Focus on Math

Helping children become mathematicians!

12 Days of Christmas: Canadian Style December 11, 2013

Screen shot 2013-12-11 at 9.57.17 AMIf you are looking for a great Christmas-themed book for the young (or young at heart), you will want to check out A Porcupine in a Pine Tree: A Canadian 12 Days of Christmas (written by Helaine Becker, illustrated by Werner Zimmerman). Of course, the math connection is in the counting and in the adding up of all the items being given during the 12 days – all totaled it adds up to quite a tidy sum.

Certainly, any book with numbers tends to make me smile, but this one more than most. It is seriously delightful! It offers Mounties frolicking, squirrels curling, moose calling, hockey players a-leaping, and more.

The hardcover version is available from (not for the bargain price of $12.26 (price valid as of date of posting). Actually, for just a bit more, you can get the gift set version with the hardcover book plus an adorable little plush porcupine (sounds like an oxymoron doesn’t it!)

I am not affiliated with the author or publisher – I was given a copy of this book last year for Chrismas and I love it! I know you will, too.

May your days be merry and bright!

Mathematically yours,



It’s About TIme December 10, 2013

7pm picIn this day of digital clocks and watches, more and more children are having trouble with telling time on analog timepieces. The clock face with its two (or three) hands remains shrouded in mystery for many of the students.

I recently had the privilege of going into a grade 5/6 classroom and doing a lesson with the students in the area of measurement. One of the parts of the lesson included telling time on an analog clock.

Each student was given a sheet of paper with a circle marked on it. Together we added the numbers 1 to 12 for the hours: first we placed the 12 and 6; then the 9 and 3; then we “filled in” between those anchor numbers. There was much discussion at that point about even that particular part of clocks. Student were quick to share that some clocks have only the four anchor number, some have Roman numerals for the hour numbers, and some clocks have no numbers at all – just lines or “ticks” marking the hours.

We discussed as a class the idea of half and quarter hours. Of course, this must be done in the context of the whole hour having 60 minutes. We also noted that in money quarters denote 25’s, but in time quarters denote 15’s. Finally we pointed out that with referring to quarters of hours we speak of “quarter past” or “quarter after” an hour, and we speak of “quarter to” or “quarter ‘til” an hour, but we do not usually refer to three-quarters past an hour.

Each student was also given a large Post-It note arrow to be used as the hour hand. I wanted to render the idea of telling time down to the most simplest elements, and thus set about showing how one can tell time with a reasonable degree of accuracy just by using the hour hand of the clock.

7-30pm picWe placed the arrow to show several “o’clock” times (e.g., 12:00, 4:00, 7:00 — see first picture) and then did some “half past” or “—thirty” times (e.g. 12:30, 4:30, 7:30–see second picture). The point of this is, of course, that when it is half past an hour, the hour hand has moved half the distance to the next hour number. Thus a 7:30 the hour hand is half-way between the 7 and the 8.

We even pointed out that we could tell the quarters fairly closely as well. If, for instance, the hour hand were slightly past the 9, the time was approximately 9:15 or quarter past 9. Alternately, it the hour hand were slightly before the 10, the time was approximately 9:45 or quarter to 10.

We did add the minute numbers around our clock and discussed that using the minute hand added accuracy to our time reading, but by focusing on just the hour hand all of the students were able to make sense of the analog clock, some for the very first time.

Mathematically yours,



Math Assessment: There’s More Than Just One Way to Grade December 6, 2013

Screen shot 2013-12-06 at 3.06.04 PMIn the world of education, not all assessment is created equally. In the area of mathematics, folks seem to think marking or assessing student work is very easy to do: things are either right or wrong – full stop. This could not be farther from the truth.

Most of us grew up in a system that was based on that kind of marking, the kind that looked for a particular math problem to be only right or wrong. Tests and quizzes were comprised of many examples of problems, and we endeavoured to get as many as we could correct, generally using the same procedures over and over to solve the items.

That, however, is a very shallow level of assessing. Students can produce an answer to a problem without knowing or understanding the underlying mathematical concepts – and it is there, in the conceptual knowledge, that the “real math” lives. So a student who has done a page of problems (say, long division or quadratic equations) may be able to follow memorized rules and procedures to come to a correct answer without having the faintest notions of the “why” behind the rules and procedures. There was no assessment of the “real math”.

My friend Katie Wagner, who teachers a number of math courses at McMath Secondary School (isn’t that a cool place to be teaching math!), has written in her blog about two different kinds of math assessment:

Standards-Based Assessment — a grading method where students’ proficiency on a specific outcome is assessed based on a set of pre-established standards.

Outcome-Based Assessment – a grading method that assesses students discretely against the particular outcomes that they are to learn in a course, a unit, or such.

I encourage you to read Katie’s posts and think about the assessments you are doing with your students in your classroom (or, if you are a parent, think about the assessments your children are experiencing). Hopefully you will have some points of discussion and/or questions after you read about the other possibilities.

We need to be helping students understand the “why’s” of mathematics, to understand the deep concepts that underlie all of the rules and procedures, and we need to ASSESS students for those “why’s” and understandings.

Thanks for being a “guest speaker”, Ms Wagner!

Mathematically yours,



Fractions All Around Us: A Photo Gallery December 5, 2013

Filed under: General Math — Focus on Math @ 3:12 pm

Screen shot 2013-12-05 at 2.16.45 PMOk, trying again here. If you got a “posting notice” with nothing in it, cyberspace ate the content of the blog post! So…..
I was in Mrs. DeGroot’s classroom this morning doing a lesson with the class around measurement. We had a grand time! Afterwards Mrs. DeGroot showed me some work her students had done recently. Using iPads the students had taken pictures of “found fractions” and then shared them with each other using a SmartBoard. Mrs. DeGroot shared images of the SmartBoard work with me showing both the photograph but the marking of the “part” of the fraction and the naming of the fractional amount.

Using phones, tablets, or digital cameras is a great idea in math. Besides taking pictures of fractions, students could capture images of angles (e.g., acute, obtuse, etc), lines (parallel, perpendicular, etc.), volume (e.g., 1 litre of coloured water poured into a variety of different containers) just to name a few.

I hope you will try having your students look at math in the “real world” and snap a picture!

Mathematically yours,


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