Promoting Mathematical Literacy through Non-Procedural Questions

Math was okay until they threw the alphabet in it.

I have heard, Math was okay until they threw the alphabet in it, so many times from so many adults, who I wish were trying to be witty, but are in fact completely serious.  And these are reasonable, intelligent people.  Why does the abstraction of a variable or an unknown become so confusing that it creates a huge disconnect?

I’d like to share with you an assignment that I believe will help students transition from the concrete application and properties of Real Numbers to the abstractions we deal with in Algebra and mathematics beyond.

The desired outcomes of this assignment are:

  • Improve mathematical literacy by encouraging students to read the mathematical meanings created by the spatial arrangement of numbers and symbols
  • Improve their understanding of the Order of Operations and to help student realize that the Order of Operations is not its own topic, compartmentalized, but rather an over-arching understanding of how math is performed.
  • Promote abstract thinking about numbers and their properties
  • Introduce some concepts that will come into play later in Algebra, like finding the x-intercepts of a quadratic once it is factored

How to Introduce the Assignment

Students must be aware that they will be dealing with abstract ideas and that there are sometimes more than one right answer.  Also, a student can be right, but not completely right, they could also be wrong, but not always.  By fostering a healthy discussion about these problems you can introduce the idea that in order for something to be mathematically true, it must always be true.  If a single circumstance is untrue, then the statement is untrue.

Consider the problem: Given that a and b are real numbers, and the following is true, what do you know about the numbers a and b?   a×b=0

A student might say, In this case a and b are both zero.

That is correct in one case, but there are many cases where that is not true.  It is true that one of them must be zero.

The First Prompt

Given that a and b are real numbers, and the following statement is true, what can you conclude at the numbers a and b?

a – b = 0

At first have them think and write on their own.  Make sure they’re all working, not avoiding this uncomfortable notion.

After a given amount of time (short, maybe one minute), instruct them to talk with two different people.  Be clear that the expectation is that they take turns, one person shares, the other listens and responds.

After the time is up, have a whole-class discussion, but avoid being the authority until the discussion is winding down.  Only be the authority on the subject to help summarize.

The Second Prompt

With the same properties of the numbers and the statement being true, provide them with the equation:

(a + 5)(b –7) = 0

Conduct conversations as you did with the first, maybe allowing more time for them to talk together as this is a more complicated situation.

The Third Act

Switching gears from properties of variables to applying some properties of real numbers will promote their understanding of when the associative and commutative properties as well as challenge their understanding of how math is written.  We are really trying to promote their ability to read and write math and their fact that the spatial arrangement has meaning in math.

Have the students try and add parenthesis, as many as they like, to the following equation, so that it will be true.  Have them try it on their own first, then provide a short amount of time for peer discussion.

3•2 – 72 + 5 = 80

 

When you conduct a whole-class discussion, make sure it’s student lead, your role is as a mediator, not a disseminator of facts.

Fourth Act

Instruct students to create a similar problem by making a statement they know is true and removing the parenthesis.   For example, they might make up:

8(5 – 3) + 11 = 27

But would only write

8•5 – 3 + 11 = 27

When they’re completed their problem, have them show you.  Once everybody has a problem, hand out 3×5 cards.  On the front of the card the student will write their problem, without the parenthesis.  On the back, they’ll write their name.  Have them pass the cards forward and you can distribute them at to another class the following day.

Last Thing for the Lesson:

The idea here is the same as with the previous activity, but we are accessing their knowledge from a different angle.  They will be given an expression with parenthesis and be asked if the parenthesis can be removed without changing the value of the expression.

For example:  (5 + 4) – 2, or 11 – (4 – 9).

Introduce these problems in the same fashion, with quiet thinking first, then small group discussions, then whole-class discussion.

The Homework:

The homework is critical here because it will challenge students to think and examine how the way in which math is written changes the meaning.  It will also force them to think about numbers in a general fashion.

Timing

The last thing I’d like to mention is that this could easily be done over two separate days depending on the aptitude of the class you’re teaching.

I hope this is helpful and food for thought.

 

 

Why Teaching Properties of Real Numbers is Important

If you are going to do a fraction review, the lesson here might be of some help.  I believe things are best reviewed in context, but this is a decent set of information that also introduces the real numbers and some other basics of math.

The PDF icon to the left has a lesson outline you can feel free to use with the PowerPoints of in any way you see fit.

The structure is all there in the lessons, but they're not over scripted.  Remember, I believe the majority of a lesson should be spontaneous.  It should be anticipated and prepared for, but how the lesson really unfolds depends on the audience.

Below you will find an overview of how and why I teach real numbers as well as two PowerPoint icons you can download and use as your own.  I only ask that you share where you found them.

Anything you purchase from Amazon.com through the banner below goes to producing more materials, and at no cost to you.

What Good Is It?

The Real Number Line has always been one of the dullest lessons I have to teach.

Click the icon to download a PPT.

 Natural Numbers are the set of numbers you can count on your fingers, beginning with one.  The Whole Numbers are the Natural Numbers and Zero...Integers are ...

Blah Blah Blah

I have to teach it because it's in the curriculum.  And I always wonder, what use is it if a student knows the difference between a whole number and a natural number?

It is hypocritical of me to complain in such a fashion because I laud the virtues of education being greater than a set of skills or a body of knowledge.  Education is about learning to think, uncovering something previously unknown that ignites excitement and interest.  Education should change how you see yourself, how you think about the world.  It should enrich our lives.

Teaching the Real Number Line can be a huge first step in that direction, if done properly.

Math is About Ideas, Not Just Computation

There are some rich, yet entirely approachable, mathematical ideas that can be introduced with the Real Number Line (RNL).  For example, a series of questions to be posed to students could be:

  1.  The Natural Numbers are infinite, meaning, they cannot be counted entirely.  How do we know that?
  2. The Integers are also infinite.  How do we know that?
  3.  Is infinity a number?
  4. Which are there more of, Natural Numbers of Integers?  How can you know, if they're both infinite?

The idea of an axiom can be introduced.  Most likely, students assume math is true, or entirely made up, but correct or incorrect, because it is written in a book and claimed to be such by a teacher.  The idea of how we know what we know and if math is an invention or a discovery can be introduced by talking about axioms.  For example:

  1. Is it true that 5 + 4 = 4 + 5 ?
  2. If a and are Real Numbers, would it always be true that b = b?  (What if they were negative?)
  3. Is it also true that b = b - a?  How do we know that?
  4. Is the following also true:  If a = b, and b = c, then a = c?  How do we know?

The idea here is not to teach students the difference between the Associative Property and the Commutative Property, but to use these properties to introduce students to math as a topic that can be discussed, and that it is not about answer getting, but instead about ideas.

For more on this topic and a few other related items, visit this page.

Why Are Some Rational Numbers Non-Terminating Decimals?

If you had a particularly smart group of students, you could pose this question.  I mean, after all, 1/3 = 0.3333333333333333...  And yet, we are told rational numbers include decimals that can be written as a fraction (the ratio of two integers).

How it works is sometimes very clear and clean.  For example, 0.7 is said, "Seven tenths." And "Seven tenths," can also be written as the ratio of seven and ten.  And the number seven tenths is of course equal to itself, regardless of how it is written.  The number 0.27 is said, "twenty seven hundredths," which can easily be written as the ratio of twenty seven, for the numerator, and one hundred, as the denominator.  And this can continue so long as the decimal terminates.  But try the same thing with the a repeating decimal and you do not end up with things that are equal.

The algorithm to convert a repeating, but non-terminator decimal into a fraction is pretty straight forward.

But that does not address why a rational number would be a non-terminating decimal.

Click the PPT Icon to the left to download a lesson on converting repeating decimals into fractions for honors students.  It includes a proof of why the square root of two is irrational.

The simple reason that some rational numbers cannot be expressed as terminating decimals has to do with our numbering system.  We use base 10 numbers.  Our decimal system provides us an easy way to write fractions with denominators that are powers of 10.

That means that we have ten numbers, including zero, that fill up one column, like a car’s odometer.  When you travel 9 miles the odometer will read 000009.  When you travel the tenth mile the odometer will read 000010.

Not all of our number systems are base ten. While the metric system is base 10, or at least translates into powers of ten, the Imperial system is base 12 from inches to feet, but 5,280 feet to the next unit of miles, and so on.

Time is another great example of bases other than ten.  Seconds and minutes are base sixty.  You need sixty seconds before you have an hour, not ten.  But hours are base 24 because 24 hours are needed to make one of the next category, which is days.

In time, 25 minutes of an hour is the ratio: 

But in base ten this is 0.4166666666666666... Our decimal system does math in base ten, not base sixty.  This is not 41 minutes!  A typical mistake would be two say 25 minutes is 0.25 of an hour.

Back to our original example of 1/3.  Not all numbers can be cleanly divided into groups of ten, like 3.  If we had a base 3 numbering system, where after the third number we moved, then 1/3 would just be 0.1.  But in our numbering system, 0.1 is one tenth.

Other numbers, like four, translate into ten more easily.  Consider the following:

The only issue remaining is that 2.5/10 is not a rational number because 2.5 is not an integer and rational numbers are ratios of two integers.  This can be resolved as follows:

Let's try the same process with 1/3.

As you can see, we will keep getting ten divided by three, forever.

This is a great example of how exploring a question can uncover many topics within the scope of the course being taught.

I hope this has caused you to pause and think of how exploring questions, relationships and properties in mathematics can lead to greater understanding than just teaching process and answer getting.

Here is a PowerPoint presentation you can download and use in your class.

The video below is a fun way to explore some of the attributes of prime numbers in a way that provides insight into the nature of infinity.   All of the math involved is approachable to your average HS math student.

Here is a link to the blog post that goes into a little more detail than offered in the video:  Click Here.

Click here to download a Power Point you can use in class.

Click here here download a PDF of the information covered in Real Numbers.

 

If you find these materials valuable, you could help me create more.