Visual Representations in Math

Without a strong emphasis on steps and procedures, students might actually begin to enjoy math because they will realize they can do math.  (Almost) nobody wants to sit and memorize steps to complete a problem, and many people will actually forget these steps at a later date.  Take the following for instance when comparing fractions:

1. Use the LCD to write equivalent fractions with a common denominator.

2. Compare the numerators: The larger fraction is the one with the greater numerator.

Here is an example from an online resource- but does this confuse anyone as much as it confuses me?

For the students that know what all of this means, those steps might appear to be easy.  But think about the ELL students in the class, or the students who have a hard time remembering the difference between a numerator or denominator.  These steps will only confuse them more.  Further, students will become to focused on memorizing these steps rather than gathering a comprehensive understanding of why each step makes sense.

This week, we looked into drawing and representing to understand math problems through a video, which put a lot of things into perspective for me.  After watching the video I started to think about my own experiences with drawing and representing and I realized it's how I have always done math.  I could never remember the steps, but always thought I was wrong because of this.  Students should never feel this, which is why being open minded to different ways of thinking and representations is vital for student success.  

During my placement, I remember my students all having some really interesting visual methods that I personally found confusing, but they found easier to understand.  For instance, the lattice method.  For anyone who has not seen the lattice method done, here is a video:

Even though I didn't understand this method at first, I would never take marks away from a student for doing it differently because it shows they understand!  In the video module this week it stated that "Drawing to represent gives access to understanding." This is a quote that really resonated with me, because that's why I personally love number talks so much- which I've talked about in various prior posts.  They allow students to show their thinking in a way that is not about the steps and procedures, but rather their own understanding of the problem.  Further, they provide other students an opportunity to see a method or technique that maybe they had never thought of before, but found easier.

This weeks topic also reminded me of a post I recently saw on Facebook, where we know everyone has an opinion, but I was particularly shocked at how against many people were on this particular picture to the left.  The person who posted the picture had captioned it with "Well.. homeschooling my kids keeps looking like a better idea."

I was mostly shocked at how closed minded many people appeared to be with new representations of mathematics that they never grew up with.

If a certain method, or drawing works for a student, we should never take that away from.  Rather, we should be encouraging these methods because it is what works for that student.  Allowing students opportunities to show their thinking will help them to value their own thought.  One way teachers can do this is through math journals.  I've never personally tried math journals, but I've made a goal for myself, that this year in placement I want to try out math journals.  I believe journaling can be a great exercise where students can record their thoughts, understandings and explanations- either visually or through written words.  For anyone else who is also interested in math journals, Teacher Vision has a great article to start teachers out, with full lesson plan ideas.

Giving students an opportunity to express their ideas in math is just as important as expressing ideas in Language Arts.  My hope is that we all become more open minded to different ways of thinking and representations of math so all students can feel confident in their own abilities!