventures into the 3D world, and how it connects this concept with a real world context.