Connections in Math

  I'm currently reading a book "Writing on the Classroom Wall" by Steve Wyborney which is filled with big ideas that apply to learning.

The first one is "Learning is about Making Connections".  Making connections is one thing my students struggle with daily.  They approach each topic as if it is totally isolated.

I constantly struggle with trying to help them make connections.  Steve includes some great examples, including ones I can use in my lowest level math.  I like the ideas he presents but the review of the book is for another day. 

The first big idea brought up the need for students to connect ideas together in math.  I came across an article from the National Council of Mathematics discussing this very topic.  The example the author provided gave me an "aha" moment because I've never seen the connection.  At some point in elementary, middle or high school, teachers cover multiplication, division, multiples and factors.  When the class used manipulatives to find the area of various polygons, a student pointed out this was just like using factors and multiples from earlier in the year.  Give it some thought and that student is absolutely correct.

Too many times, both students and teachers rely on tricks or procedures without going any deeper?  My students cannot explain why when dividing fractions, they have to flip the bottom fraction and multiply.  I've covered it in high school math because my students have never seen the explanation.

The NCTM states all students should:
1. Recognize and use connections among mathematical ideas.
2. Understand how mathematical ideas connect and build upon one another to build a coherent whole.
3. Recognize and apply mathematics in situations outside of mathematics.

This article actually reinforces the idea of showing all the steps even if the students are not thrilled with all the micro steps because they are so used to using shortcuts and tricks. 

This spring, I started looking at reorganizing all my math classes so I group similar processes together to see if I can help students see connectivity.  For instance, review exponents and rules of exponents before going on to rational exponents, logs, natural logs etc which all use the same rules rather than teaching them at various points throughout the course.

I'm thinking of reviewing regular fractions and teaming it up with algebraic fractions since the process is the same for both.  I'm hoping by starting with the simpler review, students will have a chance of extending their understanding that one situation is actually a more complex application of the other.

In the past I've reviewed the lattice method of multiplying whole numbers before showing students how its used to multiply binomials.  The process is the same.  I also have used drawings of the same type to show both the multiplication of numbers and binomials. 

I'm hoping to utilize the concept of Big Ideas next year to show the umbrella idea applies to more than the one thing they've concentrated on the past. 

I'd love to hear what you think about this.  I appreciate when I hear from readers.  Have a great week.