Ok, I admit it. I have a habit of creating titles that create a disconnect. And are a little click-baity. But to be honest it only happens because I often like to write about misconceptions, and so by definition the title will appear counter-intuitive. Today I am going to write about something that over the course of my teaching career has met with perhaps the most resistance from students and parents, but which has also met with the most success when embraced.

Scandalous, I know. And trust me, I have heard all the rejoinders. So as you can imagine, I will explain.

See, in the current system of education, grades stopped being a measure of progress some time ago. What they have become instead is currency. A commodity that is pursued, traded and leveraged with as much vigor and ferocity as the dollar, euro, or yen. And I am not using hyperbole here. Schools these days have come to be viewed by many students, parents and even teachers as a marketplace. Teachers have the grades, students want them. And in this marketplace the end goal is to get as high a grade as possible. To very many – but to be completely fair, not to all – how that happens is not nearly as important as that it happens. To this category of student, the goal of school is not to learn, but to get grades. And this paradigm shift causes a fundamental change in how the entire process is viewed. I will list just a few examples:

• Bargaining
It has become standard operating procedure now that when teachers return graded work, the immediate next phase is the negotiation. Students dissatisfied with the magnitude of their grade will question, cajole and even harass the teacher about the grade, with the common theme that since the student believes the grade should be higher, the teacher has assigned a wrong grade. There are even times when the guise of reason is dropped completely, and the student will actually say things like “I need a 97% to get into <insert elite university program here> so can you raise my mark?”
Academic dishonesty (aka cheating) is not a new phenomenon. What is new, however, is the pervasiveness of it, and the total lack of ethical struggle involved in making the decision to use it as a tool for getting high grades. After all, if the only purpose of school is to get a high grade, and if cheating accomplishes that, then where is the ethical problem? And so we see rampant use of things like plagiarism, paying others to do work that students then submit as their own, or gaming the system so that assessments like tests are skipped, then done at a later date after getting information from other students who were present at the time about what was asked.
Countless “schools” have popped up over the last decade or so who’s sole purpose is to guarantee official credits and high grades. The thinly disguised mission of these schools is to create a means by which, for a price, students can get a credit on their high school transcripts and also get an absurdly high grade. What separates a grade mill from a more legitimate private school is how accurately the student’s grade reflects their knowledge on completion of a course. I have taught many students who received a grade mill credit in a prerequisite course for the one I am teaching, with a grade of 100%, who do not possess the most basic skills meant to be learned in that prerequisite course.
• Cramming
This is definitely not a new concept in academics, but it has spread to more and more students, who in fact no longer recognize that it is not actually a means of learning. In courses where there are scheduled tests/exams, students do little to no work during times where there is no assessment looming. They attend class, possibly take notes, and otherwise devote minimal attention to the lessons, because “this won’t matter until the test.” They do not see this as an ineffective strategy at all. The belief that drives this is that the only time the subject knowledge will matter is when they are tested on it (and thus in a position to get grades), and so the plan is to study as much as possible the day – or even the night – before a test. Cramming all the information into their short-term memories just long enough to unleash it onto their test papers, to be promptly forgotten as they leave the room after writing the test.

These are not the only examples of what I am talking about, but they are the most common. And it is clear that none of these appear to give actual learning more than the slightest courtesy of a head nod. They are completely and totally about getting grades.

Sometimes, they even work. But that’s a trap. Because even when they work, they are only short-term solutions to a lifelong endeavour, and they all create stress and anxiety in the process.

• Bargaining for grades, when it works, teaches that it is not about what you earn, but about what you can badger people into giving you. It shifts the perspective about where the effort should be placed. Rather than placing effort on producing good work, the effort is placed on convincing the teacher to assign a high grade. This creates an internal tension that results in generalized anxiety, because the student ends up in a position of having to convince the teacher of something that is not actually true, and for which there is no evidence.
• Cheating works for its intended purpose (when you don’t get caught), but like grade mills, perpetuates the “appearance over substance” philosophy, and also imbues dangerous long-term values that erode at the ethical fabric of society. The stress this creates is clear – fear of getting caught, and the consequences. Additionally there is the gradual accumulation of anxiety brought on by creating an academic avatar that is more and more fraudulent and removed from the person who wears it.
• Cramming is arguable the lowest offense on this list, because in its purest form the student is not misrepresenting themselves at all. However it is fraught with disadvantages, from the fact that many students struggle to absorb and then reproduce the knowledge in a meaningful way, to the fact that when needed later – in the same course or in a subsequent one, the knowledge is no longer accessible. It also creates a great deal of stress and is a common cause of test-anxiety, which is a very real issue for many students who find they “totally knew this last night” but can not recall it when test time comes.

Perhaps most tragically, this issue causes stress and anxiety not just for the students engaged in them but for the many students who are not, because it creates an unlevel playing field that places incredible burden on the ones who are doing things the right way. Grade inflation is a real and dangerous phenomenon, where just like monetary inflation, a loaf of bread is still a loaf of bread whether the price tag says $0.75 or$2.99. The difference is that because we use percentages as grades, there is a ceiling, and so we are starting to distinguish by decimals. And that means that for any student mistakes cost much more than they ever did in the past.

Ok. So I’ve devoted the article to this point (approximately 5 minutes of reading time, if the algorithm that tells me how much I have written so far is to be trusted) outlining the issue. And maybe I’ve made it seem like hope is lost, because we do in fact live in a system where grades matter for universities, colleges, academic awards, and sometimes for that first job, and all of these vehicles by which students are getting the grades are either unethical or riddled with stress and anxiety. But hope exists! Because there is very good news.

To get good grades, all you have to do is actually learn the material!

Revolutionary, I know. It almost seems like I am joking. I assure you I am not. This simple fact is lost on more students and parents than I wish was the case. Clearly it would work though, right? Of course it would. Students, you can take all the effort you are devoting to “getting grades” and shift it to “learning material”. Immerse yourself in class. Ask questions of the teacher. Engage in discussion. Pay attention. Do work in increments (that is, homework), instead of cramming the night before a test or exam. Decide that you will be a master of the topic and use your teacher as the resource they are. Develop a love of learning – trust me, this is not as hard as you think – and as you grow into this person who legitimately strives to learn, the grades will automatically follow, as an afterthought!

Now I know from experience that this message lands differently on everyone. Some people roll their eyes, either inwardly or outwardly, and decide that the game as it is being played works just fine for them. Others hear me and know it makes sense, but feel that it’s too hard, and getting grades some other way will be the way to go. But, there is a significant portion of students I have talked to who have taken the idea to heart. And without fail they are the most academically successful, as reflected both in their grades, and in their facility with the material they have learned. These students inevitably report back to me that once they stopped their pursuit of high grades, and shifted their energy to the learning, they began earning higher grades than they ever had before. And their confidence grew as their anxiety atrophied. Because so much of the mentality of getting grades involves somehow gaming the system into awarding false credit, that when they shift into the person who actually has earned the credit they are receiving, they feel bulletproof.

And what a great feeling that is!

Rich

Depending on the perspective of the different people in my life I am many things: a son, a father, a brother; A student, a teacher, a mentor; A friend, a colleague, a training partner. And yet, in all these capacities I have learned one lesson well, and seen it play out repeatedly. Failure is critical for growth, and letting loved ones fail is one of the hardest manifestations of loving them that there is.

This is never more acute than when it comes to your child. It is one of the most painful struggles in normal parenting. Seeing a that a child is making poor choices and is therefore on a path to failure, letting it happen, and watching the fallout is heart-rending. And yet, the alternative is worse: Letting them believe that there is always a safety net.

As parents, it is our job and our duty to give our children all the tools and guidance they need to succeed as they grow. But it is not our job to give them success. It is not our job to undo their failures, or to mask them as something else. In fact, any “success” that is not earned is not a success at all, and any part we play in delivering these false successes is, in fact, failing in our duties as parents.

Repeatedly and consistently rescuing a child from failure teaches them that failure is not on the table. It teaches them that life will be fine regardless of poor choices, lack of knowledge, or lack of skill. It’s like giving someone voice lessons and using autotune to correct their pitch before playing back their singing to them and to the world, letting them believe they are the next Freddy Mercury or Whitney Houston, and then letting the world watch their train-wreck audition for American Idol. It not only sets them up for confusion later in life, it inhibits self-awareness to the point that life in the “real world” will for them will seem like walking through a minefield, where the reactions from others are so completely out of line with their internal measures (calibrated in their youth) that they can, at times, feel like they never really know what is wrong with them or why they get the responses they do. Consider those American Idol auditions where the singer is so clearly awful, and yet they think so highly of themselves that they tell the judges it’s the judges who are terrible.

The thing is though, that in the moment failure is happening, we parents want nothing more than to take the pain away. Every parent knows what I mean. Your child’s pain, whether physical or emotional, is about a million times worse than your own. My wife always talks about that moment when a toddler is running to you in the park to show you some amazing treasure they found, with that big toddler smile and enthusiasm, when they trip and fall and start crying miserably. We all want that enthusiasm and joy to be the only thing they ever feel. We never, ever, want them to fall.

But it’s unrealistic. Everyone falls. Learning what caused it, analyzing how to have avoided it, and recovering from it are the real lessons. The lessons that lead to a person who truly can be successful. And so we have to let our children fail. And then we have to be there for them to love and support them as they recover from it. We have to show them that failure isn’t the end of anything, and that even in their failure we believe in them and love them as much as we always have. That is how they will learn to grow from failure, without spending a ton of counter-productive energy and emotion on self-recrimination and shame.

In my years of teaching, I have had countless conversations with parents concerned about their children’s academic success – or lack thereof. And on more occasions than I care to count, a parent has outright asked me if there is anything I can do to raise their child’s grade. These parents are always taken aback when I tell them that there is nothing I can do, and nothing they can do either. However there is everything the child can do. We can give them the tools, we can be there to support their learning. We can be the best parents and teachers there are. But we can not “raise the child’s mark”. That is on the student. And sometimes, the student fails. When I tell the parents that I am willing to let that happen, they often think it means I am a bad teacher. There have been a few occasion where they said so. But my response is always the same:

“I am being the best teacher I can be. I will always be there to support your child. I will give them every tool I have, and the guidance to use it, in order for them to succeed. When they need me, I will be there. I will be there even when they don’t realize they need me. But I will never do the work for them, and I will never assign a grade they did not earn. And if they should fail, I will be there to support them and help them see what went wrong, and how to address that in the future. Given all that, my only hope is that your child will look back on their time as my student and realize the gift I gave them: The gift of letting them fail.”

We want the next generation to be resilient, strong, caring and educated. Failure is the path to all of these. As hard as it is, we need to let our children fail.

And then celebrate the heck out of their successes!

Rich

The global COVID-19 pandemic has been a reality long enough now that the emails and texts that start off “in these extraordinary circumstances” or “considering the difficult times”, or “unfortunately, due to the unprecedented situation we find ourselves in” now feel like the opening sentiment is redundant. It has strangely become awkward to know what to say when sending out an email with yet another update on how someone or some institution has been impacted. We get it. And yet … we don’t get it. Because we didn’t sign up for this.

I have always admired and respected people in law enforcement, the armed forces, and perhaps especially firefighters and EMT’s. I remember 9/11, which drove home the point that hardly needed making that when everyone’s instincts are screaming at them to run away, these are people run toward. And my respect for them stems from the fact that they signed up to do it. Something inside compelled them when they were younger to sign up for a career of running toward the danger. Of being in place when the shit is going down. Of being the person who was standing between the monster and the town, or being the first person the rest of us might see after getting mauled by the monster – doing their best to get us out of the danger and back into the loving arms of uneventfulness. My respect for them is unchanged.

But now we are finding out that some monsters don’t make a frontal assault. Some of them don’t take the path through the Hot Gates, so that Leonidas and his Spartans can know where to stand to get between the armies of Xerxes and the Greek civilians. Sometimes there is no bridge for Gandalf to block, because sometimes the Balrog descends like a toxic snowfall, on everyone at once.

Grocery store clerks and cashiers, essential workers in office buildings, gas station attendants … they didn’t sign up to have to go out each day and do the exact opposite of what the rest of us have been told to do. Nurses and doctors signed up to treat sick and injured people – but they didn’t sign up to fight for 15+ hours a day, 7 days a week, against an unseen and not fully understood enemy, under-equipped and underfunded. They didn’t sign up to put themselves at risk caring for ICU patients with a highly contagious virus. They didn’t sign up to hold the phone as patients who might never go home FaceTime their families who just cry as they watch their loved one in an induced coma, breathing only thanks to a ventilator.

It is sadly ironic that some of the lowest-paid occupations in our society are now clearly the backbone of it. I find myself more and more troubled by this glaring imbalance. They didn’t sign up to be brave. The didn’t sign up to run toward the trouble. They didn’t sign up to be at the front. And yet.

I also find it ironic as an educator that for years now we have been saying that we need to prepare our students for a future that we don’t understand, and yet now we find ourselves living a present that we have no frame of reference for. All the rules have changed. Societal norms are in flux. Responsible governments are scrambling to make the best decisions for the present and for the future. We’ve seen politicians completely shed their veneer, humbled into humanity by the pandemic. We’ve seen others double-down on the default political position of obfuscation, redirection, and selling the fantasy. They don’t have a rule book for this, so some are writing a new one, while others are desperately trying to make the old one work. Time will show either way that governments around the world are making many mistakes. But time will also show the wisdom of many of their decisions. It is too soon to be able to tell in each instance which is which. But as I wrote a few years ago, mistakes are just as valuable in the long term as getting things right, because time doesn’t stand still and we are only ever as good as the lessons we learn. Mistakes make excellent teachers for those willing to learn instead of criticize.

Because we didn’t sign up for this, we don’t have a playbook. There is nothing that is time-tested and proven effective. There is nothing you are “supposed” to be doing with your time. There is only what makes sense in the moment. And the biggest thing to understand is that everyone is in the exact same boat. Or to use another metaphor – we are all on the same ride, and the ride has stopped. So when things return to whatever normal will look like, we will all emerge into the same sunlit sky, rubbing our eyes and stretching our arms and legs. Standards and expectations that applied before the pandemic in many cases not be relevant.

Everyone will understand what you went through.

Rich

## Maximizing Exam Performance

As a high school math teacher and now university lecturer, one of the most common issues I see is students who under-perform on tests and exams. By this I mean that the grade they earn is not an accurate reflection of their level of mastery of the material – sometimes tragically so. I have been teaching for around 20 years, and my experience teaching, as well as my experience as a student, has caused me to form some strategies to address this. In a recent email exchange with a student who was asking for advice on how to manage studying for multiple exams, I decided to organize my thoughts and type a detailed response. With that student’s permission, I also decided to share it here on my blog.

##### Question: How would you recommend studying for finals and what are the key things that I should prioritize?

I have some tips, but there is a caveat: these are things that I’ve learned through personal experience as well as my experience teaching for about 20 years, and I have learned that while they are solid tips and work for many people, everyone needs to tailor them to suit their own learning style and personality.

First, some terminology for the purpose of these tips:

• Homework
I’ll use this term to refer to any work you would be assigned or do voluntarily throughout a course that does not directly get assessed for grades. It may be questions/reading assigned by the prof, or practice assignments. It should also be daily review of notes taken in class, which can and probably should include enhancing the notes with more thoughts or research.

• Test
I’ll use this term to refer to some kind of minor assessment that does count for marks, but not a large part of the course. These are usually assignmentstests or quizzes, and they generally happen regularly, often weekly. The knowledge they assess is normally not cumulative, focusing on one topic or unit at time, or perhaps just one or two weeks worth of material from the course.

• Exam
This refers to the major assessments in a course and are usually cumulative in the knowledge they assess. In many math classes this is a midterm and a final exam. Sometimes it is just a final exam.

• Term Mark
This refers to the grade a student earns on any tests, not including exams, as defined above.
##### Here are the tips more or less in order of priority
###### Don’t study for exams!

Sounds weird right? What I mean is that you should not be studying for exams – you should be studying for expertise. Furthermore, this studying should occur throughout the term, not in the days before an exam. The root of the word student is study. This literally means that if you are thinking of yourself as a student, then you are thinking of yourself as one who studies.

For many students this is a paradigm shift, even if they don’t realize it. Students have a habit of using the term for accumulating marks, instead of accumulating expertise. A combination of procrastination on homework, sub-optimal time management, and misplaced priorities create a situation where decisions are made to maximize a term mark while minimizing actual learning. That may seem harsh, but students usually see some or much truth in it when they inspect their own habits during non-exam times.

The shift is to start to view homework as the tool to help you grow into an expert in the course material, to the point where you could potentially even teach it. In fact, finding ways to teach the material (usually by helping others with the work) is the very best way to gauge your own level of expertise. If/when you reach this level during the homework, writing tests and exams becomes much less a matter of which questions you are prepared for, and more a matter of making sure that when it is time to write that you are rested and feeling physically comfortable.

###### Make an exam prep schedule

So once you have an exam schedule from the school, and you know when your exams will be, make a schedule for exam prep. It is almost always better to devote parts of each day to different exams rather than trying to devote all your attention to the next exam. So for example, you might have 4 exams in all, and from day one of prep time (which is the day after the last actual lecture), each day should be divided into 4 parts. At first, you can devote more time in each day to the first exam. Then, when that exam is done you only have 3 per day, and can devote more time to the second exam, etc. Make sure to include recreation/rest time in your schedule! It is unreasonable and unrealistic to think that if you have 16 hours of awake time in a day that you can spend 16 hours doing exam prep.

The puzzle of creating the schedule is mathematically pleasing and kind of fun. When you’re doing it, imagine you were creating it for someone else – someone that you are close to and care about, like a sibling or friend. Not for yourself.

Once you have prepared your schedule, put your faith in it. Don’t question your scheduling decisions after that fact. Pretend it was set for you by someone else – someone that you are close to and who cares about you, like a sibling or a friend. Someone you don’t want to let down. They took the trouble to make this schedule for you so that you would be successful. You owe it to them to follow it to the letter. Resist the temptation to over- or under-use the times they allotted for work. Resist the temptation to under- or over-use times they allotted for recreation.

###### Prepare!

If you have spent the term studying as outlined above, then most of the work is done when exams roll around. At this stage what you want to do is prepare. I like to think of it using sports as a metaphor. With a big game or event approaching, athletes don’t use the time leading up to the day to become excellent at their sport. That work has already been done, over the course of their training. What they do before a big event is prepare themselves. They manipulate their training and diet so that they will peak on the day of the event. They get ready mentally for the stress of the day. They take care to eliminate distraction that during the rest of the year they allow, more or less. In other words, they sharpen everything they have already done, so that their performance will be optimal on the day. That is what exam prep is. Here is how it might look:

1. No phone during prep time.
That means it’s not just on silent in your pocket, or upside down on the desk. It means it is literally off, and in a different room than you are in. You are allowed to use it during recreation time, and you should, if that makes you feel better.

2. Review the most recent course material first
Work your way from the end back to the beginning of the course. This is for two main reasons: One, that material is the most fresh in your mind, and so it needs the least review. Two, by working backward you by necessity end up cementing ideas from earlier in the course, since they were generally used to build the ideas that came later. This forces you to assimilate them more completely, and by the time you get to those concepts you are already quite an expert.

3. Use tests and/or homework assignments as your map to prep.
Take blank versions of them and simulate writing them from scratch. If you studied throughout the course you will find that you can mostly do this without consulting any external resources like class notes, course notes, textbooks, other students, tutors or the web. But when you find yourself struggling to recall a concept, don’t hesitate to use those resources, more or less in the order I listed them.

4. Do not focus on being ready for a certain type of question.
Instead, make sure you are expert in all the concepts you review. A litmus test for expertise is to imagine you had to teach the concept. Would you be able to field questions from students like yourself? Can you envision a way to present it to your peers that would make the concept clear for them? If you can, take opportunities to actually try. There are usually people more than willing to let you explain concepts to them!

5. From time to time, refer to any exam outline that has been shared by the instructor.
As you progress through the review, put a check on concepts you feel you are good with. When you have finished working back, it would be shocking if there was not a check on every item. But if there is, go back and find instances of that concept in your review and have another look.

6. Sleep!
There is nothing to be gained by staying up all night before an exam, and much to be lost. Any feeling of security it may give you, or any thoughts of “I’ll just hold it together then crash after the exam” are almost always self-delusion. No athlete would ever go into a major event already exhausted. It does nothing to improve performance and much to impair it. On the other hand, a solid night’s sleep will give your brain a chance to reset, refresh, and reorganize.

7. Finish studying the night before
It is almost never a good idea to do any studying the day of an exam. If you made a good schedule, you finished studying the night before. After a good night’s sleep, your job is to stay relaxed and properly fed. On the day of the exam, the first time you look at course material should be when the exam begins. Otherwise you risk spiraling on some
last-minute topic you have convinced yourself is important, and it defeats much of the organization that has taken place in your brain. Picture a well-organized filing cabinet with everything where it belongs. That is what you want to take into the exam with you. Contrast that with a mostly well-organized filing cabinet, with a few files removed and papers scattered about the room. That is a much less effective way to be able to access your knowledge during the exam.

8. When the time comes to begin writing, do not begin writing.
This is one of the biggest mistakes I see students make when the exam actually starts. They flip to the first page and start writing. That is putting your destiny in the hands of the person who created the exam instead of in your own hands. What I mean is, why should you write the exam in the order that someone else decided? And furthermore, why should you start writing without any idea of what is coming? When starting on a long road trip it is a much better idea to zoom out and see the whole route so that you have an idea of where you are going, rather than just thinking of the next turn.

What I strongly recommend is that when the word is given that you may begin, turn the first page and just read. Read all the questions. Slowly. Take the time to make sure you understand what each question is asking. Do not attempt to answer any question until you have read all the questions. Then decide what order you want to write the exam in. Choose your best questions first. Save any you don’t know how to do for the end. You will find that if you write the exam this way three things happen:
• First, your time is automatically optimized, since you are spending the least amount of time on questions at the beginning.
• Second, your confidence grows as you progress through the exam, which is much better than having a tough question destroy your spirit near the beginning of the exam.
• Third, as you work through the questions you are good with, you subconsciously are also thinking about the harder ones you saved for the end, and often get clues and ideas from the questions you are working on so that when you get to the ones you thought were hard, they no longer are – or at least they are easier than they seemed.

That’s it! As I said, you can take or leave as much of this advice as you like, but I would say that if you decide to ignore some of it because you’re afraid to try it, that may not be the best reason not to give it a go. Fear is not generally a good reason to avoid trying something new.

Rich

## Why Study Mathematics?

In my job, this question is one I get asked very often. To be honest, it usually comes in a slightly different form …

#### “When am I ever going to use this? What is it good for?”

As a high school math teacher for 15 years, this is one of the most common questions I received. When I began lecturing at university, I was surprised to find that I still sometimes get asked variations of this question. I suppose it’s a good question, if the idea is that at some point someone will say to you

“Determine

and have your answer on my desk by 5pm today. And don’t get any funny ideas about using WolframAlpha!”

Because the truth is, that rarely happens.

I often give a joking answer, and say flat out, “You won’t,” and then go on a rant about how math doesn’t need to be good for anything, because it is just good. Nobody ever stood in the Sistine Chapel, staring at the ceiling, asking what it was good for! They just appreciate the inherent beauty, because it speaks to their soul. Math is the same.

I think that’s a perfectly good answer, to be honest. But in a more serious light, I find the answer to the question is actually another question: “When are you not going to use this?”

Of course, there are direct applications of many branches of math. But those tend to be very specific, and these days depend heavily on software to do the heavy lifting, so I tend not to think of those. Instead, consider that football players perform bench press as part of their training, to the point that the ability to bench press 225 pounds for as many reps as possible is tested at the NFL combines. Yet not once have I ever seen a football player perform the bench press during a game. Why do they do it then? Couldn’t they just practice the skills they will actually use in a game? I can promise you that at no point during a football game does a player think “oh, this situation is just like bench pressing 225 pounds – I will apply that same skill now.” And I imagine there are very few football players who complain while lifting weights that they will “never use this in real life”. Of course, we know that the reason they train the bench press is that it increases strength and power, so that when the time comes that they need it, it will be there without consciously calling upon it.

Studying mathematics is the same. Math teaches so much if we are awake to the lessons. Here are some things I have learned, continue to learn, and apply regularly from my math studies, along with some examples of how they have impacted me personally.

#### Scale simple solutions to solve large problems

It is almost always the case that large problems can be effectively solved by breaking them into smaller problems, or by developing scalable solutions to simpler problems. For example, about 3.5 years ago I decided I wanted to learn to draw, so I took a piece of white printer paper and a mechanical pencil and drew a superhero-esque muscle man. It sucked. Like a lot. But I was not discouraged in the least by that. I was fueled by it. Why does this suck so much? I know how I want it to look, why can’t I make it look that way? I was excited by the fact that I could recognize how much it sucked, and by the prospect of working to slowly strip away the suckness. I spent hundreds of hours, solving small problems that were contributing to the suckyessence, and slowly scaling them up. Want to draw a heavily muscled arm? Learn to draw a cylinder. Then learn to draw little cylinders that lie on the main one. Then learn to draw “twisted” cylinders and tubing that changes diameter as it twists. Learn anatomy. Now put it all together. I intuitively understood platonic solids and how they interact with and reflect light. I applied these understandings to understand the types of skills I needed to hone with the way I held and manipulated pencils. I started looking closely at things I never paid attention to before. I still do this, and at no point during this process do I ever consciously say “Oh, that’s just like <fill in math course here>”, but at every point I feel exactly the way I feel when I am working on difficult math problems.

#### Being right also means proving you are

Math is really never about just “getting the right answer”. It’s about proving that an answer – or a result – is correct. The emphasis on proof is critical. In the real world, being right is rarely enough if you can’t convince others that you are. Careful, methodical, and audience-appropriate explanations are invaluable in this regard. Developing and writing proofs in mathematics is as much an art form as it is a science (perhaps even more so), and my studies in mathematics immeasurably improved my approach to constructing an audience-appropriate argument or explanation. This has had a profound impact on my communication skills, as well as my approach to confrontation. I have used this skill in more ways than I can list, but some examples are: when I have been in contract negotiations, when I deal with sales people when buying big-ticket items (and even when I bargain at markets), when I find myself moderating arguments between friends, family, colleagues or students, and when I used to work as a personal trainer and had to motivate and justify the kinds of exercise and diet choices I wanted my clients to make. In every single one of these situations, and more, I am really constructing proof. In fact, I would say that proof dominates almost all my communication.

#### Emotional attachment to a belief is irrelevant

Not wanting to be wrong about a belief, especially if it has been long-held, is normal. It is, however, illogical and possibly even dangerous in the face of proof to the contrary. Mathematics trains us to seek, understand and ultimately accept proof on its own merit, and not on any emotional yearning. It also trains us to be grateful when proven wrong, since it makes little sense to want to be wrong for even one moment longer than necessary. My training in math has led to a much more open-minded approach to new thoughts and ideas, and after careful consideration – which involves listening to argument dispassionately, asking relevant questions and weighing evidence – I find myself either happily embracing a new thought, or else more confident in the one I already had, having had the opportunity to test it rationally against a differing viewpoint.

#### Creativity and math are NOT mutually exclusive.

Not even close. Deep study of mathematics reveals and refines a strong creativity that aligns with and is mutually supportive of logic. This fusion is relatively rare, and people who have it are prone to what seem to be exceptional accomplishments. In truth, the exceptionality of it is not the accomplishment itself but the relative scarcity of people who can do it. One of my favourite examples is Leonardo da Vinci, who most people think of as a great artist, but who was also an accomplished mathematician and scientist, and who did not consider these as separate pursuits or modes of thinking. I find the same is true in my own life, although there are many people who wonder how a mathematician could be artistic.

#### Clarity lives just on the other side of contemplation

The journey math students regularly take from being completely mystified and often intimidated, to understanding and comfort is a lesson in overcoming that serves us well in all the challenges the future can bring. It instills a confidence that says, “I may not understand this right now, or even feel like I ever could, but I know I can do it.” General wisdom suggests that “easy” might seem gratifying in the moment, but true satisfaction comes from overcoming a challenge. Many people shy away from challenge for fear of failure, but studying mathematics teaches us that we can tackle large problems, even if they seem overwhelmingly daunting at the outset. An example that makes me laugh is the time I purchased a large and intricate piece of exercise equipment for my home gym (a functional trainer/smith machine combo). I bought it used, so it did not come with any assembly instructions, and perhaps embarrassingly, it didn’t occur to me to use Google. When I picked it up the seller had already “helpfully” disassembled it into n pieces, where n is large. I was completely baffled at how to reassemble it when I got it home. But I was not daunted. I laid all the pieces out on the floor, shuffled them around into sensible groups, and slowly assembled sections that made sense. I made mistakes and discovered them when they led to chaos. I backed up, took a different approach, and eventually put it together. The process was not “clean” – I hurt my hand trying to brace a nut while tightening a bolt, and cursed myself for not taking the time to get a wrench to hold it in place. But the result looks like it was assembled by a pro. I’ve had it for many years now, and it still works perfectly. I am fully aware that my engineer friends would consider this a trivial exercise, but for me it was a hard-fought and well-earned victory. This type of approach has stood me well time after time.

#### You don’t always have to see the whole path to the goal

How often have you been working on a difficult proof or problem, not really knowing if you were getting anywhere good, nevertheless continuing to take small, logical steps – always forward, occasionally pausing to reorient yourself to see if the direction made sense – when suddenly you found yourself having completed the entire thing? Some mathematicians call this the “follow-your-nose” principle of proof. A leads to B which leads to C etc. This might be the most important lesson of all. If you have a long-term goal that seems incredibly distant and perhaps overly ambitious, consider that if you just point yourself in the right direction and take small steps, occasionally reorienting yourself, you do eventually get where you want to go. Plus, the journey is so rewarding. In my life I have used this principle I learned from proof over and over and is in fact how I ended up lecturing at university, something that has been a dream of mine since the 12th grade.

And that concludes my very long answer to the common question! I hope you found something of value.

Rich

## Phone Addiction is NOT Okay

Hi. My name is Rich and I am an addict.

Ok. I know that’s been borrowed from AA protocol countless times. Often in jest when not in the context of an actual meeting. But I’m not joking – I am an addict. I have had a few addictions in my life, thankfully not any of the big nasty ones like alcohol or drugs, but some of the ones I’ve had have had negative impact, for certain. The two most prominent are food and smartphones.

I am going to talk a little about the food addiction later in this article. But I mainly want to talk about the subtle and not-so-subtle evils that my phone addiction has wrought. The first is an obvious one, and it has been discussed to death. Still, it needs addressing:

#### Phone Addiction Steals You From The World

There are way too many times I have belatedly realized that my son, daughter, wife or close friend made an attempt to communicate with me while I was using my phone. And after they realize it’s like talking to a projection, they give up. The worst part is they seem to just accept it – that they are not going to win my attention over whatever clever post I am reading or writing, or whatever video about people hurting themselves doing stupid things I am watching, or whatever likes I am counting. What’s worse is that so many of those times they were the ones who had my attention first, but during a brief lull in our interaction I pulled out my phone to investigate a notification and was subsequently checked out. Most heart-wrenching is that it is out of respect for me that they give up and don’t try to interrupt my phone time. Clearly I don’t need to go further on the evil of this aspect. The next few are a little more subtle, but insidious.

Humility aside, my concentration and focus used to be legendary. I could get lost in a process, with intense focus for hours on end. So much so that I would forget to eat or even go to the washroom, until the need for one or both of those became impossible to ignore and I would look at the clock and be shocked at the time. And the things I could do during that time were amazing. It’s how I got through my Masters degree with a GPA of 97%, for example (that and major support from my awesome wife and kids). Lately, my concentration sucks. I find myself looking for distraction only minutes after beginning something. At first, when I noticed this, I blamed my age. It’s no secret that aging can have an impact on brain function. But the truth is I am 48, and there are plenty of people older – even much older – than me who can concentrate on a task for a long period of time with no trouble. On a smartphone, nothing needs concentration, and everything is provided in quick bites. You flit from item to item like a politician moving through a political rally, shaking hands with everyone and meeting nobody. And you can tell yourself that this is behaviour limited to your phone time, but what you miss is that your brain is a learning machine, and it’s learning a behaviour. Your brain wires itself to adapt to your environment, even if that environment is the manufactured barrage of inanity you kill time with on your phone.

Possibly worse, is that we tell ourselves that the phone is a harmless distraction, and we use it at the weirdest times. I’ve used it while watching TV, thinking that surely the TV doesn’t care so no harm done. But while the TV doesn’t care, my brain does. Watching a television show requires focus. Not the same level as calculus mind you – but focus nonetheless. There’s a story unfolding and if you miss something the subsequent bits make less sense. So you are missing an opportunity to focus. To practice what I tell my students is the single most lacking skill in the computer generation: uni-tasking. I have heard many a person brag about their ability to multi-task, which is great, I guess. But smartphones have stolen our ability to focus on one thing for a long time. We can no longer effectively uni-task.

Yeah, this one took me a while to clue into. I’ve noticed over the past couple of years that I am making more mistakes at things that I should be able to do automatically. Frighteningly, I don’t even notice them when I make them – only once they are pointed out, or long after, do I notice. And when I do notice I have no memory of making the mistake.

Now I hear you say Wait! That’s just a symptom of aging!

Well, yes. It could be. But the thing is, there is a thought groove I find myself in when these things happen – like a road I know I’ve been on because the landmarks are familiar but I can’t quite place it. Then it hit me. Autocorrect. Autocorrect and predictive text. Both of these so-called innovations have made it so that we don’t even have to concentrate on the simple act of typing a text, email, or post. Just type close to what you want and autocorrect will fix it for you. Not always correctly, mind you, but often. So much so that when it gets it wrong we don’t even notice until later, making for some humorous/embarrassing messages getting sent. I actually hated this feature, and turned off autocorrect some time ago, but I kept the predictive text on. And it works so well that I often only have to type the first 1-3 letters of what I want and an entire phrase is suggested to me – correctly. So I select it and don’t check it. And sometimes, my iPhone does the old bait-and-switch on me so that the thing I think I’m selecting changes to something else just before I select it, and I don’t even notice and the wrong thing gets sent.

Like I said before, the brain is a learning machine. And my smartphone addiction has taught my brain that you only have to start a thought to have it completed automatically. And that is what has manifested into these increasingly frequent mistakes. It is NOT age. I can feel it when it happens. It is the same thought groove as typing on my phone.

#### At Least I Remember When It Wasn’t So

The scariest thought to me is that these things I describe are changes. As in, it wasn’t always so for me. I am actually frightened about the generations that don’t have a comparison to make to a different time. A generation that believes that being able to concentrate on one thing for a long time is an unusual, maybe even useless skill. Or that being able catch every joke in a sitcom is a freakish ability. Or that being able to spell a word correctly on purpose is a worthwhile ability. With respect to the issues I have outlined, I want to go back to the way I was. But for many younger people, this is the way they have always been. There is nothing to go back to.

#### Quitting May Not Be the Answer

So the fix would seem obvious, wouldn’t it? Just stop. But that’s not realistic, or even really desirable. Many good things come from my smartphone. I have made friends – good friends – who live in different countries from me, and I stay in touch with them using my phone. There are legitimately useful apps, like the medication reminder, the shared grocery list. Or the math puzzle game Euclidea that actually reverses some of the deleterious concentration effects I described above. I use GPS apps all the time and get lost much less often (though sadly, I still manage to get lost sometimes). I listen to so much more music variety than I did in the days when you had to buy records, tapes or CD’s, and it is so much more easy to discover new music now than then. I take a lot of photos with my phone, and some of those are real treasures that I wouldn’t otherwise have. I read some great articles, and have learned a ton of cool things. Chances are super-high you are reading this on a smartphone. I believe these are good things. So I don’t want to quit. And I don’t think anyone has to. We live in a pretty plugged in world, and more and more there are things that are designed to be done on our devices that are actually difficult if not impossible to do otherwise. So modified usage is the key.

#### Modification Requires Discipline

Or does it? I mean, yeah, it does. Almost always. If you want to modify a behaviour it means breaking a habit, and that is not trivial. You need to be disciplined. And you need to believe that you can do it. Which I think, for many people, is the main obstacle to success. That belief that you can do it. Which leads me to my food addiction:

#### How I Modified My Eating Behaviour

The first addiction in my life I had to acknowledge was food. Maybe I was genetically doomed to that, or maybe it was my upbringing, but my addiction to food is real either way. It resulted in a max weight of 250 lbs, and an adulthood of gaining-losing-gaining, until my heart attack 3 years ago. The heart attack was actually not caused by my eating, or so the cardiologist says, but rather genetics. My arteries are shaped in a more clog-worthy way than you’d set out to do if you were designing a human from scratch. However there’s no argument that my food addiction wasn’t helping. The heart attack added a new tool to my psyche though – something that’s simultaneously heard and easy to describe: a hard stop. Before the heart attack meals were a negotiation in my head. And I often didn’t come out the winner in the bargain. Now there is no negotiation. I eat clean, all the time. I don’t overeat. I actually can have just one french fry. Truly! French fries are a food I will not allow myself to order or prepare anymore, but on occasion I have had just one from someone else’s meal – even though they were finished and offered me the rest. Just one, for the taste, and I have no desire to eat another. Cardio is now a non-negotiable habit as well. I don’t negotiate whether or not I will do it. I just do.

I am as fit as I’ve ever been and certainly as healthy as I’ve ever been. People often ask me how I did it and I sadly tell them that I wish I could give them the secret, but the secret is I had a heart attack and I don’t want another one.

The picture on the left features my beautiful niece at her first birthday. You may be distracted by her cuteness – I understand. Take your time. But when you are ready, the place to look is my belly protruding under my forearm. The picture on the right is a fun shot I took after a workout a couple of weeks ago. Oh, and by the way – both those photos were taken with my phone.

So. I have managed to find a way to live a healthy life despite a food addiction, even if the way I found was not something I’d wish on anyone. But after all, isn’t it all in my head? This new tool I called the hard stop?

Of course it is. And because it is I believe I can apply that rationale to my phone addiction. I have started to, and will continue to do so.

#### What I Am Doing to Fix it

I turned off all the features of my phone that try to think for me. Any typos or wrong words I send now are all mine. And there are so much fewer of them now! It takes me longer to compose a message. But it’s worth it.

I will not look at my phone while watching TV. Not even during commercials. Because you can’t force yourself to stop the moment the commercial is over.

If I am in the middle of a legitimate text conversation with someone, I make sure the people in the room with me know it, so they know why I am checked out. And when it is done, I tell them explicitly that I am done and I put my phone away so that they can have my distraction-free attention.

When I sit down to do focus-heavy tasks, like work or drawing, I put my brain in do-not-disturb mode, meaning I don’t check notifications at all, and my phone becomes a music player exclusively. When I do this, I let anyone know who would normally feel like they have 24/7 access to me that I have gone dark. This way if they need to communicate with me they know they need to either do it in person or call. Because as we all know, nobody ever uses their smartphone as a telephone, so when there is an actual call I know I should check to see who it is and probably answer.

#### It Can Be Done

As they say, no need to throw the baby out with the bath water. You don’t have to quit if you can properly modify. And you can. I believe you don’t need a critical moment to be the catalyst. Just the knowledge that you can do it. And I really, really believe we need to.

I’d love to hear your thoughts and experiences on smartphone addiction, and how it can be addressed. Feel free to share in the comments section.

Rich

## Our Schooling System is Broken

It has been a while since my last blog – it was never my intention to go so long between posts but you know … sometimes life hands you other things. In any case I plan to start writing again more frequently, starting with a subject that has been on my mind for a while now: Education.

See, I am thinking our system might be broken. Scratch that – I know it’s broken, and in many ways. But I am talking about a fundamental issue, which is the assumption that performance in a school system with a standardized curriculum is a key measure of personal value. I will try to explain, starting with some background for context.

#### I Am a Teacher

It’s true. I am a teacher. A very happy one at that. I love my job. I teach high school math in Ontario, Canada, and have been doing that for about 15 years. Prior to that I worked as a software developer for about 10 years. When people ask me what I do for a living (something other adults seem to have a deep need to know upon meeting each other), the conversation always goes roughly the same way:

Other Adult: “So Rich, what do you do for a living?”
Rich: “I’m a teacher.”
OA: “Oh? Nice. What do you teach?”
(Rich’s note: There may or may not be a “joke” here by OA along the lines of “Oh yeah? You know what they say: ‘Those who can, do, and those who can’t, teach’ hahahahaha”)
Rich (being honest, even though it’s not what they meant): “Kids.”
OA: “Oh, haha. But seriously, what subject?”
Rich (with an inward eye-roll – here it comes): “I teach high school math.”
OA: “Oh god. I hate math. I remember I used to be so good at it until grade 8 when I had Ms. Heffernan. She hated me! And she was so terrible to the kids. She made me hate math. I never understood anything in math after that. I am so bad at math! The other day I tried to help my 7 year-old with her homework and I couldn’t even understand what they were doing. Do you tutor? I think I may need to hire you to help little Kelly with her math. Math is so confusing. I keep telling her she doesn’t need it anyway. I mean, I run a multi-million dollar business and I never use any of the math they tried to teach me in high school. Why don’t you guys start teaching useful stuff like understanding financial statements and investing? I had to learn all that stuff on my own. I don’t see why math is so important. I am really successful and I was never any good at it thanks to Ms. Heffernan …..”
Rich: “I apologize on behalf of all my brethren. Please carry on with your successful math-free life. Yes, I’d be happy to help Kelly, but honestly she probably doesn’t need any help. She’s 7. She will conquer.”

You get the point. But as I said, I seriously love my job. For many reasons. But perhaps the main one is that it keeps me connected to the fluidity of humanity. I keep getting older, but my students do not. They are the same age every year. And because I spend a large number of hours each day immersed in their culture, it forces me to keep my thinking current, so that I can continue to effectively communicate. In this way I feel less like some sort of flotsam floating on the river of time and more like a beaver dam of sorts, constantly filtering new water on it’s way downstream, being shored up with new materials as each generation passes through. I am like a connection between the past and the future, and the older I get, the larger the gap I am privileged to span. And in any case, young people are perpetually awesome. It brings a faith in humanity I’m not sure I could get in other ways to see firsthand the caretakers of the future.

#### School Is Not Really About Education

Um, what? Isn’t school by definition about education?

Well, yeah. Granted it’s supposed to be. But it isn’t. All you have to do is talk to almost any adult existing in Western society today about it and they will tell you (often with great relish, as though they are solving world hunger), they never use a single thing they learned in school in their day-to-day lives. Which is of course false. But mostly true. If you went to school in Canada you probably at some point had to know things like what year Champlain landed in Quebec (1608, in case you’re stumped – I just Googled it). I can say with a fair amount of confidence that whether or not you have that tidbit of info available in your memory banks is not affecting your life in any measurable way. You probably also had to know the quadratic formula at some point. I don’t have to Google that one. It’s $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Nifty, isn’t it? Sorry if I scared you.

Here. Check out these definitions of “education”. I Googled it.

Side note: Google is so cool. I’m 48. I have existed both with and without Google. I had to assimilate the use of Google as a verb into a lexicon that did not previously have it established that way. Your kids didn’t. They have to establish the word googol as a noun referring to the number $10^{100}$ into a lexicon that likely does not have it established that way.

See? The very first definition says I am wrong. School is about education, literally by definition. Oh, but check out definition number 2! Now that is a good one. Education is an enlightening experience! The thing is, in many cases, and for many people, school isn’t enlightening. In many cases, school is an exercise in conformity and alignment. It’s a system which simultaneously (and somewhat arbitrarily) defines success (um, I think I mean worth) and then provides measures for individuals to evaluate their success (yeah, I definitely mean worth).

#### Performance of Curriculum is a False Measure of Worth

Ok. So when you were a kid, around the age of 5, your parents sent you off to school. And almost immediately, you started to get report cards. They aren’t cards anymore. In many cases they aren’t even paper. But still called report cards. A figurative card summarizing your score on a number of predetermined criteria for success. And make no mistake – from that very first report card on, kids are sent the message that they must perform according to standards so that they get good report cards. I’m going to stay away from the early years, since that’s not my area of expertise, and fast forward to high school, which is. Let’s look at an example. This is based on a real person, whose name I’ve changed. Yes it’s anecdotal. It’s not meant as proof, only to demonstrate my point.

Sally is a young lady in grade 9 who has never particularly had an interest in math – at least in how it’s presented at school. She really doesn’t care about direct vs. partial variation, or the sum of the exterior angles of a polygon, or about how doubling the radius affects the volume of a sphere. But Sally goes to school, in a system which has not only determined that these things are important, they are also mandatory. So she has no choice but to engage in attempts to learn about these things, despite the fact that she is actually incapable of being interested in them. Sally is awesome, by the way. A genuinely caring human with deep empathy, intense loyalty, and a great sense of humour. Sally is also depressed, because she feels worthless. No matter how hard she tries, she can’t figure out how substituting $2r$ for $r$ into the formula $V=\frac{4}{3}\pi r^3$ causes the volume to increase by a factor of 8. In a test designed to determine if Sally can do these things, her performance was dismal. Her mark on that test was a 58%. The class average was 85%. When the teacher returned the test, she made comments like “Well, I know that I taught about exterior angles, but it is clear that some of you did not learn it.” When Sally brought the test home to show her parents (who, incidentally, knew that Sally had the test coming up, hired a tutor to help her prepare, and then asked if she had gotten it back each day after school starting the day after it had been written until the day a week later when it was returned), her parents were frustrated and disappointed. Sally has been conditioned to believe that her ability to understand direct and partial variation is critical. Because it is mandatory, her inability to care or comprehend is forcibly highlighted. And subsequently, her performance is recorded for posterity on her report card. Sally’s final grade in math ends up being 61%. Her parents are disappointed. Sally is devastated. She believes there is something fundamentally wrong with her, because she legitimately can not measure up to the standards set in a system she has no choice but to be engaged in.

#### Breaking Down the Process

Let’s have a look at this situation of Sally’s, and the number of places where the system went wrong.

First off, the concepts Sally has to learn have honestly been arbitrarily determined. See, someone, somewhere, decided that Calculus is an important prerequisite subject for many university and college programs. And to learn Calculus (a grade 12 subject in Ontario), you arguably have to start with concepts like direct and partial variation in earlier grades. And because grade 9 is truly too early to know if programs that require Calculus are in your future, this pre-calculus material is incorporated into the curriculum for pretty much everybody.

Second, math is mandatory. In Ontario high school is a 4-year program, and you must have a minimum of 3 math credits to earn a high school diploma. All of this has nothing – and everything – to do with Sally. Next comes the process. Sally’s teacher created quite possibly an amazing series of lessons on these topics. Sally just doesn’t have the wiring for them, so even though the teacher may be phenomenal, it will have a marginal impact on Sally’s ability to synthesize the material. Sally used to ask questions in class. Now she doesn’t, because she learned that even when she asked, it didn’t help. Sally is deeply empathic – she has seen firsthand the good intentions and effort of her teachers in the past. She can read facial expressions and body language. She has seen her teachers answer her questions and try to help her and seen how much they believe the answers and help are working, so she has pretended that it was, since that was easier than admitting that it wasn’t, because it meant burdening herself with her inadequacy instead of her teachers.

Third, Sally has been conditioned to believe that it is all being done so that she can get high grades. Because ultimately her success is defined by her report card. And because pretty much everyone is telling her that you don’t need this stuff in life anyway. You just need to learn it to get high grades so you can be successful. Which means the only purpose to learning this stuff is the test you will eventually write on it. This is exacerbated by the well-intentioned parents who put so much emphasis on the test – both before and after. You’ve heard the question “When am I ever going to use this?” Well the answer for most kids is “Next week, when you are tested on it.”

See? It’s all artificial. Sally is a high-worth kid, forced into a situation she isn’t wired for, and told that her worth is defined by her performance in that situation. It’s incredibly sad to watch.

#### The Present and the Future

So here’s where we are now. Many kids and parents these day believe that high marks are critical. The perception is that material presented in school is not inherently valuable, but instead the value is that it is a vehicle to high marks. This means that kids will often do anything it takes to get the high marks. This includes cheating, but that’s not really what I am driving at. What I mean is that they develop strategies that focus on getting high marks, as opposed to learning. Cramming for tests, paying for courses in small private schools that guarantee (implicitly or explicitly) very high grades, or negotiating with teachers (even to the extent of aggressively bullying) after the fact are all standard operating procedure. What is so terrible is that Sally may be able to get high grades using any of these tactics. But Sally will also always know that she didn’t earn them. She will always know that she is not good at something that she should be good at if she is to be valuable. And if and when she manages to gets the high grades anyway, that sense of fraudulence will haunt her. I’ve seen it. It’s tragic. Sally has a great future if she can discover her true worth. The worth she was born with and the worth that her friends probably value more than anything she can do in a math class. But she may or may not discover it.

#### What I Do About It

I love people. And that includes the kids I teach. And I teach high level math, which is honestly not for everyone. I get a lot of kids coming through my classroom doors who are not there because they want to be, and who will not be able to draw the joy from studying math that I really, really do. I have constraints. I have to teach the material as it’s laid out by governmental process. I have to assign grades that reflect students’ performance against some pretty specific criteria. But within this I make sure that each and every kid I teach knows that I value them as a person. That I care about their story. That I see them not as a 2-dimensional projection of my consciousness, but as a multi-dimensional consciousness of their own, with a narrative as rich and intricate as mine. I make sure that they understand that any evaluation I do of their abilities in math is a tiny, tiny cog in the complex machinery of their existence, and has zero impact on my impressions of them as people, or on my estimation of their worth. I show them that it is totally acceptable to love and be passionate about math, without tying that love and passion to an evaluation. It’s not about the math. It’s about giving them permission to take joy out of abstractions, and to pursue the things that they were wired to do. I’m not always successful. Some kids are too preconditioned. But I will never stop trying.