Prove it

Select 3 consecutive number (i.e. 2, 3, 4).   Multiply the first number by the third number.  Now square the middle number.  Do the same with another group of 3 consecutive numbers.  What do you notice?  Will this happen every time?  Why?  Can you prove it?

We worked through this at “Let’s Talk Math” as we examined the notion of proof.  How do we get our students to develop and evaluate mathematical arguments and proofs and use mathematical reasoning to deepen their mathematical understanding?

Things that make you go, “hmmmmm”.

Number Talks AGAIN!!!

When I hear over and over again how weak mental math skills are, it begs the question, “What are we doing about it?”  People will be quick to blame “discovery math” or “new math”, but the fact remains that it has always been a part of the curriculum for students to learn basic facts and also mental math skills.  So why don’t our students have these skills?  These skills take practice and not necessarily a drill and kill approach.  That only works for some.  What our kids need is time to make sense of numbers and the way that they work together, their relationships, how to manipulate them.  There is no better way, in my opinion, than a daily Number Talk.   I wholeheartedly believe that if all classes did 10 minutes of Number Talks daily, we would see such a significant impact on student learning in math.  Why are we not doing this?  What are your thoughts?

Grade 1/2 Number Talk

I had the pleasure of joining Mrs. Antonucci’s grade 1/2 class at W.H. Ballard for some Number Talks.  We are working on helping the students to articulate their own thinking as the solve mental math problems.  Today, in the photo, you can see we were working on understanding the associative property in conjunction with a “make 10” strategy.  The feedback from the class was that I will need to come up with something much more challenging because they are amazing little mathematicians!

Scope and Sequence

Edugains has long been a great resource for math instruction.  They have recently beefed that up even more with some great guidance for teachers to aid in their planning of math.  They have published a scope and sequence for the Ontario Mathematics Curriculum with suggested timelines for each topic.  Below is a link to the very general grade 1 – 8 sequence.  More detailed scope and sequences that are specific to divisions can be found in the corresponding tabs on my blog.  For example, click on the Primary resources to find the scope and sequence for grades 1 to 3.

Imagine the possibilities…

Many teachers and students have these types of activity trackers.  What an amazing opportunity to bring mathematics from the real world into the classroom.  Think of all the math questions and responses that you could generate with the data collected on these devices.

  • how many more steps?
  • how much longer?
  • how many steps per hour?
  • how many steps in a week?
  • how big is a step?
  • how many steps to get to …

The list could go on.  As a classroom teacher, I would use this everyday, multiple times over the course of the day.  In my own experiences as a teacher and a parent, the opportunities to engage kids in math through this are are authentic and fun.

The Power of the Model

I have long maintained the importance of students understanding conceptually before procedurally.  Take, for example, measurement units.  How many of us grew up truly understanding what a square centimetre looked like or a cubic metre?  I know that I didn’t.  On a recent visit to Sault Ste. Marie, I had an opportunity to visit a K-8 school and happened upon this:

I can only imagine how much better I would have understood measurement units, conversions, volume, and the formulae for various volumes had I had the opportunity to explore these concepts outside of the pages of a textbook.  So powerful indeed.

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.