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/


What Do Grades Really Mean?

What Do Grades Mean

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

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

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

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

A – Mastery

B – Fluency

C – Proficiency

D – Lacking Proficiency

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

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

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

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

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

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

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

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

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

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

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

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

How Do Students Earn Grades

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

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

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

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

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

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

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

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

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

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

Rewarding Effort?

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

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

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

Grades

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

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

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

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

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

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

 

The Purpose of Homework and My Response

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

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

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

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

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

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

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

 

The Smallest Things Can Cause Huge Problems for Students




preemptive


Pre-Emptive Explanation

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





f(
x
)
=3x
x
2

+1




f(
2
)



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa
qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaaaaaa@43D9@


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





f(
2
)
=3(
2
)

2
2

+1




f(
2
)
=6+4+1




f(
2
)
=11


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaeyypa0JaaG4mamaabmaa
baGaaGOmaaGaayjkaiaawMcaaiabgkHiTiaaikdadaahaaWcbeqaai
aaikdaaaGccqGHRaWkcaaIXaaabaGaamOzamaabmaabaGaaGOmaaGa
ayjkaiaawMcaaiabg2da9iaaiAdacqGHRaWkcaaI0aGaey4kaSIaaG
ymaaqaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaacqGH9aqp
caaIXaGaaGymaaaaaa@4F4E@


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

Here’s what the
students actually read:





f(
x
)
=3x
x
2

+1




f(
2
)
=3(
2
)
+
(

2

)

2

+1


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

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

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

 

How to Save Time Grading

 

How to Grade Efficiently

and Promote Assignment Completion

 

Grading papers is one of the most time-consuming responsibilities of teaching.  Hours upon hours can be, I argue, wasted, pouring over daily homework assignments.  This article will discuss how to integrate awarding credit for daily assignments in a way that saves hours of time while increasing your awareness of student progress, increases student completion rates and better informs students regarding their progress in the subject.

This routine described here is a daily variety, not how I grade quizzes, tests or projects.  However, there are some tips that apply to recording those grades later in this article.

Let’s begin with a question: What is the purpose of homework?  For me, it’s practice needed for students to gain proficiency.  Homework is about trying things, working out how to struggle through difficult problems and making, and learning from, mistakes.  Without effective homework, students will not integrate their learning into a body of knowledge that they can draw upon for application or just recall.

The breadth of the purpose of homework and how that purpose is best served is beyond the scope of this article, but I would like to suggest that homework is something done in their notes, whenever possible.  The reason being is that notes are a receipt of their learning, to be reviewed in the future to help remember observations and important facts.

Overview of How It Works:

At the beginning of class, often before the bell rings, I begin walking around the classroom stamping homework that deserves full credit.  (What merits full credit is up to your discretion, but it should be a clear and consistent expectation, known to students.)  As I circle the room, I look for common mistakes, ask kids if they have questions or difficulties and make small talk.

Students that didn’t do, or complete, the homework have to answer for it on the spot!

Then, I simply mark those that did not receive credit for the homework on a student roster I keep on a clip board.  (For a video of how this works, visit the link here: )

Quick Notes:  This method has students ready for class because they have their notes.  They’ve also asked me questions if they had any, so I can begin with meaningful review.  I also have forced students that are remiss to account for their actions and done so in a way that applies positive peer pressure.  The scores are recorded by leaving blanks for completion and only marking those that do not get credit (which will be very few).

Credit:  I award full credit or zero credit when checking homework.  If a student attempted all problems, with evidence of attempt demonstrated by work shown and questions written, they get full credit.  Those that fail to receive full credit have the opportunity to reclaim 80% (the percentage is arbitrary but again needs to be consistent, clear and known by all), the students must see me during tutoring time by the Friday of the week of the assignment to show that they’ve fulfilled the expectation.  Students that did not attempt the homework can also see me during tutoring time (before or after school, not between class times or lunch), and receive partial credit.

But the rule of being due the Friday of the week assigned is big.  The purpose of homework is practice.  Without proper practice skills and knowledge are not developed.  Homework is not about compliance and fulfilling an expectation with a grade as a reward.  Students that are hustling to complete homework from two months prior are likely not promoting their understanding of current materials.  Plus, by having the time requirement applied to the homework policy, students are not enabled to fall too far behind.

The added bonus is that you will not be buried with make-up work the last week before grades are due to be reported!

Work to be Turned In:  If the nature of the work is not something that can be kept and must be turned in, have the students pass their work forward by row.  As you collect each row’s stack, count them.  If a row’s stack is incomplete, ask who in the row didn’t turn in the work.

If students can NOT fulfill the expectation and only receive a bad grade from it, and that reprimand comes well after the unwanted behavior, they will quite happily go along thinking nothing bad is going to happen.  Having to answer, publically, for their lack of work, especially when the vast majority will work, is a powerful deterrent!  Just as when checking the work of students and asking those who failed to complete for an explanation, this keeps them accountable and will increase the amount of students completing their work.

When collecting the papers, alternate the direction of the stacks and do not mix them up when grading.  This will allow you to quick return the papers after you’ve been done.  If it is a daily practice type of work turned in, I’d suggest awarding full or no credit and only recording, again on the printed class roster, those that were awarded no credit.

Recording Grades:  Whether you’ve collected daily practice or are carefully grading quizzes and tests, how you record those grades can either waste your time, or greatly reduce the amount of time spent.

By recording each grade as it is calculated by hand on the student roster it is quick and easy to transfer them to the computer.  This is a huge time-saving practice.  You don’t need to hunt on the computer screen for each student, and do so for each assignment.  When they’re recorded by hand, you can simply enter the column of numbers in the computer.  When the last name lines up with the last number that you entered, you know they’re all entered correctly.

By following this method, the data entry side of grading is done in a few moments of time instead of over hours, working through those stacks of papers, again!

Final Thoughts:  By looking at, and discussing, homework with students on an individual basis, very briefly, you gain insight into their progress.  They get a chance to ask questions.  Students that need a little bit of motivation receive it as an immediate consequence for poor behavior, rather than waiting until the end of the quarter, when a lot of pressure will be placed on you to help them bring up their grades.

This routine has proven to be a cornerstone of my classroom management.  It gives me a way to set the expectation that we are here to learn and that learning is done through work and reflection.  Students that need discipline receive it immediately and in a way they find uncomfortable, but it is done so with an invitation that guides them to the desired behavior (of completing their work).

 

Math Can Not Be Taught, Only Learned

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

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

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

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

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

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

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

The Bearded Math Man