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/


Best Practices

Philosophy and Best Practices

Cart Before the Horse

The easy part of teaching any curriculum is the curriculum.  The hard part of learning to teach a new curriculum is figuring out how to teach it most effectively.  For the sake of clarity, effective teaching will develop student conceptual understanding and problem solving.  Earmarks of quality teaching include retention, connection, engagement, and thoughtful discussion among students, as well as high test scores of course.

One of the problems with modern education and education research is the need to quantify everything.  The adage, If it can’t be measured it cannot be improved, has bound us with conclusions based on extremely soft data, nonsensical results, and has engaged us in an insidious pattern of behavior.  That same pattern is easily seen in students, and it drives teachers crazy.

If a student is in a class, then that class should cover new material for them.  If a student gets a good grade, they will have learned the material.  Grades measure learning and gained ability.  Students that focus on the grade in lieu of learning struggle to get good grades.  They have confused the marker of success with success.

Is there a more frustrating question to a teacher, from a student, than, “What can I do to bring my grade up?”  Isn’t the answer almost always, “… just about anything will work, so long as it involves learning.”

That is exactly what is happening today in education.  In order to conduct “scientific” research we quantify markers of success and measure them.  We lose sight of what is really important and chase what has been measured.  Teachers jump through hoops to create word walls, daily and in-depth content and language objectives, the good old SWBAT, and perform remediation that teaches students how to get quiz questions correct.

All of these actions are designed to show an increase in the markers of success but potentially do so in direct conflict of the true goal of education.  Our job as educators is to train the minds of young people.  Many best practices ease the cognitive strain to promote short-term positive results.

The end result is that between 40% to 60% of first year college students need to take remedial courses in English or Mathematics, or both.  Let us explore how this happens.

Use retention as an example.  Suppose a teacher is being coached that good teaching requires that students remember the material in say, four months.  That teacher will take steps to make sure that students remember.  There will be tricks, rhymes, and rewards for getting good test results on retention.

The problem is, this will not be the type of retention we are trying to develop.  The retention our students need is a consequence of conceptual understanding, not brute-force memorization.  With conceptual understanding comes the ability to reconstruct lost memories.  That is a great tool!  No matter what, the likelihood of losing information over time is astronomically likely.  Besides that, if a student only remembers the fact or procedure, their ability to use that fact or procedure will be bound by how they’ve been taught to use it.

Retention is just an earmark of conceptual understanding.  There’s more that comes along with conceptual understanding, like the increasing pace of integration of new material due to connections made, and improved engagement and problem solving skills.  And, perhaps most importantly, when students conceptually understand, they’re not bound by what we have taught them, they can take what they know and go further than we did!

Not everything in education has artificial restrictions placed upon it by inappropriate data seeking.  Climate and culture, for example, are two areas of focus for administrators.  Yet, climate and culture would be incredibly difficult to measure … but you can feel it the moment you walk onto a campus!  The same goes for relationship building between a teacher and the students.

Take a minute and imagine how artificial markers of determining the existence of a quality relationship between a teacher and student could be made.  Now if the efficacy of your teaching was being evaluated on this false metric, and if that metric had been engrained in the culture of education, it is likely you would be engaged in the best practices of the day that would increase the frequency of that marker.  Yet, would you really be building relationships?  What if you believed you were and you were not, and the system told you that you were doing well?  How would you know you were wrong if the system told you that you were right?

Baby and Bath Water

Now it is likely that quality outcomes are realized by chasing faulty markers of success.  For example, if you learn how many siblings each of your students has, you’ll likely open the door to conversation and find connection.  If a student attends tutoring to bring up their grades and they participate, they’ll likely learn.  By learning their grades will improve, even if they believe the act of attending tutoring is what brought about the improvement.

There is nothing inherently wrong with a Content and Language Object, or Learning Target, or SWBAT, or whatever it may be called at your school.  In fact, objectives are really needed when designing and implementing a lesson.  Without the target the lesson will meander.

Before deciding that a particular best practice is a false metric and deciding to abandon it or treating it as a matter of compliance, first see if that false metric can be employed to foster the desired outcome.  What should this best practice really build and why?  Can you make it serve that end?

A great example is allowing students to re-take a test.  If a student takes the same exactly test, especially after teacher-led remediation, of course the student will improve their grade.  But, did they really learn the content, did they develop a conceptual understanding?  Did they improve their problem solving skills?

Most likely, the conceptual understanding remained unchanged but the problem solving skills would have damaged.  It is a bad outcome when students learn that the way they solve problems is by seeking someone else to solve the problem (unless they’re bound for politics).

That does not mean that allowing students to re-test is a bad practice.  But it is a bad practice when done as described.  What if, in order for a student to re-test, the student had to demonstrate their conceptual understanding and problem solving before they re-tested, and then the second test was different than the first? (Changing the numbers on a test does not make it different.)  This would serve the student’s needs and encourage them to perform well.  It would provide you with data that could be used to more accurately assign a grade.  It is all around a great outcome!

How and Why

It is how and why we do what we do that makes it worth-while.  That adds an additional layer of complexity to the profession of teaching the United States today.  A teacher can follow all of the directives and engage in all the best practices as assigned by their leadership team.  They can have wonderful reteach and differentiation strategies, the best posted objectives, standards referenced lesson plans informed by classroom data, and formative assessments published by respected companies.  In their classroom you may see varied questioning techniques, students getting up and performing tasks, and the teacher “owning the room.”

I was that high school math teacher.  I was energetic, my students loved my class and proclaimed I made math fun for them.

I was also teaching remedial courses at the local community college.  When I began seeing some of those same students in my classes, I realized I was not effective.  The reason I wasn’t effective is I was putting on a show.  Sure, I followed the best practices of the day, but I failed to see what was important.

In the next few blogs, I will break down what I have discovered to be best practices that promote student conceptual understanding and improve problem solving skills.  I will try to explain why I believe they work so that you can adapt them to your needs.

Student Skills and Tools

The biggest hurdle in transitioning from Middle School to High School is the lacking set of student skills possessed by incoming Freshmen.  Students come in failing to appreciate the importance of homework, struggle to think independently, cannot communicate mathematical thinking, and are easily frustrated to the point of quitting.

This observation is not a knock on the students’ experiences in Middle School.  It is entirely likely that the brain of a 12 to 14 year old cannot develop these skills.

In the upcoming 2019/20 school year I will be running an experimental program where I use SMART Goals focused on student skills to hasten the development of those lacking student skills.  The pay-off could be huge…the development of quality student skills would transcend the classroom, even school.  Ultimately, student skills are goal-oriented problem solving and personal management skills.

Here’s how it is going to work.  During the first week of school I will teach students what SMART Goals are (read about them here if you don’t know: https://www.yourcoach.be/en/coaching-tools/smart-goal-setting.php).  We will practice setting small SMART Goals in order to learn what is required, and how to foster them.

During the first week I will also teach students what quality student skills are.  I’ve made a reference sheet of what they are, what they look like in action, and how they’re beneficial.

Perhaps the most important thing taught in the first week will be how motivation drives engagement.  If a student is deeply engaged in their studies, they’ll persevere and be successful.  The two types of motivation, intrinsic and extrinsic, are directly related to the quality of engagement.  A student motivated by reward, or fear, from grades is extrinsically motivated.  They’ll easily give up and will engage in their work at a shallow level.  Their mindset is to complete required work.  A student that is intrinsically motivated is motivated to learn.  They engage deeply and seek learning.  They persevere and find learning opportunities in their work.

At the end of the first week of school students will draft individual SMART Goals that focus on student skills.  I have created a four-week long form where students will be guided through the reflection, monitoring and fostering required to have those goals come to fruition.

If you’d like to see the documents I’ve created, they are here.  Here is the Student Skill Sheet:  https://drive.google.com/file/d/1TCDKiwWrU-Ycoc1JILbOVd_y0f5_VkxR/view?usp=sharing

Here is the Smart Goal Planner:  https://drive.google.com/file/d/1mrZtEM3sAUcYkU5pxLdatnyn9DjVqT4k/view?usp=sharing

If you’d like to follow along with how this goes, you can read my blog:  https://thebeardedmathman.com/home/blog/

 

Math is Hard

Math is Hard

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

A common conference would go something like:

Parents:  Why is my child failing math?

Me:  Well, let’s ask your child.  Why are you failing math?

Child:  Because math is hard.

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

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

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

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

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

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

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

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

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

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

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

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

Accessing Prior Knowledge in a Way That Uncovers Misconception – A Lesson

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

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

Click to download

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

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

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

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

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

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

An example is:      4  4  4  4  = 3

Students may add:  + - × ÷ [  ]

So an attempt might be:  -[4÷4]+4÷4

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

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

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

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

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

 

Teaching Conceptual Understanding Flow Chart for Educators

Focus on Conceptual Understanding
Flow Chart
for
Educators

Teaching by concept alone will lead to inefficiencies in students.  They will, in effect, be reinventing a large part of the wheel at every turn.  (See what I did there?)  We have all witness what focus on procedure alone does.  It leaves students will a bunch of isolated skills that they do not recognize out of context.  Out of context here literally means changing the font or using a different set of variables.

An example is the topic/skill of finding the lowest common multiple of greatest common factor.  Students are well versed in many procedures, yet of course, mix the two up.  That is, they’ll claim a GCF (greatest common factor) is a LCM (lowest common multiple).  This is NOT their fault.  They don’t understand the difference between a multiple and a factor.  They don’t see how those two are applied in other mathematical calculations, even though in order to perform the majority of operations with fractions, those are required.

The focus in education has shifted, and like large bodies do, they swing too far.  More than likely the focus has been too great on concept and avoidance of procedure and rote memory of basic math facts.  That’s a discussion for another time.

I’d like to help you, the teacher, strike a good balance.  Unlike big publishers or professional development companies, I am in the classroom, trying these methods with all of my topics and a wide variety of students.  It is highly successful.

One key component of the success is removing yourself from the role of, “The Human Wikipedia,” in the room.  Think of yourself more as a coach than a teacher.  The knowledge you possess cannot be possessed by the students simply by you telling or explaining what you know to them.  They must experience it themselves and grapple with the misconceptions to make sense of things.  You’re a facilitator of discussions and explorations, and quite importantly, you’re a guide.  No need to chase too many rabbit holes.  When a level of understanding is achieved it is up to you to help bring closure, probably through a discussion and writing activity where students write down their explanations of what they’ve learned.  Then, that’s when homework changes from uncovering misconceptions to solidifying understanding and making efficient processes that are repeatable.

I’ve harped on many of those things in the past.  If you have questions about any of those ways in which homework is used to help learning, please feel free to leave a comment or send an email.

With all of that said, let’s get into it.  The chart at below is a general idea of how concept can be established and explored, how procedures can be introduced as a way of generalizing patterns and features of the concept, and last, how that concept can be used to introduce a connecting concept, or consequence of that concept.

Here’s the idea.  The rectangular shapes are lessons, or whole group discussions.  Everything with an arrow is student work where your job is to encourage and direct.  Typically, it is a bad idea to explain things during this time.  Instead, encourage students to find other students in the room that they trust that might be able to explain what it is that’s confusing them.

Another big idea during this time is to encourage students to articulate what it is that is confusing them.  When students say, “I don’t get it,” they’re helpless.  They’re not even thinking about what is causing trouble.  By forcing them to reflect on what’s causing the trouble, they’ll likely find their way through the confusion.  For you to step in and let them off the hook will only make them have to face that point of confusion later, and it will be bigger and the nature of the confusion will be less clear to them.

A great topic to use an example of this works is exponents.  All of the “rules” of exponents come from the idea that exponents are repeated multiplication, of the same number.  The difficulty in exponents comes from students inability to read the notation properly, especially when groups are involved.

Let’s briefly explore how this chart can help guide your planning with something like exponents.
Concept:  Introduce the notation, perhaps tying it in to how multiplication is written to describe repeated addition of the same number.

3 × 5 = 3 + 3 + 3 + 3 + 3

35 = 3 × 3 × 3 × 3 × 3

Some conceptual questions would be things like providing three different expressions written with exponents and having the students pick the two that are the same.  Another way to do this is to give the students an expression and then give them a choice of five other expressions, often which may contain more than one equivalent expression, and have the students pick which match.

During such matching activities keep in mind that the students having the right answer is not necessarily a reflection of understanding.  Without the proper explanation, accurate and concise, they likely do not know.  Their results of being right will not be repeatable.

Also, when exploring things like this, tell the students that they should write down the examples, but students that will learn will focus most of their notes on their thoughts and questions.  This is especially true since we are NOT discussing procedure.

(If you’d like to see some examples of these types of conceptual questions you can find them in the PowerPoint attached here.)

During the questioning of concepts you should chase misconceptions and show how they do not match up with what is true.  Always focus on the fact that it is through mistakes that students are learning.  Thank students, praise them for participating even when they’re not sure they’re right.  We all hate being wrong, and students are often insecure and fear being judged harshly for being wrong.

After exploring the misconceptions and then finding patterns and developing some procedure it is a good time for them to practice what they’ve learned, AKA, homework.

When reviewing the homework the next day make sure things are determined right or wrong by referring to the concept, not finding mistakes in procedure.  Of course some refinement of procedure is appropriate when reviewing homework, but that should be for the sake of efficiency, not understanding!  This is likely a huge shift for teacher and student!

An in-class, open note pop-quiz is a good follow up, depending on the ability of the students and complexity of the topic.  If I were to do such an activity, I would make sure the grades are not too punitive, providing credit to those that correct errors, or perhaps grade it like homework, on completion, not correctness.

If that in-class pop-quiz doesn’t work, a subsequent, more complicated homework assignment is in order.  This next assignment should change the language of what’s being learned.  Rephrase instructions or change some of the look of the problems so that students are not finding false clues by recognizing patterns in the problems themselves that have more to do with you, or the author of the work, than the concept at hand.

It is also a good idea to throw a few problems that tie into the next topic in, stretch problems, you could call them.  Use reviewing these problems to introduce the next concept.  I often do this without telling the students the new lesson has begun.  It works well because students should be taking notes on their homework assignment in pen (not erasing mistakes but instead annotating them).

Two observations about these practices.

  1. Student involvement is key.  Of course, students don’t learn if they’re not involved, but their involvement is less needed for a tradition, stand up and lecture while students take notes, type of classroom setting.  These methods are truly student focused and student driven.

    As the teacher you must anticipate the questions and points of confusion.  Do not have answers at the ready, but perhaps simple problems that students can explore so they can discover clarity. Be ready to show a consequence of their misconceptions or perhaps a problem that simplifies their misconception so they can see it.

  2. Textbooks are woefully inadequate as a resource here. You need many books and resources in order to provide students with exposure to concepts, conceptual problems, and different levels of practice problems (the last practice problems can often come from books).  The last set of problems, the stretch problems that connect what they’ve learned with what is coming next I have never seen in a textbook.

    You’re going to have to be creative.  I am trying to publish my materials and questions as I go through this year, but even so, they relate closely to my interpretation and view of the topic, the heuristic framework I developed.  Yours is likely different and so the ways in which you can stretch understanding or expose misconception will vary slightly.

I hope this has been helpful.  It is something I hope to explore more fully and deeply.  Whenever I have been able to employ these methods the results have been powerful. Students learn and they retain their learning.  I’ve been refining these methods over the past six years or so and my students have realized great success from it.

I thank you again for reading and hope this helps.  Please let me know what questions you have, just leave a comment.

Philip Brown

 

Why Remediation Fails

Why Remediation Fails

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What can we do?

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

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

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

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

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

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

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

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

 

Philip Brown

 


Try to Solve This Problem … without Algebra

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

Given:  3ak

And:  ax = 4k

What must x be?

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

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

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

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

 

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

Given:  3ak

And:  ax = 4k

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

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

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

Let’s look at the second statement now.

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

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

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

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

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

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

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