Favorite Technology

My Favorite Technology

With the invention of the radio came claims that, “This will revolutionize education, forever.”

Then came television, and more claims that, “This will revolutionize education, forever.”

Then came the VHS player.  You guessed it, more, “This will revolutionize education, forever.”

Then the internet came along, and louder than ever were the claims that, “This will revolutionize education, forever.”

In truth, all of those pieces of technology have revolutionized education.  Education is now, more than ever, about coming up with new ways to make information increasingly accessible and more engaging.  And, more kids than ever are starting college.  What’s not to love, right?

Well, there is plenty not to love.

The reason all of those pieces of technology were destined to change education forever was because they were going to allow experts in particular fields to communicate with students.  The thinking was that books and stuffy teachers were making learning unnecessarily difficult.  By allowing students to bypass the texts and teachers to gain access to the content to be learned, they’d learn better and faster.

It makes sense to me.  As an adult, if I want to learn about writing a blog, for example, I do a search on the internet and find some self-proclaimed blog expert.  I watch their videos, read their blogs for advice, and give it a shot!  Or, if I want to learn to change the air filter on a new car, and I can’t seem to figure it out myself, I look for videos on YouTube.  Technology like the internet has provided me with so much greater access to information that has enriched my life than was afforded before the internet.

That’s how it is supposed to work with students, too.  A kid might be stuck in Algebra 2; logarithms killing my grade, mister!  They look up “logarithms,” on the internet and there are tons of helpful videos.  The student learns how to do logarithms, and their grade is saved.

It sure seems like it is all on the up and up, right? Well…

With up to 60% of college freshmen needing remedial math classes, I’d say these revolutions have not had a positive outcome for students.

At this point you might be thinking, here’s another doomsday message: Kids these days are horrible, fear for the future.  I promise you, this is not a doomsday message.  Education needs to improve, and that’s what this blog is about.

Let’s take a step back and look at the example where I learned to change an air filter from a YouTube video.  Was I educated?  Was I trained?  What’s the difference?

There is a huge difference between training and education.  Training equips the trained with specific skills and knowledge that the trainer knows the learner will need, when they will need it, and how they will apply what they’ve been trained to do.  Training is what happens when you get a new job.

Training could be said to equip a person with a specialized tool.

Education is different.  People often complain why they weren’t taught certain practical skills in school.  The message is that education is worthless.

Education equips a person with the ability to find the specialized tool they need and then figure out how to use it.  While training prepares someone for a known task, education prepares someone for an unknown task.

When a student watched a video on the internet about logarithms are they being trained or educated?

If the intent of the video is to help a student complete homework and pass a quiz, then the person knows exactly what the student will need to be able to do, and when they’ll need to do it.

This is a seemingly subtle difference.  The difference between training and education is anything but subtle.  It is of massive consequence.  Why?

One attribute of an educated person is that they quickly incorporate new, more effective approaches.  By contrast, a trained person resists new methods, regardless of efficacy.  Education makes a person adaptable.

The reason that the radio, television, videos, and the internet have failed to improve educational outcomes is because they have not addressed the short-comings of a textbook.  All of these sources provide the same information, and use the same approach.  They disseminate information.

A good teacher entices curiosity, finds what motivates students to learn, and provides educational experiences for students.  That quality human connection is what makes education happen for students that are otherwise uninterested in being educated (which is an overwhelming majority).

There is such a massive push, with some much inertia behind it, to focus on comprehensible input, scaffolding, all of the components of teaching examined in isolation and treated with a leaning towards training a teacher instead of educating them about teaching, that it feels like quality teaching is becoming a lost art.  Maybe that’s a skewed perspective having only taught in Arizona, which by nearly every metric, is the worst state for education in the US.

What students need is a reconnection with their instructor.  The instructor needs to get in-tune with the needs, pace, and interests of the students.  PowerPoints, videos, SmartBoards, Chrome Books, and the like focus on the dissemination of information.

That is why my favorite piece of technology is the document camera.

Wait … hear me out.  I believe that it can be the most powerful piece of technology for a student in a math class.

The first reason why I love the document camera deals with how mathematics is a written, not spoken language.  The spatial arrangement of characters conveys meaning.  The way math is printed on paper, or a PowerPoint, and the way it is written on the board, is different than how math is written and performed on paper.  The physical parameters change the way we write.

In the image below you can see a lot of repeated information.  Some of the information is written mathematically, some of it is written in English, and there are arrows and annotation that connects the two.  These annotations are done in real time in response to questions from students and answers by students to my questions.

 

The way we write math greatly impacts how we perform the math.  This is an overt example, but I think it will make the point.  The first expression below is extremely difficult to deal with, while the second has the same meaning and is quite easily understood.

A bright student might realize to rewrite the first expression as the second.  But an average student will realize with the second that they only need to add the exponents, and they’re done.  This is not an example of how the interaction of math is different when writing on paper, versus typing.  What it does show is that how math is written greatly impacts the interpretation of the meaning.  That interpretation and translation occurs more naturally when written in real time compared to being typed.

What the example above does show is how writing mathematics drastically changes our interpretation of what is written. In effect, it rephrases the information.

What the document camera does is allow the teacher to show students, in real time, the mechanics of the mathematics, while allowing for discussion and annotation of the theory of the mathematics.  It does these things at a writing pace.

Here is a picture of a lesson in Algebra 1.

In this picture what you see is how a problem can be broken apart in response to what it is that the students in the classroom, at that very moment in time, are struggling with.

While this could have been addressed while writing on the board, or even in a PowerPoint lesson, it was more apparent to me, as the teacher, because I was going slower.  I was asking more questions, students were asking more questions.  Teaching with the document camera really can improve the dialogue between teachers and students, changing it from speaking to conversing.

While a conversation can be had over a YouTube video with students, or during a PowerPoint presentation, it is more difficult.  The pace is different; the engagement of the students is different.  When watching a video, or watching a PowerPoint, students are … watching.  If they begin writing, it is often dictation that’s being performed.

There is certainly a measure of dictation happening by students when engaged in a lesson delivered through a document camera.  However, the switch to addressing a question or point of confusion during a lesson in a way that students incorporate that response as a natural part of the lesson, happens naturally when using a document camera.

Consider a lesson about exponents.  No matter your teaching experience, you cannot anticipate all possible misconceptions, prior or actively developing, and dispel them pre-emptively.  Along the way there will be confusion and misunderstanding.  It is when the confusion is discussed, and properly addressed, that learning really takes place.

When that confusion is brought forward by the students, in a lesson delivered through a document camera, the question can be written, explored, answered and summarized in a way that feels natural for the students.  They’ll recognize this as part of the lesson, not a tangent.

In the picture below you will see a refocusing of a concept learned the day before.  In the day before this lesson, students really struggled to identify separate bases in one expression.  They could not distinguish between things like

Of course confusion is exposed and can be properly addressed in other delivery forms.  The message here is not that other methods are ineffective.  However, students typically view a diversion from the script as tangential to the lesson objectives.  They do not recognize that the diversion is the most important part of the learning.  How could it be when it doesn’t have pretty animations and bold, underlined font?

The last benefit of a document camera is pacing.  Students need think time.  The pace of delivering a message is slowed when you, the teacher, are essentially taking notes with the students.

This allows them to think about what is being written while they write it.  After all, you won’t be reading what you’re writing.  Instead, your writing will be a summary of what’s been said!

With the slower pace, which has a higher engagement because students are using the time to carefully take notes, comes better questions from students.  In response to these questions you can naturally annotate the notes throughout the lesson, highlighting the source of the confusion for the students continually.

What all of this means is that by using a document camera, a lot of the elements of quality teaching are naturally accessible.  The pace is naturally improved to match the needs of students, the dialogue is improved, the exploration of misunderstanding is seamlessly incorporated into the lesson itself, without feeling tangential to the learning.

And all of that, especially the exploration of misunderstanding, provides the teacher with opportunity to provide for students what technology cannot do.  It allows you to easily step into a role that you must carve out for yourself when using more advanced technology.  The most important function of the teacher is to entice interest in students, to discover their motivations and to teach them instead of cover material.

How many times has this happened:  You teach a lesson.  The lesson is organized, complete, you’re proud of how it is constructed and delivered.  The students seem okay.  But when they test, the results are horrible.

This is what happens when we focus too much on the material, too little on the students.  For me, anyway, the document camera really helps me to focus on the students.  This is especially true with low-achieving students.  They need more help, a slower pace, a more responsive teacher.  Low-achieving students are less adaptive, flexible, and have less inclination to explore and challenge their understanding independently.

I am not saying, of the document camera, “This will revolutionize education.”  The document camera, like all technology, is only as good as it is used.

What I have tried to show here is how the document camera naturally offers you opportunity to perform what cannot be scripted, what cannot be programmed into a computer, what need an expert on a video cannot fulfill.  Your role as a teacher is to teach students, not cover material.

Whatever technology you use will fail to be effective if it is not used in a way that furthers that connection between students and content.  If the technology only improves exposure to content, does not help students to engage with the content in a way that is challenging and builds conceptual understanding, then it, too, will be ineffective.

The take away is, there is not replacement for a good teacher.  Tools that are used to enhance what a good teacher provides for a student are great.  Tools that lose sight of what quality teaching is, ultimately, hamper the educational process and harm students.

 

Teaching Square Roots Conceptually

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.

24=62   or 4=2 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIYaGaaGinaaWcbeaakiabg2da9iaaiAdadaGcaaqaaiaaikdaaSqabaGccaqGGaGaaeiiaiaabccacaqGVbGaaeOCaiaabccacaaMc8UaaGPaVlaaykW7caaMc8+aaOaaaeaacaaI0aaaleqaaOGaeyypa0ZaaOaaaeaacaaIYaaaleqaaaaa@479C@ 

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.

 

1.2 asks, "What squared is 2?"2. If you square the number, 2, the product is 2. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8145@

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.

How is 9 like x2=9. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeisaiaab+gacaqG3bGaaeiiaiaabMgacaqGZbGaaeiiamaakaaabaGaaGyoaaWcbeaakiaabccacaqGSbGaaeyAaiaabUgacaqGLbGaaeiiaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaI5aGaaiOlaaaa@4620@ 

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

2×2. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIYaaaleqaaOGaey41aq7aaOaaaeaacaaIYaaaleqaaOGaaiOlaaaa@3A82@ 

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:

m×m. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGTbaaleqaaOGaey41aq7aaOaaaeaacaWGTbaaleqaaOGaaiOlaaaa@3AEE@

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. 

Given that the number k2=m, what is m? MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4raiaabMgacaqG2bGaaeyzaiaab6gacaqGGaGaaeiDaiaabIgacaqGHbGaaeiDaiaabccacaqG0bGaaeiAaiaabwgacaqGGaGaaeOBaiaabwhacaqGTbGaaeOyaiaabwgacaqGYbGaaeiiaiaadUgadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaWGTbGaaiilaiaabccacaqG3bGaaeiAaiaabggacaqG0bGaaeiiaiaabMgacaqGZbGaaeiiamaakaaabaGaamyBaaWcbeaakiaac+daaaa@571D@ 

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.

Is this true: 24=26 ? MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeysaiaabohacaqGGaGaaeiDaiaabIgacaqGPbGaae4CaiaabccacaqG0bGaaeOCaiaabwhacaqGLbGaaeOoaiaabccadaGcaaqaaiaaikdacaaI0aaaleqaaOGaeyypa0JaaGOmamaakaaabaGaaGOnaaWcbeaakiaabccacaqG=aaaaa@479A@ 

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.

2=1.4 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIYaaaleqaaOGaeyypa0JaaGymaiaac6cacaaI0aaaaa@3A09@ 

3+2=4.4 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgUcaRmaakaaabaGaaGOmaaWcbeaakiabg2da9iaaisdacaGGUaGaaGinaaaa@3BAB@ 

3×2=4.2 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgEna0oaakaaabaGaaGOmaaWcbeaakiabg2da9iaaisdacaGGUaGaaGOmaaaa@3CDE@ 

  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:

3+2  or  3×2. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgUcaRmaakaaabaGaaGOmaaWcbeaakiaabccacaqGGaGaae4BaiaabkhacaqGGaGaaeiiaiaabodacqGHxdaTdaGcaaqaaiaaikdaaSqabaGccaGGUaaaaa@414A@ 

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:

Is this true: 24=26 ? MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeysaiaabohacaqGGaGaaeiDaiaabIgacaqGPbGaae4CaiaabccacaqG0bGaaeOCaiaabwhacaqGLbGaaeOoaiaabccadaGcaaqaaiaaikdacaaI0aaaleqaaOGaeyypa0JaaGOmamaakaaabaGaaGOnaaWcbeaakiaabccacaqG=aaaaa@479A@

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

(26)2=24. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIYaWaaOaaaeaacaaI2aaaleqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGOmaiaaisdacaGGUaaaaa@3D46@ 

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,

26×26=2×2×6×6. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaakaaabaGaaGOnaaWcbeaakiabgEna0kaaikdadaGcaaqaaiaaiAdaaSqabaGccqGH9aqpcaaIYaGaey41aqRaaGOmaiabgEna0oaakaaabaGaaGOnaaWcbeaakiabgEna0oaakaaabaGaaGOnaaWcbeaakiaac6caaaa@468F@

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. 

2×2×6×6=4×6. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgEna0kaaikdacqGHxdaTdaGcaaqaaiaaiAdaaSqabaGccqGHxdaTdaGcaaqaaiaaiAdaaSqabaGccqGH9aqpcaaI0aGaey41aqRaaGOnaiaac6caaaa@44CB@

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:

Show that mnm=m3n2. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4uaiaabIgacaqGVbGaae4DaiaabccacaqG0bGaaeiAaiaabggacaqG0bGaaeiiaiaad2gacaWGUbWaaOaaaeaacaWGTbaaleqaaOGaeyypa0ZaaOaaaeaacaWGTbWaaWbaaSqabeaacaaIZaaaaOGaamOBamaaCaaaleqabaGaaGOmaaaaaeqaaOGaaiOlaaaa@4737@ 

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.

Simplify 48. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4uaiaabMgacaqGTbGaaeiCaiaabYgacaqGPbGaaeOzaiaabMhacaqGGaWaaOaaaeaacaaI0aGaaGioaaWcbeaakiaac6caaaa@4056@ 

48=2×24,3×16,4×12,6×8. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaaiIdacqGH9aqpcaaIYaGaey41aqRaaGOmaiaaisdacaGGSaGaaGPaVlaaiodacqGHxdaTqqa6daaaaaGuLrgapeGaaGymaiaaiAdapaGaaiilaiaaykW7caaI0aGaey41aqRaaGymaiaaikdacaGGSaGaaGPaVlaaiAdacqGHxdaTcaaI4aGaaiOlaaaa@529A@

48=16×3 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaI0aGaaGioaaWcbeaakiabg2da9maakaaabaGaaGymaiaaiAdaaSqabaGccqGHxdaTdaGcaaqaaiaaiodaaSqabaaaaa@3D31@ 
Write the square root of the perfect square first so that you do not end up with
34, MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIZaaaleqaaOGaaGinaiaacYcaaaa@3847@ which looks like 34. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIZaGaaGinaaWcbeaakiaac6caaaa@3849@ 

48=4×3 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaI0aGaaGioaaWcbeaakiabg2da9iaaisdacqGHxdaTdaGcaaqaaiaaiodaaSqabaaaaa@3C4F@.

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:

Simplify  18. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4uaiaabMgacaqGTbGaaeiCaiaabYgacaqGPbGaaeOzaiaabMhacaqGGaGaaeiiamaakaaabaGaaGymaiaaiIdaaSqabaGccaGGUaaaaa@40F6@ 

18=9×2 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaaGioaaWcbeaakiabg2da9maakaaabaGaaGyoaaWcbeaakiabgEna0oaakaaabaGaaGOmaaWcbeaaaaa@3C75@

18=32 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaaGioaaWcbeaakiabg2da9iaaiodadaGcaaqaaiaaikdaaSqabaaaaa@3A33@

(32)2=9×2 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIZaWaaOaaaeaacaaIYaaaleqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGyoaiabgEna0kaaikdaaaa@3EAD@

9×2=18 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiabgEna0kaaikdacqGH9aqpcaaIXaGaaGioaaaa@3C10@

18=18. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaaGioaaWcbeaakiabg2da9iaaigdacaaI4aGaaiOlaaaa@3ACE@

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 12 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaaGOmaaWcbeaaaaa@3789@ is not 32. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaakaaabaGaaGOmaaWcbeaakiaac6caaaa@3847@  It is not “three root two,” because (32)2=18, not 12. MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIZaWaaOaaaeaacaaIYaaaleqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGymaiaaiIdacaGGSaGaaGzaVlaabccacaqGUbGaae4BaiaabshacaqGGaGaaeymaiaabkdacaqGUaaaaa@4508@ 

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 12=32, MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaaGOmaaWcbeaakiabg2da9iaaiodadaGcaaqaaiaaikdaaSqabaGccaGGSaaaaa@3AE7@ 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 5238, MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaakaaabaGaaGOmaaWcbeaakiabgkHiTiaaiodadaGcaaqaaiaaiIdaaSqabaGccaGGSaaaaa@3AD8@ 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
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.

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.

Confuse Them So They Learn

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

While preparing all of this information I was thinking:

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

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

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

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

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

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

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

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

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

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

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

 

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

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

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

Once again, thank you for your time.

How to Be a More Effective Teacher

How to Teach Well

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

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

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

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

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

Wrong.

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

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

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

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

What can be done?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Or perhaps: Is zero odd or even?

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

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

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

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

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

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

Practice this phrase:  Go ask another student.

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

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

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

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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

 

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

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


The Purpose of Homework and My Response

The purpose of homework is to promote learning.  That’s it.  It’s not a way to earn a grade or something to keep kids busy.  It’s also not something that just must be completed in order to stay out of trouble.  Homework is a chance to try things independently, make mistakes and explore the nature of those mistakes in order to better learn the material at hand.

If students are not learning from the homework, it is a waste of time and effort.  There are a few things that could cause students not to learn from the homework.  Even if the assignments are of high quality, without the reflection and correction piece, students will not learn much from homework.

Reflection and correction go together.  It’s not about getting right answers, but thinking about what caused mistakes, identifying misconceptions or procedural inefficiencies and replacing those.  To reflect a student should NOT erase their incorrect working but instead should write on their homework, in pen, what went wrong and what would have been better.

It is quite possible more can be learned when reviewing homework than any other time.  It is certainly a powerful experience.

Textbooks and videos, tutors and peer help offer little appropriate support to help make homework, or practice, meaningful.  Textbooks only provide correct answers, YouTube videos usually do similar treatment to topics as textbooks offer.

I wish to help students learn and believe that reviewing work that has been done is too powerful of an opportunity to pass.  The trick is, how can I provide reflection and insight when to someone I am not sitting with and talking to?  I think I can help provide this reflection piece by doing all of the practice problems myself on a document camera and discussing pitfalls and mistakes, as well as sharing my thinking about the problems as I tackle them.  Further, I can share typical mistakes I see from students as they are learning topics.

So as I develop the Algebra 1 content I will be working on adding videos and short written responses to the assignments to help students think about what they’ve done, its appropriateness, correctness and their level of understanding.