# Focus on Math

## Helping children become mathematicians!

### Calgary City Teachers’ Convention: Seeing DotsFebruary 10, 2016

The 100 Dot Array remains one of my favourite tools for helping students visualize numbers. This session at the CCTC focuses mainly on its use with students in grades 2 and 3, although it can be used at many other grade levels. We will be talking about the best way to introduce the tool to students, showing an early activity to help with general number sense, and using the number in problem solving situations. A variety of problem are included to show its diverse use.

Please let me know how it goes with using the 100 dot arrays with your students! I love to hear about kids using tools and strategies in math.

Mathematically yours,

Carollee

### Making a DifferenceNovember 25, 2015

Filed under: General Math — Focus on Math @ 12:40 pm
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It is always wonderful to hear when what you do makes a difference, and I wanted to share with my readers the wonderful email I recently received from a student who took a university course (Designs for Learning Mathematics: Elementary — a “how to teach math to kids” course) with me a few years back. I love the “ripple effect” that happens when I can help teachers and student teachers change their approach to teaching mathematics, which in turn can make a great change for how students understand mathematics!

Thank you, Laura, for taking time to share this! Here is Laura’s letter with only a couple of edits for clarification. Woohoo!

Hey Carollee,

Don’t know if you remember me, but I took your Simon Fraser University course a couple of years ago. My husband and I relocated to Ontario and I just started my first teaching contract 3 weeks ago in a grade 2/3 French immersion class. My students are very weak in math… but since I have  started teaching the way you taught me to, I can already see the ideas flowing in their heads! They are really starting to get it!!!

Two days ago we did our first word problem… they were blown away when I put the answer on the board and told them I didn’t care about the answer, but instead how they got to it! The first day they were a bit shy to try and fail, but by the 3rd day boy were they trying everything! number  lines, dot array, hundreds chart, blocks, pictures!!! I felt such joy!!!!

So… I guess  I just wanted to say thank you… and to let you know that  you are changing children’s understanding of math… EVERYDAY!

THANK YOU, THANK YOU… THANK YOU!!

Laura Fusco

### GAD Workshop, Surrey, BCOctober 23, 2015

Thanks to the teaching staff of GAD Elementary in Surrey, BC, for their warm welcome and heartfelt participation as we delved into problem solving, math tools and strategies, and math processes (especially communication). Changing our teaching practice is not an easy feat, but if we commit to some small changes, practice them regularly, add more changes, practice those regularly, and keep on going in that manner, we can end up making a significant and lasting change that will benefit students greatly.

Remember, “math talk” does not just happen. We have to plan ways to incorporate it into each math lesson. It is a good idea to create math partners so students are responsible to talk to someone about their math thinking. Modeling (letting students hear YOU talk through a demonstration problem) is always a good idea. Responding to students with proper math language/vocabulary (when they have not used such) is helpful. Posting “sentence stems” is a great way to give them an easier start in speaking math. Additionally, try creating a “math words” chart with the students that they can use as an on-going reference in both their speaking and writing (click here to see an example of a “math words” chart.)

As promised, I am adding links from this post to the handouts from today’s session (see bottom of the post) and some that we just talked about.

I would LOVE to hear from any of the GAD staff of how things go in your math lessons in the next weeks. You all listed something that you could begin to do right away in your classrooms, and I hope you will share what you are doing and the effect it is having on the students.

Remember, understanding “lives” in the processes! Reflect on your teaching regularly to see if you are embedding those processes into math classes. It will make a big difference in students’ understanding if they are immersed in the processes!

Mathematically yours,

Carollee

100 dot array (teacher size)

100 dot arrays 4 per page

100 dot arrays 6 per page

100 dot arrays 12 per page

break apart number sheet – 2’s

break apart number sheet – 3’s

problem solving assessment rubric

10 frames (teacher size)

10 frames (student size)

10 frames blank mini’s

### OLOL Problem Solving WorkshpSeptember 2, 2015

Filed under: General Math,Ideas from Carollee's Workshops — Focus on Math @ 10:19 am
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A big thanks to the staff of Our Lady of Lourdes in West Kelowna, BC for the attention, participation, and “buy-in” at the Problem Solving Workshop yesterday. I hope you went away inspired and equipped to do problem solving regularly with your students in this coming school year.

As promised, I am posting links to the items we discussed. Some of those are available elsewhere on the blog (and may have more explanation about using them in lessons) so you may want to use the “search” feature on the right side of the page to find other posts about particular resources. I certainly hope you will let me know how it is going in you classroom. Send me photos of your students’ work, your Tools and Strategies posters, your “sharing pages”. It’s going to be a great year!

Mathematically yours,

Carollee

100 dot array large

100 dot array – 12 per page

100 dot array 6 per page

100 dot array 4 per page

student rubric for PS

ten frames – student

ten frames – teacher

mini blank ten frames

percent circles

### Ecole Muheim Elementary, Smithers, BC: Math WeekMay 20, 2014

My appreciation to the staff at Ecole Muheim Elementary in Smithers, BC, for your great participation in the workshop last Friday! With the weather as beautiful as it was, and the fact that it was the beginning of the Victoria Day long weekend, your focus and participation is doubly appreciated!

One of the things we talked about at the session was the idea of breaking away from the traditional focus of teaching mathematics in units that are seldom revisited. You know, the three-week unit on fractions, after which we do very little with fractions for the remainder of the school year. I showed you the “math week” I worked with as a classroom teacher, and promised I would post a copy of that.

Please note that the main focus everyday was on number and operations. I think particularly in elementary school that this is often neglected. We will do units on patterning, on geometry, and such (which, of course, are valid topics in mathematics!) that go for weeks at a time while leaving behind number. I personally believe that this is not the best way to cover the topics, and so propose a weekly plan where the other strands/topics are addressed in mini-lessons.

I have had teachers tell me that the idea of juggling all of the different topics at once was overwhelming – and to that I offer the plan of keeping the mini-lessons the same for a week or two. Even doing that you will be covering all of the topics each grading term, which I believe is a very good thing.

You do not need to re-invent the wheel so to speak when it comes to the mini-lessons. For example, as you consider what you will do in the patterns & relations mini-lessons, you can go to your regular textbook source and use lesson ideas from there – you will just chunk the lessons into smaller bits and do them in successive lessons.

I coached one teacher regarding the Math Week system, and she created for herself a series of thin binders to keep track. Her Monday binder had the unit content photocopied for the topic for each Monday mini-lesson, and she just highlighted the chunks she was going to do and numbered them in the order she was going to do them, and thus in one planning sitting she laid out for herself the next 8-10 mini-lessons on that topic. She did this same thing for each of the days of week. The system worked very well for her in that manner.

For the Muheim folks I will add links here to a number of the handouts we used so you have access to clean copies of those:

–100 dot arrays (1 large)

–100 dot arrays (4 per page)

–100 dot arrays (6 per page)

–100 dot arrays (12 per page)

–number of the day level 1 (English)

–number of the day level 1 (French)

–number of the day level 2 (English)

–number of the day level 2 (French)

–number of the day level 3 (English)

–number of the day level 3 (French)

–base 10 grid paper (larger)

–percent circles

I would love to hear about the ideas you are trying in your classrooms! Please comment on the blog or send me an email!

Mathematically yours,

Carollee

### Seeing Dots: NCTM 2014 New Orleans PresentationApril 11, 2014

I am excited to be here in New Orleans at the 2014 NCTM conference. Yesterday was a great day of sessions for me, and I am delighted to be presenting a session in just a couple of hours! “Seeing Dots: Using Arrays to Add, Subtract, Multiply and Divide” will focus on all the different ways the 100 dot array can be used to help students visualize and represent numbers — something which leads to a deeper understanding of numbers.

I am posting the handout from the workshop as well as links to 100 dot arrays is the different sizes.

I hope you try using the 100 dot array in your elementary classroom!

Mathematically yours,

Carollee

### Number of the Day – Level IIIMarch 10, 2014

Today I am posting the third Number of the Day sheet. I cannot overstate that I believe that elementary school students should be involved with numbers everyday they are in school!

Level III is one to primarily use with numbers to 100. The section “100 chart tic-tac-toe” will not be familiar to most. I had devised that math game based on the positioning of a number on the 100 chart. For instance, if 26 is written in the centre of the chart, then the middle line is to show one more and one less than 26. (25, 26, 27 across). Above the middle number is 10 less, in this case 16. Below 26 is 10 more, 36 in this case. The corners can then be filled in using the horizontal or vertical relationships already established. (For more on the use of 100 chart tic-tac-toe, see my previous blog post.)

When using 100 dot arrays, I have students use highlighters to colour the numbers. I also stress marking efficiently – we do NOT colour each individual dot; rather a line or partial line is coloured with a swipe of the marker.

At every level breaking apart the number of the day is an important component of the sheet. Quoting John Van de Walle once again, “To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers.” [Van de Walle, J. and Folk, S. (2005). Elementary and Middle School Mathematics: Teaching Developmentally (Canadian Edition). Pearson: Toronto.]

I did have one teacher ask a question about the breaking apart section. She was used to having students only break apart numbers according to tens and ones. Thus 26 could be broken apart as 20 and 6 or 10 and 16. But sometimes it is easier to work with numbers when we break them in ways other than ten and ones. Consider the thinking that might happen when adding 26 + 27. If a student knows that 26 comes apart as 25 an 1 and that 27 comes apart as 25 and 2, it is easy to put the 25′s together to get 50, then add the 1 and the 2 —total 53. Students who use the 100 dot array often get especially comfortable with 25′s. Also consider adding 97 and 36. If a student notices that 97 is just 3 away from 100, it makes sense to split 36 as 3 and 33. Breakng apart in tens and ones are definitely useful, but so are other “break-aparts”. If students do not practice this kind of thinking they are not likely to ever do it!

I had one teacher here in my district that was using this sheet and her students were getting tired of making tallies for large numbers. So I am including a second English version of the sheet asking for equations for the number instead.

Again, a French version is offered as well with thanks to my friend and colleague Lynn St. Louis for her translation.