This Is The Title Since I Don’t Know How To Make A GIF The Title Or If You Can Even Do That

This GIF is what I wanted as the title of the post.

I am currently pursing a graduate degree in Pure Math at Bowling Green State University.

I won’t go into a long story of how or why like I usually do when I overblog. Just wanted to share a few thoughts.

Something I have noticed that I really like but I did not experience as much as an undergradute studying mathematics and secondary education is how my fellow students and I tackle problems together.

We use the phrase “I’m not convinced,”.

A lot.

When I say this to a fellow graduate student or someone says it to me, we generally mean one of two things:

  1. I don’t understand your argument. Can you explain it again? Convince me. 
  2. I agree with your conclusion, but I don’t know if your argument is sufficient. Convince me.

I think both of the meanings for this phrase are incredibly useful. The first allows us to express a lack of knowledge without saying “I don’t understand” or “I don’t know.” This may sound like math students being overly concerned about saving face, and I can’t say that it isn’t, but we are all generally secure in our mathematical abilities. If we admit that we don’t know, it is okay. It is not really a reflection on our intelligence or ability or personhood.

On the other hand, to our own students who may not have that confidence and who view their mathematical knowledge as not independent from their intelligence and worthiness as a person, this phrase is great. It gives them a way to express a lack of understanding that relates back to the communication of mathematics. “I don’t understand what you are saying,” not “I don’t understand this” (And the subtext in the latter option of course being “And I never will so why try?” or “Because I am not smart enough” or “Because I am not a math person” etc.).

The second option is great for my fellow graduate students and myself because it gives us a tool to challenge each other to be better mathematicians by making better arguments without telling someone their argument isn’t good. (It may have even been good, but was missing a minor detail here or there.) It says “I’m not sure if that is how you argue that. Can you walk me through your thought process” rather than “Man, you suck. That isn’t how you prove that.” This friendly challenge to make a better argument makes us better mathematicians together not only because we increase our understanding, but it also our skill at communicating that understanding gets better.

This second meaning is also completely applicable to our own students. The phrase “I’m not convinced” not only give students a way to communicate mathematics to each other in a positive way, it inherently shifts the focus of the conversation to how to best express mathematical ideas instead of how a certain person understands those ideas.

I think this phrase encourages students to practice the Communication and Reasoning and Proof Mathematical Process Standards much better than a “I don’t know” or a “I don’t understand”.

In the future I will be encouraging my students to say “I’m not convinced” whenever they want to say the other things.


The Two Cliffs (An Allegory)

In Which I Ramble On Like I Always Do

Daniel Boone National Forest in Kentucky has a beautiful area called Red River Gorge. It’s a great place to go hiking, camping, and climbing. I have done the first two there many times, but I have never gone climbing there.

I’m not very good at rock climbing. I have a few buddies who are very into it. One whom the gorge has sent to the ER several times and another who is trying to start a climbing business in the middle of Cincinnati. (No, he didn’t pay me to promote that. Besides, what’s the going rate to advertise to all ten readers of my blog?)

These guys enjoy rock climbing a lot. Enough to take return to trips to the ER for hernias, broken bones, jammed fingers and who knows what else. Enough to invest time and energy and risk money to start a climbing gym. Enough to have those weird pieces you hang above the door in your room to practice grips and holds. Climbing a rock face for the sake of it is a passion for these guys. It isn’t my passion, but I do share an enjoyment of the activity.

Just, you know, none of the skill. I think my passion for food might get in the way.

I have never gone into the wilderness, found a rock face, and trusted my life to the knots I tied in the rope that connects me to metal spikes I drove into the cliff myself. I have never been bouldering (I’ve tried telling my buddy that it would ruin my guitarist’s fingers, but even I knew that was a BS excuse.) My experience climbing is exclusively in the summer camp context. A 30 foot tower has been constructed in the middle of the woods and when–if–you get to the top your group for the week all cheers and then you turn around do the hardest part, which is rappel backwards off the wall. Or, if you are lucky, there is a zipline off the top of the tower. So you get your extreme sports adrenaline rush for the week and you go home that Saturday to tell your mom and maybe write about it in your stereotypical “What I Did Over Summer Vacation” essay in September.

I experienced that thrill several times as camper and later on when I would regularly volunteer my time at a local summer camp I cheered on many young pre-teens to the same.

So what does this have to do with math teaching?

During my time as an undergraduate majoring in Secondary Education, I was given as assignment by one of my favorite education professors to write a “teaching metaphor”. You may have written one yourself or read one at some point. “A teacher is a like a gardener”, “A teacher is a like a coach”, “A teacher is a like tour guide”, and so on.

You might have already guessed what metaphor I wrote about, but before I say it, picture something with me for a moment. I want to tell a story. It is a little long, so bear with me.

The Two Cliffs

The First Cliff

Imagine you are in the bottom of a deep gorge. Steep walls of red rock rise up on both sides of you. The gorge itself is very long, you can’t see the end of it before you or behind. If you are having flashbacks to the death of Mufasa then you are doing a decent job imagining the gorge. If you are imagining Fezzik preparing to carry Princess Buttercup, Vizzini, and Inigo Montoya up the Cliffs of Insanity then you are doing an excellent job imagining the gorge.

Suppose that you don’t know how you ended up in the bottom of the gorge and all your supplies are at the top of the cliff lining the gorge. And it’s getting dark. And you’re hungry. Now imagine–or maybe like me, you don’t have to imagine–that you are not a good climber. Maybe you have never climbed anything except the stairs to your apartment. And that scares you. You aren’t even really afraid of heights, but generally if someone came up to you and said,”Hey, how would you feel about having the strength of your three longest fingers on each hand be the only thing between you and a 140 foot freefall to your death?”, you would say, “Pass”. Your lack of experience and the fear itself lead to further anxiety. But if you want food and shelter you will have to climb that cliff. No choice. You will have to climb or starve.

Now, there is some good news. It just so happens that there is someone else down in the gorge with you. And, better news, they are very good at climbing. Heck, they climb cliffs twice as high as the one in front of you just for funsies. They love it and they want you to love it to, but they are content to just help you get to the top. They have ropes, harnesses, that weird chalk dust; everything you need to succeed in climbing the cliff safely. They grip your shoulder reassuringly and say, “Don’t worry, friend, I’ll be there to help every step of the way.”

But here’s the thing. They can’t climb for you. You have to do it. You know that is a fact, but you still wish it wasn’t true. The safest and best thing for them to do would be to stay on the ground and belay for you. They would stay on the ground and be an anchor for you, holding the rope that keeps you alive as you climb. From their vantage point below, they would scout out good paths for you to climb as you ascend the cliff face, giving advice and direction, but generally allowing you to make your way up the wall of solid rock.

But you are apprehensive and unsure of yourself. They end up climbing just ahead of you, saying “Do exactly as I do. Put your hand here. Now put your foot there.” In this manner, you progress very slowly up the cliff face. Maybe even for some of you, the person becomes Fezzik himself. They put you on their back and you cling tightly as they carry both of you up the cliff, hand over hand over hand over hand.

You and the climber finally reach the top of the cliff. Both you and your friend the climber are glad to be there. There is food, there is shelter. Heck, there is even Netflix. But where your friend seemed to oddly enjoy the experience, you are just thankful to be done and you hope you never have to do anything like that again.

Inexplicably the next day your find yourself at the bottom of another gorge just like the first. No, not like the first. This is one is maybe a little taller, a little steeper. Your friend the climber asks if you are ready to start and seems oddly enthused to begin. “You ready for me to stay on the ground and belay this time?” she asks excitedly. You shake your head. “Oh no. I need you with me. I don’t know what I am doing.” Your friend reluctantly agrees and, like the day before, the two of you make slow progress as you grab the various hand and footholds as they are pointed out to you. Without the direction, you would probably just cling to the cliff tightly. After a frustrating time for both you and your friend the top finally arrives. You are once again relieved.

And once again the next day you find yourself at the bottom of another gorge. You are irritated. You didn’t ask to be placed in the gorge. You don’t enjoy climbing. It is very irritating that this seemingly insurmountable obstacle is constantly placed in between you and the things you need like food and shelter. But you have to climb the cliff, so with the help of your climbing enthusiast friend you do so each day. Although this routine is a grind to you, your friend seems to genuinely enjoying the climb each time, and he always asks if you are ready to attempt it on your own. Some of you turn him down because you are confident in your lack of confidence in your skill. Others of you turn her down because you have confused the act of you climbing with her telling you what you holds you need to make. You think you have become good at it and how else would climbing look?

There eventually comes a morning when you wake up and are pleasantly surprised to find that you have not somehow been placed at the bottom of an endless gorge with a cliff to climb. You friend is still there. She smiles at you, “I heard about this great, super-crazy-tall cliff a few miles east! Wanna go try it out? Maybe you can try climbing on your own this time and I will belay for you. Come on! It’s great fun.” You are, of course, incredulous.

“Are you crazy? I’m never doing that again. I only climbed because I had to in order to reach what I wanted and needed. Why would I do it just for fun?” You friend smiles again in a understanding manner, but you can tell that they are disappointed that you do not see climbing the same way that he or she does.

The Second Cliff

Imagine you are watching Netflix. This is probably not difficult.

You are lounging in the comfiest couch you have ever owned and eating your favorite snack food. (I am imagining chocolate cake fresh from the oven and a tall glass of cold milk.) You have never been forced to climb any daggone cliffs to get these things and aren’t even sure why I’m mentioning cliffs because, “Uh, dude, you can buy a smart TV like this anywhere. It’s not like Best Buy is on the top of Everest. What are you talking about?”

Your climbing friend comes over to visit. You know he enjoys climbing, but you’ve never gone with him before. He asks if you want to go try it out. You agree since you just watched the last episode of Ink Masters on Netflix and are feeling a little guilty about the sheer amount of cake you just ate.

You and your friend drive to the state park nearby. She sets up equipment near a modest cliff. It’s pretty high, but you can see the top. Your friend says it was the first cliff she climbed and it’s as good of a place to start as any. You strap on your harness and walk up to the rock face with your friend. She gives you a few brief pointers, shows you a good place to start, and reminds you to take a deep breath and relax because it’s okay to be nervous. She reminds you that she will be at the belaying for you the entire time and you will be safe because she is holding the rope attached to your harness.

Your friend tell you that he is ready and you start climbing. It is slow going. You get stuck a lot. Sometimes it is because you aren’t sure where to grab next. Other times it is because you went up a path that was too difficult and you had to back down and try going up another way. Your friend points out a good place to try and you make progress from there. A few times you fall, but your friend has a good grip on the rope. You only drop a few feet before your safety harness catches you. Soon you are back on the wall. Your limbs are very tired. (If you have ever climbed in real life then you know that rock climbing is very exhausting.) You don’t make it to the top, but your friend grabs your shoulder and says, “You did a good job today. Want to try again tomorrow?” You agree on the condition that you get to watch Netflix and eat more cake tonight. Which of course, you do.

The next day you do not reach the top, nor do you reach it the day after. But each day you get a little closer. You enjoy the activity very much. You enjoy the challenge and you appreciate that your friend is there to give help and support, but allows you to do the climbing.

A few weeks later, you reach the top of the cliff. You friend joins you there. She points to the east. “Want to try that gorge over there? I hear its walls are killer.” You smile and say yes.

Teaching is like Belaying

I don’t imagine myself as an expert writer and I don’t believe my little allegory is very opaque so I won’t insult you by elucidating every detail. However I would like to explain it a bit and how it relates to the teaching metaphor I mentioned in the beginning.

The first is the teaching metaphor I wrote for that education course as an undergraduate. I think good math teaching is like belaying for a climber. A belayer’s role is not to tell the climber exactly what to do to scale the wall, but to provide help when appropriate and support when necessary. The presence of the belayer makes an otherwise fatal mistake a short drop and, at worst, a sore butt from the jerking in the harness. With the vantage point from below the belayer can see the bigger picture and can give advice to the climber on direction or a certain step. Sometimes they may give explicit instructions if necessary. The belayer does not say “Hey, watch me scale this wall!” or “Here, get on my back and I will carry you up the rock face.”

Likewise, a good mathematics teacher does not tell the mathematics student exactly how to solve a problem. The teacher picks a problem that is appropriate in scale and difficulty and then provides guidance as the student attempts to solve the problem. Sometimes the teacher may give a hint or make clear a certain concept, but the teacher does not do the problem for the student nor does the teacher give explicit step-by-step instructions on how to do the problem. If the student is in danger or making a mistake or giving up on the solution, the teacher is there as a safety to keep the student on the way to the top.

This metaphor for what I envision good math teaching looks like led me to this story of the two cliff faces. It is not so much about what good math teaching looks like, but what I think math education looks like. Countless students are told that they must pass a math class for their degree: their high school degree, their college degree. Math is a frustrating obstacle between them and the things that they want. Why should or would they enjoy being forced to do math every day when all it is to them is a hoop to jump through or a box to check off. In the survey I give to all of my students at the beginning of every class I teach, I ask, “Why are you taking this course?” I don’t have hard data, but I would say from experience that 98 times out of 100 the answer is “This course is required for my major”.

Now, I don’t expect how mathematics is viewed and taught in the US to change. (If you have any experience at all with discussing the Common Core then you know that mathematics education reform becomes very heated and very political very fast.) And there are definitely skills and degrees that need a certain level of mathematical fluency. But I still want to communicate what I would like to see. We need to stop imposing arbitrary math requirements for the sake of requiring math. I believe as long as we continue to force students to take math courses because they “need” to take math courses they will see themselves climbing the first cliff instead of the second one.

They will always be people fighting to get out of the gorge to survive instead of people who climb for the sake of climbing.