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.

Follow your exam prep schedule

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 and 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.

Thanks for reading,

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.

Thanks for reading!

Rich

Open-Mindedness

Recently I have been thinking a lot about why so many people seem inconvincible of certain things which I hold to be true. And while I could certainly make a list of some of these things, that is not the intention of this blog entry. Instead, I have been reflecting on open-mindedness and wanted to share.

Many people – myself included – often enter into discourse with someone of a differing opinion with the intention of convincing them to change their mind. For example, maybe your friend Paul thinks all trees in your neighborhood that are taller than 12 feet should be pruned to 12 feet or less, so as not to obstruct anyone’s view of the lakefront. You know that he’s clearly wrong! You get into a discussion. Only it’s not really a discussion – it’s an argument each of you is trying to win. Maybe out of frustration you start incorporating personal attacks. Maybe you get so angry at Paul’s refusal to capitulate, as well as the horrible things he is saying about you, that it ends your friendship. Maybe in the middle of the night, Paul prunes all of your tall trees. Maybe the next night you erect a 30 foot statue on your lawn directly in Paul’s line of sight to the lake … and so on.

It’s sad, and you don’t even like the statue, but what choice is there? Paul must be taught a lesson.

I wish this was hyperbole. Sadly, it is not. And the conclusion is clearly suboptimal.

Well … let me construct a basis for discussion with some (hopefully) fair assumptions. In doing so I’m going to have to use a little bit of math terminology, and it occurs to me that some people might not know precisely what I mean, or even be put-off by some of my more mathematical references. If you think this might be the case, I ask you to bear with me. The concepts and symbols I use are the best way for me to illustrate my point, and I’ve included here a bit of a math lesson, in case it is not something you’ve encountered in your life – it will clarify some of the words and concepts I use for the rest of this article. Of course, if you feel there’s no need for you to read this section, by all means scroll past it and keep reading (I won’t feel bad).


Some Math concepts

Sets
Mathematicians like to talk about collections of values that are somehow related, and when they do, they use the word set. We use curly brackets to list the objects (known as elements) of a set. So for example the set F=\{apple, orange, banana, kiwi, peach, nectarine\} is a set I have named F, and just so you know, it is the set containing all the fruits I might bring to work with me in my lunch. A subset of a set S is another set that only contains elements from S. So for example M=\{apple, kiwi\} is the set of fruits I brought to work in my lunch on Monday, and is a subset of F. On the other hand, A=\{apple, pineapple, banana\} is not a subset of F.

A Little Bit of Algebra (Apologies to the Arithmophobic)
Consider this simple algebra equation:
\displaystyle 3x+4y=7
The x and y are understood to be symbolic of numbers, but the use of symbols mean that they vary – which is to say, they are variable. The equation is a statement. In this particular statement,
x = 1, y=1
would be a valid solution (i.e., the equation becomes true), since
3\times 1 + 4\times 1=7.
So would
x = 5, y=-2,
since
3\times 5 + 4\times (-2)=7.
However
x = 5, y=2
would not be a solution (i.e., the equation becomes false), since
3\times 5 + 4\times 2=23,
which is not 7.

Statements
In math and philosophy, a statement is a sentence that must either be true or false (but not both, and not maybe). Very often the truth value (i.e., “true” or “false”) of the statement depends on values for variables contained in the statement. The algebra equation above is a statement. Another example is the statement “I like cheese”, which contains two variables: “I”, and “cheese”. If the “I” refers to “Rich Dlin” (i.e., it is me speaking and not you), and the “cheese” refers to “Havarti”, then the statement is true. If the “I” is “Rich Dlin”, and the “cheese” is “Cambozola”, the statement (I promise you) is false. Notice that if the “cheese” were to refer to “gingerbread cookie” the statement would be nonsense, since “gingerbread cookie” is not a cheese – even though it is true that I like gingerbread cookies, it is irrelevant in the context of this statement. A mathematician would say “gingerbread cookie” is not an element of the set of all cheeses. Going back to the algebra example, {(1,1),(3,-2)} is a subset of the set of solutions to the equation given. The actual set has an infinite number of solutions in it, but that’s more than I need to talk about here. What I will say is that the truth value of the statement “Three times John’s favorite number plus four times Gail’s favorite number will yield seven” is:

True if (“John’s favorite number“, “Gail’s favorite number“) belongs to the set of solutions of 3x + 4y = 7,

False if (“John’s favorite number“, “Gail’s favorite number“) does not belong to the set of solutions of 3x + 4y = 7, and

Nonsense if, for example, John claims his favorite number is “cinnamon“. Be on the lookout for nonsense – it is surprisingly common.



The Assumptions

Ok. Welcome back. Here are the assumptions I was talking about:

All questions have a right answer
… when the answer is justified properly with a well framed statement.
The truth value of the statement may be subject to variables that change which answer is correct, but with a fixed set of values for the variables, there is a right answer. For example, the question “Should all trees taller than 12 feet in our neighborhood be pruned?” could be answered “Yes”, justified with the statement “It is unacceptable for some trees in our neighborhood to block sight lines to the lakefront”. Note that here the answer to the question is “yes” if the statement is true, and “no” if the statement is false, and may reasonably depend on whether or not the tree is also so wide, or part of a grove, as to make it impossible for a resident to see the lakefront from a different angle standing on the same property. It may also depend on whether 12 feet is a reasonable height with respect to whether or not sight lines get blocked. In this case these variables need to be introduced into the statement, or else agreed upon as not being variable.

The right answer may well not be knowable
 … even with the variable values fixed – which doesn’t mean there is no right answer!
As an example, consider the question “How many humans are alive on Earth right now?”

  • The number changes many times in a short span of time. So the truth value of the answer depends on what time it is indexed to.
  • The answer is subject to a definition of “alive”, and the answers to whether or not some organisms are living humans are in dispute.
  • “On” Earth needs to be defined. If I am in an airplane, am I on Earth? What if I am in low orbit?
  • However there is an answer, if we fix the variables.
  • There is currently no way, even with the variables fixed, to know the answer.

Knowing the truth is inherently valuable.
This is a big one. Many people demonstrate by their behavior that they do not adhere to this assumption. A simple example is the person who refuses to go to the doctor about a problem because they are afraid of what they might find out. In some ways, not wanting to know the truth is a human quality, especially in situations where a false belief has spawned an entire tree of values and beliefs we have been living by. If the root belief is false, what happens to the tree?

When it Comes to Truth, What We Want Doesn’t Matter
So with these assumptions, my position is that for any belief I hold, I am either right or wrong, and that I may not be able to tell. So then what am I to make of someone who disagrees? Can I immediately conclude that they are wrong? Clearly not. However I freely admit I want them to be wrong, so that I don’t have to be. After all, being wrong has some negative implications. On a fairly benign end it means I have been somehow deluded, which injures my pride. On an extreme end it may mean I have to discard an entire tree of conclusions that were premised on my error, leaving behind a buzzing hive of uncomfortable questions and observations about my previous behavior. But if the root belief is actually wrong, what choice do I really have? Since it is rooted in falsehood, the whole tree is an illusion anyway.

Here is a hard truth: What we want has nothing to do with what is true. I want there to be peace in the Middle East. But there is not peace in the Middle East, and no amount of wishing on my part, no matter how fervent, can alter the truth value of this or any other statement. On the other hand, what is true can and should definitely impact what I want. What we all want.

Ok. Here is another statement that is tautologically true: In the set of things I hold to be true, some might be false. And from a probability perspective, I am also comfortable saying that in the set of things I hold to be true, some are true, and some are false. I want to say “most are true and some are false”, but I am honestly not sure I have a reasonable argument to claim that, so we’ll leave it there as a desire more than a fact.

Shades of Gray
Now I will focus on statements where the truth depends on fixing values for the variables in the statement., which to me is the core of the shades of gray argument: In cases where there is a continuum of possibilities between true and false, almost everything in the set of things I hold to be true lies somewhere within the boundaries of the continuum, rather than on one of the ends.

Here a philosopher or mathematician might (and should!) argue that there can be no continuum between true and false, since those are binary options. My response is that I am talking about a sphere of reasonable answers centered on the truth, where anything outside the sphere is easily demonstrated to be false (or worse, nonsense), but things get a little more touchy inside the sphere. This is a consequence of my point about the truth of a statement depending on fixing values for variables the statement depends upon. To elaborate on this, I am going to define something called an assumption set.

Assumption Set
Suppose a statement depends on a set of variables. For example, consider the statement “Running is good for you.” The truth of this is not absolute. It depends on some variables:

  • How much running (the quantity of the running)?
  • How intense (the quality of the running)?
  • What preconditions do you have that running would exacerbate (e.g, bad knees, asthma, heart problems)?
  • Where do you plan to do your running (road, track, beach)?
  • and many more.

So before we could discuss whether the statement is true, we would have to fix values for these variables. I call these fixed values the assumption set. So for example an assumption set for this statement could be
R=\{45 minutes per day, at 80\% of maximum heart rate, \{sensitive to sunlight, plantar fasciitis\}, track\}.
Notice that one of the elements (the preconditions) in this assumption set is itself a set – that’s completely acceptable. On the whole, I would judge this assumption set to be a reasonable one – which is to say, the elements of the set have a probability associated with them that makes them not unexpected in the context of discussing the claim that “Running is good for you.”
Another assumption set could be
S=\{15 hours per day, at 120\% of maximum heart rate, \{multiple hip replacements, torn Achilles tendon\}, Interstate Highways\}.
On the whole, I would judge this assumption set to be very unreasonable – which is to say, it is highly improbable that this would be an assumption set on which the claim “Running is good for you” would be a relevant discussion.

Reasonable Answers (Approximately True?)
A reasonable answer to a question can be defined as a statement that is true when evaluated with a plausible assumption set. That is to say, the assumption set is comprised of elements that have probabilities high enough that if we observed them we would not be surprised. In situations where the variables are in constant flux, the approximate truth value of a statement may be argued as the one that holds given the most likely assumption set. In cases like this, we may generalize a statement as true, while being willing to challenge it in the face of a game-changing assumption set. We maybe won’t talk about who gets to define “plausible”, even though there are times when that becomes the most relevant thing.

Arguing(?) With an Open Mind
Here I have chosen to use the word “arguing”, even though in truth I prefer the word “discussing”. That’s because most people seem to think that discussions between people in disagreement need to be arguments. I disagree. Remember the assumption that we are not right about everything? And remember the assumption that knowing the truth is inherently valuable? These two should premise every discussion we enter into. So when discussing the answers to questions, or the truth about statements, we need to do our best to remember that what we are trying to do is get as close to the center of the sphere as possible, because that is a good thing to do, and because we may not be there yet.

Of course, we all think we are closer than an opponent. If not, we wouldn’t be having the discussion in the first place. But keeping in mind that if two people are in disagreement, one of them must be wrong, a productive conversation is one where at the end of it the parties have converged on something they both hold to be as close to true as they can see getting. When this happens, the world gets a win. I’ll list some techniques for true open-mindedness.

Discussing With an Open Mind

  1. Remember that you might be wrong.
    Put another way, be willing to change your mind, or adjust the approximate truth of what you believe.
    See, you believe that you are probably right. You may even believe that you are certainly right (although for the truly reflective, certainty is a pretty difficult thing to attain). But your opponent has the same thoughts. Both of you probably have many reasons for these. And they probably have a lot to do with assumption sets, and which one of you is applying the most plausible set. Sometimes the discussion is not about the truth of the statement but on the plausibility of the assumption set. Keep that in mind. Yours may be the less plausible. Or maybe both assumption sets are equally plausible, in which case the statement can be split into two (or more) more detailed statements that include some of the differing assumptions explicitly. But keep in mind that emotional attachment to an assumption set can and will blind you to the plausibility of an alternate set, and ultimately cause you to refute a statement with unreasonable (even fanatical) obstinacy.
  2. Have higher expectations for yourself than you do for your opponent.
    This means you need to challenge yourself to inspect the assumptions and claims of yourself and your opponent objectively, even if they are not doing the same thing. When you do this – and do it out loud – they hear that. Look at elements of the assumption sets and objectively evaluate their probability. Also evaluate whether they change the truth value of the statement or not. And be prepared to evaluate whether or not they render the statement as nonsense – this happens surprisingly often but it’s not obvious until it is isolated. Discussing things this way models a behavior that is necessary for the two of you to converge on a conclusion you both agree with. And if you are consistent with it, your opponent will often adopt the same style, if only because they think this is the way to convince you they are right.
  3. Thank your opponent, regardless of the outcome.
    I don’t mean this as a politeness. I mean this in the most sincere sense. Any opportunity we get to reflect on our set of beliefs is valuable. Sometimes your opponent and you will converge. Sometimes you will not, and they leave the exchange completely unmoved, perhaps even claiming “victory”. This is sad, since the only true victory would be a convergence of opinion, but ultimately it is not relevant to your own experience. Make it so that if you have moved on a topic, it is because you discovered something you were not considering, or were considering incorrectly, and now you are closer to the center of the sphere of truth. If you do not move, make it because you were not presented with any strong evidence that you needed to. In either case your beliefs will have been strengthened in some way, either because you changed to something as a result of new insight, or because you were challenged in some way, and it was unsuccessful. For this you have your opponent to thank.

How to Spot Real Open-Mindedness
Many people claim to be open-minded. It may be true, or it may be a trick (some people say it so that when you fail to convince them of something it will prove they were right). True open-mindedness doesn’t mean you are ready to believe anything. It means you are willing to change your mind when presented with evidence that objectively compels you to do so. If you know of (or are) someone who has changed their mind in the moment, during rational discourse, but who was fairly slow to do so, they are probably the type of person I am describing. This goes back to my point that we are probably not right about everything we believe. Which means mind-changing can occur. Which means if you’ve seen it occur, it occurred in someone with an open mind.

Thanks for reading,

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

I can hear your thoughts.

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.

Definition of Education
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.

Thanks for reading,

Rich

Thoughts From a Dance Dad

Lately it seems as though the amount of time I have to write is inversely proportional to the amount of ideas I have to write about. But today’s entry is about something I’ve been thinking about for years: Dance.

Early Years

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My Little Ballerina (age 5)

My daughter is a competitive dancer. She’s 15, and has been dancing since she was around 3. She’s gone to countless dance camps, workshops, and of course classes. So I have been a dance spectator for about 12 years. Prior to that I knew next to nothing about dance, save for the fact that I have never been any good at it.

At first, dance was pretty much entirely about how cute the kids all looked executing choreography. They got to wear these elaborate costumes and perform for friends and family at recitals. The teachers and teaching assistants are always on stage at the same time as the kids, and the kids essentially never take their eyes off them, mimicking the movements they’ve all spent months in class learning. It’s exceedingly adorable, and naturally every person who comes to watch immediately rushes to them afterwards to tell them how wonderfully they danced. In short, it’s a typical exercise in getting kids involved in an activity that provides some structure around working toward a goal, and then the kids get congratulated on essentially existing for the duration. And it’s awesome.

As the years progressed, we saw less and less boys involved. I won’t attempt to analyze that or comment on why it might be, but political minefield notwithstanding, it is true. This meant that as the dancers grew, it became – for my daughter’s group at least – a girls-only activity.

Emerging Talents
1042Starting around the age of 8 or 9, and lasting for 3-4 years, it starts to become obvious which of the girls are well suited to dance and which are not. This obviousness is not lost on the girls. Dance becomes a micro-society where “Haves” and “Have-Nots” start to identify, and the behaviours that result are what you would expect. In a way it mirrors what is happening at that age in school, but from where I sat it was definitely magnified at dance. These can be pretty difficult years for the girls, and perhaps more so for the parents. As I watched from the sidelines, I always told myself that whether a Have or a Have-Not, there are very valuable lessons to be learned from these dramas, and whether my daughter was receiving or giving grief (it certainly seemed she was receiving a lot more than giving, but nobody ever accused a dad of being impartial), my wife and I always did our best to ground her in reality and look for the long-term life lessons that could be taken. I do think, subjectivity aside, that I can safely say my daughter began to show real talent for dance during this time. I can also say, objectively this time, that she emerged from this phase with an inner-strength and confidence that is astounding. As I watch her navigate the social quagmire of the tenth grade, I am exceedingly proud and awed at how well she manages to stay true to herself and her friends, while gliding above the drama that can consume most kids of that age. She never judges others, and always stays honest in helping her friends deal with whatever the current issue is. In and out of the dance world I have watched her handle victories with honest grace and compassion, and failures with resolute determination. She’s my hero, and I firmly believe we have the “emerging talent” years of the competitive dance program to thank for that.

From Girls to Women
As the girls mature into women, things change at dance in a way that I could never have understood if I were not so immersed in it. This phase is not something I came to understand only as my daughter entered it – the nice thing about being a dance dad is that every recital and dance competition you attend features dancers from all the age groups. So that long before my daughter was in high school I have been observing this stage of a dancer’s development. I also have the added advantage of being a high school teacher, and so for my entire career have had the pleasure of seeing how dancers take the lessons from dance into seemingly unrelated arenas, like a math classroom, which really is my domain. Having a daughter in dance always made me pay attention to how older dancers behaved, kind of as a way of glimpsing my daughter’s future. Here are some observations I’ve made over the years, and observations I have now had the pleasure of seeing manifest in my own daughter.

Dance is a Language
This is not metaphor. Dance actually is a language. It took me some time to fully appreciate that. Because of my daughter’s involvement in dance, our family has been marisa-dancewatching So You Think You Can Dance since season 2. It’s a great show to be sure, but I admit at first I was too absorbed in marveling at the physicality of it to understand what it communicates, despite the fact that the judges on the show really do a great job emphasizing this (I always assumed they were saying it metaphorically). But like a child that learns to speak simply from hearing the spoken word and contextually absorbing meaning from the sound, I began to absorb meaning from the movement. The first thing I realized was that unlike languages that use words, dance doesn’t translate to any other language, and communicates things which can’t be communicated any other way, with the possible exceptions of fine art, or poetry. Really good fine art will enthrall and speak to the viewer through infinite contemplation of something static. Really good poetry succeeds at using words which individually can be quite linear, by combining them in a way to create depth and consequently say something the language the poem is written in was not necessarily designed to say. Really good dance? A different thing entirely. It speaks to our humanity on multiple levels, and the fluidity of it allows the choreographer/dancer to tell us stories no written word could approach.

Words are discrete, and a picture is static. But motion is a continuous medium, and the very continuity of it results in an infinity of expression within a finite frame of time and space. It has been said that dance is poetry in motion, but I honestly have come to see it in the reverse. Poetry is dance stood still. I can’t find words to describe this any better, because words will fail here. If you want to know what I mean, watch dancers. And in the same way that second and third languages improve thought processes and imagination, so does dance – but it does so in a way that is magnified a thousandfold because of its unique method of delivery, and because of the world of thought and emotion it opens up for communication. It also is unique in that you don’t have to be able to speak it to understand it. You only have to watch.

Dancers Make the Best Actors
Because of my passion for theatre, I have had the immense pleasure of being both actor and director in various musicals. And here is what I’ve noticed – not all great actors are dancers, but all dancers are definitely great actors. To me there is no mystery as to why this is. Many actors focus on the words they’re saying or singing, trying to pour all of the character they’re portraying into the delivery of the lines or lyrics. Physicality is often an afterthought, or a simple by-product of the emotion they are feeling about the performance. For dancers it’s entirely different. Because of their fluency in dance, they are simultaneously vocalizing and dancing the performance. By dancing I don’t mean the choreography that often accompanies musical numbers, although naturally a dancer excels there. Rather I mean that they are speaking to us in two languages simultaneously. And even those of us not able to communicate with dance can still understand it. So I have often found myself thinking of a dancer “It’s not a je ne sais quois she has. It’s a je sais qu’elle est une danceuse” (yes, you have to speak some french for that one  😉 ).

Dance is Empowering
Okay. So obviously dance results in physical fitness. You can always spot a dancer by their muscle tone, posture and grace in simple movement. The importance of this can’t be underplayed. But the kind of empowerment I’m talking about here is more than that. I flying-marisaremember once at a dance recital there was a senior acro small group number about to start (see what I did there – Dance Dad knows the terminology). It was clear from the opening positions that one of the dancers was going to execute a crazy trick to start the dance. Before the music started there were hoots and hollers from the wings and from the audience, and one dancer’s voice from the wings rang out with “You GO girl!”. I was momentarily taken aback. I think maybe I had just read an article or watched a show where that phrase was called into question as demeaning to women. And then I looked around. The stage and venue was dense with strong, confident young women, certain of themselves and certain of their power. And the dancer who called it out numbered among them. I couldn’t see anything demeaning at that point about what she had said, but on a deeper level I realized just what dance had done for these kids. It showed them what inner strength, determination and dedication could do. And so naturally I began to think about the dancers I’ve taught in my math classroom and I had this moment of revelation. It is exactly that quality that has always made them stand out to me in that setting. Not that they all excel in math, because not all kids do. But that regardless of their abilities in math, there is always an inner strength and peace that says “I know who I am, I know what I can do, and I know how to commit to improving.” Where many students in high school still need the explicit motivation that our culture seems to thrive on too often, the dancers have internalized their motivation in the best way. I can’t say enough how important that is for success in life.

unsung-marisa

Thanks for reading,

Rich

Arithmophobia (Fear of Math): My Thoughts

I’ve been a high school math teacher for 11 years now, and I’ve also been tutoring students privately for even longer than that. Consequently I’ve seen the whole spectrum of math students. Everything from the freakishly gifted to the astonishingly weak, For the most part I think this is fine. Some people are wired for certain things and some are not. I am not wired to be a sprinter. I could train my butt off for years and still not qualify for a track team. I’ve made peace with that.

What I don’t think is fine however, is the growing number of math-phobic students I am seeing. Students whose deep fear of math is so intense that it is almost impossible to determine where their strengths and weaknesses in the subject are. To understand what I mean think of a person who suffers from stage fright so severely that every time they sing in front of even a small group of people their throat closes up and all they can manage is a pathetic croak. Anyone listening would conclude this person is a terrible singer. Yet it may not be true. The neurosis camouflages the talent and it’s impossible to know what the person’s singing ability really is. What makes it worse is that it is extremely difficult to evaluate the cause of poor performance. After all, some people just can’t sing.

In math these students whose fear interferes with their performance very often conclude that they have no ability in the subject, which further feeds the phobia. A seriously vicious cycle that is difficult to break, even after it’s been recognized. So my question as of late has been, what’s causing the increase in math-phobic students?

I don’t have research to back my conclusions. It’s all purely anecdotal. However these observations have been made from the trenches. I see these students every day, in a classroom setting and one-on-one, for over 11 years. Here are my thoughts.

Poor Evaluation Criteria
More and more I am becoming convinced that this may be one of the single biggest causes of arithmophobia. I am talking about the alarming tendency for students’ grades to not reflect their ability, due to poor evaluation criteria. I’ll give you an example.

A simple linear equation to solve.
A simple linear equation to solve.

here is one student’s work, graded:

A flawed solution to the simple equation.
A flawed solution to the simple equation.

and here is another student’s work, also graded:

The equation answered correctly but not traditionally.
The equation answered correctly but not traditionally.

The first student received a mark of 2/3, which rounds to 67%. What are we to take from this? Imagine the student coming home with a report card that says 67% in math. What would the parents conclude? What would an independent observer (like a university) conclude about a grade of 67% in math? The easiest and most likely answer is that this is a student who grasps roughly 67% of the concepts covered in math. With respect to this question and the topic it tests, it means the student grasps only 67% of the concept of solving linear equations. Now based on their work, do you believe that is a true assessment? What would we have this student believe? It’s disturbing to say the least.

But significantly more disturbing is the grade of 0/3 assigned to the second student. This student answered the question correctly, however the traditional approach is to assign one mark per step in the question, and since the student did not show any of the expected work, he lost all marks. Now he has 0%. What would that say to parents and universities? Most disturbingly, what does it say to the child?

Stop and ask yourself what it means to solve an equation. The above equation, translated to English, states that

“There is a number which is multiplied by 4 and then the product is reduced by 3, for a result of 29.”

The instruction “Solve for x” means

“Tell me what the number is.”

Student 2 has successfully done just that. Period. End of discussion. Not only has he correctly answered the question, but in doing so has demonstrated that he understands the question and has the higher level thinking skills to answer it without employing any traditional algorithms. And we work in an educational system which has evolved to tell this student that he is so bad at math he gets a zero. Shame on us. Shame on us all.

So what happens to this student? Well from my experience he either dismisses the subject as “a stupid bunch of rules” (and who can blame him? When the answer is so obvious what value is there in writing down a bunch of steps that do nothing more than add tedium?), or he “learns” that to be good at math you have to suppress your instincts and replace them with the all-important STEPS. And let me tell you something. By the time you get to senior math in high school, there are a lot of steps! There’s no way most of us – myself very much included – could memorize all those steps, know precisely when to apply them, and do so with complete accuracy and precision every time.

Enter fear.

Imagine for a second you are a dog. A puppy. You mean no harm to anyone and in fact are a bouncing bundle of happiness and joy. Unfortunately you have an owner who has anger issues. You’ve discovered that your owner hates it when there is pee on the carpet in the house. The reason you know this is because every time he discovers any he loses his temper and yells. So in order to help, you begin peeing in hiding places around the house. To a dog this makes a lot of sense and is very considerate. Unfortunately all this does is make your owner even more angry, to the point where he smacks you every time he discovers the hidden pee. Result? You are now afraid of the owner, and afraid of peeing. Nothing productive comes of this because despite your best efforts, and despite the fact that you are doing what you think is right, you are still getting in trouble. That is a recipe for fear. And that is what happens to students who do what they think/know is right, but get rewarded with marks like 0/3 for their efforts. How can a person continue with a positive attitude under those kinds of circumstances?

What also happens to a large number of students is that over the years, as they fail more and more to memorize the right “rules”, they become more and more disillusioned with themselves. The mathematics becomes totally obscured by the algorithms, to the point where students believe that the algorithms are the mathematics, and can hardly be convinced otherwise.

I tutor a student named Randy and she is in grade 7. Here is a question from a test she wrote recently.

Sam has answered the question “7 – 3 ½” with “4 ½”. Sam says this is because seven minus three is four, and then there’s an extra half to make four and a half. Is Sam correct? Explain.

Here is what Randy wrote:

Sam is not correct. To answer the question you have to convert 7 and 3 ½ to improper fractions, then subtract the numerators, then convert your answer back to a mixed number. This is what Sam should have done:
7 – 3 ½ = 14/2 – 7/2
= 7/2
= 3 ½
So the correct answer is 3 ½

For this answer Randy received a “2+” which is a mark out of 4, with these comments from the teacher: “What was wrong with Sam’s thinking? How could he modify his strategy so that it would work? Expand on your answer.”

Hmmmmmmm. My thoughts as a teacher were immediately “Those comments would have made good questions for students to answer on the test instead of criticisms of Randy’s answer”. In any case let’s have a look at how this result impacted Randy.

So marks out of 4 like this one can be roughly converted to percentages, which they ultimately will be for reporting purposes. A mark of “2+” converts to around 65%-70%. I implore you, dear reader, to tell me just exactly how Randy has shown her capabilities in subtracting mixed numbers from whole numbers to be 30% less than perfect. The message to Randy?

Because you were unable to extrapolate from the word “explain” that I, your teacher, was expecting you to delve into the mind of a person who, unlike yourself, can not subtract mixed numbers from whole numbers, I conclude that you, Randy are a mediocre math student, at best. Despite the fact that the question was in two parts (“Is Sam correct?” and “Explain”), and that you addressed both correctly, you should have known that what I was really looking for was for you to help Sam understand why his thinking was wrong, despite the fact that it did not say this anywhere in the question and despite the fact that Sam is a fictitious person. Please work harder from now on so that you may become a better math student.

Randy was in tears over her results. She said she was sure she understood the material going into the test but she’s just bad at math and she hates it and she is never going to be good at it. It took quite an effort on my part to show Randy that she completely and perfectly understands subtraction of mixed numbers from whole numbers and that the real flaw is the question. I’m not sure she is totally convinced and her grade in math will certainly not reflect what I know to be true so it will be a difficult pill for her to swallow. Randy is developing a serious case of arithmophobia based on experiences like this. She is not wired to “know what the teacher means”. She reads instructions and takes them literally, and then answers them as best she can, usually correctly. But since there is more wrapped up in the evaluation criteria than is revealed in the question itself, Randy is rewarded for her efforts with marks like “2+”. To her this makes math incomprehensible, and who can blame her? To her math is now a mysterious subject with weird expectations that you have to “just know”, and what hope does she have of being able to do that?

So what can we do? The answer is as simple to state as it is difficult to implement in today’s education environment:

Let’s start teaching MATH again. And when we grade a student’s work let’s stop comparing what they did to some sort of “template of perfection” and instead evaluate what the work we see says about the student’s fundamental understanding of the mathematical concepts. Solving an equation means finding the values of the variable that make the equation true. The fact that we have algorithms for solving equations is wonderful and essential for very difficult equations, but let’s not punish students who are able to understand and solve without the algorithm! Let’s celebrate those students because they are the ones who really get it. The algorithms can be introduced and reinforced later when the equations get harder, but it serves no purpose to tell a student like that they are bad at math, for they are truly not. And for students like Randy? Let’s throw away the rubrics and fancy words and assess what their work tells us about their abilities. If we want Randy to extend her knowledge to be able to help Sam modify his strategy so that it will work let’s help her with that, but there is very little value in tying her grade in math to that ability, unless that ability is very specifically what we are trying to teach and assess, in which case we need to ask ultimately how much is that worth and how should it be reflected in the grade that she will use to determine her performance?

Arithmophobia is real and it is getting worse each year. We must change what we are doing if we want to reverse the trend.

Thanks for reading,

Rich

The Death of the Mistake

(Disclaimer: NOT ALL PARENTS are guilty of what I describe in this blog, so please don’t take it personally. But many are. Far too many.)

I’ve made a lot of mistakes in my life. I imagine you have too. Some of mine are pretty indelible for me.

Like the time my friend and I were playing with fire (literally) and almost burned down a hotel. We were scared to death and actually grateful we got caught.

Like the time I decided I could get through that intersection before the car coming from the left got there. Result? One totaled car that wasn’t even mine. It was my girlfriend’s car.

Like the time I thought I could get away without studying for my final exam in STAT 331 and still scrape a credit in the course. I earned a 42. I asked the prof to remark the exam which he did. The mark went down.

I didn’t like any of these or the countless others when they happened. Actually they felt pretty miserable. Yet each one has had an impact on decisions I made later in life and each one of those failures thus resulted in bigger successes. This is not a revolutionary concept. The phrase “we learn from our mistakes” is not new. But have you ever considered exactly how true it is? We aren’t born with an abundance of knowledge. In fact we are born with almost none. We know how to do things like breathe and cry and fill diapers, but we can’t even control those actions much. At first, all the knowledge we accrue comes from our innate curiosity and willingness to take risks. What’s funny is that as babies we don’t even know we are risking anything. We’re just really curious. Watch all the stuff a baby is willing to put in his mouth. You’ll know exactly what I mean. So babies try things and sometimes the result is pleasant or satisfying and other times its not. Each experience whether success or failure goes into the data bank and both guides subsequent experiments and imbues us with confidence regarding our ability to reproduce successes. In this way it is completely correct to categorize both “success” and “failure” as positive outcomes. And yet we have attached so much negativity to the word failure that it has become a thing to be reviled and feared … avoided at all costs. Which is a true tragedy, because success in the absence of failure is yin without yang.

As time passes the culture we grow up in imposes a formalized education, mainly because there is value placed on certain nuggets of knowledge. This happens in the form of school. And that’s where I come in.

Sort of. I teach high school so I guess I come in about 10 years later. After 10 years of a system which has sadly killed the mistake. Kids are no longer encouraged or allowed to experiment. They have to “get it right” the first time. Many parents spend insane amounts of energy making sure their children never experience failure, defeat or mistakes. A friend of mine calls this phenomenon “the snowplow parenting” model. The parents walk ahead of the child, plowing all obstacles out of the way, frantically making sure that no failure is ever experienced. As the child grows and the potential obstacles increase, parents run themselves ragged continuing to pave a smooth way. The effect this has on the kids is incredibly frightening. The child grows up not ever really experiencing a failure, but watching parents become more and more neurotic making sure this “failure monster” never has a chance to get near their kid. It’s a doubly-bad edged sword. First, no failure is ever experienced so there is no chance for the best kind of learning and second, the children pick up on this deep fear of failure and when they find themselves faced with the potential for a mistake they freeze in terror at the possibility. In short, they are simply not equipped to deal with anything but a smooth road, and lack the understanding and confidence that comes from having failed.

So what I see in my math class is kids who are petrified of assessment. I have seen kids cry when they earn a mark in the 90’s on a test. Worse, I’ve seen kids with marks in the 90’s crying when they come in to write a test because of how afraid they are that they might make a mistake. And I’ve had to defend marks to parents who insist to me not that their child earned a higher mark, but that their child needs the higher mark. In grade 9. A grade that no university or college even remotely cares about. A grade for which no scholarships are awarded. Yet the child needs the higher mark. Lest they experience failure.

This phenomenon may actually be the single biggest threat to our culture. Thanks to Snowplow Parents we are raising a generation of kids who have never had a chance to experiment and fail. Never a chance to pursue curiosity, which is the spark for innovation. So what we get is anxiety-ridden underperformers with huge self-esteem issues, fostered by parents who have made it clear that the child is not capable of fending for themselves and thus needs the parents’ involvement every step of the way.

Parents, please. Take a step back. Watch them do it themselves. Watch them fail and celebrate the failure. Mistakes are critical for evolution. Let’s bring them back. Let’s start the Mistake Revival.

Thanks for reading,

Rich