# Teaching Square Roots Conceptually

teaching square roots

How to Teach Square Roots Conceptually

If you have taught for any length of time, you’ll surely have seen one of these two things below.

Sure, this can be corrected procedurally.  But, over time, they’ll forget the procedure and revert back to following whatever misconception they possess that has them make these mistakes in the first place.

I’d like to share with you a few approaches that can help.   Keep in mind, there is no way to have students seamlessly integrate new information with their existing body of knowledge.  There will always be confusion and misunderstanding.  By focusing in on the very nature of the issues here, and that is lack of conceptual understanding and lack of mathematical literacy, we can make things smoother, quicker, and improve retention.

Step one is to teach students to properly read square roots.  Sure, a square root can be an operation, but it is also the best way to write a lot of irrational numbers.  Make sure you students understand these two ways of reading a square root number.

Students are quick studies when it comes to getting out of responsibility and side-stepping expectations.  Very quickly, when asked “What does the square root of 11 ask?” students will say, “What squared is the radicand?”

When pressed on the radicand, they may or may not understand it is 11.  But, they’ll be unlikely to have really considered the question for what it asks.  Do not be satisfied with students that are just repeating what they’ve heard.  Make them demonstrate what they know.  A good way to do so is by asking a question like the one below.

Another way to test their knowledge is to ask them to evaluate the following:

$\sqrt{2}×\sqrt{2}.$

We do not want students saying it is the square root of four at this point.  To do so means they have not made sense of the second fact listed about the number.  An alternative to using a Natural Number as the radicand is to use an unknown.  For example:

$\sqrt{m}×\sqrt{m}.$

Step two requires them to understand why the square root of nine, for example, is three.  The reason why it is true has nothing to do with steps.  Instead, the square root of nine asks, “What squared is 9?”  The answer is three.  There is no other reason.

Once again, students make excellent pull-toy dolls, saying random things when prompted.  Once in a while they recite the correct phrase, even though they don’t understand it, and we get fooled.  It is imperative to be creative and access their knowledge in a new way.

Before I show you how that can be done with something like the square root of a square number, let’s consider the objections of students here.  Students will complain that we’re making it complicated, or that we are confusing them.

First, we’re not making the math complicated.  Anything being learned for the first time is complicated.  Things only become simple with the development of expertise.  How complicated is it to teach a small child to tie their shoes?  But once the skill is mastered, it is done without thought.

The second point is that we are not confusing them, they are already confused.  They just don’t know it yet.   They will not move from being ignorant to knowledgeable without first working through the confusion.  If we want them to understand so they can develop related, more advanced skills, and we want them to retain what they’re learning, they have to understand.  They must grasp the concept.

So how can we really determine if they know why the square root of twenty-five is really five?  We do so by asking the same question in a new way.

Another way to get at the knowledge is by asking why the square root of 25 is not 6.  Students will say, “Because it’s five.”  While they’re right, that does not explain why the square root of 25 is not six.  Only when they demonstrate that 62 = 36, not 25, will they have shown their correct thinking.  But, as is the case with the other questions, students will soon learn to mimic this response while not possessing the knowledge.  So, you have to be clever and on your toes.  This point is worth laboring!

Step three involves verifying square root simplification of non-perfect squares.  This uncovers a slew of misconceptions, which will address. Before we get into that, here is exactly what I mean.

Have students explain what is true about the square root of twenty-four.  There are two ways they should be able to think of this number (and one of them is not as an operation, yet).

1.      What squared is 24?

2.      This number squared is 24.

The statement is true if “two times the square root of six, squared, is twenty-four.”  Just like the square root of 9 is three only because 32 = 9.

The first hurdle here is that students do not really understand irrational numbers like the square root of six.  They’ve learned how to approximate and do calculation with the approximations. Here is how they see it.

$\sqrt{2}=1.4$

$3+\sqrt{2}=4.4$

$3×\sqrt{2}=4.2$

What this means is that students believe:

1.      Addition of a rational number and an irrational number is rational.

2.      The product of a rational and irrational number is also rational.

a.       This can be true if the rational number is zero.

This misunderstanding, which naturally occurs as a byproduct of learning to approximate without understanding what approximation means, is a major hurdle for students.  It must be addressed at this time.

To do so, students need to be made to understand that irrational numbers cannot be written with our decimal or fraction system.  We use special symbols in the place of the number itself, because we quite literally have no other way to write the number.

A good place to start is with π.  This number is the ratio of a circle’s diameter and its circumference.  The number cannot be written as a decimal.  It is not 3.14, 22/7, or anything we can write with a decimal or as a fraction.  The square root of two is similar.  The picture below shows probably over 1,000 decimal places, but it is not complete.  This is only close, but not it.

Students will know the Pythagorean Theorem.  It is a good idea to show them how an isosceles right triangle, with side lengths of one, will have a hypotenuse of the square root of two.  So while we cannot write the number, we can draw it!

The other piece of new information here is how square roots can be irrational.  If the radicand is not a perfect square, the number is irrational.  At this point, we cannot pursue this too far because we’ll lose sight of our goal, which is to get them to understand irrational and rational arithmetic.

This point, and all others, will be novel concepts.  You will need to circle back and revisit each of them periodically.  Students only will latch on to correct understanding when they fully realize that their previously held believes are incorrect.  What typically happens is they pervert new information to fit what they already believed, creating new misconceptions.  So be patient, light-hearted and consistent.

Once students see that the square root of two is irrational, they can see how they cannot carry out and write with our number system, either of these two arithmetic operations:

This will likely be the first time they will understand one of the standards for the Number Unit in High School level mathematics.

Students must demonstrate that the product of a non-zero rational and irrational number is irrational.

Students must demonstrate the sum of a rational and an irrational number is irrational.

Keep in mind, this may seem like a huge investment of time at this point, and they don’t even know how to simplify a square root number yet.  However, we have uncovered many misconceptions and taught them what the math really means!  This will pay off as we move forward.  It will also help establish an expectation and introduce a new way to learn.  Math, eventually, will not be thought of as steps, but instead consequences of ideas and facts.

Back to our question:

Just like the square root of nine being three because 32 = 9, this is true if:

${\left(2\sqrt{6}\right)}^{2}=24.$

Make sure students understand that there is an unwritten operation at play between the two and the irrational number.  We don’t write the multiplication, which is confusing because 26 is just considered differently.  It isn’t 12 at all (2 times 6)!

Once that is established, because of the commutative property of multiplication,

$2\sqrt{6}×2\sqrt{6}=2×2×\sqrt{6}×\sqrt{6}.$

There should be no talk of cancelling.  The property of the square root of six is that if you square it, you get six.  That’s the first thing they learned about square root numbers.

$2×2×\sqrt{6}×\sqrt{6}=4×6.$

As mentioned before, students are quick studies.  They learn to mimic and get right answers without developing understanding. This may seem like a superficial and easy task, but do not allow them to trick themselves or you regarding their understanding.

A good type of question to ask is:

To do this, we students to square the expression on the left of the equal sign to verify it equals the radicand.  This addresses the very meaning of square root numbers.

Last step is to teach them what the word simplify means in the context of square roots.  It means to rewrite the number so that the radicand does not contain a perfect square.

The way to coach students to do this is to factor the radicand to find the largest square number.  This is aligned with the meaning of square roots because square roots ask about square numbers.  When they find the LARGEST perfect square that is a factor of the radicand, the rewrite the expression as a product and then simply answer the question asked by the square roots.  Here’s what it looks like.

$48=2×24,\text{\hspace{0.17em}}3×16,\text{\hspace{0.17em}}4×12,\text{\hspace{0.17em}}6×8.$

$\sqrt{48}=\sqrt{16}×\sqrt{3}$
Write the square root of the perfect square first so that you do not end up with
$\sqrt{3}4,$ which looks like $\sqrt{34}.$

$\sqrt{48}=4×\sqrt{3}$.

At this point, students should be ready to simplify square roots.  However, be warned about a common misconception developed at this point.  They’ll easily run the two procedures into one.  They often write things like:

$\sqrt{18}=\sqrt{9}×\sqrt{2}$

$\sqrt{18}=3\sqrt{2}$

${\left(3\sqrt{2}\right)}^{2}=9×2$

$9×2=18$

$\sqrt{18}=18.$

The moral of the story here is that to teach students conceptually means that you must be devoted, diligent and consistent with reverting back to the foundational facts, #1 and #2 at the beginning of this discussion.

This approach in no way promises to prevent silly mistakes or misconceptions.  But what it does do is create a common understanding that can be used to easily explain why $\sqrt{12}$ is not $3\sqrt{2}.$  It is not “three root two,” because

This referring to the conceptual facts and understanding is powerful for students. Over time they will start referring to what they know to be true for validation instead of examination of steps.  There is not a step in getting $\sqrt{12}=3\sqrt{2},$ that is wrong.  What is wrong is that their work is not mathematically consistent and their answer does not answer the question, what squared is twelve?

If a student really understands square roots, how to multiply them with other roots, and how arithmetic works irrational and rational numbers, the topics that follow go much more quickly.  After this will be square root arithmetic, like $5\sqrt{2}-3\sqrt{8},$ and then cube roots and the like.  Each topic that you can use to dig deep into the mathematical meaning will, over time, quicken the pace of the class.

In summary:

1.      Square roots have a meaning.  The meaning can be considered a question or a statement, and both need to be understood by students.

a.       This meaning is why the square root of 16 is 4.

2.      Square roots of non-square numbers are irrational.  Arithmetic with rational and irrational numbers is irrational (except with zero).

3.      To simplify a square root is to rewrite any factor of the radicand that is a perfect square.

a.       When rewriting, place the square root of the square number first.

4.      The simplification of a square root number is only right if that number squared is the radicand.

I hope you find this informative, thought-provoking, and are encouraged to take up the challenge of teaching conceptually!  It is well worth the initial struggles.

For lessons, assignments, and further exploration with this topic, please visit: https://thebeardedmathman.com/squareroots/

## This

Let’s talk about THE question in a teacher’s life … the baleful, “When am I going to use this in my real life?”

Yeah, that one.

The honest answer is probably, never… and they know it.  Why else ask, if not to subvert and diminish your role and purpose?  They don’t really want an answer.  What they want ... is just to watch you squirm, or to hear what B.S. you might spit out.  Either way, your class is now off the tracks!

However, it is a valid question, in its essence.  What is our purpose here?  It is a fair question, and one that needs to be answered.  And don’t be one of those people that thinks all content is applicable to daily life … it is NOT, nor should it be.  The purpose of education is not to train kids for every possible situation in life, but to equip them with an education so they can adapt to any possible situation.  The purpose of education is the development of the mind.  Sometimes we learn tricky things for the same reason some accountants lift heavy weights at the gym.

So let’s talk about how to change the answer to THE question.  Let’s turn it on its head.  Let’s answer the question: When am I going to use this in my real life, it in a way that swings the pendulum back in your direction.  Let’s answer the question in a way that stops the division and animosity that fosters the question in the first place.

See, the reality is, when a kid asks that question, they’re doing you a favor.  They’re providing you with insight that perhaps you’re serving the wrong this.  And if you’re not, whatever purpose you serve is not apparent to that student!

Let’s back up and take a larger scope view of the situation.  Especially in today’s educational climate, teachers stay in the classroom to be of service to students.  That’s it.  Teachers are blamed for all of the woes of society, for the failings of education, called lazy, and everybody seems to know what they’d do to fix it, if they were a teacher!

That sentiment, why teachers stay in the classroom, is the gateway to changing the answer to THE question.  Teachers are only in the classroom for the benefit of the students.  And surely, a teacher wouldn’t take on the sacrifices they do to stay teaching only to waste the time of their students, right?

Well … no, but kind of yes, too.

Let’s talk about job pressures … failing percentages in your classes, average scores on “high stakes tests.”   Those are big deals!  The test results are used to evaluate schools and teachers.  Administrators can be rewarded or fired on the basis of such things!

After all, good test results must be the sign of a great school.  Bad test results, well, that is really the teacher’s fault!  Yet, if a teacher holds a standard aligned with test results, the class failing rate will be too high, unacceptable, and a sign of bad teaching.  There will be unpleasant parent meetings; counselors, and administrators asking what’s being done to help the student, as if the student is hapless, a victim of the inevitable.

What am I doing to help this student?  I don’t know … showing up to work every day with lesson plans, a warm welcome and words of encouragement?  Oh, and I tell him to pick up his pencil fourteen times an hour, that’s a start, right?

Through either threat or blarney, bean-counters and pencil-pushers outside the classroom press hard to ensure that the teacher is performing due diligence to achieve high test grades.  Parents hover, students object, and through it all teachers are led to one inevitable observation: this is not why I teach.

Do those test scores really matter for students?  Sure, maybe ACT or SAT, AP, IGCSE or IB tests matter.  But those are the culmination of years of work.  Does it matter, to the student, if they pass their local state’s Common Core exam?  Not one bit.

In my real life, everybody will be impressed that I got a 3 on my Common Core State Test in English.

The purpose of an education is not to be able to pass a test.  The purpose of education is the development of the mind.  An educated person should be adaptable, thoughtful, able to communicate and appreciate various points of view other than their own.  An educated person should have perseverance and confidence, creativity and curiosity about the world around them.

A person that is educated should have an enriched life as a byproduct of their education.

When am I going to use this in my real life?

Well, that depends on what you mean by this.

Are you, as an educator, teaching this to help kids pass a test, or get a good grade in your class?  Are you teaching this to help them to know how to do their homework?  If so, there’s no judgement or blame.  Sometimes you have to make concessions just to get through the day.  We want kids to be successful.  The problem is, what are we using as markers of success?

But what if you could make this align with why you come back every year?

If you teach students about factoring polynomials so they can complete a worksheet, and maybe pass a quiz, your this is not powerful.  That is not why you teach.  Why make it what you teach?

The trick is to devise ways to teach kids how to think, to encourage creative problem solving and develop communication … to give them an education, while they learn how to factor a polynomial.

Personally, I never want a student to say to me, “Mr. Brown, you’re the only reason I got through math class.”  That’s too low of a standard.  That is exactly the this that makes THE question so damning to our efforts.  I don’t show up every day so my students can pass a quiz or test, or get a minimal passing grade in math class.

I show up every day to provide a challenge to my students, so they can test themselves and be better tomorrow than they were today.  And by better, I don’t mean greater proficiency at rationalizing the denominator. I mean of better mind.

To me, the best compliment a student can give a teacher is, “You taught me how to learn.”  In learning to learn, all of the pieces of an education are there.  To know how to learn you must be a problem solver, a critical thinker, be reflective, confident, and resourceful. A student that knows how to learn is prepared for an unknown future.

Ask yourself:  By teaching this, what’s being learned?  Are you just rehashing the same old lessons, just giving the same information the students could get on Khan Academy?  Are you asking them questions that can easily be answered by PhotoMath?  Are you printing another worksheet off of KutaSoft?

Challenge yourself to raise the bar.  Forget the bean-counters.  They’ll be happy when they see the results because when a student that knows how to learn takes a silly test, they do well!  Not only that, they’ll stand out when compared with students who were taught the content of the course only.

Unfortunately, if you’ve been dragged to the point where you realize, this is not why I got into teaching, and it consumes your day, you cannot answer THE student’s question honestly without using the word never.

You’re unlikely to find much guidance in the education industry that will change the this in THE question.  The industry sells books and professional development designed to get students to pass the test.  Their livelihood is generated from keeping the this we don’t want in THE question.

It is on us, educators dealing with students daily, to change the this.

The next time a student asks THE question … ask yourself, why?  Why did they ask?  Which this am I serving, the one designed for test scores, or the one educating students?

## Thoughts on Teaching

Foundation

1. The goal: If the question, "When am I going to use this in my real life," derails your class, there's a problem with your purpose and goal. The truth is, almost nothing after taught 5th grade is knowledge used daily. The purpose of education is not to teach MLA formatting or how to factor a polynomial.

The goal is to develop a careful, thoughtful and resourceful young person that is adaptable, a problem solver and has perseverance. That's the destination. The particular subject serves as (1) the vehicle to arrive at the destination, and (2) an exploration into potential aptitude and interest, (3) as well as a foundation of reference knowledge.

2. Autonomy: When students understand they're in charge of education outcomes and find value and validation from their efforts, they'll perform.

In other words, when they do it for themselves and receive appropriate praise and feedback for progress, their potential and performance will increase.

3. Letting Go: Some kids aren't ready. I barely passed Algebra 1 as a freshman in HS...in fact, I'm sure that 60% final semester grade was rounded generously. Yet, I ended up with a BS in Math.

You, the teacher, cannot reach them all. Leave the door open, realize every misstep is a chance to teach them, but learning is done on their end, not ours.

If a kid fails, let them. Work with them to succeed, but hold firm to the standard. If you falter, and pass a student that didn't deserve it, the value of the accomplishments of other students will be discounted.

Why I'm sharing this is to color this short story:

The last three years I had 100% passing rate by all takers, not cherry picking, on IGCSE, around 10% passing rate in AZ. This year I'm pretty sure at least one student will fail. They earned the first F grade I have assigned in six years in that class.

That student just wasn't ready. At the end, the student came begging to get a passing grade. I explained to the student that while they were close to passing, to change their grade would be a grand insult...it would say that I did not believe they were capable of performing as well as their peers.

The next day the student approached me. I thought, ut oh, more grade grabbing negotiation...but to the student's credit, they just thanked me, said they're glad for the F and will do better in the future. No more crying, no hang-dog look...but instead a confidence because the student was capable and will be in the future. Perhaps now, the student is ready.

I don't want students to say, I only got through math because of you, Mr. Brown. That would make math the destination, not the vehicle. Best compliment a teacher can get is, you taught me to learn.

## #REDforED – Arizona’s Working Poor – Arizona Educators United

In the year 2000, Arizona voters said that education was important the sustainability of Arizona’s economy and society.  They voted in Prop 301 which promised to keep teacher salaries competitive by providing cost of living increases and performance pay, among other things.  The state legislature has failed to exact the will of the voters and has instead acted on “their will.”

A lawsuit was filed to restore the missing funding for education and the state’s response was to propose Prop 123, which would borrow money from the land trust.  This was spun as a way to “pump money into education,” but in fact would settle the bill for \$0.07 on the dollar owed to the state’s voters, in order to fulfill Prop 301.  The ruse worked and the proposition passed … but was ultimately determined illegal by a federal judge.

Now we find ourselves with a teacher shortage, one that threatens to be a true crisis.   The short version of the story is that teachers are making less take-home today than they were 5 years ago.  Adjusting for inflation, teachers made substantially more a decade ago, and more than that a decade before that!  Below is a short video that lays out the situation today:

To learn what the #REDforED movement is all about, here’s a short video, less than a minute long:

To get involved, here are a few links.

The first link is a nonprofit that I have started, which is why content here on The Bearded Math Man has slowed.  (We are pretty well up and running, and I have a big project ahead for BMM).
Arizona’s Working Poor

Arizona Educators United

Save Our Schools – Arizona

AZED101

## Our Youth Deserve Better – Computer Based Learning

There has been a push for computer-based learning in public education for about a decade or so now.  The thinking is that students can go at their own pace, have optimally focused and differentiated remediation and instruction, and thus, students will perform better.  That’s the sales pitch, anyway.

I teach remedial math courses part time at a community college (the observations made here pertain to all of education not just math), the shift was made so that 100% of these remedial math courses were taught on such computer programs.  Students take placement tests where their strengths and weaknesses are accurately identified and they then work their way through lessons and assignments, with help along that way that addresses their specific short-comings.  If students grasp something easily they can move quickly through the curriculum.  Students that need more time can go at their own pace.  At the end of the section (or chapter), students take a test and must show a predetermined level of accuracy before they’re allowed to move forward.

It sounds great, but it doesn’t work.  Even if it did work and students could pass these classes in a way that prepared them for higher level classes, it would be a failure.   The purpose of education is not future education.

The ugly truth here is that we’ve lost sight of the purpose of education.  Education has become a numbers game where schools receive funding based on graduation rates and percentages of students passing multiple choice tests that have mysterious grading schemes behind them (70 multiple choice questions will be graded on a scale of 450 points, for example).  We lull ourselves into believing we are servicing our students if they graduate or our school surpasses the state average on these tests.

The truth is that the quality of education is rapidly decreasing, seemingly in direct response to the remedies that seek to reverse this trend.

The question often asked by students, in minor rebellion to the tasks at hand in class, “When am I going to use this in my real life,” needs to be carefully considered, with honesty, by the public and by educators.

The particular skills and facts being tested are of little to no importance.  What is important is the ability to be teachable, the ability to learn, which requires a lot of maturation, determination, focus and effort.  The purpose of education is to create an adaptable person that can readily latch onto pertinent information and apply previous learning in new ways.  An educated person should have the skills to adapt to an unknown future, a future where they are empowered to make decisions about the direction of their own lives.

Absolutely none of that happens in a computer course.  The problems are static, scripted and the programs are full of basic “If-Then” commands.  If a student misses this question, send them here.  There’s no interpretation of why a student missed.  There’s no consideration of the student as a sentient being, but instead they are reduced to a right or a wrong response.

What do students gain from computer courses?  They gain those specific skills, the exact skills and knowledge that will serve little to no purpose at all in their lives after school.  But, they’ll gain those skills in a setting with a higher student-teacher ratio (fewer teachers, less students), and where the teachers need not know the subject or how to teach.  That’s right, it’s cheaper!

But the cost is enormous.  Students will be trained how to pass tests on the computer, but will not be receiving an education. They will not develop the interpersonal skills required to be successful in college or in the work place.  They will not develop as people.  They will miss the experiences that separate education from training.  They will be raised by computers that try to distill education down to right and wrong answers, where reward is offered for reciting facts and information without analysis, without learning to consider opposing points of view, without learning how to be challenged on what it is they think and believe.

Our youth deserve better.  They deserve more.

Not only that, our young teachers (and we have an increasingly inexperienced work force in education), deserve better support from within education.  Here in Arizona the attitude from the government is that the act of teaching has little to no value, certainly little to no skill, and that anybody can step in and perform the duties of teaching in a way that services the needs of young people.

And while those in education throw their hands up in disgust, they follow suit by finding quick, easy and cheap solutions to the ever-expanding problem of lack of quality education, especially here in Arizona.  Instead of providing meaningful professional development and support for teachers, teachers are blamed for their short comings.  Instead of being coached and developed, they are being replaced by something cheaper and quicker, something that is fully compliant.

I fully believe that a teacher that can be replaced by a computer should be.  I also believe that a computer cannot provide the inspiration, motivation, the example, mentorsing and support that young people need.

The objection to my point of view is that teachers aren’t being replaced, they are still in contact with students.  This is true, the contact exists, but in a different capacity.  Just like iPads haven’t replaced parents, the quality of parenting has suffered.  The appeal of having a child engaged, and not misbehaving, because they are on a computer, or iPad, is undeniable.  But the purpose of parenting is not to find ways for children to leave them alone.  Similar, the role of education is to to find ways to get kids to sit down and pass multiple tests.  Children are difficult to deal with.  Limiting that difficulty does not mean you are better fulfilling your duty to the young!

The role of a teacher in a computer-based course is far removed from the role of a teacher in a traditional classroom.  While students are “learning” from a computer, the role of the “teacher” is to monitor for cheating and to make sure students stay off of social media sites.  Sometimes policies are in place where teachers quantitatively evaluate the amount of notes a student has taken to help it seem like a student is performing student-like tasks.

Students learning on computer are policed by teachers.  The relationship becomes one of subjects being compliant with authority.

The most powerful tool a teacher has is the human connection with students.  That connection can help a student that sees no value in studying History appreciate the meaning behind those list of events in the textbook.  A teacher can contextualize and make relevant information inaccessible to young learners, opening up a new world of thinking and appreciation for them.  None of that is tested of course.

A teacher inspired me to become a math teacher, not because of her passion for math, but because of how she conducted her business as a teacher.  Before that I wished to work in the Game and Fish Department, perhaps as a game warden.  That would have been a wonderful career.  Consider though, over the last decade, I have had countless students express their appreciation of how I changed their thinking about math, how I made it something dynamic and fluid, something human.  Math went from a barrier, in the way of dreams, to a platform, upon which successful can be realized.  Those things happened because of human connection.

We owe our youth more.  They deserve better.

It is time to unplug.

## Teaching Conceptual Understanding Flow Chart for Educators

Focus on Conceptual Understanding
Flow Chart
for
Educators

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).

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.

Philip Brown

## What Do Grades Really Mean?

The following is highly contentious.  Many of the situations discussed here should ultimately be considered on an individual basis.  The purpose of this is not to create a rubber-stamp solution to all problems that arise with grade assignment and student ability and or performance, but is to provide a general framework so that those individual decisions can be made in fairness and with respect to what is best for the student.

In a previous post I asked about a student in summer school that obviously knew Algebra 1 (he earned 100% on his quizzes and tests), but failed during the year because he didn’t do his classwork.  The question is, Does he deserve to fail Algebra 1?

When you flip the situation around it is equally interesting.  There are many kids who work hard, but do not really understand or learn the math.  Do they deserve to pass based on the merits of effort?

The real issue with both of these situations is what grades mean, or what should they mean.  When I worked at Cochise Community College I adopted their definition of letter grades which is described below:

A – Mastery

B – Fluency

C – Proficiency

D – Lacking Proficiency

Those are clean and inoffensive definitions of grades.  A student with an A has mastered the material.  To be fluent means you can navigate the materials but not without error.  To be proficient means you can get the job done, but there are some gaps in ability, but the student can demonstrate a measurable level of command of all of the objectives. Students who earn a D are not able to demonstrate proficiency.

A student who struggles with the material does not deserve an A, even if they worked harder than those who earned an A.  This might seem unfair, but unless the objective of the class is to teach the value of hard work, to reward the hardworking, but barely proficient, student with a label of mastery is to cheat the student and cheapen the merit of your class.

Do these definitions mean that a lazy kid that get 95% on the final exam deserves an A, but that a hard working kid that gets a 52% on the same final deserves an F?  I say, with a few qualifications, yes.

Is this really fair to the student who works hard but has not yet realized an appropriate level of mastery to be awarded a passing grade? (I used the phrase, “has not yet,” instead of, “cannot,” to acknowledge the belief that students can learn, and if they are motivated and working, the only question will be the time scale of when they learn the material.)

I would say, for a math class, that the best thing that can happen is they are awarded the appropriate grade, an F.  Consider if this student is given a passing grade and the class is a prerequisite course?  They’re truly set up for failure in the subsequent class.

There is perhaps no worse example of bad teaching that remains within legals bounds than to inappropriately assign grades to students.  If a student deserves a C based on ability, but is given an A based on effort, they will believe they are doing everything right and do not need to improve in order to achieve similar success in subsequent courses.

But to give a student who possesses mastery a failing grade in a class because of lack of work ethic is to teach the student that passing classes is a matter of compliance.  Behave and you’ll be rewarded.  Those kids are taught that grades are not a reflection of knowledge or ability, and that means that education is not about learning.  To me, this is an injustice.

I do not believe in the efficacy of these objective lessons.  That would be, failing a student based on the notion that they do not deserve to pass because they are lazy. I believe that given meaningful and challenging opportunities, most of these highly intelligent, but seemingly lazy, students will show themselves to be hard working with amazing focus and direction and incredible capacity for quality work.

What about percentages.  Is it appropriate that an 80% is a B, if a B means fluency?

When I first began teaching I would have said, absolutely, a student does not deserve an A if they scored an 87% on their test.  Since then I’ve changed my mind.  Some topics require higher than 90% accuracy to be awarded an A, while with other topics, mastery might be far below 90%.

The level of complexity, variability of solutions and length of assessment all must be considered.  This is why sometimes a grading rubric is far superior to assigning grades based on a percentage of correctness or completion.

I teach a curriculum that is designed and tested by Cambridge University, the IGCSE test is what students take.  They have a very different way of assigning and defining grades than we use here in the United States.  Without going into details about how they do the specifics, they assign large portions of credit based on evidence of appropriate thinking.  In other words, if a student demonstrates understanding they will receive passing credit.  But, to achieve a high grade, mastery is truly measured.  And yet, in math at least, the percentages of correctness for mastery are usually in the mid-70’s.  This is because the nature of the questions asked are often non-procedural and the method of solution is not clear, students cannot be trained on how to answer the questions they face on IGCSE exams.

How a student can earn a grade varies, or should, depending on subject and age, and perhaps even minor topic within the subject.  I believe that separating student work into weighted categories is an appropriate method of helping make transparent to the student how their grade will be assigned.  It also by-passes the tricky question of, “What is a point?”  For me, a homework assignment is worth 5 points, they’re assigned daily, except Fridays, for a total of 20 points for the week.  Yet, a quiz might only be worth 12 points, but will be a far more accurate representation of student’s ability on the topic.

By assigning weights to the categories, this can be easily balanced.  This begs the question, how do you weight the categories?

But what about the student who works, performs all assigned tasks, but can only demonstrate a level of understanding best described as “Lacking Proficiency?”  Shouldn’t hard work be rewarded?

And whatever your beliefs on these questions, would your opinion change depending on the age of the student, or perhaps the subject?  Should a Chemistry student be rewarded for effort in the same way they’d be rewarded for effort in a Dance class?

At some point, nobody cares about potential or effort.  If a child’s mother wants his room clean, she knows he has the potential to clean it, but if he fails to do so, the potential matters not.  And if he’s really trying to get it done, but cannot master the discipline to carry through the task, does the effort really matter?

Here is how I set up my grades for high school.  It is nuanced and complicated, but I’ll give the outline.  Note that for college classes I use a different system.

In high school I weigh categories of grades and have changed the percentages and categories over time until I settled on what seems to work best.  These work for my students because it seems to motivate the lazy-smart students and also rewards the hardworking – low aptitude student, because if they remain persistent, they will learn.

Tests – 40%
Quizzes – 25%
Homework – 25%
Other – 10%

I believe extra credit should be awarded for students that perhaps help others, or for extraordinary performance.  However, a student should NOT be allowed to raise their grade through extra credit.  That is, at the end of the term a student is given a pile of work, that if performed, will raise their grade.  This is bad teaching!

The difference between a quiz and a test is similar to the difference between a doctor’s check-up versus an autopsy.  The quiz is a chance to see how things are going and adjust accordingly.  The test is final.  In high school I award credit for homework based on completion, but do not accept late homework.

Rewarding Effort?

While I wish that effort equaled success, it doesn’t always work that way…depending on how you define success.  For example, I can try as hard as possible to paint a world-famous landscape, but will likely fail if my measure of success is producing a world-famous piece of art. That said, I believe there is a reward beyond measure only discovered with true effort.  Our potential, our best, is not static, it changes.  It changes in respect to our current level of effort.  We can never fulfill our potential, you see.  It is always slightly above how hard we are trying.  So, if you’re not really trying, your potential decreases, but if you’re pushing your limits, the limits themselves stretch.  That is the real downfall of those with an inherent talent that never learn to push themselves.  Their potential decreases, dropping down to just higher than their level of effort.

I greatly reward effort, encourage it and makes positive examples of how effort promotes success.  However, I do not assign grades to effort.  How hard someone needs to try in a given subject to be successful varies entirely upon the student’s aptitude.  And suppose you have a truly gifted student, they could be great, if they learn to work hard, right?

Well, perhaps, but there’s more than work ethic involved in greatness.  What role does passion play?  Take a great young musician and over-structure their training and practice, they’ll burn out.  You’ll snuff their passion.

I asked the boy whose situation started this whole conversation if he felt he deserved to be in summer school.  Before he answered I explained that I didn’t have an expected answer, I didn’t really know if he belonged in summer school or not.  Without hesitation, he said he did deserve summer school, because, he said, he was lazy.

So maybe the kid will learn that if he’s lazy he gets punished.  But he also learns that grades are arbitrary, with respect to ability.

I do not like objective lessons, do not believe them to be effective.  I prefer a punishment that fits the crime, but also one that redirects the offender, allows them to correct their action.

I cannot say in this child’s case specifically, I was not there and I am not judging his teacher, but perhaps a quicker punishment that redirected him could have also taught him that being lazy was unacceptable and at the same time also allowed him to see grades as a reflection of his abilities.

All that said, this is highly contentious and varies incredibly depending on particular situations of students.

Let me know what you think, agree or disagree.  Leave me a comment.

## The Smallest Things Can Cause Huge Problems for Students

preemptive

Pre-Emptive Explanation

It is often the case,
for the mathematically-insecure, that the slightest point of confusion can
completely undermine their determination.
Consider a beginning Algebra student that is learning how to evaluate functions
like:



f(
x
)
=3x
x
2

+1

f(
2
)

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa

A confident student is
likely to make the same error as the insecure student, but their reactions will
be totally different. Below would be a
typical incorrect answer that students will make:



f(
2
)
=3(
2
)

2
2

+1

f(
2
)
=6+4+1

f(
2
)
=11

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaeyypa0JaaG4mamaabmaa
aaikdaaaGccqGHRaWkcaaIXaaabaGaamOzamaabmaabaGaaGOmaaGa
caaIXaGaaGymaaaaaa@4F4E@

3, and the mistake is that -22 = -4, because it is really subtract
two-squared. And when students make this mistake it provides a great chance to
help them learn to read math, especially how exponents are written and what
they mean.

Here’s what the



f(
x
)
=3x
x
2

+1

f(
2
)
=3(
2
)
+
(

2

)

2

+1

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa
ZaWaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacq
GHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa
ey4kaSIaaGymaaaaaa@4E85@

A confident student
will be receptive to this without much encouragement from you. However, the insecure student will completely
shut down, having found validation of their worst fears about their future in
mathematics.

There are times when
leaving traps for students is a great way to expose a misconception, and in
those cases, preemptively trying to prevent them from making the mistake would
actually, in the long run, be counter-productive. Students would likely be mimicking what’s
being taught, but would never uncover their misconception through correct
answer getting. Mistakes are a huge part
of learning and good math teaching is not about getting kids to avoid wrong

But there are times
when explaining a common mistake, rooted in some prerequisite knowledge, is
worth uncovering ahead of time. This -22
squared is one of those things, in my opinion, that is appropriately explained

## Math Can Not Be Taught, Only Learned

Math is something that cannot be taught, but can be learned.  Yet, math is taught in a top-down style, as if access to information will make a student successful, and remediation is rehearsal of that same information.  Earnest students copy down everything, exactly like the teacher has written on the board, but often still struggle and fail to comprehend what is happening.  I argue that if copying things down was a worthy exercise, why not just copy the textbook, cover to cover.  Of course such an activity would yield little benefit at all because math is a thing you do more than it is a thing you know.  Math is only partly knowledge based and the facts are rarely the issue that causes trouble for students.  I’d like to propose that you, either parent, student, administrator or teacher, considers math in a different light and perhaps with some adjustment the subject that caused such frustration will be a source of celebration.

There are many things that cannot be taught but can be learned.  A few examples are riding a bike, playing an instrument, creative writing and teaching.  Without question knowledge is a key component to all of these things, but it is rarely the limiting factor to success or performance.  Instead, the skill involved is usually the greatest limiting factor.  I argue that to learn these things a series of mistakes, incrementally increasing in complexity, must be made in order to learn.  Let’s see if this will make more sense with a pair of analogies.

First, watching someone perform something that is largely skill-based is of little use.  Consider driving a car.  A fifteen year old child has spent their entire life observing other people drive.  And yet, when they get behind the wheel for the first time, they’re hopelessly dangerous to themselves and all others on, or just near, the roads!

Second considering learning to ride a bike.  Sure, the parts of the bike are explained to the child, but they have to get on and try on their own.  The actually learning doesn’t really occur until the parent lets go (letting go is huge!) and the child rolls along for a few feet until they fall over.  Eventually they get the hang of the balance but then crash because they don’t know how to stop.  After they master braking they crash because they don’t know how to turn.  And then speed, terrain, and other obstacles get thrown in the mix.  Each skill must be mastered in order.  Preemptively explaining the skills, or practicing them out of context does not help the child learn to ride a bike.  They must make the mistakes, reflect, adjust and try again.

What a math teacher can provide is the information required, but more importantly feedback, direction and encouragement.  If a student understands that making mistakes isn’t just part of learning, but that a mistake is the opportunity to learn (and without it only imitation has occurred), and a teacher helps provide guidance, encouragement and feedback, then both parties will experience far greater success.  When a math teacher completes a problem for a student it is similar to an adult taking the bicycle away from the child and riding it for them.  When a student gives up on a problem, it’s as if they stopped the car and got out, allowing the adult to drive them home.

The job of math teacher is perhaps a bad arrangement of words.  Coach, mentor or sponsor is perhaps more appropriate.  There is no magic series of words, chanted under any circumstance, that will enlighten a struggling student.  The frustration making mistakes should be cast in a different light, a positive light.  The responsibility of learning is entirely on the student.  They cannot look to teachers, friends or tutors for much beyond explanation of facts.

In a future post I will explain how too much direction and top-down teaching of math promotes failure of retention and inability to apply skills in new applications.  But for now, please consider that math cannot be taught.  A teacher cannot teach it, but can help a student to learn.