During my orientation, we read the article “Want kids to learn math? Level with them that it’s hard.” by Jordan Ellenberg, a math professor at the University of Wisconsin. I bring it to your attention, because it was the final piece that I needed for motivation to write *something* for math again. I wanted to unpack this complicated sentence of “math is hard” and how it can be as detrimental to learning as saying “math is easy”, if used ineffectively.

This past summer, I had the opportunity to assist in teaching Math 804T “Experimentation, Conjecture and Reasoning”, a class for current K-12 math teachers looking to earn their masters in teaching. After the first day, I realized that the “heart” of the class was to put these teachers in an uncomfortable position, mathematically. These are teachers who have had years under their belt teaching their respective topics and curriculums. You could say they’ve gotten comfortable with it.

To my limited perspective and interpretation, this class was meant to reintroduce them to “productive struggle” through the use of much more abstract ideas, ideas that they’re not comfortable with. For example, the class teaches the idea of “infinity” and how there are different types and sizes of infinity. As a K-12 teacher, you’re more than likely to never have to teach this topic, so why learn it? It’s because it’s “hard”. It was a topic they haven’t familiarized themselves with for years or ever. The goal was never to walk away from the topic as a master of infinities. That would be nice, but the more pertinent learning experience was to remind ourselves that the confusion we faced learning about infinities is not that far from the confusion that a first grader will face when learning about fractions.

We should remind ourselves that this extends beyond K-12 mathematics. Professors teaching their area of expertise should not forget or underestimate the struggle for new students to wrap their mind around their new “infinities”.

So we remind ourselves that learning math is hard. Now what? Similar to what 804T secretly teaches, we should be more empathetic. An 804T student-teacher was worried about asking me for help, because they had nothing to show for the time they’ve spent just understanding the problem statement. We should be kinder to students who may not be able to show that they’ve been working at a problem. Being more empathetic is a difficult task. It is also not as evidently clear that you succeeded. It is unlike being able to say “Thus, the theorem holds” at the end of a challenging proof. But it is similar to a challenging proof in a way that you should be motivated to see it through. We ask students to try and try, because we know an answer waits at the end. So, we must ask ourselves to try and try, because we believe that a better learning experience awaits at the end.

Ellenberg specified that math is hard but you (the learners) can do it! My father used to describe learning math as a series of steps on a staircase. While he used it to emphasize that skipping ahead and not fully learning a step will cause you to tumble down, I understand it more as each step is hard, but not forever and that the next step may also be hard. It is a better promise to students, a more realistic view of how learning any subject goes.

This ultimately goes back to my belief that math is not only about learning these techniques to solve *math* problems, it is also about building character and grit to solve, or at least deal with, *all* problems. This is because math is one of the best subjects to exploit the learning process of productive struggle. If you become comfortable with the struggle but knowing that an answer lies ahead, then you’re more well-equipped to tackle future problems. In other words, each step ahead may be more difficult, but you’re able to power through it easier. It is less so that math becomes easy, but that you have become friends with the difficulty of it.

Ellenberg explains wonderfully why saying “math is easy” is bad. In summary, it creates an environment where students don’t feel comfortable asking questions. If it’s easy, you should get it immediately, no questions needed.

However, I believe that this mantra of “math is hard” can still create this “no questions needed” environment. There have been studies on the effect of a parent’s attitude towards math on how their children will view learning math. One such study is by Nicole Kerkhof. Kerkhof states that a parent’s fear of math “negatively effects not only their children’s current math achievement and attitudes toward math, but also their opinions on math and math skills later in life” and “expressing these negative beliefs and attitudes toward math could be demotivating to children”.

This is how the mantra can be ineffective. When we get stuck in the mindset that math is hard and will always be hard, we lose the opportunity to grow. To continue the staircase analogy, it’s as if we sat down at the current step we’re on, blinded by the difficulty we’re facing. As this math anxiety grows, we could abandon the staircase altogether. Kerkhof mentions this with how high math anxiety levels correlates with beliefs that math is not useful.

Therefore, if saying “math is easy” leads to “no questions needed” because you should already know it. Merely saying “math is hard” (and not qualifying it) leads to “no questions needed” because you shouldn’t even try to learn it.

To conclude this, I would just like to reiterate that every student is different, you will have students that come in believing math is the easiest thing to them, a born genius you could say, but you could also have a student who’s on the brink of abandoning their staircase. As instructors, as people who argue for the importance of learning math, we have a difficult mission of being versatile and fluid on top of all our other responsibilities. I’ll be teaching my first ever class this semester, and I am frightened by this infinity. But just like how I learned math in steps, I’ll learn how to be an effective instructor in steps.

** August 2022 Edit:** It’s been about a year and I’ve taught two classes on my own already. During orientation this year, we had sessions that made me sit on this idea again. One, I want to reiterate the point that saying “math is hard” or “math is easy” and letting the sentiment end there is what endangers the growth of people. You can absolutely still believe that the subject is hard, like me and my other graduate student peers, but avoid staying in that thought without a path to grow.

Two, the adjective “hard” has been under some scrutiny this past week. I don’t think we should make a villain out of the word. When I say a problem is hard, it’s an honest statement of its *current* difficulty, not a life sentence. We tend to turn off our brain when someone says something is hard or difficult, but I think something I’m working on is allowing myself to sit in that discomfort. Learning can be uncomfortable, it sucks having to feel the weight of knowledge to get to the next step in your staircase. But, it is all temporary. Crawling, walking, biking, or driving *used to be* difficult. Another graduate student put it nicely, math isn’t an innate ability, it is a skill. Math is something we can train and develop.

I like to have a little picture at the end of these writings, so here's a picture of my future classroom where I'll be teaching Intermediate Algebra! It fills me with excitement and a little bit of anxiety!

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