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.