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!
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.
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.
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.
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!
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.
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?
I’m really excited to be back at work after a lengthy absence with a terribly awkwardly broken leg! I’m happy to report that I’m feeling about 80% recovered. Despite this, I’m hitting the ground running (well figuratively). I will be travelling all over the province over the course of the next few months to deliver workshops to Occasional Teachers. I did this last school year as well. While the previous workshops were very well received, I want to make certain that I’m bringing current and relevant content to the teachers that I will work with. That being said, what is it that OT’s need? What can I bring to them to ensure that they are getting the best bang for their buck on their day of PD? I’d love to hear from you.
If you are part of HWDSB, and haven’t already heard, you’ll begin hearing rumblings about Number Talks by Sherry Parrish. It’s not a new concept if you are familiar with Strings by Cathy Fosnot or Cluster Problems by Van de Walle. It is, however, a very teacher friendly resource that maps out the process beautifully and simply. The focus of such a program is mental math skills which is something that teachers frequently voice as being an area of student weakness. The beauty of such a concept is that it requires 10 minutes a day and has proven to be very engaging for students. With the new mandate of 60 minutes of math per day (previously 300 week, but didn’t specify that had to be 60 minutes each day), I got to thinking. Most of us have 50 minute periods on balanced day. My idea is that a teacher use a Number Talk outside of the 50 minute math period. For example, teachers may wish to begin each day with their Number Talk for 10 minutes, or maybe after nutrition break each day, or whenever it fits in their schedule. The regular math program can still exist within the regular time slot and voila, there’s your 60 minutes and it’s 60 minutes of real, meaningful math. Doing a Number Talk will give you far more bang for your buck than having your students do calendar or drill sheets. To read more about this amazing resource, see below:
Here I am, on summer vacation, 12:39 a.m. on a Monday night, and I find myself working on a math workshop that I am doing next week. I don’t need to be doing this now, but I got lost in it. I started digging up some good problems to use, watching clips, reading articles, and actually solving some math problems (the most fun). The minutes turned into hours, and frankly, if common sense wasn’t sending me to bed shortly, I’d happily do this all night. I was concerned at one point that I didn’t have hobbies like many of my friends, no real interests or passions. It has occurred to me recently that math is my hobby. I am always happy to do it, learn about it, read about it, talk about it, promote it (shamelessly at times), just bask in it. So, I’m off to bed, but only because I want to get up in the morning and start at it again. It is true love.