Focus on Math

Helping children become mathematicians!

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,