Focus on Math

Helping children become mathematicians!

Early Counting: the Foundation of Math May 12, 2014

Screen shot 2014-05-12 at 9.55.18 AMThe meaning attached to counting is the most important idea on which all other number concepts are developed.

Counting Involves at Least Two Separate Skills:

  • A child must be able to produce the standard list of counting words in order: “one, two, three, etc.” This must be learned by rote memory.
  • The child must be able to connect this sequence in a one-to-one manner with the items in the set being counted. In other words, each item must get one and only one count. This important understanding is called one-to-one correspondence.

Meaning Attached to Counting:

There is a difference between being able to count as explained above and knowing what the counting means. When we count a set, the last number word used represents the magnitude or the cardinality of the set. When children understand that the last count word names the quantity of the set, they are said to have the cardinality principle.

Give a child a set of objects and ask, “How many”? After counting, if the child does not name how many are there (as, “There are 7 of them,”), then ask again, “How many?” If a child can answer without recounting, it is clear he or she is using the cardinal meaning of the counting word. Recounting the entire set again usually means that the child interprets the question “How many?” as a command to count.

Almost any counting activity will help children develop cardinality.

  • Have the child count several sets where the number of objects is the same but the objects are very different in size. Ask the child to talk about this.
  • Have the child count a set of objects, and them rearrange the objects. Ask, “How many now?” (If the child sees no reason to count again, likely the child has a good sense of number and has developed cardinality.)

Happy counting!

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,



“What Do I Do When My Kids Don’t Get the Math?” February 15, 2011

Filed under: General Math,Ideas from Carollee's Workshops — Focus on Math @ 9:26 pm
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This weekend is the annual parent conference here in Fort St John, BC, put on by School District #60 and some other community partners.

I have the privilege of presenting a session, the title of which is the title of this post. It is an important topic, one which is of concern for both parents and teachers alike. What do we do when our children or students are not “getting the math”? And what does it actually mean to “get the math”?

Because of the way that math as been traditionally taught (for many generations), most of us think that doing pages of problems all of which follow the same formula or algorithm is “doing math”. Indeed, we have little experience doing anything else in math. But once someone has correctly done their 50 problems (for instance, multiplying a 2-digit number by a 2-digit number), what can one see about the person’s learning? I propose that such a page of problems tell us only two things: first, that the person knows their basic facts, and second, that the person knows how to follow directions. There is nothing in a page of 50 problems to show that the learner understands anything about the concept of multiplication or knows when multiplying is the appropriate operation. The same can be said for any page of practice problems where there is no context given, and no questions which address the mathematical concepts which underlie the rule-based practice.

One way to change this is to ask students to do a word-based question through strategies which show their understanding of the math. Most of us, again because of our educational experiences, think there is only one “right way” to add, one “right way” to subtract, etc. In fact, there are many ways to do each of these operations, and almost all of the other non-traditional ways are more meaningful to students (and adults!).

If you will be in FSJ on Saturday, I encourage you to attend the parent conference (pre-register at the school district website and, of course, come see me!

Mathematically yours,