Student Created Tools at Hillcrest

One of the most powerful lessons for fractions is having students create their own fraction strips.  This is usually greeted by “We did this last year”; however, what I’ve found is that it’s a good review of basic fraction concepts and a good launching off point for comparing fractions and for finding equivalent fractions.  Many students have created the strips but never actually used them as a tool.  You can see from below how they can be used so students can develop their conceptual understanding before the procedural is ever introduced.

Maddison, in grade 5 at Hillcrest, compares 1/4 to 1/2.  It’s clear from the photo that 1/2 is greater, though a common misconception is that 1/4 is greater because 4 is greater than 2 (as seen in the denominators).  We had a great discussion on why the pieces are larger for the halves than the quarters.


Moving from comparing unit fractions, we now use Maddison’s strips to compare 2/3 to 2/4.  Again, this provides a fabulous visual for students.


Here, Maddison shows me all the ways she can create equivalent fractions to 1/2 with her strips.

I’d like to thank Mr. Weaver for a great start to my day in his amazing classroom.  I’m hoping for another visit some time soon!

The Importance of C – P – A in Math Learning

Many people (teachers, parents, and students) view manipulatives as a tool only for students who are weak or need extra help.  This, in fact, couldn’t be further from the truth.  There is a significant amount of research to support the necessity of the CPA progression for learning math – that is first concrete learning, then pictorial, then the abstract.  A very important skill to possess to think mathematically is visualization.  We need a model from which to visualize.  If I tell you to picture something you’ve never seen before, it’s far more difficult than if you’ve actually seen that object or situation. This is the purpose of manipulatives – to provide the basis for our visualization.  Many people will argue that manipulatives confuse their students.  I would suggest that students don’t have a solid understanding of a concept if manipulatives are confusing them.  This may happen with a concept like division.  Often times, students are instructed immediately at the abstract phase with things like the long division algorithm and phrases like “how many times does 3 go into 8”, drop down, arrows, subtract, etc.  For students who are able to carry out the algorithm accurately, many of them don’t know what the numbers represent and whether the division is partitive or quotative.  Do you know the difference?

As we entered double digit division in Mr. Cowan’s awesome grade 4 classroom at Viscount Montgomery, you can see how this concrete learning supported the students understanding of the situation.  Some students felt comfortable operating at the pictorial stage while others had built models with manipulatives that they could use to help them communicate their thinking both orally and in writing.  Skipping these developmental steps will lead, if you are lucky, to short term success and almost guarantees lack of retention or understanding in the long run.

Visualizing Fractions at Viscount Montgomery

I always try to have my posts revolve around anything innovative that could support student math learning.

I had the pleasure of working in Ms. Weston’s 6/7 room during which she was helping them to visualize fractions and use models to show their thinking.  I worked with a group that used erasable markers on the desk top.  I loved this as they felt comfortable taking risks and making alterations to their work.  What seems like such a small measure actually contributes to an environment that is inviting and safe for students.

In addition to this, I witnessed the textbook being used in an effective manner.  Ms. Weston used one problem from the textbook to reinforce the learning and consolidate student understanding.  I could see from my work with the students that they had a good understanding of the lesson from the day by observing and listening as they worked on only ONE question.  Do you need to have students do pages and pages of the same types of questions?  Thoughts?