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,



Welcome to SUCCESS! July 31, 2014

Screen Shot 2014-07-31 at 12.36.30 PMAs I write this the summer is half over — at least for students and teachers in BC, Canada. Schools are generally out for July and August, and then begin after Labour Day in September. I know in other places school will resume in mid- or late-August.

Whenever it begins for you, my question is this: what tone do you set in those first days/weeks of school? What is the most important message that you relay to your students?

For me it was simply this: WELCOME TO SUCCESS!  I had cut the letters for that saying out of construction paper 12 inches high (one letter for each page) and I stapled the message above the chalk board at the front of the classroom.

I talked about student success many times each day for the first few weeks. I basically inundated the students with the message that they would succeed in my classroom because I would not let them fail. I would do whatever it takes to work with them to be successful throughout the year. Failure was NOT an option — this was a classroom of successful students! I even went so far as to tell them that it was their lucky year getting  me for a teacher! Oh, there would be work involved along with lots of learning, talking, thinking, wondering, solving, thinking, testing, proving, thinking, recording, demonstrating, thinking… But we would be working together as a class and each and every student would be successful.

I was especially vocal about success in mathematics. I was teaching grade 6/7 in the early years of my “WELCOME TO SUCCESS” campaign, and it was clearly evident that a large proportion of the class came to me telling me they did not like math and that they were not good at it. I knew the real story was that they did not UNDERSTAND the math and they were not good at remembering all the rules. My plan was to work continually with the concepts in the mathematics knowing that once they understood they would get better and be more confident. My promise was to help them be successful even in an area of study they thought they could not be successful in.

The wonderful things was, of course, that my statement of declaration proved true year after year. All of the students WERE successful! They believed me when I declared it (I guess I said it and they read it so many times that they could not help but believe it!) and ultimately their personal belief regarding their personal success was a turning point for them.

I will ask my question again: what tone do you set in those first days/weeks of school? What is the most important message that you relay to your students?

Mathematically yours,


Host a Parent Night in Math March 9, 2014

Parent night pic

As I have worked with teachers both in this district and in other districts regarding changing their math practice, there is often another element that needs to be addressed. Parents of the students in the class begin to wonder and ask questions about how things are being done in the classroom. Parents notice that instead of a page of problems all done using the exact same “formula” or algorithm, a lesson may be structured quite differently, possibly around a single question! It seems so foreign and strange, and parents cannot help but ask, “What’s going on in math? Why does it look different than when we went to school? The other method worked for me – why, I passed math, so shouldn’t things just stay the same?”

One of things I do to support both teachers AND parents is to hold a “Math Night” for the parents of a given class or school. This is NOT meant to be a fun “Family Math Night” that is set up like a carnival with a variety of stations, all with activities centered on math topics. Those are wonderful events and can be an exciting way to expose parents and children to many interesting components in math, and they certainly have their place. I would encourage any class or school to host such an event!

However, there is a need to actually address mathematical issues with parents, so I am talking about a parent meeting that is meant to be something deeper, something to challenge the “why?” of how we have long taught mathematics. Such a meeting is meant to invite parents to think about what it means to “do math” and why it is “better to do one problem five ways than five problems one way” (Polya). I am asking parents to challenge the notion that just because they were taught a certain way does not make it an effective method of teaching.

Knowing that we are all busy, I keep the time frame to a minimum, but I usually plan for about an hour.

My Math Night plan looks something like this:

  • Welcome and other necessary starting info (e.g., washrooms for young children)
  • Introduction of me – who I am and how I am involved with the class/school/district (done either by the teacher/principal hosting the meeting or by me). If you are hosting for the parents of your own students, this step is, of course, unnecessary!
  • Posing a problem: how many ways can we find to solve a problem
    • Doing the problem (parents actually doing the kind of work I ask students to do!)
    • Sharing our methods for solving the problem
    • Drawing conclusions about the thinking that was taking place
    • Rethinking philosophy about the teaching and learning of mathematics: why it is better to really think in math class and not just do pages of (usually) meaningless problems
    • Questions and Answers

Parents just want what is best for their children, and we want to help parents understand something about mathematics curriculum, and in so doing, grasp a vision of deeper mathematical understanding for their children.

I’d love to hear from you if you host your own event!

Mathematically yours,



Student Participation: Using Technology to Choose the Next Speaker January 30, 2014

Screen shot 2014-01-30 at 9.57.22 AMI do an extensive amount of problem solving with students, and part of each such lesson is devoted to sharing strategies for solving the problem. As teachers, many of us have looked for ways to give “equal opportunity” for all during sharing (which also ensures that no students are “coasting” and never choosing to share an answer, a strategy, or an opinion).

There are some “low-tech” ways to accomplish that, such as to write the name of each student on the end of a popsiscle stick, place all the sticks in a jar or can, and then pull the sticks out of the container one at a time. When a name is called the student is asked to share something.

I have recently learned of an app for iOS devices that can replace the popsiscle stick jar (or any other low-tech method you may be using).

iLeap Pick A Student is a simple app designed specifically to help teachers pick students to help or participate in class. It supports multiple different classes and various options to choose students. Choosing a student randomly will pick any student from the class, and using turn based selection every student will be picked before any student is picked again. It requires the teacher to input the class list (or multiple class lists), and the rest is easy.

Pick a Student is available to download for free on the  iPhone App store.  It can be used on iPhones, iPads, iPods, and other such iOS devices. You can find out more about the app here:

Happy Problem Solving!

Mathematically yours,



Developing Math Vocabulary September 12, 2013

math words pic Mathematics, like many subject areas, has some terms specific to discipline. Additionally, there are words that have uses in everyday language but a specific meaning in math (like “product”, “root”, and “obtuse” just to name a few). Within mathematics itself are some strands that are particularly vocabulary rich, such as geometry and measurement.

There is also the issue in most schools where some portion (in some cases a large portion) of the student population are English Language Learners , ELL, (or termed English as a Second Language students, ESL).

Clearly there is a need for teachers to be proactive regarding helping students learn the various terms that we use regularly in the mathematics classroom.

One easy way to support math vocabulary is a make a Math Words chart that hangs in the classroom, always visible to students. Now, some teachers, particularly in the elementary grades create word walls of general vocabulary terms for young learners, and this is a great idea. Many that I have seen have individual words written on cards and placed alphabetically on the wall. That is a great idea, but I must confess one that for me was not very easy to keep up with on a regular basis.

I am suggesting, instead, that you give math its own sheet so you can add words easily at any time. You need only start with a few words at the beginning of the year and ask your students for suggestions of words to be included. As new words come up in the course of the year, add them. I have often had students in my class prompt me to do just that – they would stop me during our math work and inform me that a certain word needed to be added to the chart. Students used the chart regularly when writing about their thinking. In fact, many times I would see a student sitting, not knowing what to write, scanning the Math Words chart. Finally one term would spark something for him, and the writing could begin.

Having the words posted also reminded us all to use the words in our math discussions. Instead of calling a blue or tan Pattern Block a “diamond”, we would use the correct mathematical term “rhombus”.

The picture of the chart posted here is clearly one used in an primary grade, but the chart is easily adapted to any level. If a phrase is used (such as “ten frames”) one colour is used to show it is a phrase. Otherwise words are written in any colour, multiple words to a line. If possible a small symbol or “cue” is added beside a term to prompt students to remember the meaning of the term.

Screen shot 2011-06-10 at 11.09.25 AMIn particular units of study (such as angles) where there are many new terms, it may be helpful for students to do deeper vocabulary work with the various terms. Using a Frayer Model is helpful for that. (Click here for information about that.)

I hope you will put up a Math Words chart today if you do not already have one up!
Mathematically yours,


Step Out of Your Comfort Zone January 21, 2013

Where the magic happens pic

Most of us tend to teach mathematics in the same manner as it was taught to us. I think of that as our “default setting”. We are comfortable with it.; it “feels right” to us. Unfortunately, it is often not the best way to teach math (which is why most of the North American population does not understand mathematics!).

To teach otherwise, to use strategies and approaches that we did not experience in or school years, requires real effort to change. It makes us uncomfortable; it does not “feel right”.

I believe that when we teach mathematics meaningfully, we need to have students doing more than just following our instructions. When we show them how to do a particular computation (e.g., 27 x 46), demonstrating each step of the computation that leads to the answer, their subsequent work (i.e., the 50 problems to do on the page) only shows to us whether or not the students could follow all of the necessary sub-steps in order to arrive at the final answer. Such work does not show any understanding of multiplication, nor does it show that the students understand why the sub-steps produced the answer.

I contend (again!) that “understanding” lives in mathematical processes. The National Council of Teachers of Mathematics (NCTM) lists 5 math processes, namely these:
• Communication
• Connections
• Problem solving
• Reasoning and Proof
• Representation

If we regularly incorporate these processes into our mathematics teaching, students cannot help but build mathematical understanding!!

I will add one caveat: you cannot add the processes for a week, examine the results, and say, “this doesn’t work!” The truth is, we must help students build skills in these areas. If they have not been talking and/or writing about their math thinking already, such communication will take time to build. If students have not been problem solving (in the truest sense of the word) then they will need to learn some strategies and approaches to help them solve problems. A similar case can be made for making connections, reasoning and proving, and representing.

But building competency in the processes is worth the time that it takes! When students are doing the hard thinking in math (and not just following rules that are meaningless to them) you will find you and your class enter a new place of teaching and learning!

Will you step out of your comfort zone so the students can go “where the magic happens”?

Mathematically yours,


Communication: Recording a Problem Solving Discussion June 29, 2011

One of the things I learned this year was the advantage of recording student ideas as they shared strategies and solutions after solving an “rich” math problem [and I will restate my personal definition of a “rich” problem as one with a) many solutions; b) one solution but many strategies for finding it; or c) both many solutions and many possible strategies] .

For most of the school year as I would be working with students, I would record the students’ strategies and solutions on the chalkboard. My general rule of thumb is to have students tell me what they did, and I would do the recording. This was done very specifically so that the students had to practice verbalizing their thoughts — they usually found it easier to write things down with pictures and symbols that to tell me how to write things down.

I carefully recorded all that the students would tell me, but then, before my next class of students would come, I had to erase the board and get ready for another round. It occurred to me during the year that I was missing out on an important scaffolding step for students: if I were to record the work on large chart paper rather than the chalkboard, then the work could be hung for all to see could be referred to in later classes.

I should point out that in the particular case depicted in the photo, the T-chart on the right was done after the students had shared various solutions. It is important that, toward the end of the discussion, the teacher pose questions that can help the students move to a “bigger picture” — in this case I was moving toward an “n-rule” with the class.

So, that is how I began recording the discussion. I even got to the point where I printed off a large copy of the problem the students were working on. (NOTE: Thanks to Sharlene K’s brilliant idea, I now always write the student problems on the computer and copy and paste to fill a page. I usually only have to print off a few sheets, then use a paper cutter to cut them apart. The students begin each session by gluing the question strip into their exercise books — no one has to take the time to copy out a question, while it ensures that there is a copy of the question on the page.) Making a large copy of the problem and gluing it onto the recording sheet was easy since I was typing out the question anyway. I hung the chart paper on the side wall where it could be viewed. Each week, on Wednesdays, as I worked with the students, I taped the new work for a class on top of that class’ previous week’s work.

It was soon apparent that recording the discussions of solutions and strategies was a good idea. Students referred to solving previous problems (knowing the solution was still visible) making connections between one problem and another. I also referred to the previous problems, reminding them of strategies they could not clearly recall.

Even though I was doing this in an elementary setting, the principle of recording student solutions would work at ANY grade level. I HIGHLY recommend recording the class discussions on chart paper! I think you, too, will find it valuable for students and yourself.

Mathematically yours,

PS:The problem on the page, used for a grade 3 class, is this:

Jacob, Charlie, Kara, and Heather shared a bag of Skittles.
They each ate the same amount. There were 2 left over, and
they gave those to Jacob’s little sister. How many Skittles could
have been in the bag?