Friday, October 10, 2014

Visual Thinking: Using our eyes as well as our minds

One way to help capture student interest, and to create better access to math is to express mathematical ideas by using images or visual representations.  In Algebra classes, we have been exploring visual patterns as one lens to identify and study linear, quadratic, and even cubic patterns.

If this image shows steps 1, 2, and 3, how many squares would you predict in step 43? 

These patterns are easy for many students to approach.  Even though they present the same ideas as an equation or a table of values, for many kids, they are much less intimidating.  They also help to cement understanding by asking students to express ideas in multiple ways.

Dividing these shapes up into sections helps us to see the x term, the x squared term, and the constant term in this trinomial pattern.

...and if you think that these are a way to water down the math, try to figure out how many penguins are in the nth term of this friendly looking, but difficult pattern!
The patterns shown here come from a site put together by master teacher Fawn Nguyen.  Check out some more of the patterns she's accumulated, and work with your kids to try to create a general rule for the nth term HERE.

Saturday, October 4, 2014

Using our hands as well as our heads

One of the toughest steps for Algebra students to make is from the concrete to the abstract.  In math language, this is often when we try to generalize a result - or to create a rule, which works for a general case rather than just for a specific case.

One example might be multiplication of numbers versus multiplication of variables.  By the time they get to Algebra, students have a good conceptual understanding of 3 times 5, of 46 times 250, or even -32 * 25.  But moving from these concrete examples to an abstract concept like x times y, or 5x+2 times x-5 is often very challenging.

One way to help students to make this leap is by implementing some kinaesthetic activities.  These can help to bridge the gap from the specific to the general and can give students some "muscle memory" to help them remember how to make this leap.  Examples that we have used this year include using algebra tiles to represent variables and numbers, or actually acting out with our bodies some function transformation.

We need to use all of our tools and ideas to assist students in climbing up the ladder of abstraction!

Wednesday, October 1, 2014

Scholarship and Kindness

At the start of the year, it is important to articulate what we need to have a successful learning environment.  The class is a better place for everyone when students feel that they have a voice in how the classroom will function - and we do include their ideas about how our class will run.

When we collaborated to create our Algebra class learning agreements, there were two themes that kept coming up: the high value that this group places on learning, and on compassion for each other.  We brainstormed all of our ideas with sticky notes, prioritised and categorized these ideas, and put together a poster, which synthesised and summarized these ideas.  You can see our classroom agreements all fall under the umbrella of scholarship, but are rooted in kindness.  Students read over the summary and signed their names to the poster before we hung it up in the classroom.