# 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/

## 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.

## Math is Hard

Math is Hard

A typical conversation with a failing math student, with a failing math student’s parents, or with a counselor or administrator about a failing math student either directly sites this, or is pulled in a direction like driftwood in a tide by the fact that math is hard.

A common conference would go something like:

Parents:  Why is my child failing math?

Child:  Because math is hard.

Parents and other interested parties accept this as sufficient reason and place the onus back on me as though I can alleviate the very nature of the subject.

I am completely fed up with the observation that math is hard.  And while refraining from profanity in response to this excuse should award me man of the year, I get it.  Math is hard.  No kidding!

It doesn’t matter what innate abilities someone has in math, eventually it will become difficult, confusing and … well, hard!  It is something everybody that learns math must face.  They must learn how to learn something that is hard, demanding and elusive.  That whole experience of, “Oh, I got it,” and then ten seconds later, “Wait a minute, I don’t get it anymore,” is something we all suffer.

When I was taking math courses in college I was certainly challenged.  At one point a formal proofs/topology class was really destroying me.  It was designed to be a bit of a gate-keeper of a course.  If you failed to posses the ethic and fortitude required to be successful in mathematics, this class would ferret out such things.

While taking this class my birthday rolled around.  I am the oldest grandchild on my father’s side and share my birthdate with my grandmother.  I am the oldest grandson.  So birthday parties are kind of a special thing for the two of us!

At the party I showed up with a small dry-erase board, a marker, rag for erasing and my book.  I didn’t have an assignment, no test coming up soon, but was well aware that I “didn’t get it.”  While friends and family hang out enjoying themselves I sat in a room with the door closed and practiced.

To be clear, I wasn’t struggling for mastery, I wasn’t fighting to get an A.  I was struggling just to get by, just to get a C in the class.

So yeah, math is hard.  Education changes you, or it should.  I’d argue if it was easy and didn’t change you, what is the purpose?  Sometimes you have to fight to get things done.

Think math is hard, try beating addiction.  How about facing cancer?  Raise children.

The difference between those that get math and those that don’t is a simple one…some are fighters while others site difficulty as sufficient reason to surrender and quit.  While that may sound harsh, there’s a little more to it than just that.

Fighters have faith and patience.  They have faith that through perseverance they will overcome.  They have the patience to persevere through hard times, knowing that it will pass and the result will be worth the endeavor.

By facing the struggles presented in math that perspective can be gained.  If math is hard for you it offers you an opportunity to learn that if you persevere, keep faith and have patience with yourself you will overcome.

## Accessing Prior Knowledge in a Way That Uncovers Misconception - A Lesson

If what we are doing in class today does not promote our understanding tomorrow, we have wasted our time.

The order of operations and arithmetic with signed numbers combine to be the downfall of many high school math students.  As with many things, remediation does not work.  While exposing mistakes with these topics in applied contexts can be powerful, that also sometimes leaves students with the wrong impression that their new conceptual understanding is flawed.

For example, say a student is graphing a polynomial, factorable, function and they make sign errors.  Their roots might be off and they’ll likely consider their new understanding to be flawed when in fact they’re just adding incorrectly.

With careful discussion and in an environment that encourages exploring mistakes, this is a fantastic way to shore up weaknesses that are prerequisite in nature.  However, with a shy student or even a new student, this often just leads to frustration.

I’ve attached a link to a free PowerPoint activity/lesson that is designed to get students to explore combinations of arithmetic operations and the order of operations to arrive at an answer.  It challenges their understanding of both integer operations and the order of operations in a way that does not just leave them wrong, but empowers them to change what they’re doing and make use of their wrong answers.

The activity uses the scary clown from the Saw movies.  He wants to play a game.  If you’ve never seen those movies they’re likely not your taste, but teens love them!  The, playing a game, with this reference is powerful to them.

The game is that there are four-fours with spaces between, and they equal a number, like 1.  Students can add in whatever operation signs and or parentheses they want in order to make the total equal 1.

An example is:      4  4  4  4  = 3

Students may add:  + - × ÷ [  ]

So an attempt might be:  -[4÷4]+4÷4

This is of course wrong, but there will be more gained from discussing why it is wrong than simply sharing what is right.

There are often many proper solutions, and if a student gets an answer quickly, have them see how many they can find.  Another thing to do with a student that gets it too quickly is have them help another student, but not in a way where they share the answers but instead their thinking, helping the struggling student to arrive at their own answers.

Depending on the ability and enthusiasm of the class you can take one of the problems in the lesson and write four or five of that problem on the board and have people come up to the board and write in their arithmetic notation and parentheses.  Then when all of the slots are filled, you can discuss what’s right and wrong, and why, as a class.

This would also make a fantastic white-board activity, although recording thoughts and realizations in their notes is very important, because:

If what we are doing in class today does not promote our understanding tomorrow, we have wasted our time.

## 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

## Why Remediation Fails

Why Remediation Fails

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What can we do?

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

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

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

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

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

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

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

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

Philip Brown

## Try to Solve This Problem … without Algebra

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

Given:  3ak

And:  ax = 4k

What must x be?

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

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

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

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

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

Given:  3ak

And:  ax = 4k

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

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

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

Let’s look at the second statement now.

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

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

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

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

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

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

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

## Vestiges of the Past Making Math Confusing

Something in Math HAS to Change

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3 + 3 + 3 + 3 + 3 = 3 × 5

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

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

3 + 3 + 3 + 3 + 3 = 15

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

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

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

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

3 + 3 + 3 + 3 + 3 = 15

The second step is six.

Another way to see this is shown below:

3 →6→9→12→15

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

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

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

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

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

Exponents:  3⁵ = 3 × 3 × 3 × 3 × 3

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

Pretty cool, eh?

Just FYI, 3 times itself 5 times is 243.

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

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

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

3 × 3 × 3 × 3 × 3 = 243

3→9→27→81→243

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

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

## Confuse Them So They Learn

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

While preparing all of this information I was thinking:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Once again, thank you for your time.

## 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.