Focus on Math

Helping children become mathematicians!

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,



Pi Day is Approaching: Get Ready to Celebrate March 5, 2015

Filed under: General Math — Focus on Math @ 10:12 am

Screen Shot 2015-03-05 at 9.08.25 AMWith Pi Day coming up on 3/14, I thought I would share some images that are out there celebrating this important number.

What other poetry or pi-ku might your students write about pi?

Mathematically yours,



“Number of the Day” Sheets: Choosing the Number January 5, 2015

Num of Day III eqn picI recently received an email from Stephanie, a grade 2 teacher in Newfoundland, inquiring about choosing the number for the Number of the Day sheets:

“I really love the Number of the Day sheets you have produced and the opportunities for differentiating the instruction. Just wondering how you set this up? Do children do this everyday or on designated days? How do you decide on the number for the day?”

I thought others might be asking this same question, so have decided to post an edited version of my response to Stephanie’s question:

As for setting up the Number of the Day sheets, things are really flexible. There is no one right way — you want what works best for your students and your time constraints. That being said, I have found that if you are able to have the kids do them really regularly (daily if possible) over a good number of weeks, the students are able to really get into the meat of them. By sharing about them after they have worked them, students get to hear what others have tried and will often stretch themselves to try to match what others are doing. They have a chance to really play and explore the number relationships that are brought out in the sheets’ activities.

The number can be picked in a variety of ways — everything from you choosing, a student choosing, drawing a number from a jar or dropping a bean on a 100 chart! Sometime I have chosen specific “repeats” (e.g., every number that week has a 9 in one’s place) sot the kids to see and compare what happens in such cases. What is the same as before? What is different? Or I might pick several numbers within a “decade” (e.g., 33, 37, 31, 35, 38) and again have students compare/contrast over those days.

No matter what number is chosen, one question that is really great to ask is “What do/did you notice?” When that is asked often in the math classroom, students get in the habit of paying attention to details, looking for patterns, making comparisons, and such.

I am happy with random numbers, too, but sometimes choosing numbers with a particular relationship is good so you can really draw out the depth of the relationship.

I hope you will try the Number of the Day sheet(s) with your students. More information about the sheets as well as the downloadable versions, can be found in the links below.

Mathematically yours,


Number of the Day Sheets to Download:

Level 1 (English and French)

Level 2  (English and French)

Level 3 (English and French)  (pictured above)





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

Screen Shot 2014-12-31 at 2.59.05 PM

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,



Dot Cards Revisited November 19, 2014

Filed under: General Math — Focus on Math @ 2:29 pm

Screen Shot 2014-11-19 at 1.17.10 PMNumber sense is an important thing for students to develop. Some do it naturally, but most do so as the result of activities and lessons that are designed to have students think about numbers and their relationships with each other.

I recently had an email from a teacher asking how she might use dot cards to help the students in her grade 2/3 split class develop number sense, so this post is specifically to address that question of Amanda’s.

I use dot cards extensively in K and grade 1 classrooms, but they can be used for grade 2 and 3 students as well, especially if those students have not been exposed to many visual representations of numbers. If students are rather unfamiliar with the visuals, going back to the “basics” and working with them regularly for short periods of time is profitable. There are many ideas for such activities in the handout from my Dot Plate Workshop. (Here is a link for my other posts that refer to using dot cards.)

Here are a few ways to use the dot cards for grade 2/3 students:

Estimation: have the dot cards in a pile, a basket, or other way to easily access them. Have students draw three or more cards and then estimate the sum. It is important to have students discuss why they made the estimation that they did. It is also important to not force them to get really close to the actual sum. Rather, the skill is to practice the thinking about the numbers to get a reasonable sum. There are many ways students might do this: they might look for pairs of cards close to 10; they might decide that every card is either over or under 5 by a bit, and thus just add 5 for every card; they might realize that no card has more than 10 dots, so the answer cannot be more than the number of cards x 10; they might use a combination of methods to get as close to the answer as they can. Again, it is the process of thinking that is important here.

Mental Math: again, have the dot cards ready to be accessed, but also have some “decade cards” marked with 20, 30, 40, 50, 60, 70, 80. 90, 100 (and higher if you wish, depending on the ability of the children.) Have the students draw a dot card and a decade card, and subtract the dot card from the other number: e.g., if a student draws the 60 card and a 4 dot card, he would do the question 60 – 4 = 5. We have students practice subtracting in basic facts (including 10 – x) but often we do not have them practice the mental math of subtracting from other two-digit numbers. This is a great place to begin that practice.

Match and Add: place the dot cards face up. Have students scan the cards for two that have the same visual someplace on the card. For instance, a student may pick two different cards that each have “5-on-a dice” on them, and then add the two amounts (In the illustration, the student would be adding 9 + 7 but may see the strategy 5 + 5 + 4 + 2).

Once students are quite familiar with the dot card/dot plate visuals, it is fine to do the exercises with numerals cards.

Mathematically yours,


Responding to a FB Post… October 29, 2014

Filed under: General Math — Focus on Math @ 10:40 pm

Screen Shot 2014-10-29 at 10.25.01 PMI just finished posting an opinion to a thread on FB that had this graphic with it. It began here:

Do you know how to do Common Core math? Confusion over the standards has some calling for their removal. NBC’s Rehema Ellis reports tonight.”

One reader had addressed me specifically in that thread, asking for my opinion, but as I responded to her under my personal FB account and not my Focus on Math account, I decided to copy and paste my thoughts here on the blog (and thus to my FoM FB account) so they are officially “on the record”. Here is my post:

Louise Cook, This is about learning to think about numbers deeply. Most adults, if asked to add or subtract two two-digit numbers in their heads, cannot do it. They have to to the “carrying” or “borrowing” formula (on paper usually, or with great difficulty in their heads), and have no alternative ways to think about the numbers. If children are truly exploring numbers regularly it is AMAZING how they learn to manipulate numbers. More importantly, they understand deeply about the number system and why things work the way they do. The true mathematics is not just in memorizing methods that spit out an answer, but understanding the concepts that under lie those methods. Adults may be able to divide fractions (say, divide 1/2 by 3/8 using the “invert and multiply” method, but very, very few understand and can explain WHY they get an answer of 1 1/3. They think it just “magically” appears as an answer, but they don’t know why. I think that moving numbers around in operations (adding, subtracting, multiplying, dividing) without understanding why the methods work is like allowing a student to “read” without having any comprehension of what the words said. In fact, is it really reading without comprehension? Part of the problem is that we have been teaching math in the “spit out an answer way” for so long, that we believe it is right just because it feels comfortable. (I will also add that many teachers have been asked to teach for deeper understanding, such as with multiple strategies, without having developed an understanding for themselves. That is rather like trying to teach a foreign language that you don’t speak yourself — a difficult task to say the least. No wonder is does not work!) It was George Polya, a pioneer in problem solving, who said, “better to solve one problem five ways that five problems one way.” (end of that post)

I believe deeply in letting children really think about math, not spitting out a page of fifty numerical problems on a page without a context, ones they are expected to solve in a prescribed manner. If this method was doing such a great job we would have all of Canada and the US loving math and feeling confident about it. Instead, the vast majority of folks will say they hated math, were not (and are not) good at it, etc. Math usually wins as the “least-liked subject in school”. Oh, dear, I am still fired up about this and could go on and on, but it is late and I must stop. I am happy to continue chatting about this.

Mathematically yours,



What’s Important to Have in a Grade 1 Classroom? October 2, 2014

Screen Shot 2014-10-02 at 10.02.55 AMI was recently contacted by a former colleague, Dawn, regarding what manipulatives a grade one classroom might need to have on hand to support effective learning math. It seems a friend of Dawn’s is in a classroom which really has nothing for the children to use for hands-on math learning and they were wondering what was needed.

First off the classroom needs counters — counters in different shapes, sizes, etc. They can be purchased ones (such as mini plastic teddy bears) or ones gathered from home (such as bread tags, but†ons, etc.). But the need to be abundant and available.

Students need a way to count efficiently, especially in tens and ones. Egg cartons cut down to 10 holes, blank 10-frames printed on paper or card stock, or commercially produced 10-frames can all be used. I even like using cookie sheets (non-aluminum) and marking them with coloured tape as a giant 10-frame for use with magnets.

Base 10 blocks are also great for young students. These a generally in the form of small 1 cm cubes for “ones”, sticks for the “tens”, and flats for the “100’s”. I do want to make a critical point here: students may be engaged in a game of trading 10 cubes for a stick, or 10 sticks for a flat with every appearance of understanding the “ten-ness” of our base-10 number system. But be careful here. Student can be following your rule of trading 10 for 1 without that understanding. They might be just as happy to trade 8 for 1 or 12 for 1. The manipulatives give a opportunity for students to develop that important base-10 understanding, but moving blocks around correctly does not necessarily indicate that the understanding has been built in the student’s mind.

I think a grade one classroom needs “pop cubes” (multi-link cubes) — those blocks about 1inch in each dimension that can be attached together. I like to store them sticks of 5. If students need a particular amount for an activity, say 18, we discuss how many sticks each student will need, and then go get them. I also use these in many quick number-sense building activities. If I have students hold up a certain number of blocks, I want them to do so to model a ten frame. If I ask for 9 blocks and a student were to hold up a single stick of 9, I, as the teacher, cannot tell from a distance if the student is holding 8, 9, 10, or 11 blocks. But if he holds up a five stick beside a four stick, I can tell at a glance that he has the correct number. Pop cubes can be used in a multitude of math activities and should be on-hand for regular use.

Another must-have in my book are pattern blocks. They are particularly great for patterning activities for exploring symmetry, not to mention the creativity factor! I love them!

There are a number of things that I think should be in the classroom that are “make-able” such as dot cards, dot plates, printed ten-frames, even printed dominoes (click for more info on these)— all useful in exploring numbers, in building number sense, and  in helping students develop the skill of subtilizing. Students need to SEE the numbers in math, and these materials can help develop that “seeing” in the children.

Of course there are many other things that are fun to have in the math classroom, such as dice, dominoes, blocks, playing cards, geoboards, plastic coins, bingo chips, square tiles, Cuisenaire rods, and two-colour counters, to name a few. But lots of math learning can take place with some thoughtfully crafted lessons and activities and just the basics.

I hope you will focus on the math understanding with whatever materials you have at your disposal!

Mathematically yours,



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