WPP: Part A

I am having a problem.  The problem is that my students are not making lasting connections with the hierarchy of numbers.  Lets tart from the beginning.

The Hierarchy of numbers is a way of classifying numbers.  When you are very young you learn about some very simple numbers like 1, 2 3, 4, 100, and so on.  Even though there a lot of these types of numbers (infintly many), they are very easy to understand.  You can see one car, you can touch two apples, you can even hear 1,000 football fans all cheering for their favorite team.  These number are called the Natural numbers.  That makes sense because these are the number that come most naturally to us. 

Then, Things start to get more complex.  The all mighty zero is added to the mix.  This is a very hard concept to understand because it is not easy to see, touch, or hear zero.  It takes time to let that concept sink in.  If you think about all of your natural numbers AND you also include zero then you have the set of numbers that are known as the Whole numbers.  Some people think they should actually be called the "hole" number instead but that's just a bad math joke (get it?).

Then another big step takes place when negative numbers are introduced.  These are another challenging concept because once again, it is harder to show people what a negative number looks like.  When these negative numbers are added to the Whole number (The Natural numbers and zero) you have a collection that is known as the Integers.  And the process continues as you add in fractions, decimals, and even stranger types of numbers. 

Students seem to do well with the first few tiers of the hierarchy but as things become bigger and more numbers are added to the mix they lose their connection to the material.  One of the problems is the one that I have already mentioned.  It becomes harder and harder to see and touch the different kinds of number.  Its not possible to show the square root of two apples.  The other problem is the way the number sets are defined.  As the sets get bigger and include more numbers they tend to stop being defined by what they are and start being defined by what they aren't.  Its kind of like trying to explain what it means to be alive.  "Alive" and "living" are very general terms that apply to a lot of different things.  You can't say that all living things have bones, or hearts, or leaves.  You will miss a larger portion of life by using these terms so it is sometimes easier to define "living" by stating what it means to NOT live.

All of this means that students don't particularly care about the hierarchy of numbers and even when they do learn it the information doesn't stick with them for very long.  I have tried to teach this topic a number of different ways in my classroom but I still struggle.  It is as if there just aren't enough examples in the world.  My students don't leave my classroom seeing the same things I see.

I have tried using diagrams and they work well but they are far from perfect.  They require a lot of examples.  I feel like I have to come up with hundreds of different examples of numbers that fit into each tier and that is impossible.  I can;t write all of the numbers that I need to write.

I have also tried interactive lessons like the one described HERE.  In the lesson the kinds have to put number they have been assigned into certain locations.  This is good but it lacks the concept that some sets of numbers exist within other sets.

I want to remedy this issue that I am having and the only way I can think to do it is to differentiate my instruction in a lot of different ways so that at least one method clicks with each individual in the class. 

My proposed technique.

I would like to start the first day with what I am comfortable with and that is the diagrams. I fully know that the diagrams are flawed but they still hold some value for some students so I would still like to use them.  I would also like to create a video version of this diagram so that it stops being a static item and becomes dynamic.  I would like to have students create short videos of themselves finding the different kinds of numbers in the world.  Natural numbers will be very easy and irrational numbers will be much harder but I believe it is possible.

To continue with the diagram, I would like student to create their own "trick" to help them remember the the hierarchy.  My initially thought for my "trick" would be that my finger tips would represent natural numbers.  Then if you touch your thumb to your index that creates a "0" so my hand would be Whole numbers.  Then my elbow would be integers because it can swing back and fourth representing positive and negative numbers. And so on.  another great idea would be a song or rap.

I would also like to work on the interconnected nature of the different tiers by having students create a collection of yes/no questions that could be answered to determine where a number belongs.   we could then turn the questions into an online quiz or a program (graphing calculator or C++).  I think this would be very beneficial for student because it would really force them to understand exactly how each of the number sets were interconnected.

I have definitly identified a problem in my class and come up with a number of different techniques to help but I won't know if it works until much later down the road.  As stated earlier, students usually retain information about the hierarchy long enough to do well on a chapter assessment but they lose it after that. My indicator for success would occur at exam review time or even at standardized testing.  I am trying to create activities that will help my students hold onto what they have learned for more than just a couple of weeks.  I want to see student recall this knowledge months later when they really need it.

Have you ever taught the number hierarchy or number systems?  What have you tired that I am not thinking about?