Focus on Math

Helping children become mathematicians!

Calgary City Teachers’ Convention: PS February 10, 2016

It is my pleasure to present this session “Power Up Your Problem Solving” to the participants of this session.

Regular problem solving is a powerful way to help students develop conceptual understanding in the various strands of mathematics. Since there is a tradition in North America of “teaching by telling” (the “here’s-how-to-do-it-go-practice-50-of-these” method), it may take many weeks to develop a culture of deeper thinking in a classroom. Students need a variety of thinking tools and strategies to work with, as well as skills and practice in talking about math problems, but the time it takes to help students gain these needed things is time well spent. The payoff is huge!

I hope many of you will be encouraged to begin building a regular problem solving program with your students. It works at every grade level!

All the here are the downloads for the problem solving session:

I would love to hear from you how it goes in your classrooms!

Mathematically yours,



Making a Difference November 25, 2015

Filed under: General Math — Focus on Math @ 12:40 pm
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Screen Shot 2015-11-25 at 11.25.37 AMIt 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!


Laura Fusco


Ten Things about Teaching Math I Wish I Had Known as a New Teacher September 22, 2015

Filed under: General Math — Focus on Math @ 6:09 pm
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Screen Shot 2015-09-22 at 6.10.24 PM Every teacher has a first year of teaching, one  which often seems overwhelming. Even the next few years can be taxing as one tries to deal with all the planning, the lessons, the assessment, the special needs, the routines, the supplies, the parents, the responsibilities, etc. Whoever says teaching is a piece of cake needs to step into the classroom for a while.

Besides spending years as a classroom teacher, I have been a math coach, a teacher of university math education courses, and a full time district math coach. My math journey has been exciting, but looking back I wish I had known in those early years of teaching some of what I know now.

Here are some thoughts along that line:

  1. My attitude toward mathematics matters. If I don’t like it, the students won’t either. If I love it, they will, too.
  1. Every child can learn math
  1. Concepts need to be developed beginning with concrete materials (manipulatives).
  1. It is important to use many kinds of visual/pictorial representations regularly.
  1. My students need my help to learn how to “talk math” and need time in my lessons to do it.
  1. There really are many ways to solve a problem, not just one “right way” (that was demonstrated by me, the teacher).
  1. It is better to do one problem five ways than five problems one way (Polya).
  1. Students need to develop strategies for solving problems, strategies that they understand and can explain.
  1. Adults use mathematical estimation daily. Kids need to practice it often.
  1. Doing algorithms requires no mathematical understanding, just a knowledge of how to follow rules.

What do you think? Let me know.

Mathematically yours,



Is the Task Worth the Time? September 18, 2015

Filed under: General Math — Focus on Math @ 4:42 pm
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blog picAll the learning minutes of the school day matter — in fact, if you are like me, often what I had planned for a given school day was not accomplished. Things took more time than I had expected, or there were interruptions to the day. The learning time seemed to go by so quickly.

Because learning time is a precious commodity, it is critical that we, as educators, think carefully about what mathematical tasks we are giving our students to do. Is the task worth the time it takes to do it? Are we getting good value for the time spent? Are we getting “bang for our buck”?

This is true for teachers at every grade level. But I am going to be a little bold here and say this is especially true for primary teachers: those who are teaching students in Kindergarten, grade 1, grade 2, and grade 3. There are so many activities available at the touch of a computer key, but certainly not all of them give us good value for the time spent.

Two examples of such activities are pictured here. In one, the students are asked to colour the picture based on the sums of the problems on the page. Can you see the problem with this task? The students are likely to spend more time colouring than engaged in math thinking. There is nothing wrong with colouring — I know it helps develop fine motor skills in these little people. The problem is that much of the time set aside for math is usurped by the colouring part of the activity. In the second picture there is a similar conflict. There are five math questions to solve, but the cutting and pasting of the numbers for the answers will take much longer. Again, cutting is a great skill, one which students should develop. I just think it is sad to have students cutting and pasting during math time in which I can have them developing number sense through more meaningful tasks.

I want to encourage you to jealously guard the math learning time that you have with your students. Use your critical thinking skills to evaluate prospective tasks (no matter where they come from!) and choose those that are truly valuable, those that don’t waste math time. By all means, have your students cut, colour, and paste. Just do it in art time, not math time!

According to Howden, number sense “develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Arithmetic Teacher, NCTM, February, 1989, p. 11.) Is your activity one that allows students to explore numbers, visualize them, and make connections to other math concepts? Is the task worth the time?

Mathematically yours,


blog pic 2


Processes are Important September 3, 2015

Filed under: General Math — Focus on Math @ 11:22 am
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Screen Shot 2015-09-03 at 11.14.33 AMIf students are to understand mathematics, they must DO mathematics – the kind of doing that is much more than just working through algorithms comprised of meaningless steps for them.

Students must get their hands dirty, so to speak, and muck around with numbers, with number concepts, with problematic situations. They must try out strategies and ideas, talk about them, test them, prove them, and represent them. They must look for alternate strategies and process those in the same ways as the first strategies. They need to consider the ideas and strategies of others and compare those to their own. They need to look for generalizations among the ideas. They need to consider what new questions arise from their work.

Math can be “messy” as students work to make sense of problems. In fact, it can be argued that if it is not “messy”, then there really was no true problem at all. If one immediately knows how to solve something, there is no problem, just some figuring to be done. A true problem requires one search for a solution.


This all brings me to the mathematical processes that the NCTM has been promoting for many years: communication, connections, problem solving, reasoning and proof, and representation. When students are engaged regularly in these processes, they cannot help but build true understanding of mathematical concepts. Without the processes, there may be an ability to do calculations, but students will likely be devoid of any depth of understanding.

“Understanding ‘lives’ in the processes.” Norris C., S. Chorney, D. Wright, & T. Thielmann. “Communication on Communication”. Vector, Volume 53, Issue 1 (Spring 2012).

I hope you will put these processes into practice with your students.

Mathematically yours,



OLOL Problem Solving Workshp September 2, 2015

Filed under: General Math,Ideas from Carollee's Workshops — Focus on Math @ 10:19 am
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Screen Shot 2015-09-02 at 9.52.23 AMA 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,


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


Simple Definitions Too Simple? August 10, 2015


Math definitions matter! There are many words we use in mathematics that have one meaning in that discipline and another in ordinary life. Take for instance the word “difference”. In regular conversation, if I ask you to find the difference between two things you are looking for some way in which the items are not the same. However, in mathematics, finding the “difference” specifically refers to finding the answer to a subtraction problem.

But we as teachers might be sending some confusing messages to students, sometimes even when we think we are right on track with our definitions. One example of this is the seemingly easy-to-define term “even”. How would you explain to a young child what an even number is?

There are two popular ways this property of even is explained to primary students: First, many teachers suggest that we can do an “even check by examining whether a particular number of items can be split into two equal groups. Armed with this definition, children should see that six is even since there can be two groups of three, but five is not even because there are two groups of two and one left over.

Alternately, teachers often suggest that students look at the value of the one’s place digit of the number in question. If there is a 2, 4, 6, 8, or 0 in one’s place, then a number is even. Using this method children should conclude that 74 is even since there is a 4 in one’s place, but 73 is not since there is a 3 in one’s place.

The problem is that both of these simple definitions are not fully correct. There are exceptions to them that, in fact, that are incorrect.

Concerning the “two equal groups” definition, young children figure out quickly that when sharing 5 cookies between two friends, each can have 2 ½ cookies. There are two equal groups, but the number 5 is still an odd number. What important detail have we failed to communicate here?

Concerning the “one’s place digit of 2, 4, 6, 8, or 0” definition, a student can declare that 74.3 is even since it fulfills the definition stated. Again, what important detail have we failed to communicate here?

Some might argue that these exceptions above the student; that we need not muddy the waters, so to speak, by giving extraneous information that we think is above our students’ heads. I disagree. I feel that if we just mention the restrictions as we talk about the definition, that it becomes part of the language the students are used to. We often underestimate how much students can understand, and we “dumb down” the language as a result. I would love to challenge your thinking along that line. In the case of primary children in particular, they love to learn what I refer to as “27-syllable” dinosaur names, yet we are afraid of using good math language with them!

I hope you will stop and think about the simple definitions you are using with your students and reflect on whether or not there are some hidden exceptions that need to be teased out and exposed.

Mathematically yours,