mathematical literacy

Mathematical Literacy

Integrating Literacy Development
Opportunities in Your Instruction

 

A few years ago I attended a professional development seminar designed to help American math teachers integrate best practices and strategies required for their students to be successful with the Cambridge IGCSE program.  I was a guest “reference-source,” in the seminar because of the success my students experienced in the program over the prior six years.

The IGCSE program is, in short, a college preparatory program.  By passing the end of course examinations students can demonstrate college readiness.  In my school they’re even given a high school diploma at the end of their 10th grade year, upon successful completion of the program of course.  Some have even exited the school to attend college during what should’ve been their 11th grade year.

At the end of the seminar participants were invited to ask questions.  A teacher, quite frustrated, asked, “How am I supposed to get my freshmen prepared for calculus by their senior year?  There are too many things to teach and not enough time.”  (What she was getting at is that the Cambridge curriculum is appears sparse compared to typical American curricula.  In 9th and 10th grade there are a total of 10 topics for math.)

The presenter asked me to handle the question.  I knew the answer, but could not articulate my thinking in a concise fashion.  She and I were speaking different languages.  I tried to explain that she didn’t have to teach everything.  It is better to have a solid foundation that can be applied to all of the tangential and “one-off” topics in math, than it is to have brief experience with all of those various topics.  We do not have time for both.  We cannot develop deep understanding of the fundamentals of Algebra and have students exposed to all of the iterations and applications.

All she heard was a know-it-all teacher bloviate about some theoretical ideal.  She needed practical advice.  While I tried to provide that advice, I failed, miserably, to do so.  I realized after writing this article that this information, in this article, is what I should have shared with that teacher.

Her question had a specific context, but I believe it hit the heart of one of the biggest issues faced by mathematics teachers, world-wide.

 

How do I get my students to acquire and retain mathematical thinking?

 

I’m going to offer a two-word solution:  Mathematical Literacy.

If we want our students to really learn mathematics and be flexible enough to apply that knowledge in their futures, they have to be mathematically literate.   Mathematical literacy, for our purposes here, is (1) the ability to decode information from mathematical text and (2) the ability to encode contextually relevant information in mathematical text.

A mathematically literate person can understand mathematics as it is written, but also realizes countless associations, contextual meanings, and tangential ideas.  When a mathematically literate person sees mathematical text, they don’t wonder what should be done.  They read it as information, which is decoded and analyzed.  They are able to articulate appropriate, contextually relevant mathematical responses to information provided.  A student that has developed this literacy is prepared for whatever type of math their futures may hold.  They’re not bound by our efforts, they’re not reliant on what we have directly shown them.  Instead, they’re empowered with the ability to think and communicate mathematically.

The prior two paragraphs are entirely insufficient for defining mathematical literacy.  This article is about developing mathematical literacy, not defining it.  If you’re interested in learning more about what is meant by mathematical literacy, consider listening to the On Teaching Math podcast on mathematical literacy.   You can access the podcast with this link.

The development of mathematically literate students involves two components.  First, students must make sense of problems they’ve never seen, and problems that often expose a misconception created by a person overly reliant on procedural proficiency.   Then, students must apply something they know that is contextually relevant to the problem at hand.  The key component here is they must identify the relevant concept and understanding.  They must make the association.  They cannot be following a mapped-out procedure or following instructions. 

In order for this to happen, students must have a certain degree of conceptual understanding and procedural proficiency.  However, marginal proficiency with both is sufficient.  Through developing literacy, they will improve their conceptual understand and their procedural proficiency.

Warning:  Carefully acclimate students to answering questions designed to improve literacy.  If you do the thinking, instead of teasing it out from them, you can destroy the possibility of developing literacy in students.  As we dive into a few examples we will discuss, in detail, how this works.  But, for now, understand that if you demonstrate how to solve the problem, or answer the question, students will not develop literacy.  In order to develop literacy, students need to bring in relevant conceptual understanding (may or may not be directly related to the topic being taught), and then devise a plan and monitor the appropriateness of their approach as they work through it.

If we, the teachers, make all of the connections and do all of the decision making, we’re the ones exercising our own literacy.  Literacy will not be developed through imitation.

Let’s get into how we can set up experiences for our students that will promote the development of mathematical literacy.  We will use solving simple polynomial equations in Algebra 1 as our initial  testing ground.  View these examples in their spirit, not specific application so that you can begin to craft your own questions and design their implementation.

Suppose your students have been taught how to combine like terms, and then solve simple equations, like 3x + 3 + 4x – 5  = 23.  You can run them through countless pages of practice where they’d see every possible iteration of this type of problem.  But, you’d not be increasing their literacy or developing a deeper conceptual understanding.  That would only promote procedural proficiency, which is of course not well retained over time.

Instead, you could give students a problem like Problem A. 

The problem on its own will not promote literacy.  How you introduce the problem and your expectations of students will promote mathematical literacy.  If you work a similar problem, by changing the numbers, the students will latch onto the procedure.  They will not be pulling in various mathematical understandings they possess that are contextually relevant. 

However, without support, at least initially, students will likely be unable to even approach this type of problem.   The level of performance and thinking required of your students is likely brand new, and foreign to your students.  They will wait for you to show them how it goes, and then try to recreate what they’ve witnessed.  That is exactly what we do not want. 

If this was the first opportunity for my students to develop mathematical literacy, I’d explain my expectation and goal to them first.  The purpose of the problem is not to find an answer, but to develop the ability to understand what is written and draw in previously held understanding.  Once the understanding and associations are complete, students are practicing articulating their thinking mathematically. 

The purpose of this problem is to provide students with experiences that will prepare them for unknown futures.  This is practice that will help make them adaptable by teaching them how to think mathematically. 

A good way to start is to show students the diagram and the information, but not the question.  Ask students to brain storm about what they see, what they know, what comes to mind.  They’ll often be hesitant to state the obvious things, but those obvious things are sometimes the most difficult to see and are sometimes the most important things to notice! 

Once students have collaborated, through whole-class discussion collect and list ideas and observations on the board next to the diagram.  Many kids will have forgotten how perimeter works.  This will be a great time to shore-up that issue.

Then, after all of the observations have been recorded and discussed, show students the question.  Remind them that the steps to be followed are not what is important here.  Creating the steps to be followed is what’s important.   We want students to write mathematically, in appropriate contextual response to information provided.

Unfortunately, once this introduction has been completed, the opportunities to develop literacy with this style of problem are long gone.  The road is familiar.   Students will be remembering the process instead of making mathematical connections.  In response to this, teachers need to have two things at the ready. 

  1. Students coached to fully engage with the problems.  They cannot sit back and wait for the path to be clear.  Finding the path amidst uncertainty is the pursuit.  Once a problem has been explored, the path is found and the goal is no longer attainable. 
  2. You need a bank of problems at the ready!

Here is another, similar, but fundamentally different problem that could be used to follow Problem A.

Of course helping students develop the habits of thinking that will lead to literacy takes time.  You could easily teach students to “do” this problem in a handful of minutes.  Then, you could try to back-fill some meaning.   But then, students are learning how to “do,” the problem.  They’re not getting practice learning how to thinking mathematically.

The pay-off, however, is worth the time spent!  By learning how to make sense of mathematical information, and how to identify contextually important prior knowledge, then articulating their thinking mathematically, students will, over time, learn much more quickly.  They will also strengthen the prior knowledge through these experiences because these experiences provide opportunity to create connections between topics.  All of these benefits together result in greater retention of the new, and old, mathematical concepts.

Let’s see an example that would be appropriate for students at this level that does not involve Geometry.  Again, we are considering a group of students who can distribute and combine like terms, and solve equations in one variable. 

 

There are two boys, John and Bob.  Both boys like to collect colorful rocks.  Bob puts his rocks in his left pocket, which has a hole in it.  John finds half of the rocks that Bob drops.

If Bob found 36 total rocks, and one third fell out of his pocket, how many of Bob’s rocks did John find?

 

There is nothing special about this problem, or the previous two.  What is different is how you introduce the problems and how you coach students to approach the problems.  Encourage brainstorming, making sense of the problems.  Set the expectation that students will need to develop mathematical literacy in your class to be successful.  If it is a true expectation, and you are unwavering, but encouraging, students will develop literacy over time.

Questions that are not directly related to the topic at hand can also be used.  In my podcast, On Teaching Math, I start each episode off with a question like this.  They’re typically easily understood and involve solutions that are within reach of most people, regardless of mathematical prowess.  Also, it is often the case that the answer or discovery made by exploring the question is of little consequence.  But, what is important is that students must create a hypothesis and test it, either through independent exploration or collaboration.  As they test their hypothesis, through reflection they must decide to adjust their or approach, or through validation, continue on. 

A typical question will be:  How many times in a 24-hour period will the hands of a clock create a 90-degree angle?

Another question that is simpler is: Why is 5 the only prime number that is the sum of the previous two prime numbers?

One more example is:  What number less than 100 has the greatest amount of unique prime factors?

These types of problems are a great way to give students experiences that develop mathematical literacy.  The way a person must engage with those problems is the same way a mathematically literate person can engage with our last example.

One last positive outcome from these questions is that a lot of meaning will be exposed. Students will likely discover things you never thought of.  That is a great outcome and a great way to include activities that promote academic discussion into your classroom.

This final example is a favorite question that can be used to develop literacy.  An ancillary benefit is realized for students who failed to obtain the solution.  In review, students will have a deeper understanding of exactly what the concept at hand with this topic really means.

Suppose you’ve taught your students the mechanics of functions.  They can read and perform operations from examining the notation, they can perform function arithmetic, maybe composition of functions, and they can find inverse functions.  I selected the words, can find, here because they indicate procedure, not concept!

Here’s the question:  Given that f(x) = 2x, what is the value of x when f -1(x) = 4?

When I first saw this question on a Cambridge IGCSE examination I thought the question was entirely unfair!  In fact, I was asked by a person outside of my district how kids could solve this.  The students taking the test had no experience with how to find the inverse of the function!  

When the test results were released I was shocked to see that the majority of my students answered the question correctly.  I could not believe it.  Upon questioning, students explained that the question was easy because the input and output for a function and its inverse are reversed.  For example,  if g(2) = 3, then g-1(3) = 2.  So, if the output of the inverse of function f is four, then the input for the function f is four.  Then, f (4) = 24, which is 16. 

Because the students understood the concept and had practice applying concepts in new ways, they were successfully able to answer a difficult question correctly!  To make it even better, they answered a question that I had never dreamt of before.  This is a great example of the power of mathematical literacy.

Let’s pull it all together here.  To develop mathematical literacy students must apply conceptual understanding in non-routine applications.  This will likely be a shift in engagement for students and teachers.  As such, we, the teachers, must orchestrate a series of experiences that will help students make this shift.  We start students off with simple to understand questions that are non-algorithmic in nature, and gradually move to more complicated application of the concepts at hand.  All the while, we increasingly move students to more independent thinking, where they collaborate AFTER they've have created and executed a plan. The pay-off is well worth the time and effort required!  This is absolutely a case where going slow early can speed things up over time! 

Your devotion and consistent application are required to help students develop mathematical literacy.  You will need to incorporate these style of problems and the appropriate pedagogy into your lessons.  Students will need opportunities to practice their literacy on homework, quizzes and tests.  Many of the students will require continual encouragement and reiteration of the relevance of their efforts (why it is important for them, that they develop literacy). 

If your students develop mathematical literacy under your tutelage, then you will have served the future needs of that student well.  They will be prepared for an unknown future because they will be empowered with the ability to think, and communicate, mathematically.

If you are looking for questions that can be used to promote mathematical literacy within the application of a specific topic in math, please leave me a comment below.  I have a large collection of these types of questions built over the years.  

 

 

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?

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.

Why Remediation Fails

Why Remediation Fails

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What can we do?

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

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

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

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

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

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

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

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

 

Philip Brown

 


Try to Solve This Problem … without Algebra

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

Given:  3ak

And:  ax = 4k

What must x be?

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

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

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

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

 

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

Given:  3ak

And:  ax = 4k

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

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

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

Let’s look at the second statement now.

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

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

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

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

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

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

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

For every dollar you have, Bobert has four.

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

Vestiges of the Past Making Math Confusing

Something in Math HAS to Change

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3 + 3 + 3 + 3 + 3 = 3 × 5

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

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

3 + 3 + 3 + 3 + 3 = 15

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

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

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

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

3 + 3 + 3 + 3 + 3 = 15

The second step is six.

Another way to see this is shown below:

3 →6→9→12→15

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

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

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

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

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

Exponents:  3⁵ = 3 × 3 × 3 × 3 × 3

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

Pretty cool, eh?

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

Just FYI, 3 times itself 5 times is 243.

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

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

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

3 × 3 × 3 × 3 × 3 = 243

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.

Why Does the Order of Operations Work?

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

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

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

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

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

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

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

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

For example:

4 baskets of 7 vials each

4 × 7 = 28

or

7 + 7 + 7 + 7 = 28

or

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

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

4 × 7 + 6

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

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

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

Exponents are repeated multiplication, of the same thing!

Consider:

3 + 6

4 × 7 = 7 + 7 + 7 + 7

74 = 7 × 7 × 7 × 7

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

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

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

So, let’s look at:

9 + 4 × 74

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

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

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

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

(9 + 4) × 74

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

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

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

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

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



Is Infinity Real?

How Many Primes are There
Is Infinity Real
Part 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The gaps between these prime numbers are below.

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

Another string would be:

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

The gaps here are listed below.

4, 8, 10, 8, 4, 6

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

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

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

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

Why all the fuss? What good are they?

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

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

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

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

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

 

The differences here are as follows.

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

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

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

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

Fact 1:  5! = 5×4×3×2 = 120

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

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

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

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

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

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

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

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

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

Then x! + a is composite.

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

Crazy, right?

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

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

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

3,756,801,695,685 × 2666,689 – 1
and

3,756,801,695,685 × 2666,689 +1

Those numbers are too large to be written out!

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

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

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

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

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

 

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

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

Gaps between prime numbers: https://youtu.be/vkMXdShDdtY

The largest prime number:  https://www.youtube.com/watch?v=lEvXcTYqtKU

Infinite Primes:  https://www.youtube.com/watch?v=ctC33JAV4FI

Large Gaps Between Primes:  https://www.youtube.com/watch?v=BH1GMGDYndo

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

www.patreon.com/beardedmathman