Focus on Math

Helping children become mathematicians!

10 New Year’s Resolutions for the Math Classroom December 31, 2014

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1. praise effort, not correct answers
2. make sure my students know their intelligence is not fixed: hard work pays off
3. make my classroom a safe place for students to take risks
4. encourage students to take risks
5. give my students rich problems that require they engage in problem solving
6. build a class repertoire of strategies
7. have “thinking tools” handy
8. give regular attention to basic facts (for students who do not know them)
9. give students lots of opportunity to talk to each other when solving problems
10. support math vocabulary learning with a word wall chart

Mathematically yours,



Use What You Know to Figure Out What You Don’t Know March 7, 2014

Screen shot 2014-03-07 at 10.36.15 AMI was working with some students this week who were learning their “basic facts” in multiplication. These are generally considered to be those one-digit times one-digit problems that we use when we figure out the products of multi-digit problems. I was going over some different strategies and ways of thinking that can be used to help students learn those facts.

There are a number of strategies that can help in the learning of basic facts, but one phrase sums up many of those individual strategies: “Use what you know to figure out what you don’t know.”

This phrase actually applies to FAR more than just the learning of basic facts. The truth, however, is that often we condition students to NOT think for themselves in mathematics. We have a long tradition of teaching by telling: the “here’s how to do it now go practice 50” method. In reality, that kind of math lesson programs students to think that unless someone has told them “the way” to do something (and, of course, they must remember exactly how to follow the directions of “the way”). If they forget, they are stymied and cannot know how to proceed. They remain in their “stuck” position until someone comes to rescue them with “the way”.

It is far better to regularly encourage students with the idea that when they are stuck, they need to stop and think about the things they DO know that can be applied. We might ask questions (and teach them to ask themselves) such as these:

  • What might be something similar that you do know?
  • If the problem had smaller or simpler numbers, how would you try to solve it?
  • Why did you choose to do it that way?
  • What is important in the question?
  • Is there a pattern?
  • Is there a way to record what you have done so far so you a pattern might be noticed?
  • Can you think of another way to do that?
  • Does this remind you of another problem you have done?

In the case of basic facts, “Use what you know to figure out what you don’t know,” might look like this: a student cannot remember 6 x 8. But 5 x 8 is known. So, knowing that 5 groups of 8 is 40, he need only add one more group of 8 to have the needed 6 groups of 8; thus 40 + 8 = 48 is the solution to the unknown fact.

Students may need practice in doing such strategies, but the important thing is that there ARE strategies to help. It removes the case of having to rely solely on memory and sitting there stuck if memory fails.

What are you doing in your classroom today that encourages students to help themselves when they are stuck? Maybe post the title phrase for them (and model for them how it looks): “Use what you know to figure out what you don’t know.”

Strategies make a difference in student learning!

Mathematically yours,




Strike it Out: a Primary Game from NRICH March 5, 2013

Filed under: Basic Facts,General Math,Parents,Primary Math Ideas & Problems — Focus on Math @ 12:06 pm
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Screen shot 2013-03-05 at 8.43.25 AM One thing that I came across recently on the NRICH math website is an quick primary students’ game for practicing addition and subtraction with numbers to 20. On the particular webpage for “Strike it Out”, they offer a poster (a picture of which is posted here), a short video clip of the game, and a power point file all which give the instructions for the game.

I had each of my grade two math classes playing this game recently as a warm-up activity, and they loved it! The games go quickly for the most part – of course, some of the pairs of students were slower at the game, but those students were still engaged and trying their best.

The rules, simply, are these:
• Using a number line marked 0-20, one student begins by creating and recording an addition or subtraction equation, e.g., 4 + 10 = 14. On the number line he crosses out the 4 and 10 and circles the 14. His turn is over.
• The partner must now create a new addition or subtraction equation, but it must use the number 14 as one of the first two numbers, e.g., 14 – 6 = 8. She would crosses out the circled 14, crosses out the 6, and circles the 8. Her turn is over.
• The 8 must be used now by the first partner in his new equation, with the recording and crossing out and circling continuing.
• Play continues until one of the partners cannot make a correct number sentence, and the player who made the last correct equation wins.

Although there are many possibilities for equations near the beginning of the game, there are fewer possibilities as the game progresses. I watched students doing a lot of mental math trying to come up with appropriate equations. The students who needed support had a set of ten frames on the table to use to help the visualize and calculate.

I am including for download the game board page I made for students to use. (Cut the page in half to use.) Two students play on a single game board at a time.

There are lots of other great ideas on the NRICH math site for many different levels. I hope you will take some time and explore what is there!

Mathematically yours,

Link to game on the NRICH math site here

Download the game board here.


Metacognition using an Addition Strategy Math Mat February 28, 2013

Filed under: Basic Facts,General Math,Parents,Primary Math Ideas & Problems — Focus on Math @ 9:14 am

add strategy mat DM Recently a teacher shared with me a math mat she had created listing a huge variety of addition strategies (thanks, Doreen M.!) Now, the idea of such a strategy mat is not brand new. Indeed, many such mats are circulating on Pintrest and other sites.

What delighted me about Doreen and her mat was not the uniqueness of the mat, but how she was using it. Doreen told the story of having given the mats to her students after they, as a class, had talked about each of the strategies. The math problem-solving lesson was structured with the students being given a block of time to solve the day’s problem using as many strategies as possible. Doreen hoped the students would use the strategy mat to prompt their thinking as they were solving word problems.

However, Doreen observed that the students rarely referred to the mats during their actual problem solving work. The students basically ignored the mats, even those who particularly needed the scaffolding.

Her response to this was to have the students do some metacognition regarding the strategies they had actually used in their personal solutions – but to do this once they were done solving the problem. Each student already had a laminated copy of the mat, but now she gave each student a marking pen. Students were asked to look over their work, and each time they noticed that they had used a particular strategy on the mat (e.g., breaking down a number into smaller parts or using doubles), they marked it on their own mat.

This, then, become the “norm” when doing problem solving. Over time, the students in the class became much more aware of which strategies they were using as well as the ones they weren’t using. This metacognitive thinking provided a great starting place for discussions when sharing solutions for the particular problem of the day.

So I invite you to try the adding strategy mat with your students. But more than that, I hope you will also try Doreen’s method of having your students do some “thinking about their thinking.” There is great power in metacognition!

Mathematically yours,

download a copy of the addition strategy mat here


Basic Facts: Mental Math as a Foundation for Multiplication Fact Strategies March 16, 2012

Basic Facts are still very important. Although newer curricula put a greater emphasis on problem solving, communication, reasoning, and representation of numbers, basic facts are still an integral part number sense in students. If a student is a good “memorizer”, then learning the multiplication facts will not be difficult. However, for many students the random bits information we call “facts” don’t stick well in the brain (the brain tends to remember information that is personally meaningful!), and thus it is important that we support those students in their learning by teaching and rehearsing thinking strategies.

Before we look at those particular strategies that are useful for learning the multiplication facts, there are some “prerequisites” to consider. Many of the strategies I will be suggesting use some kind of mental math to help students go from a known fact to an unknown fact.

The mantra for students is this: “Use something you KNOW to get to something you DON’T KNOW!” This is a comforting thing for students, particularly those who have struggled with learning their facts. They tend to feel that there is no hope for them. In some cases they have worked for a very long time, even several years, to memorize these facts, and at this point they feel like it is a hopeless task. We need to offer hope in the notion that they can begin their learning with things they do know, and build from there.

Consider working with your students to build these kinds of skills, remembering to tie them to concrete and/or visuals (such as ten frames or 100 dot arrays):
• Subtracting a single digit number from a multiple of ten (e.g., students use the known fact of 10 – 6 to solve 60 – 6. Tie into ten frames).
• Subtracting a double-digit number from a multiple of ten (e.g., build on the previous skill and have students solve 60 – 16 by subtracting first 10 and then 6.)
• Doubling any 2 digit number, using whole-part-part strategies if necessary. It may be easy to double 12 (think 10 + 10 + 2 + 2), but it will be harder to double 16. Students might consider 16 as 15 + 1, then double each of the two parts, and add back together (think 15 + 15 + 1 + 1 — look for “friendly numbers ending in 5’s or 0’s).
• Adding any single digit number to a double-digit number, particularly when the sum of the one’s place digits is greater than 10. E.g., 35 + 7 can be considered as 35 + 5 + 2; 48 + 6 can be considered as 48 + 2 + 6. (Pull apart the number to be added in a way that makes a group of ten.)
• Subtracting any single digit number from a double-digit number, particularly when “regrouping” would be required. E.g., 54 – 8 can be 54 – 4 – 4. (Again, break apart the number being subtracted into parts that make the work easier.)

It is well worth the time that you invest with students doing mental math. In As well as being a great life-skill, mental math allows students to be flexible with numbers and use powerful thinking strategies.

Mental Math and Basic Facts — don’t skip these important things!
Mathematically yours,


Ten Frames for Solving Problems September 14, 2011

Earlier I wrote about using 10 frames to help students learn basic facts. Using those pre-made ten frames, students used strategies to help them solve equations such as 9 + 7 = ?

Blank ten frames can also be useful in solving problems, those which involve numbers in the “basic facts” category as well as those which involve larger numbers. In the first case, students might use large blank ten frames and put blocks or other counters on them to work out the problem. Egg cartons can also be cut down to replicate a 10 frame and used with blocks or counters.

When students do problem solving with me, and am usually interested in having them document their thinking using pictures, numbers, and/or words. I want students of any age to learn how to record the mathematical thinking they used in solving a problem.

One of the ways I facilitate this recording is to provide mini versions of visual tools we use in the classroom. These sit in small baskets in the room available for students to come and take and to then glue into their exercise book. Mini blank 10 frames (27 per page)  (40 per page) are useful in such situations, particularly for primary students. If a student has used larger 10 frames (or egg cartons) with blocks, he can record what he has done by gluing on however many little blank ten frames he needs, and then drawing circles on them (or colouring in the squares) to record the solutions. Some students are happy to not use the larger version of the 10 frames, and just use the mini version to work out the solution to the problem.

I use mini 100 dot arrays and mini 100 charts in this same way. Baskets of each of these sit at the back of the classroom (cut apart and ready for the student to “grab and glue”). Students who have solved a particular problem in more than one way may have used several of these tools (or the same tool but with different thinking strategies shown).

Thanks to Charlene K. for sharing these mini 10 frames with me. She made the original sheet and has allowed me to share it with others.

Remember, students even in Kindergarten and grade 1 can learn to represent their mathematical thinking, and providing tools for them can make it easier.

Mathematically yours,


“Each Orange Had 8 Slices”: Exploring Multiplication September 7, 2011

This is a great book to help students develop an understanding of multiplication. It is important that students internalize that multiplication and its inverse operation of division are always, always, always about groups of things. One of the factors in the multiplication problem must name the grouping mechanism.

When I used this with my grade two and thee classes last school year, I read the book to them. After that I set the kids the task of using the pattern of the book to create their own grouping page.

Here are several examples of the students’ work. I created the template for them to use (with prompts beneath the blanks to help them make their statement properly). As a bonus, the finished pieces made a great hall bulletin board! I was always looking for interesting math to post for public viewing. Note: the “teacher” line is because I was doing this with students from 4 different classroom teachers and needed to keep classes straight!

It is an easy lesson to do, but it helps develop the concept of multiplication. Give it a try!

Mathematically yours,