Why Make Math Models?

Why make math models?

Making math models allows you to understand the mathematical processes that occur when working with numbers or shapes.

Making the models takes time, but that time builds meaning.  Also making models is a differentiated activity that creates the opportunity for terrific coaching/mentoring opportunities. Making math models moves learning from passive recitation and response to multi-modal interactivity.

What are the considerations with regard to making math models?

First, start with the standard.  What is the standard describing?  Take the standard apart with students.

For example, today I worked on a lesson associated with this 4.NBT.1:

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. 

After that, ask the question, What does this standard look like in pictures, numbers, and words?  As an organism, how does this mathematical process work?  In what ways can we create models to show the mathematical process in action or in image?

Next, create models to demonstrate the standards' meaning.  As I created the model below, I thought about order, shading, size of text in order to make the model as meaningful and representative of the standard as possible.

Note that the size of the font increases as the value of the place increases (as I think about it, it would have been great to have the cells increase in size too--next step).  Also notice that places in the same periods are shaded with differing shades of the same color--richer, darker shades for numbers with greater value.  The period's column is white demonstrating that it is the dividing point between whole numbers and decimals/fractions.

I started to create another version with exponents and a triangle to show the way the numbers increase in value depending on place:

I used Sketch-up to improve this model representing that values increase.
(Not in proportion, however--the Eames do that well in the film below).

I also want to create this model on SCRATCH or with other coding as well so that it has interactivity, but at this time I'll leave that choice to my students at this point since their coding is better than mine. 

After that I used symbols to show what happens as you move up and down the place value chart:

I will add the division notes as I work with the students, and I will also demonstrate how it's X100 or X1,000. . .depending on how many jumps you make on the chart.

Further I'll use the Everyday Math online eTool to create models using base 10 blocks to show the distributive property and what is happening with quantity as we multiply by multiples of 10.

As I create these models, my understanding of the standard increases 10-fold!  I will probably also reshow the Eames Power of 10 movie as well, a movie that makes a powerful impact as we study the powers of 10--an example of using film to create a powerful model of mathematical processes. 

I may revisit the eduCanon assignment we did earlier in the year too.

Most of all, I'll give students the chance to make these models and interact with the content and meaning of the standard.  This will provide the essential meaning, meaning that David Culberhouse emphasizes with this quote, "Whether it’s the Kelley Brothers with building creative confidence, the DuFours and Eaker on improving education, or Pfeffer and Sutton for initiating business success…they all expound on the value and necessity of doing." in his post, "Mindshifting: From "Try" to "Do."

Hence, my investigation with model making and math continues--a journey I'm invested in at the moment.  I welcome your links, thoughts, and questions. 

Helpful Links
Beyond Traditional Math Website 

Thoughts to Come
School-wide Math Model Galleries Online/Off
Using Math Models to Teach Younger Students: The Student as Teacher
Maximizing the Use of Online Tools to Develop Math Learning and Understanding
Playing with Math