Teaching, Learning and Habits

How Habits
Education Collide


The best definition I have come across for a habit is, “action without thought.”  A quick search on the internet says that a habit is, a settled or regular tendency or practice, especially on that is hard to give up …

We certainly need habits, especially in education.  Students, in order to be successful, need to be in the habit of being on time, having their homework done properly, whatever the classroom norms and expectations are need to habitual.  In other words, the day to day activities of school should be done automatically, without thought or the student needing to be reminded.

And certainly we want students to be in habits when it comes to performance.  For example, putting their name on their paper, showing appropriate work, employing effective questioning strategies and the like all end in higher levels of academic performance.

But what about what they’re learning.  Are we teaching them habits, that is, the action without the thought?  I say that we definitely are, and that is in direct conflict with the purpose of education.  That purpose is to give people the opportunity to learn how to think in a safe environment where the messiness that comes from the process of learning to think does not have major consequences.

As with most thinks related to teaching, this is highly nuanced and subjective, and there are certainly times where teaching a kid a habit that leads to a right answer or desired outcome is best.  That’s part of what makes education so powerful is that you can learn from what others before have done and take the next step, right?

What makes this double tricky is that we grade the results of habits.  Can a student see a prompt and spit out an appropriate output?  If so, they’ve obviously learned, right?

If you’re an expert in the field you’re teaching, you most likely approach problems at the level you’re teaching habitually.  Little reference to the ideas at play is required for you to arrive at a solution.

If you’re not an expert but have enough background to teach the topic, you’ve probably brushed up with some Khan Academy videos or the like, where you were shown those efficient methods and techniques that are the ways the expert acting habitually would do.

If a student is able to pass a standardized test they must also possess these habits.  However, if they’re taught the actions without thought, the process alone, they have no way to connect what they’re doing to other things.

Let’s consider how thinking and problem solving really works.  After all, learning to think is the purpose of education, right?  It’s highly unlikely that any student will have a practical use for 90% of the materials learned in your class.  But the learning that takes place, that is entirely useful and practical!

In thinking and problem solving the issue at hand must have a level of novelty.  If not, a habitual approach will be successful and little thinking will take place.  The problem must first be grappled with and understood and then the person dealing with this task can generate some ideas.  These ideas are the conceptual understanding of the task at hand.  From these ideas come the actions, the steps taken.  Upon completion review of the entire undertaking is performed and if the outcome was desirable, success can be claimed.

Often it is the case that not only is success claimed, but all similar problems now have a heuristic background.  Upon further review and generalization and actual procedure can be articulated.

Since the procedure is the measured and share-able portion of this entire development, that is what is written in books and what is measured on tests.

Yet, it all came from a conceptual understanding, an idea.  The idea initiates the procedure.

To not allow students access to the time and level of involvement required to explore ideas and develop heuristic approaches to problems is to rob them of the very purpose of education.  They do not learn how to learn when they are trained to follow steps given a particular input.  That’s training.  Sit Ubu, sit. Good dog!

It is certainly a challenge and uncomfortable for all parties involved to have students develop this level of understanding and explore without explicit direction.  However, it is the absence of such things that has education in the United States in such a terrible predicament.

My challenge to you, the reader, is to pick an overarching, big idea in your topic, something that is coming up next, and develop an activity/problem that will require a lot of thinking and little direction from you.  Make it something where the student result can be assessed as correct or incorrect based on the concepts at play, or by reverting back to the original question itself.

What you’ll find is that the students uncover connections that you have forgotten or taken for granted, or maybe never realized at all.  Over time, with regular activities/lessons like this they will begin to adjust to what is expected of them and they’ll increasingly enjoy actual learning!

Let me know what you think by leaving me a comment.


Thank you once again for reading.

Philip Brown

Teaching Conceptual Understanding Flow Chart for Educators

Focus on Conceptual Understanding
Flow Chart

Teaching by concept alone will lead to inefficiencies in students.  They will, in effect, be reinventing a large part of the wheel at every turn.  (See what I did there?)  We have all witness what focus on procedure alone does.  It leaves students will a bunch of isolated skills that they do not recognize out of context.  Out of context here literally means changing the font or using a different set of variables.

An example is the topic/skill of finding the lowest common multiple of greatest common factor.  Students are well versed in many procedures, yet of course, mix the two up.  That is, they’ll claim a GCF (greatest common factor) is a LCM (lowest common multiple).  This is NOT their fault.  They don’t understand the difference between a multiple and a factor.  They don’t see how those two are applied in other mathematical calculations, even though in order to perform the majority of operations with fractions, those are required.

The focus in education has shifted, and like large bodies do, they swing too far.  More than likely the focus has been too great on concept and avoidance of procedure and rote memory of basic math facts.  That’s a discussion for another time.

I’d like to help you, the teacher, strike a good balance.  Unlike big publishers or professional development companies, I am in the classroom, trying these methods with all of my topics and a wide variety of students.  It is highly successful.

One key component of the success is removing yourself from the role of, “The Human Wikipedia,” in the room.  Think of yourself more as a coach than a teacher.  The knowledge you possess cannot be possessed by the students simply by you telling or explaining what you know to them.  They must experience it themselves and grapple with the misconceptions to make sense of things.  You’re a facilitator of discussions and explorations, and quite importantly, you’re a guide.  No need to chase too many rabbit holes.  When a level of understanding is achieved it is up to you to help bring closure, probably through a discussion and writing activity where students write down their explanations of what they’ve learned.  Then, that’s when homework changes from uncovering misconceptions to solidifying understanding and making efficient processes that are repeatable.

I’ve harped on many of those things in the past.  If you have questions about any of those ways in which homework is used to help learning, please feel free to leave a comment or send an email.

With all of that said, let’s get into it.  The chart at below is a general idea of how concept can be established and explored, how procedures can be introduced as a way of generalizing patterns and features of the concept, and last, how that concept can be used to introduce a connecting concept, or consequence of that concept.

Here’s the idea.  The rectangular shapes are lessons, or whole group discussions.  Everything with an arrow is student work where your job is to encourage and direct.  Typically, it is a bad idea to explain things during this time.  Instead, encourage students to find other students in the room that they trust that might be able to explain what it is that’s confusing them.

Another big idea during this time is to encourage students to articulate what it is that is confusing them.  When students say, “I don’t get it,” they’re helpless.  They’re not even thinking about what is causing trouble.  By forcing them to reflect on what’s causing the trouble, they’ll likely find their way through the confusion.  For you to step in and let them off the hook will only make them have to face that point of confusion later, and it will be bigger and the nature of the confusion will be less clear to them.

A great topic to use an example of this works is exponents.  All of the “rules” of exponents come from the idea that exponents are repeated multiplication, of the same number.  The difficulty in exponents comes from students inability to read the notation properly, especially when groups are involved.

Let’s briefly explore how this chart can help guide your planning with something like exponents.
Concept:  Introduce the notation, perhaps tying it in to how multiplication is written to describe repeated addition of the same number.

3 × 5 = 3 + 3 + 3 + 3 + 3

35 = 3 × 3 × 3 × 3 × 3

Some conceptual questions would be things like providing three different expressions written with exponents and having the students pick the two that are the same.  Another way to do this is to give the students an expression and then give them a choice of five other expressions, often which may contain more than one equivalent expression, and have the students pick which match.

During such matching activities keep in mind that the students having the right answer is not necessarily a reflection of understanding.  Without the proper explanation, accurate and concise, they likely do not know.  Their results of being right will not be repeatable.

Also, when exploring things like this, tell the students that they should write down the examples, but students that will learn will focus most of their notes on their thoughts and questions.  This is especially true since we are NOT discussing procedure.

(If you’d like to see some examples of these types of conceptual questions you can find them in the PowerPoint attached here.)

During the questioning of concepts you should chase misconceptions and show how they do not match up with what is true.  Always focus on the fact that it is through mistakes that students are learning.  Thank students, praise them for participating even when they’re not sure they’re right.  We all hate being wrong, and students are often insecure and fear being judged harshly for being wrong.

After exploring the misconceptions and then finding patterns and developing some procedure it is a good time for them to practice what they’ve learned, AKA, homework.

When reviewing the homework the next day make sure things are determined right or wrong by referring to the concept, not finding mistakes in procedure.  Of course some refinement of procedure is appropriate when reviewing homework, but that should be for the sake of efficiency, not understanding!  This is likely a huge shift for teacher and student!

An in-class, open note pop-quiz is a good follow up, depending on the ability of the students and complexity of the topic.  If I were to do such an activity, I would make sure the grades are not too punitive, providing credit to those that correct errors, or perhaps grade it like homework, on completion, not correctness.

If that in-class pop-quiz doesn’t work, a subsequent, more complicated homework assignment is in order.  This next assignment should change the language of what’s being learned.  Rephrase instructions or change some of the look of the problems so that students are not finding false clues by recognizing patterns in the problems themselves that have more to do with you, or the author of the work, than the concept at hand.

It is also a good idea to throw a few problems that tie into the next topic in, stretch problems, you could call them.  Use reviewing these problems to introduce the next concept.  I often do this without telling the students the new lesson has begun.  It works well because students should be taking notes on their homework assignment in pen (not erasing mistakes but instead annotating them).

Two observations about these practices.

  1. Student involvement is key.  Of course, students don’t learn if they’re not involved, but their involvement is less needed for a tradition, stand up and lecture while students take notes, type of classroom setting.  These methods are truly student focused and student driven.

    As the teacher you must anticipate the questions and points of confusion.  Do not have answers at the ready, but perhaps simple problems that students can explore so they can discover clarity. Be ready to show a consequence of their misconceptions or perhaps a problem that simplifies their misconception so they can see it.

  2. Textbooks are woefully inadequate as a resource here. You need many books and resources in order to provide students with exposure to concepts, conceptual problems, and different levels of practice problems (the last practice problems can often come from books).  The last set of problems, the stretch problems that connect what they’ve learned with what is coming next I have never seen in a textbook.

    You’re going to have to be creative.  I am trying to publish my materials and questions as I go through this year, but even so, they relate closely to my interpretation and view of the topic, the heuristic framework I developed.  Yours is likely different and so the ways in which you can stretch understanding or expose misconception will vary slightly.

I hope this has been helpful.  It is something I hope to explore more fully and deeply.  Whenever I have been able to employ these methods the results have been powerful. Students learn and they retain their learning.  I’ve been refining these methods over the past six years or so and my students have realized great success from it.

I thank you again for reading and hope this helps.  Please let me know what questions you have, just leave a comment.

Philip Brown


Why Remediation Fails

Why Remediation Fails

Students that struggle unwittingly do two things that ensure they continue to struggle with concepts and procedures.  Students can go to tutoring over and again, and sometimes it works, but it’s a long and frustrating journey.

I’ve fallen victim to these two habits myself, we all have.  How students learn in school is not any different than how adults learn outside of school.  Learning is identifying something that’s wrong and replacing it with something that is right, or at least more efficient.

It is the act of identifying something that is wrong that is the hitch here, the hold up.  The first of the things students do when presented with remediation, that is review materials or a review of what went wrong before, is they morph what they’re seeing to fit what they know.  Of course if they did that the other direction, things would be great.  But that’s not how we learn.

It is imperative to recognize that we develop new learning by relating it to old knowledge.  We don’t just replace all that we’ve developed over time with this new thing.  Instead, we create connections between what’s already in our noggins and what is new.  The more connections we have, the stronger the new learning is and the more quickly it happens.

Consider someone learning to cook.  Say, they learned that Worcestershire sauce is yummy and delicious on steak.  Some spills over into potatoes and that’s not too bad either.  It’s not even unpleasant when it mixes with green beans or broccoli. With some experimentation we can learn that it’s good with chicken, rice and mushrooms.

What’s the thing we know?  Worcestershire sauce makes things taste good.  Not wrong, but not a very deep understanding, right?

Now let’s say this person want to make some desserts.  Someone hands them some cream and tells them to whip it up, so it can top a pie.  Why, they might ask.  Well, to make the pie better, of course.

This whipped cream is new information, it’s something different than what they know.  It’s fundamentally different than Worcestershire sauce.  Yet, whipped cream is supposed to make food better, just like Worcestershire sauce does.  So what students do, in effect, is say, oh, whipped cream is the same as Worcestershire sauce, and I’m used to Worcestershire, so let’s just use that instead.  Same thing after all, right?

A similar thing happens when trying to train someone to use the computer.  They know how to do a set of things and try to use those processes to manipulate this new software.

That is, instead of seeing the new protocol for interfacing with the software as completely new, they instead relate it to what they had done in the past.  They fail to replace old knowledge with new.  Instead, they see the new information as the same thing as what they already have at hand.

How do we, as teachers, combat that phenomenon?  Well, we have to expose what they believe as fundamentally different than what’s right.  We have to expose their misconception as being, well, a misconception that is not aligned with reality.

That’s a tricky thing to do, especially in math, for two reasons.  The first reason is that often in math we are dealing with abstractions.  We can’t have them taste Worcestershire topped cherry pie.  The second reason, especially for math, is that when students see a procedure performed, they feel they understand if they believe they’re able to follow that procedure. (That is not that they are able to perform the procedure themselves.)

That second reason that it is tricky to expose misconception is the second thing that students do, they latch onto procedure.  It makes them feel grounded, even if they are obviously off-base!

How many times has this happened?  You, as the teacher, review a quiz question with students.  They sit there, take notes as you work through a problem.  They all exclaim they can’t believe how dumb they are, how could they have missed that?  They get it now, right?

No.  They don’t.  They followed what you did, you doing all of the thinking along the way.  A large percentage of students will be no better off than before the review.  In some ways, some will be worse because they’ll now think they understand.  Before the review, they just knew they were wrong, probably had no idea why.

What can we do?

This is a tricky thing to answer, dependent on too many variables to articulate a clean protocol.  However, I think I have some ideas that will help in general.

First, when developing a review lesson, test or quiz review, or remediation lesson, you need to have students confront some mistakes.  Maybe they need to try a problem and get it wrong.

Once the misconception is exposed, address why it’s wrong, what’s wrong with it.  Don’t discuss what is right immediately, they’ll translate that to fit what they believe (and that is wrong).  Expose why the misconception is in fact wrong, on a fundamental level.

Next, if possible, arrive at the right conclusion without process or procedure.  Is there a way to think through the conception at play and arrive at what is right?  If so, that’s beautiful.

The last thing is that this new learning will be soft in their heads, a fragile thing.  They need to make a record of what they’ve learned, in their own writing, preferably on the old quiz or next to the thing they used to believe was true.  It’ll be a reminder, because they’ll go for that Worcestershire sauce again when they shouldn’t!  Old habits, they die hard!

I tried something along these lines in a video I prepared for a remedial math class at a community college.  The topic is fractions.  I tried to show how common denominators work without treating them like they were stupid, because they’re not, they just never had to learn fractions, and tried to do so without use of a process.

As I explored the inner workings, and why various things were wrong, I began describing what needed to be done, but the focus was conceptual.  The video is posted here at the end of this article.

This is a topic I hope to explore more in detail, how to help promote the efficacy of remediation and tutoring.  I am working on some experiments I’d like to try to determine more closely the behind the scenes workings here.  Until that time, thank you for reading, thank you for your time.


Philip Brown


Try to Solve This Problem … without Algebra

Can you solve the following, without doing any Algebraic manipulation?  Just by reading and thinking about what it says, can you figure out what x is?  (The numbers a, x, andare not zero.)

Given:  3ak

And:  ax = 4k

What must x be?

If you’re versed much at all in basic Algebra you will be tempted to substitute and solve.  After all, this is a system of equations.  But that will bypass the purpose and benefit of the exercise.

The intended benefit of this problem is that it promotes mathematical literacy, in particular, seeing relationships between terms.  It’s not a complicated relationship but it is of utmost importance to this problem.  Once you read and make sense of what the mathematical relationships are you can talk your way through the problem.

Once again, I believe the purpose of homework is learning.  Sure, sometimes it is practice and familiarity, but those are the only times that answer-getting is important.  Without understanding, having the right answer is often of little to no use.  If it were, then copying the answers from the back of the book would be sufficient for learning, right?

If you’re ready to see the solution, you can watch the video or read the text after the video.


I understand that sometimes it’s appropriate to read, but not listen or watch a video.  So here’s how this works.

Given:  3ak

And:  ax = 4k

The first statement says that the number k is three times bigger than the number a.  We don’t know what or k are but we know how they’re related and can think of lots of numbers that fit this relationship.  One number that’s three times as large as the other.

The number k is three times as big as the number a.

Think of this relationship one more way, for a moment.  The number k has two factors, 3 and a.  Whether is composite or prime is irrelevant really, it won’t change the fact that we could write k as the product of two numbers.  I mention this, not because it helps solve this problem but because it might.  Without knowing the path, sometimes it is a good idea to brain storm for a  moment and list as many things you know about the information given, before seeking an answer.  Sometimes, doing so, makes the answer apparent to you!

Let’s look at the second statement now.

Another number times a is four times as big as k.  This is perhaps a bit distracting, but the key information is there.  Remember, k is three times as big as a.  Now we have something four times larger than k.

Let’s look at this a different way.  The number 4k is not k at all, but instead, k and 4 are factors of new number.

If this new number is four times larger than k, and k is three times larger than a, how much larger is this new number than a?

You have three times as much money as me.  Bobert has four times as much as you do.  How much more money does Bob have than me?

For every dollar I have you have three.  For every dollar you have, Bobert has four.

Still don’t see it?  I know…picture good, word bad.  Here you go.

You have three times as much as I do. For every one dollar I have, you have three.
For every dollar you have, Bobert has four.

If 3ak, and ax = 4k, then is 12 because

Vestiges of the Past Making Math Confusing

Something in Math HAS to Change

Convention is a beautiful thing.  It allows us to use symbols to convey little things like direction or a sound.  We can piece those things together to make larger things, and eventually use it to create something like what you’re reading now.  There are no inherent meanings to these shapes we call letters, or the sounds we use when speaking.  It all works because we agree, somehow, upon what they mean.  Of course, over generations and cultures, and between even different languages, some things get crossed up in translation, but it’s still pretty powerful.

The structure of writing, punctuation, and the Oxford comma, they all work because we agree.  We can look back and try to see the history of how the conventions have changed and sometimes find interesting connections.  Sometimes, there are artifacts from our past that just don’t really make sense anymore.  Either the language has evolved passed the usefulness, or the language adopted other conventions that conflict.

One example of this is the difference between its and it’s.  An apostrophe can be used in a conjunction and can also be used to show ownership.  Pretty simple rule to keep straight with its and it’s, but whose and who’s.  Why is it whose, with an e at the end?

According to my friendly neighborhood English teacher there was a great vowel shift, which can be read about here, where basically, people in around the 15th century wanted to sound fancy and wanted their words to look fancy when written.  So the letters e and b were added to words like whose and thumb.

Maybe we should take this one step further, and use thumbe.  Sounds good, right?

But then, there’s the old rule, i before e except after c, except in words like neighbor and weight, and in the month of May, or on a Tuesday.  Weird, er, wierd, right?

All said, not a big deal because those tricks of language will not cause a student to be illiterate.  A student can mix those things up and still have access to symbolism and writing and higher level understanding of language.

There are some conventions in math that work this way, too.  There are things that simply are a hold-over of how things were done a long time ago.  The convention carries with it a history, that’s what makes it powerful.  But sometimes the convention needs to change because it no longer is useful at helping making clear the intentions of the author.

One of the issues with changing this convention is that the people who would be able to make such changes are so well versed in the topic, they don’t see it as an issue.  Or, maybe they do, but they believe that since they got it right, figured it out, so could anybody else.

There is one particular thing in math that stands out as particularly problematic.  The radical symbol, it must go!  There’s a much more elegant method of writing that is intuitive and makes sense because it ties into other, already established ways of writing mathematics.

But, before I get into that exactly, let me say there’s an ancillary issue at hand. It starts somewhere in 3rd or 4th grade here in the US and causes problems that are manifested all the way through Calculus.  Yup, it’s multiplication.

Let me take just a moment to reframe multiplication by whole numbers and then by fractions for you so that the connection between those things and rational exponents will be more clear.

Consider first, 3 × 5, which is of course 15.  But this means we start with a group that has three and add it to itself five times.

Much like exponents are repeated multiplication, multiplication is repeated addition.  A key idea here is that with both we are using the same number over and again, the number written first.  The second number describes how many times we are using that first number.

Now of course 3 × 5 is the same as 5 × 3, but that doesn’t change the meaning of the grouping as I described.

3 + 3 + 3 + 3 + 3 = 3 × 5

Now let’s consider how this works with a fraction.

15 × ⅕.  The denominator describes how many times a number has been added to itself to arrive at fifteen.  We know that’s three.  So 15 × ⅕ = 3.

3 + 3 + 3 + 3 + 3 = 15

Three is added to itself five times to arrive at fifteen.

Let’s consider 15 × ⅖, where the five in the denominator is saying we are looking for a number that’s been repeatedly added to get to 15, but exactly added to itself 5 times.

In other words, what number can you add to itself to arrive at 15 in five equal steps?  That’s ⅕.

The two in the numerator is asking, how far are you after the 2nd step?

3 + 3 + 3 + 3 + 3 = 15

The second step is six.

Another way to see this is shown below:

3 →6→9→12→15

Step 1: 3 → Step 2: 6 → Step 3: 9 → Step 4: 12 → Step 5: 15

Thinking of it this way we can easily see that 15 × ⅘ is 12 and 15 × 5/5 is 15.  All of this holds true and consistent with the other ways we thinking about fractions.

So we see how multiplication is repeated addition of the same number and how fractions ask questions about the number of repeats taken to arrive at an end result.

Exponents are very similar, except instead of repeated addition they are repeated multiplication.

Multiplication:  3 × 5 = 3 + 3 + 3 + 3 + 3

Exponents:  3⁵ = 3 × 3 × 3 × 3 × 3

Do you see how the trailing numbers describe how many of the previous number there exists, but the way the trailing number is written, as normal text or a superscript (tiny little number up above), informs the reader of the operation?

Pretty cool, eh?

Now, let’s see some fractional exponents.  They mean the same thing with one change...instead of asking about repeated addition they’re asking about repeated multiplication.

Just FYI, 3 times itself 5 times is 243.

15 × ⅕ = 3, because 3 + 3 + 3 + 3 + 3 = 15.  That is, three plus itself five times is fifteen.

2431/5= 3 because 3 × 3 × 3 × 3 × 3 = 243.  That is, three times itself five times is two hundred and forty three.

You might be thinking, big deal... but watch how much simpler this way of thinking about rational exponents is with something like an exponent of ⅗.  Let’s look at this like steps:

3 × 3 × 3 × 3 × 3 = 243


Step one is three, step two is nine, step three is twenty-seven, the fourth step is eighty one, and the fifth step is 243.  So, 2433/5is asking, looking at the denominator first, what number multiplied by itself five times is 243, and the numerator says, what’s the third step?  Twenty-seven, do you see?

Connecting the notation this way makes it simple and easy to read.  The only tricky parts would be the multiplication facts.

Confuse Them So They Learn

I recently did a lesson on the basics of reading and writing in Geometry.  You know, dry, dull stuff...what's a point, line, ray, segment, how do you write an angle, what types of angles are there, and so on.

While preparing all of this information I was thinking:

How can I expose misconceptions about such material so they learn it?

Remember, just seeing the facts is comfortable for students, but not only do they not learn, they somehow find confirmation that their held misconceptions are in fact correct.  It's not as wild as you think, and it's not limited to kids.  I took a psychology class in college and was unknowingly part of an experiment.  I was asked a question, a seemingly throw-away type.  But it's trickier than it looks and nearly everybody answers wrong.  But it was of such little consequence that I did not remember my answer (you weren't supposed to).  Then, I was shown the correct answer and asked if that's what I had said.

Turns out the vast majority of people mis-remember that they answered correctly.  That is, they answered it wrong, but it's hard for us to imagine we're wrong, and they latch on the to the idea they were right...even when it's quite obvious they weren't.

This is so powerful that to be wrong and be aware of it, being confronted with things we don't understand and such, is very uncomfortable and unpleasant.  Yet, that's what is needed for learning to occur.  (And I'm talking the type of knowledge where understanding is paramount to success.)

My assertions are that what Derek Muller has unconverted here goes beyond science and film.

Students are not void of knowledge in your content.  They have ideas.  Teaching them is more like part repair work on the frame of a house before roofing.  Presenting students with correct information will not shore up their misunderstandings.

Also, students need to experience some level of cognitive discord.  In education, nearly all of the "best practices" work hard to do the opposite of this.  There are things like Content/Language Objectives, or SWBAT, word walls and graphic organizers.  I'm not saying those things don't have their place, but that's it, they have a place when balanced with quality instruction that explores misconceptions and such.

When you can deliver a lesson that explores the misconceptions the students will be confused.  But if it is student lead, they won't be lost.  The amount of mental effort required is much higher than a typical delivery of information and note-taking style.  However, they'll learn!

So, how to create this tension and expose misconception over some pretty dull information?

Start by asking questions and exploring answers.  Do not use your authority in the subject to state if an answer is right or wrong, initially.  Instead, have students share their thinking on what other students are saying.

For example, a particularly nasty question that dealt with the boring definition-based lesson I just gave was, "What is an angle?"  To someone versed in geometry, this isn't a big deal.  But to a kid who hasn't taken geometry, this is monumentally difficult to describe.   The best response I received was, "Measuring the space between two lines."  So, of course, I drew to parallel lines and asked for explanation.


Now, this is just something I wonder, but is it possible that on these boring, just the facts, type lessons that exposing misconception is more important than ever?

Regardless of how that fleshes out, challenge yourself to challenge the thinking of students by exposing misconception through dialogue.  Be brave enough to explore misconception and encourage students to seek understanding by challenging the think of themselves and others.  If students understand the purpose of your methods, they'll play along.

Give it a shot, let me know how it goes.

Once again, thank you for your time.

How to Be a More Effective Teacher

How to Teach Well

Why do students struggle so much?  Let’s break it down and see how perplexing this really is.  If you’re teaching High School or higher you’re presumably an expert in your content area.  You know what you’re teaching upside down, inside out, front, back, and so on.  Not only that, if you’re an experienced teacher, you know how to disseminate that information in clear, concise and easy to follow.  You also know exactly what the hang-ups will be for students and how to remediate in response.

As an expert teacher you can lay out the path to understanding clear for all to see.  And yet, they struggle.

You might think, well, the students are probably at their threshold, their potential is being pushed here.  Maybe they lack background knowledge, they forgot the prerequisite knowledge required for this new learning to occur.

Well, let’s step back a little here.  How do we know if they learned it anyway?  I mean, yeah, they passed the previous class with another teacher, maybe it’s the teacher’s fault.  Surely, that doesn’t happen with your students, when you teach them, right?  You know when they know it, don’t you?

If they can pass a test, or some sort of formal evaluation, they got it, right?  If kids pass your class, they got it, right?


Go back to one of our original contentions about why students struggle…because of prerequisite knowledge.  How many of your students move on and struggle because they do not really know what they should know from your class.  I am not a betting man but I would lay down a lot of money that it is a higher percentage than you believe.  Only those with the pre-emptive disappointment outlook would be unsurprised to find out how many of their students passed their class, with good marks, only to struggle with that same material in the future.

There’s good reason that happens, even to the best of us teachers and with our best students.  It happens because when they’re passing a test, it’s your test.  They’re demonstrating they know what you want.  They know how to show proficiency in the markers you’ve set up that should reflect understanding and knowledge.  They hacked you.

It’s not with ill-intent, it’s well within the structure of education today, the world-around!  It is not the fault of the student, our system made them this way.  It’s not our fault either, the system made us this way!

I say that if a student cannot readily apply what they learned in my class in a future event then they don’t know it.  How then, can I assign an appropriate grade?  Grades should be a reflection of what they know.  We must assign grades regularly, without the perspective of time that provides such insight to future application and adaptation.

What can be done?

There is a YouTube channel, Veritasium.  The host of that channel earned his PhD by researching the effects of learning through video.  Students would take a pre-test, then watch a video that discussed the information on the test.  Students would take a post-test.

Students, actually I’d like to call them observers, reported that the videos were clear, concise and generally good.  They liked the videos.   When they took the post-test, there was no significant growth.

With another group he did the same pre and post-test, but the video was different.  The video addressed and exposed misconceptions.  Students reported the video was confusing and unpleasant, unenjoyable.  The post-test scores doubled the pre-test scores with this group!

I’ve said it a million times before, students do not need us to be resident experts, the on-site answer-spewing reference resource.  It is easy for us to do that, it is comfortable for them.  But they don’t learn that way.

I tried to put this together in that same spirit:  Expose misconception before proposing a solution.  Otherwise, it is likely you would just latch onto the proposed solution as though you already knew that whether you actually did or not.

All of education, it seems, pushes hard to relieve confusion, to make the path to learning clear and clean, and most importantly for the stability of schools, repeatable.  But the more we push in this direction, the deep we dig our hole.

There are nods towards creating interest and the power of cognitive dissonance in education texts and professional development.  But, they’re pretty empty words because they’re given in a way that is poor teaching.  The best teachers, with the best ideas and the most experiences epically fail to teach others because they do not employ the same quality teaching strategies when teaching other teachers.

Here’s the information, make it your own, doesn’t work.

I hope that I have sufficiently exposed the nature of the problem with teaching so that my solutions will find a home in those exposed gaps.  You see, in teaching, in person, the way this is done is very important, but a video or blog post does not allow someone like me, with limited resources and an even smaller collection of talent, to demonstrate.  I can only describe.

To teach well students must have their misconceptions exposed.  The anticipatory set (bell work) is drivel if it does not contain a twist that either incites curiosity or exposes a conceptual flaw held by the students.

This is key, it’s the first step.  The thing you want them to know cannot be tackled head on.  If the objective of the unit was to have them paint the wall blue, for example, you could not just tell them to pain the wall blue.  They might get it done to your standard, but all of the thinking and discussion amongst peers that makes them understand (which leads to retention) is stifled.  Instead, they’ve been taught protocol, they’ve been programmed, trained.

An example of a good question to introduce a topic that seems, well, goofy, might be:  Which came first, goofy the word or the cartoon character?

Another would be: Why does the dictionary say that a verb is a noun?

Another example might be:  Water freezes at 32 degrees F, and 0 C, and boils at 212 F and 100C.  Why are those numbers different?

Or perhaps: Is zero odd or even?

Then there is: Is it an evolutionary advantage to taste like chicken?

A non-sequitur can be effective:  People died of cancer before cigarettes were around, therefore, smoking doesn’t cause cancer.

Be careful with these questions as you judge them.  It is how they are received by the audience, not by you or your peers that is important.  Don’t judge the quality of the question based on your knowledge, but based on whether the question leads to curiosity and uncovers misconception or not.  And questions that are tangent to the topic at hand are great because they can flesh out connections in unanticipated ways!

Now students shouldn’t be expected to reinvent the wheel at every turn, there are appropriate times to introduce concepts fully.  However, do not for a minute believe that no matter how well you taught that material, that the students understand it.  They need the opportunity to play with it, uncover misconceptions and so on.

So you have an introduction that reveals misconception or creates curiosity to begin, and then perhaps you dispel misconceptions or introduce the material, but then what happens next, on your end, can drastically limit the efficacy of the previous work done.

They need quality tasks.  They need a question or challenge that is approachable but also exposes common misconceptions.  And here, your role is very important.

Practice this phrase:  Go ask another student.

Say it nice, explain that the more you say on the subject the less they’ll learn, at least right now.  But it is key that they are talking to each other.  I advise against assigning groups, water finds its own level.  It is okay if the smart kids all get together and get it right away, you can ask them something about their reasoning that they’ve assumed is true, but they don’t know why it is true.  Or, you could instruct them to go around the room and observe the points of confusion of others and have them guide others in the right direction without giving it away.  (They can do that, but you cannot.)

A quick word on groups.  Groups should be no larger than four, but should be self-selected.  I’ll make a future post about how to pull this off and keep kids on task, but it’s easier than it might sound.  The rule is that if a group gets stuck, a member can go on a re-con mission and ask any group in the room questions and then report back to their own group.

What you’ll find is often no student, or group will have the answer or will have mastered the task.  However, between all of the people in the room, the information is there, it just hasn’t been put together.

After an appropriate amount of time, have the students return to their individual seats and you facilitate a class-wide discussion as follows.

Ask a student a question or have a volunteer share their findings, complete or not.

After the student speaks, you say, sometimes cleaning up their language a bit, what they had said for the whole class to hear.  Make sure to ask the student if that’s what they meant.  If not, have them clarify.  If you got it, ask the class the following, and this is probably the most important phrase/question in teaching:


I am not asking you if you agree or disagree with the statement, but do you understand it?

And again, the statement is spoken by you but the authority behind the statement is a student.

Whether that statement is right or wrong is irrelevant.  The fact that it reflects where they are and what they’re thinking is why it’s powerful.

However, depending on if it is right or wrong, you can steer the direction of the conversation.

If it is wrong it might be a good idea to ask who agrees and see if someone can clarify further.  Repeat what the student said in the same fashion as before.

More than likely, as students clarified and showed supporting evidence for the misconception, more and more students that originally disagreed with jump ship and latch on to the misconception.  This is actually good.  Just because they agreed with the right belief doesn’t mean they understood.  This jumping ship is them challenging their understanding, finding holes in it and latching onto something better.

Then ask if someone disagrees.  Have them explain, you parrot their explanation and again explain that whether the students agree or disagree, do they understand what’s been said.  If the student that share is wrong, ask who agrees and have them see if they can find more supporting evidence, or different explanation as to why.

But, you are not giving away what you believe is right or wrong.

If the student is right, it would be best to see who disagrees and why.  Explore the misconceptions, allowing students to challenge these lines of thinking.  Eventually, they will arrive at the correct answer or understanding.

Through this type of discussion and explanation the truth will be revealed.  But, most importantly, it is revealed by your facilitation of discussion, not because of your authority!

The best compliment I ever received about my teaching came from a student.  It was unplanned and was not intended to be a compliment, just an observation.  She said:

Mr. Brown, you don’t really teach us but we learn when we’re with you.


I will write more about this in the future.  There are some growing pains and specific techniques for managing behaviors and expectations that are different than in a typical classroom setting.

All that said, I hope this has been informative, stirred some thought and challenged you to reconsider your role in the learning of students.

Back in Session

I’m trying a few new things this year in math.  I will try to summarize how each week goes throughout the year and highlight successes and failures.

This week I really tried to introduce the honors freshmen to “real” math.  That is, some basic proofs, how generalize things in math and expose them to some difficult questions that are easy to approach, but difficult or surprising in their answers. But all of it was done in a way that is accessible to the students and with high levels of participation.

I really like when a student shares a thought and I repeat their thinking outloud and ask if the others understand, not necessarily agree, but understand.  This seems to really get them thinking and communicating.

Some of the questions we explored were, Is 0.999… less than one or equal to one, why/how we know the square root of two is irrational (we actually did the proof in class, carefully), is zero odd or even, why can’t we divide by zero?

There is also a challenge question posted, in two parts.  Part 1:  Given that a and x are natural numbers, and a is less than or equal to x, and is greater than 1, could the following number be prime:  (x! + a)?

Part 2:  If = 99, how many values of  would be composite.

The purpose of all of it was to challenge their thinking, hopefully incite some curiosity and promote deeper understanding.

One thing I really wanted them to understand is that rational numbers could be expressed as a ratio of integers. The old way I would’ve quizzed them on this knowledge would be to say, Write the following numbers as fractions.  The better question is, Express the following as a ratio of integers, as that addresses the definitions of rational and irrational numbers.

The number zero was a little difficult for some, but since we discussed that division is best thought of as a question, the denominator times what is the numerator?, that went well for most.  For example, instead of reading 8/2 as “eight divided by two,” it is better to ask, “two times what is eight?”

This is most effective when showing why you cannot divide by zero.

I was pleasantly surprised by the show of knowledge and understanding when I asked them, on their quiz, to express the square root of two as a ratio of integers.  Many said that the square root of two could not be expressed as a ratio of integers, that’s why it is irrational.

The depth of that understanding is of little consequence really, but it is a big victory that they have taken something that they always memorized in the past, and now truly grasp.

How well this translates to them owning their learning will remain to be seen, but I think we’re off to a good start.

Textbooks Should Be Resources, Not Curricula

Textbooks are Resources

Teach like a good doctor practices medicine.  Be prepared for a wide variety of issues, but always seek for the root of the problem so that your remedies may be most effective.  

One of the biggest hurdles in education, especially in High School, is getting those that wish to help to understand the nature of the problems faced in education today.  The education industry has been down the path that has arrived at this stop for over a century.  Those guiding the bus have landed us in the wrong part of town!

And those guiding the bus wish to help get us out of this part of town, but they seem to misunderstand what got us here in the first place.  It is the direction of their help that has landed us here.  The majority of support and research and resource we find within education is just more of the same old stuff, because it has gotten off of the same bus.  It’s not much good.

Consider the new wave of high school mathematics text books being published.  There is a series called Big Ideas, which is highly ironic.  The title of the book suggests it teaches the big ideas, the concepts.  The authors claim that is the aim of the book.  Each lesson and topic has a BIG IDEAS bullet and icon and yet, every single one of them is a procedure.

An example of this could involve graphing linear equations.  Students can learn all about graphing linear equations, can even be highly proficient with this, without ever knowing what a graph is.

The Big Ideas book, which does not stand out as particularly horrible compared to its competition, will say something like:

Big Idea:  To graph y = mx + b, plot the b term on the y-axis and then count the slope, m.

That’s not even an idea, much less a big one.  That’s a protocol.  If the problem looks like this, then do that.  Ever have students see a problem like (3x + 5) – (2x – 5) and what they do is distribute, use FOIL?  That’s because they’re accustomed to such protocol.  The problem looks like (3x + 5)(2x – 5) and since they just learned how to multiply, that’s what should be done here.

The best American textbook we have found so far is by Pearson.  It contains a few decent thinking strategies and some good “bell work” type activities.  But it contains far more things like this:

Write addition, subtraction, multiplication or division, to describe the operation displayed below.

  1. 2 + 5 = 7       3 · 6 = 18 …

I kid you not!  That is a high school level math book, supposedly.

I teach a non-American curriculum designed and tested through Cambridge University.  Students that come through that program witness all of the things educators say they want to see happening with students.  They are confident problem solvers who can think on their feet, make use of information, they latch onto more efficient methods than what they previously understood, and they have high levels of retention of materials (recall).

The books Cambridge University approves to be used for their curriculum are entirely different in nature than our books here in the US.  They’re thin, don’t have silly eye-catching icons and are intended to be used as a resource.  The books are not designed to go “cover to cover,” but instead can be used in any order desired.  Math is connected in many different ways.  Depending on how it’s unfolded one topic we consider “higher” can be taught first and then our supporting, foundational topics can be introduced later.

The point is, the curriculum is not the book.  The books for Cambridge are written to support the curriculum.  The book is a resource.  Like resources, sometimes the more and the greater the variety, the better.  That’s why some formats like YouTube are so powerful for students.  There is a variety of techniques, styles, approaches and flavors all addressing the same thing on YouTube.  Students can navigate their way through, seeking understanding, which of course leads to more questions, which when chased leads to more understanding, and the cycle starts over again.

Textbooks, no matter how fantastic, do not do this.  I argue that no entity, be it a company, group or individual can offer such a service.  (Yes, YouTube and others host means of expression and information sharing, but their intent is not to address one topic or genre.)

But textbook companies have marketed themselves as providing such services.  There is a greater need for such a thing in Elementary Schools where teachers teach all subjects and topics.  Having a resource that a teacher can use, cover to cover, that structures their day, weeks, and year, balancing all of the various tasks they must cover, is a huge boon, especially for inexperienced teachers.

In High School, the game is entirely different.  Each subject is taught by an expert in that field (in an ideal world anyway).  Such a tool is not only not needed, it is impossible to use.

If students are learning a topic in English, say some writing component, and they lack some foundational skill, a textbook will not identify and then address that.  They try with remedial support materials, but they’re of the nature in the picture I shared earlier.  And remediation of such a fashion, in isolation, not embedded in new content, is completely ineffective.

The majority of what happens in a good high school lesson is unscripted.  Much of it will be anticipated and the teacher will be prepared for these things, but until the specific issues themselves crop up, the remedy is unknown.

It’s not unlike what a family doctor experiences.  The doctor can anticipate what is going to happen in an interview with a patient, but until they see it unfold, they don’t really know what the best remedy is.  But whatever the issue they either have a remedy or know where to turn.  Unfortunately, doctors don’t always do a good job identifying problems, and like bad teaching, they have remedies in mind before understanding the issue at hand.  The remedies, of course, fail, causing frustration while allowing those problems to fester.

Teaching high school math is similar to this.  A good teacher will have some beliefs about why students are confused, or what they don’t know and do know, but they explore a little bit to verify their beliefs.

It is my firm belief, though this statement is pirated for Sir Arthur Conan Doyle, that a teacher that can be replaced by a computer should be.

The educational industry has been moving to compartmentalize and create modular components of education itself.  That way, if one part of the system fails, that modular can be pulled out, and a replacement piece can be inserted.

But education is organic and to teach is to perform an act of charity.  A textbook, no matter how well written, can do perform organically and cannot itself be charitable.  A book, however, can be used as a resource, especially if the intentions of the authors is to create a resource.

The education industry does not want books to be a resource.  Many good teachers have left, positions are filled with unqualified or inexperienced people.  Those that are experienced and qualified are overwhelmed with the call to help support others.

If only we could fix all of that with a book that anybody could open and use, then our students would have a fighting chance.


Why Does the Order of Operations Work?

Why does the order of operations help us arrive at the correct calculation?  How does it work, why is it PEMDAS?  Why not addition first, then multiplication then groups, or something else?

I took it upon myself to get to the bottom of this question because I realized that I am so familiar with manipulating Algebra and mathematical expressions that I know how to navigate the pitfalls.  That instills a sense of conceptual knowledge, but that was a false sense.  I did not understand why we followed PEMDAS, until now.

By fully understanding why we follow the order of operations we can gain insight into the way math is written and have better abilities to understand and explain ourselves to others.  So, if you’re a teacher, this is powerful because you’ll have a foundation that can be shared with others without them having to trip in all of the holes.  If you’re a student, this is a great piece of information because it will empower you to see mathematics more clearly.

Let’s start with the basics, addition and subtraction.  First off, subtraction is addition of negative integers.  We are taught “take-away,” but that’s not the whole story.  Addition and subtraction are the same operation.  We do them from left to right as a matter of convention, because we read from left to right.

But what is addition?  In order to unpack why the order of operations works we must understand this most basic question.  Well, addition, is repeated counting, nothing more.  Suppose you have 3 vials of zombie vaccine and someone donates another 6 to your cause.  Instead of laying them out and counting from the beginning, we can combine six and three to get the count of nine.

And what is 9?  Nine is | | | | | | | | |.

What about multiplication?  That’s just skip counting.  For example, say you now have four baskets, each with 7 vials of this zombie vaccine.  Four groups of seven is twenty-eight.  We could lay them all out and count them, we could lay them all out and add them, or we could multiply.

Now suppose someone gave you another 6 vials.  To calculate how many vials you had, you could not add the six to one of the baskets, and then multiply because not all of the baskets would have the same amount.  When we are multiplying we are skip counting by the same amount, however many times is appropriate.

For example:

4 baskets of 7 vials each

4 × 7 = 28


7 + 7 + 7 + 7 = 28


[ | | | | | | | ]   [ | | | | | | | ]   [ | | | | | | | ]   [ | | | | | | | ]

Consider the 4 × 7 method of calculation.  We are repeatedly counting by 7.  If we considered the scenario where we received an additional six vials, it would look like this:

4 × 7 + 6

If you add first, you end up with 4 baskets of 13 vials each, which is not the case.  We have four baskets of 7 vials each, plus six more vials.

Addition compacts the counting.  Multiplication compacts the addition of same sized groups of things.  If you add before you multiply, you are changing the size of the groups, or the number of the groups, when in fact, addition really only changes single pieces that belong in those groups, but not the groups themselves.

Division is multiplication by the reciprocal.  In one respect it can be thought of as skip subtracting, going backwards, but that doesn’t change how it fits with respect to the order of operations.  It is the same level of compacting counting as multiplication.

Exponents are repeated multiplication, of the same thing!


3 + 6

4 × 7 = 7 + 7 + 7 + 7

74 = 7 × 7 × 7 × 7

This is one layer of further complexity.  Look at 7 × 7.  That is seven trucks each with seven boxes.  The next × 7 is like seven baskets per box.  The last × 7 is seven vials in each basket.

The 4 × 7 is just four baskets of seven vials.

The 3 + 9 is like having 3 vials and someone giving you six more.

So, let’s look at:

9 + 4 × 74

Remember that the 74 is seven trucks of seven boxes of seven baskets, each with seven vials!  Multiplication is skip counting and we are skip counting repeatedly by 7, four times.

Now, the 4 × 74 means that we have four groups of seven trucks, each with seven boxes, each with seven baskets containing seven vials of zombie vaccine each.

The 9 in the front means we have nine more vials of zombie vaccine.  To add the nine to the four first, would increase the number of trucks of vaccine we have!

Now parentheses don’t have a mathematical reason to go first, not any more than why we do math from left to right.  It’s convention.  We all agree that we group things with the highest priority with parentheses, so we do them first.

(9 + 4) × 74

The above expression means we have 9 and then four more groups of seven trucks with seven boxes with …  and so on.

Exponents are compacted multiplication, but the multiplication is of the same number.  The multiplication is compacting the addition.  The addition is compacting the counting.  Each layer kind of nests or layers groups of like sized things, allowing us to skip over repeated calculations that are the same.

We write mathematics with these operations because it is clean and clear.  If we tried to write out 35, we would have a page-long monstrosity.  We can perform this calculation readily, but often write the expression instead of the calculation because it is cleaner.

The order of operations respects level of nesting, or compaction, of exponents, multiplication and addition, as they related to counting individual things.  The higher the order of arithmetic we go, up to exponents, the higher the level of compaction.

Exponents are repeated multiplication, and multiplication is repeated addition, and addition is repeated, or skip counting.  We group things together with all of these operations, but how that grouping is done must be done in order when perform the calculations.