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

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.

The Toaster Problem in Education

It’s easy to talk about shifting education towards a more concept based approach.  But it’s hard to see what that really means in practice.  I’m not a betting man, but would be willing to bet that upon inspection there are many things you think you understand conceptually in your topic, but you just feel that way because you understand procedure well enough to always arrive at correct answers.

I can offer an example in math:  Why does the order of operations work?  Why does the structure in the order of operations guide us to the correct calculation?

Let’s use a mathematical way of thinking to approach this problem of understanding what conceptual approach education looks like, compared to our current procedure based approach.  (Tell-tale sign that you’re procedurally based is if your students cannot remember how to do something big a year later.  Or, do you consider the work assigned before your lesson?)

Imagine you want your students to know how to make toast.  You could introduce them to a toaster.  Then, demonstrate how the bread-item is dropped in the slots on the top, the little knobby is turned to select the desired level of darkness, the button with the picture of the type of bready-material being toasted is pushed, and the lever is depressed.

If it’s an advanced class, maybe some discussion is given to what to do if the toast gets stuck, and why you should always unplug the toaster when finished with it because toasters have notoriously cheap circuits that short out, causing a fire.

Oh, one last thing.  All toasters are good toasters.  There are no bad toasters.  Some make light toast, some dark toast.  If you show preference to one kind of toaster, you’re then the exception to the rule because we tolerate everything except intolerance.

That’s a very typical American style of teaching something.  We cover how to use a tool and throw in a little social justice message to boot.  (That is not a comment on the need for awareness of social issues except to say that math textbooks are inappropriate platforms for them.)

Imagine that instead of wanting your students to know how to make toast, you wanted them to know about toast.   You teach them what it is, previously cooked bread that is now slightly, but evenly, burned on the cut-faces making a slightly stiffer, crunchier piece of substrate for the delicious spreadable material of your choosing.

For the sake of this thought experiment, let’s say you also show them a toaster, but that’s it.

Now consider a pair of students.  One who learned the first method, and the second learned about toast, but spent little time with a toaster.

Which student could make toast if the toaster broke?  Which understands what a toaster really does?

Teaching how to use a toaster is procedural, while teaching what toast is would be conceptual.

Education is a HUGE industry with an enormous amount of inertia to overcome before change is realized.  There are jobs at stake if responses to changes go wrong.  Companies invest millions to supply the desires of schools.  And what do schools want?  They want to be like everybody else, because it’s safe!

We have these methods, that if not effective, are at least safe because we have used them for a long time, so has everybody else.  So if we’re close to the average, we’re okay.

But don’t get me wrong, things in education will change.  Pretty soon curriculum will be all conceptual.  Kids will be reinventing the wheel at every turn.  We see some of that in the elementary levels right now.  That’s truly a shame because it’s harmful.  Young kids do not yet possess the faculties for abstraction!  They need to know how to use a 3rd grade toaster, if you will.

I am NOT a doomsday preacher here, but I do not believe education cannot fix itself.  It is so established in the way it operates that the path we are on will remain until something really big from way up high changes.  The likelihood of that being a good change is slim because politicians aren’t educators.  Even if the idea is good, from above, the execution will be poor because it’s ideas, not how they play out, that gets people elected.

But, the change from teaching how to use a toaster to teaching what toast is, well, is needed.  Even for students to pass the new style of standardized testing they need to know what toast is.

Beyond that, for them to be successful in college, the nature of toast must be understood.  To change math from a hurdle to an opportunity, they’ve got to know all about toast, not just how to use the toaster.

It is these last two things, the belief that the education system cannot right itself, and the need for conceptual understand, that has motivated me to step outside of education for my project.

Why Good Lessons Fail

Ever had a lesson you were THRILLED about?  You loved it, it was fantastic, interesting, crisp, approachable and ... wonderful in every possible fashion.  And yet, when you delivered that lesson, it flopped!

What gives?  What was wrong with the lesson?

In reality, there was probably nothing wrong with the lesson.  Sure, all can be improved, but the lesson wasn't the problem, the delivery was.  It seems there exists an inverse relationship between how much I love a lesson and how well received it is.  The more I love it, the more students hate it!

What it really boils down to is engagement.  We are so sure that what we have to say will blow minds, that we forget our number one task ... making sure we are teaching students, not just covering material.  We assume that because we find it interesting and fascinating, and because we had such a grand time putting the lesson together that they'll gravitate towards it.

But gravitate towards it in favor of what?  What captures the attention of students?  Drama at lunch, fights with family members, changes in weather, they might be tired from staying up and watching the new season of Stranger Things on Netflix ... we don't know.  But whatever has their attention, we must wrestle it away.

In a normal lesson we are usually vigilant and on top of distractions and such.  We work hard to make the lesson itself interesting.  But in a lesson that needs no such adornments, we fail to sell it.

So regardless of whether you think it's great, they need to be sold on the fact!

There are a couple things that you can do, at any point in time, if they're not engaged.  These work for average and poor lessons, not just the great ones that we hope will inspire a future generation of (whatever it is you teach).

Before I share with you three ways to quickly grab their attention, let me say that once you have it, you can just jump right back into the lesson.  You'll have their attention, they'll not even notice that suddenly they're learning stuff!

My favorite, go-to, method of grabbing attention is with a quick, cheesy, usually Dad-Joke.  I sometimes look up a bunch of them, print them off and have them at the ready.  There are a few that I have on the ready at any given moment, but since I don't often tell them outside of the classroom, I forget.

Make it short and dumb, they'll be captured, even if they complain.  Then, back to the lesson.

And with all of these, you just jump right into the attention getting performance, you can do it mid-sentence if you please.

The second method is with a quick story about something interesting.  It can be that you wanted some cereal for breakfast and there was only a splash of milk left in the fridge!  So you couldn't even have dry cereal, just slightly less than soggy junk -- How FRUSTRATING!?!?!  Get some feedback and jump right back in.

The last method I use is direct.  I simply tell them they're distracted and that they need to do their best to focus.  I'll sell why (perhaps the material is dry but will be very important and interesting in context later, or some other reason).  I'll share that I feel the same way, burned out and tired, but explain that we all have a job to do.  "Let's just get through these next few parts and we're done for the day, if we do them well.  If not, we'll have to revisit this again in the near future."

Whatever methods you use, mix it up.  If you become too predictable with these they'll fail to gather attention.  So, "Stay frosty," like the line in Aliens suggests.

Anyhow, I hope these are helpful tips.  Just remember, no how great your lesson is, engagement is still the most important part of the lesson.  Without it, they'll not learn anything!

What is Algebra?

This past month has been very busy here for The Bearded Math Man.  I’ve learned a lot about things I have merely taken for granted and have shared most of them with you here on my site.  And while I have a goal and a mission, the methods of achieving that goal are still forming.  I’m learning what works best and what doesn’t work.  One such thing I’ve discovered is the purpose of this blog.

This blog is meant for two audiences.  Those interested in math and those teaching math.  Now that I have that defined, I’ll keep a more focused range of topics.  I just thought that was worth mentioning.

Now, for today’s topic, Algebra.  I do not intend to teach you Algebra, but would like to share something I did not know about the subject.  Algebra means to make complete, or to resolve.  I knew it was named after a Persian mathematician in the early 9th century, but that the branch of mathematics goes farther back in time than the name itself, even the Babylonians used Algebriac concepts.  But I thought the name was just that, a name.

It is stunningly powerful to recognize what Algebra means.  Everything operation we perform in Algebra is to meet this end, to complete or resolve an equation!  That’s what we do when we’re solving an Algebraic Equation.

One other thing you may not have known about Algebra is the equal sign.  The symbol itself never appeared until the 16th century and it traveled the entire width of the page.  It is hard to imagine how this would be a more efficient way of describing the equality present between two things, but it was.  Over time it was shortened to what we have today.  This is more than just an interesting factoid, too.  It goes to show that sometimes great ideas are so revolutionary that they seem obvious in hindsight.  First, we have a symbol that means equals, then we have, over time, an easier way of writing that symbol.

In many ways, isn’t that what makes mathematics so difficult, the jargon and abstraction?  That’s why one of my main points of focus is instilling mathematical literacy in students.  If they can read the math for what it says, not just as a funky collection of shapes and symbols, the mathematical ideas present themselves in a sensible and approachable fashion.

That’s what I’ve tried to do with my introduction to Algebra as a branch of mathematics, which is taught in Algebra 1, the class.  Here’s the link to the page.

As always, I thank you for reading and hope I’ve stirred some curiosity in you.

PS:  If you are interested in some of the history behind Algebra, the following book is highly recommended.  If you purchase it through this link you will help support the mission here of changing math from a hurdle in the way of young peoples’ dreams to a platform upon which success is built, and at no additional cost to you.

Is Infinity Real?

How Many Primes are There
Is Infinity Real
Part 1

Teachers: The following is a discussion that can be had with students to create interest in mathematics by discussing two very easy to understand, but perplexing problems in mathematics.  First, the nature of infinity.  The second is the lack of pattern and order in the prime numbers.

The number of primes is infinite.  Euclid proved it in a beautiful, easily understood proof by contradiction.  Paraphrasing, he said that there are either infinitely many primes, or a finite number of primes.  So let’s pick one and explore it.  Say there are a finite number of prime numbers.  If you were to list them all, then take their product you would have a very large number.  But if you just add one to that number, it would be prime because none of the other prime numbers would be a factor of it.  It would have exactly two factors, one and itself.

In case you don’t believe this works, let’s say we can list all of the primes, but there are only four.  Let’s say the entire list of primes was 2, 3, 5, and 7.  Their product, 2 × 3 × 5 × 7 = 210.  This number is composite because all of the primes are factors of it.  Add one to it, arriving and 211 and none of the prime numbers are a factor of it…making it have the factors of 211 and 1.  That means it is prime.

So it is false that there are a finite number of primes. Therefore, the are infinitely many prime numbers.

Beautiful, right?  Case closed. … or is it?

The case is closed, if you believe infinity exists.  To be clear, infinity is not a number, it’s a concept.  A set can only approach infinity, nothing ever equals infinity because it’s an idea.  The idea behind infinity is that the collection of things just keeps growing and growing.

We, as humans, have a very big problem with very big numbers, even large groups of things.  For example, there are some things that we only have a plural word for, we do not possess a singular word for these things.  A few examples are rice, sand, hair, shrimp and fish.  You can have a single hair, a grain of sand (or rice), and so on.  They are so vast in quantity they become indistinguishable.

And yet, they’re finite. You could conceivably collect all of the sand in the world and count every grain.  More sand does not magically appear once it is all collected.

What about stars in the sky?  What we call the observable universe is how far we can see.  We don’t know if it goes on forever, or if it is somehow contained.  Perhaps the word, universe, is misleading.  Perhaps there are multiples of it, maybe as many as there are grains of sand on the earth.

Before we chase that rabbit down its hole, let’s get back to earth.  Euclid’s proof that there are infinitely many prime numbers is beautiful.  But is he right?  Surely his proof is flawless, but what about infinity.  We have no examples of infinity, it might just be a human construction.  Now, if mathematics can discover things that are real and applicable from such a thing, that’s all the more powerful the tool it is, but what if we’re wrong about infinity?  There are two things I want you to consider as we explore prime numbers and their relationship with infinity.

The first thing is:  There’s an axiom (a statement we just accept as truth), called the Axiom of Infinity.  It basically says that there are infinite sets of things, like natural numbers.  We just say it’s true and roll with it until we discover a problem.  Then, we either adjust our axiom or start a new one.

The second thing is:  In the early 20th century a man named Kurt Gödel showed that we cannot actually prove any system of mathematics is true without assuming some supporting evidence is true.  We have to assume something is true in order to know if other things are true, roughly speaking.  In order to know if the thing we assumed to be true is actually true or not (like infinity), we have to assume that something else, more basic, is true.  So, and I’m taking some liberties here to make my point, but a conclusion, like the number of primes being infinite, is only as worthy as the presupposition (infinities exist).

Let’s look at a few strings of prime numbers and see if we can’t get our heads around this whole infinity thing.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

The gaps between these prime numbers are below.

1, 2, 2, 4, 2, 4, 2, 4, 6

Another string would be:

907, 911, 919, 929, 937, 941, 947

The gaps here are listed below.

4, 8, 10, 8, 4, 6

They are still relatively close.  Many mathematicians have tried to find a pattern in prime numbers.  After all, if you can find a pattern, then you can find the next one.  How cool would that be, right?

You might be thinking, uh, why would that be cool?

Well, there’s big money being paid if you can find the next prime number.  There is a project called GIMPS (Great Internet Mersenne Prime Search), where you can participate in the search.  And if your computer finds the next prime, you get some cash!

The last prime found with GIMPS was in 2013.  (At the time of this being written, it is 2017.) The number is massive.  The text file of the digits in the number is 7.7 MB.  That’s more data that a song and this is just a list of numbers.  The number is 257,885,161 – 1.  The number is huge that to verify that it is prime takes massive super computers days upon days to perform the calculation.  Finding the next prime number is a huge undertaking, very complicated and difficult, requiring computers all over the world working together before one is discovered.

Why all the fuss? What good are they?

Well, they keep you from being robbed, for one.  Internet security uses prime numbers to encrypt (code) your banking information.  The merchant will have a huge number that they multiply your card number by (kind of).  The huge number is the product of two of these gigantic prime numbers.  It’s so big that even though everybody (would be thieves) know it’s the product of two primes, they can’t figure out which two numbers.  The encrypted number is sent to your financial institution, who knows which two primes were used, which is basically like a key.

It’s also weird, and cool, that some bugs have a life cycle that only occurs in prime numbers!  Cicadas only come out and breed, and then die, in prime number years.  Incredible.

Back on track, forgive me.  It feels there are infinitely many tangents I can follow with math!  We have not been able to find a pattern in the prime numbers yet and let’s take a look at why.  You see, as these primes get huge, the gaps get larger and larger…approaching infinity!

Let’s take a look at one more string of prime numbers.

10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099


The differences here are as follows.

28, 2, 22, 6, 10, 12, 2, 6

No discernable pattern, right?  If you can find one, you stand to make significant history, no one has found one yet.  We have some approximations that work within certain constraints but they all break down eventually.

But, to be clear, if you could find a pattern in the gaps between the primes a formula could be created that would generate prime numbers.  We can generate natural numbers by just adding one to the largest we have come up with so far.  But primes, as you’ve seen with the GIMPS project, aren’t so easily discovered.

And here’s one of the issues.  The gaps between prime numbers can get huge, perhaps infinitely huge.  Consider this.

Fact 1:  5! = 5×4×3×2 = 120

Fact 2:  120 is not prime because it is divisible by 5 and 4 and 3 and 2.

Fact 3:  5! + 5 is not prime because it is divisible by 5.  (When we add another 5, it’s like skip counting when you first learned multiplication.)

The same is true for 5! + 4 being divisible by 4, because 120/4 = 30.  5! + 4 is 4 × 31, there’s one more four.

The same holds true for 5! + 3 being divisible by 3 and 5! + 2 being divisible by 2.

Fact 4:  What all this means is that there after 5! + 1 there are four consecutive numbers that are composite.

This would also work for 100!  The number 100! + 100 would be composite.  For that matter, 100! + 37 would be composite also.  100! Plus all of the numbers up to and including 100 would be composite, (except possibly adding 1).

This means there is a gap of 99 after 100! + 1.

This goes on forever, arbitrarily large numbers, like 1,000,000,000,000!  There would be a gap of 1,000,000,000,000 – 1 numbers after this number that are composite.

We could write this in a general sense.  Let a and x be a whole numbers such that a is less than or equal to x.  (a x).

Then x! + a is composite.

Since x is a whole number and whole numbers are infinite, then there are infinitely large gaps between the large prime numbers, themselves being infinite.

Crazy, right?

So if the gaps between primes gets infinitely large, how can there be infinitely many prime numbers?

Well, there’s one more piece of information to be considered.  Twin primes are prime numbers that are just two numbers apart.  The primes 2 and 3 are only one apart, but all others are an even number apart, the smallest gap being a gap of two, like 5 and 7, or 11 and 13.

There’s a conjecture (not as strong as an axiom), that is yet unproven, but we’re getting closer, that states that there are an infinite number of twin primes.  The largest known pair of twin primes is below:

3,756,801,695,685 × 2666,689 – 1

3,756,801,695,685 × 2666,689 +1

Those numbers are too large to be written out!

While we do not yet know, with a proof, that there are infinitely many twin primes, we do know that there are infinitely many primes that have a maximum distance between them and it might be as low as a difference of sixteen.  This is all being discovered and explored and fought over at the moment.

So on one hand we have infinitely large gaps between prime numbers, but when they do pop up, they will do so in clumps and groups?

If all of this makes your head spin, then I have succeeded.  I am not trying to convince you that infinities do not exist, or that they do.  I am trying to show that math is contentious and changing.  As we learn and discover new things math is changing.  Math is just a language we use to describe the world around us.  So powerful is math that we are not even sure if it is a human invention at all or rather a discovery!

What are your thoughts?  Please share them in the comments below.

As always, thank you for your time. I hope this has stirred some thought, maybe even sparked a passion for mathematics!


At the time of the making of this video the world’s largest prime number is not the last one found by the GIMPS project.  However, they’re likely to find another even larger one, sometime soon.  There’s a video below (Largest prime number) that discusses that number and prints it out … it takes up as much paper as three large books!

For some fascinating and approachable treatment of prime numbers, consider the following videos:

Gaps between prime numbers:

The largest prime number:

Infinite Primes:

Large Gaps Between Primes:

If you found this helpful and would like to help make these videos possible, to help break down the obstacle that math presents itself as to young people, please consider visiting my patreon site: