Focus on Math

Helping children become mathematicians!

One More/One Less (#4 Young Learners Series) April 13, 2020

Filed under: General Math — Focus on Math @ 4:48 pm

Understanding numbers requires some understanding of number relationships. For little people, there are several relationships that can be explored that will lead to a richer understanding of the number system.


Today we will talk about the “one more/one less” relationship which can later be extended to the “two more/two less” relationship.


The relationship is exactly as it sounds. We want children to easily identity for any given number which number is one more that the original number and which number is one less than the original number. Thus, when considering the number 4, we want a child to know that 5 is one more and 3 is one less.


It is great to make sets of things to explore this. If there is a pile of 5 blocks, ask the child to make a pile that has one more or one less (or fewer). Expect the idea of less/fewer to be trickier at first. From the time children are little “more” is in their vocabulary. We ask them if they want more water. They tell us they want more candy. They have built an understanding of “more”! We do not use “less” and “fewer” as frequently when speaking to them and often it is a less known concept. But as with any vocabulary, as we use words in context, children learn them. It applies here as well.


It is important to have a visual reference for this concept, and I often talk to children about all of the numbers holding hands in a line. In general, a number line is a great aid and one I use in a multitude of ways with students exploring math at many levels. This is an early use of such a number line.


You might have magnetic numbers placed in order on your fridge or onto a cookie sheet. You might print out numbers from your computer and tape together a large number line. Or you might want to take the time to cut out a string of paper dolls holding hands and write numbers on them. A ruler is a ready-made number line, but at this level the numbers tend to be too small and thus the ruler is not a good representation to use at this point. We just want the children to begin to recognize the numbers and understand the “beside each other” relationship. The question then becomes, who does 4 “hold hands with” in the line of whole numbers?


This topic certainly ties into the counting exercises we have been talking about in recent posts, for the “one more” number is the next counting number higher and the “one less” number is the next number counted “down” or “backwards”. (Although the backwards idea can be a tricky one. I once asked a young boy to count backwards for me and he promptly turned around, with his back facing me, and counted, “One, two, three, …” Clearly he was counting backwards!)


The extension of this relationship is the “two more, two less” one, which again, is exactly what it sounds like. Thus for the number 4, 6 is two more while 2 is two less. We are looking for the number two away on the number line, not the immediate “hand holder” but the number we find when we skip the hand holder and identify the one on the other side of the hand holder. Obviously the extension is more difficult for young learners, with practice it will come.


I want to point out here that all of the relationships that we will talk about are not just for young learners. In fact, these relationships extend to all number in different positions of the place value system.


For example, we want students to know other one more/ one less relationships and be able to think fluidly about them:

  • one ten more and one ten less of a number (e.g., 50 is one ten more than 40; 30 is one ten less than 40; 53 is one ten more than 43; 33 is one ten less)
  • one 100 more or one 100 less
  • one 1000 more or one 1000 less
  • one 1/10 more or one 1/10 less (in the case of decimal place values)
  • the same with all place value amounts


Of course all of the above can and should be extended to the two more/two less relationships.


We are endeavoring to build a foundation for understanding numbers. What they learn about numbers in an early context is to be built upon as they encounter numbers of greater and smaller magnitutude.


In closing, don’t just talk about the more/less quantities. Build them. Compare them in rows to show the relationship. Make it a game. Keep it fun!


Mathematically yours,




Patterns Everywhere! (#3 young learners series) April 3, 2020

Filed under: General Math,Primary Math Ideas & Problems — Focus on Math @ 4:01 pm
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patterns pic

Finding and making patters is a fabulous math activity to do with your young children. Patterning is tied to the study of mathematics in a number of ways:

  • Patterns regularly occur in mathematics.
  • Patterns can be recognized, extended, and generalized.
  • The same pattern can be found/created in many different forms.
  • Patterns can occur in the physical environment, and in geometric situations as well as in numbers.

Patterning lays the foundation for algebraic thinking, particularly in representing, generalizing, and formalizing patterns. Algebra is considered to be one of the main content standards in mathematics, so helping children navigate the waters of patterning when they are young lays a good foundation for the algebra that most of us identify with in later schooling.



We will begin talking about simple patterns which are appropriate for young ones. Children love to work with materials to create patterns and for early learners is should definitely be “hands on”. There are dozens of items in your home which can be used to make patterns: blocks, buttons, silverware, shoes, pens, toys, candies (e.g., M&M’s or Skittles), toothpicks, bread tags, shapes cut from paper, and cans to name just a few. I used a number of different items to create patterns in the picture above.

Some terms to know:

Each item in a pattern is considered to be an element.

The core of a repeating pattern is the shortest sting of elements that repeat.

Each repetition of the core can be referred to as an iteration.


An important aspect of patterning for young children is to be able to recognize or identify that a pattern exists. Just saying the pattern out loud (e.g., red, blue, red, blue, red, blue…) is a part of this recognition.


From there, extending a pattern is an important process in algebraic thinking. When patterns are built with materials, children are able to test their ideas for extending a pattern and make changes without fear of being wrong.


Make lots of different patterns with lots of different materials!



Eventually you can compare patterns and begin to name them. Patterns are identified by using letters, one for each different element in the core of the pattern.

Looking at the picture, let’s name each pattern.

  1. AB (A for the pink paperclip, B for the white paperclip)
  2. AABB (A’s for each red circle, B’s for each yellow circle)
  3. ABB (A for the square, B’s for the triangles)
  4. ABC (A for a stack of 3, B for a stack of 2, C for the single)
  5. AB (A for the vertical tee, B for the horizontal tee)
  6. ABBC (A for the white paperclip, B’s for the red paperclips, C for the green paperclip)


You can “read” the whole pattern across with the letter names. For example, the pattern of circles can be read A-A-B-B-A-A-B-B-A-A-B-B

Did you notice that there are two patterns of the same name? You can practice making patterns with the same name out of different material.



Patterns can me in other ways besides building with concrete materials. Consider making up patterns using these:

  • Physical movements such as (clap clap snap – AAB; hop squat twirl – ABC;

or touching shoulders, head, shoulders, knees –ABAC)

  • Singing tones of the scale (do mi mi –m ABB) or making general sounds (ooh aah – AB)



It is important to note that just giving the beginning of a pattern does not define it (unless you STATE that that is the core). In fact, when I have worked with kindergarten and grade 1 students in classrooms, one of the activities I did with them was to see how many different patterns that we could make given a certain beginning. For instance, with one group I told them that a pattern began “red, blue, green” and asked them to make a pattern that started that way. Their responses included these:

  • red blue green red blue green red blue green
  • red blue green blue red blue green blue red blue green blue
  • red blue green green blue red red blue green green blue red red
  • red blue green green red blue green green red blue green green
  • red blue green yellow red blue green yellow red blue green yellow

You can see that just by giving a beginning, the pattern is not clearly defined.



Be sure to look for patterns in your home or neighborhood. Maybe someone is wearing a shirt with a striped pattern. Is there a pattern in the tiles of the backsplash? What about the fence outside? And of course you can make them in everyday life (with fruits and/or veggies maybe?


Patterns are everywhere and can be made with just about anything! Have fun finding and making patterns with your young children!

Mathematically yours,



Other Kinds of Counting (#2 young learners series) March 30, 2020

Filed under: General Math,Primary Math Ideas & Problems — Focus on Math @ 6:54 pm
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In the previous blog post I talked about the most fundamental ideas about counting and how we can help children develop cardinality. Until children have grasped this understanding, there is nothing else to be learned in mathematics. Nothing. They need to know the names of numbers, the order they come in, and that the last number names the quantity. However, we don’t want a child’s mathematical development with numbers to get stuck in counting. It is important in the beginning but we want them to develop a rich understanding of numbers apart from counting. I will share more about that later.


Here are some other kinds of counting that you can do with your children as their knowledge of numbers develops.


Counting Up and Back

Counting up to and than immediately back down from a target number can be a rich exercise. It can be done with movement (such as clapping, jumping, air fist-pumping, etc.) in a rhythmic manner. Or you might move a set of objects from one line to another as you chant, “ 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, …”.


Another version is to count up and back between two numbers such as 8 and 13. As before, keep a rhythm.


What about counting between numbers such as 28 and 37? That raises the difficulty level again.


Counting On

Counting on is the practice of starting the counting sequence from a number other than 1.


WITH COUNTERS: Give your child a collection of small counters and have them line them up. Then have them count a few of them, hide them under their left hand and count from that number. For instance, if they have 10 counters, you might have them hide 4. Point to the hand and ask how many are there under the hand. (Four.) Then say, “Let’s count like this: four (pointing to the hand), five, six, …”.


Instead of covering the counters with a hand, you can use a piece of paper, a small bowl turned upside down, or something else that is handy. The important idea is to know how many are hidden (because they were counted first) and continue “counting on” from that number.


Older children can practice counting by 1’s from other larger numbers such as 37, 142, etc.


Other ideas for counting:

Most kindergarten-aged students can also practice counting by 5’s and 10’s. I have found it very useful to use hand motions for these. For instance, when counting by 10’s you can flash all 10 fingers every time the next multiple of 10 is said, thus illustrating the increase by 10 each time.


Similarly, when counting by 5’s, flash the fingers of one hand each time a number is said. Students may or may not have full understanding of the concept of “tens and ones” here, but still practicing the counting sequence of words is useful.


Counting by 2’s and 3’s are both useful exercises. These can be done using counters and/or by looking at numbers on a 100 chart.


IMPORTANT: Count past 100!

When you are counting up to 100 by 1’s, 2’s, 5’s or 10’s do not stop counting (as may be your instinct) at 100. I have asked dozens of children what comes after 100, and their answer is almost always “200” no matter what they were counting by. It is very important to continue the counting sequence past 100:

…97, 98, 99, 100, 101, 102, 102, …

…96, 98, 100, 102, 104, 106, …

…85, 90, 95, 100, 105, 110, …

…70, 80, 90, 100, 110, 120, …


Happy counting!

Mathematically yours,



Counting: the Beginning of all Mathematics (#1 young learners series) March 29, 2020

Filed under: General Math — Focus on Math @ 2:53 pm
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I am inspired to begin a series of posts about math for younger children. For each topic we will discuss it on several levels, so wherever your child is on the math journey, hopefully there will be some ideas for you to engage with him or her.


It is important here to recognize that mathematics is reasoning and not memorization. While it is useful to commit to memory certain facts and procedures, it is essential that these facts and procedures develop with understanding. We want children to build networks of ideas that allow them to cope with novelty and solve problems they have not previously considered or encountered. Just memorizing facts and procedures can actually be debilitating, as it bypasses the important activity of building inter-connected mathematical ideas.


We begin this series with counting which is, indeed, the first foundation in math.


Counting involves at least two separate skills: saying the numbers in order and connecting this sequence in a one-to-one manner with the items being counted.


First, the child must learn the counting words in order, so say them over and over and over with the child. Make if fun, of course; a game, if you will, but it is in the repetition that the child will learn them (as is true for so much of her early learning).


Often we start with counting to 10, but don’t hesitate to count farther. (Of course this may vary with the age of your child. One may want to count a shorter sequence with a two year old than with a four year old.) In English there are no patterns at all in the first 12 numbers, and even in the teens the pattern is difficult for many children to recognize it. The obvious pattern of numbers does not really become apparent until we get to the twenties. Children must just memorize the counting sequence of numbers (yes, one of the places memorizing comes in handy!).


Saying the order is important but equally important is that the child is able to count one item for each number. Most of us have seen a child “counting” some items like blocks and randomly touching the blocks either more slowly or more quickly than actually saying the numbers. They seem to know touching is part of the process but have not understood the “one touch, one count” idea, otherwise known as one-to-one correspondence. How do they develop this? By doing it over and over. Experience and guidance play major roles in the development of counting skills, so counting often with your child benefits him greatly.


Matching the spoken counting words one by one to objects is generally easier if the objects are such that can be moved compared to those that cannot be moved. Movable objects allow the child to actually slide the object away from the ones yet to be counted. If objects to be counted are in a picture then a set that is ordered in some way is easier to count than a randomly displayed set. In the case of the latter a child often either misses items or tries to count them again.


There is one more important aspect here to consider, and that is that there must be meaning attached to the counting process. There is a difference between being able to count and knowing what the counting conveys. When we count a set of items, the last number word we say is used to represent the magnitude or the cardinality of the set.


When children understand that the last counting word said names the quantity of that set, they are said to have developed the cardinality principle. How can you tell if your child has developed the cardinality rule? Give them a set of objects and ask them, “How many are there?” If they emphasize the last number they likely have developed this. For example, they might say, “One, two, three, four, five. There are five.” I have asked children to tell me how many items there are and had them count the items, look at me, and when I repeat the question, “How many are there?” have them begin to count again. In that case the child seemed to interpret the question as a command to count rather than a request for the quantity in the set.


The only way to develop cardinality is by counting! Two activities are helpful here.

  • Have the child count several sets of items that all have the same number of things but the items themselves are very different in size, e.g. six apples verses six grapes. Discuss how they are alike and how they are different.
  • Have the child count a set of items, then rearrange the items and ask, “How many now?” If the child sees no need to count them again they likely have connected the cardinality to the set regardless of the arrangement of the items. If they want to count again, discuss why they think the answer is the same.

More on counting next time!

In the meantime, count often! Count everything!

Mathematically yours,




SD#71 Workshop August 27, 2019

Filed under: General Math — Focus on Math @ 3:34 pm

I am excited for the opportunity tomorrow to present a primary math workshop here in the Comox Valley. This is the perfect time for professional development for teachers as the new school year is about to begin. We will be looking at four important number relationships that we should help young children develop.

One of those relationships is Whole-Part-Part. I know that others refer to this as Part-Part-Whole, but I rearrange those terms quite deliberately. The emphasis is on the WHOLE being pulled into the parts rather than the parts joining together to make the whole. Is is an important difference. For instance, if children take 8 counters they can pull them apart in a number of ways: 4 and 4, 3 and 5, 5 and 3 (which for them is different!), 1 and 7, 6 and 2, etc. Later on when those students begin to add numbers, it will not be surprising to them that those combinations add to 8 — they will have become familiar with them as they decomposed numbers.

No matter what grade your students are in, decomposing numbers is a great way to build up their number sense!

Mathematically yours,



Lost in Translation September 6, 2017

Filed under: General Math — Focus on Math @ 3:17 pm

Screen Shot 2017-09-06 at 12.11.47 PM.png

What is a student really saying when he or she says, “I hate math!”?

To hear someone make the statement, “I hate math!” is, unfortunately, not uncommon. In fact, it is not restricted just to students, but parents, neighbours, and, yes, even educational colleagues say this, too. But what are people really saying with that declaration? For all my experience I believe that what they really are saying is “I don’t get math.” And of course, when folks do not “get math”, then they are generally not good at it, only increasing and solidifying their dislike.

That means the onus is on those of us who teach math to do so in such a manner that allows learners to truly make sense of numbers. Math is supposed to make sense for everyone! Never allow the words to come out of your mouth, “You don’t have to understand this — just do it like this,”  while showing an algorithm that is meaningless to students. That is how the “not getting math” is perpetrated and we need to stop that now. If you want to stop the “I hate math” mentality, then you must teach math for understanding. Don’t settle for anything less.


Calgary City Teachers’ Convention 2017 Power Up your Problem Solving February 16, 2017

Filed under: General Math,Ideas from Carollee's Workshops — Focus on Math @ 7:59 pm

screen-shot-2017-02-16-at-7-42-50-pmOne of the best ways to begin improving problem solving in your math class is simply this: Just do it! Of course. you can learn some techniques for making that problem solving time more of a success than it would otherwise be. You can teach students some strategies for thinking, and help them to use mathematical tools (which I think of as anything concrete or pictorial that helps students build conceptual understanding), but ultimately problem solving is a skill that students master by engaging in it regularly.

Linda Gojak, past president of the NCTM, talks about using “rich tasks” in math lessons, and she defines a rich task in this way:

— A situation in which an appropriate path to a solution is not readily apparent
— Can be adapted to maintain high cognitive demand while meeting individual needs
— Requires students to do more than remember a fact or reproduce a skill
— Encourages investigations and deep thinking
— Has multiple entry points, solution paths and at times multiple solutions

I encourage you to look at the tasks you are asking students to do in your lessons and see how they stack up against such a criteria list.

For those attending the session tomorrow, I am not able right now to create a link for downloading the handout, but if you email me ( in the next 10 days I will see that you get a copy of it.

Mathematically yours,



Calgary City Teachers’ Convention 2017 Big Ideas for Little People…

Filed under: General Math — Focus on Math @ 7:39 pm

screen-shot-2017-02-16-at-7-16-25-pmWe will be talking about number relationships at this session of the convention tomorrow. Once early learners know their numbers in order, have one-to-one correspondence for counting, and understand that the last number in the counting sequence names the set they are ready to develop some number relationships that will serve as a basis for understanding number relationships in other kinds of number.


Since Dropbox has changed its policies on sharing, I am not able to add a link to download the handout. However, if you email me ( in the next 10 days I will send you the handout.

Mathematically yours,



Beginnings… September 10, 2016

Filed under: General Math — Focus on Math @ 10:45 am

screen-shot-2016-09-10-at-10-26-29-amSchool has just begun for a new year here in British Columbia, and the question for all teachers is this: Do you know where your students are? (I confess this particular way of wording the question comes from a public service commercial from many years ago. It went, “It is eleven o’clock. Do you know where your children are?” Does anyone else remember those announcements?)

My version of the question does not refer to their physical whereabouts, of course, but where they are in their learning. This definitely applies to all subject areas, but, as usual, I am most concerned with teachers knowing the learning levels of their students in math. It is easy to just jump into the textbook or materials for the grade group and not take time to do some initial assessment, but it is far better to take the time for those assessments and learn just which students are behind in particular strands/topics of mathematics. Don’t assume all of your students are on the same level. If a student is behind at the start of the year, it is very hard, even nigh unto impossible, to ever catch up without some intervention.

Find some assessments in your school or make some simple ones of your own, but your year will go better if you take the time to know where your students are.

Mathematically  yours,



“Why am I doing this?” August 25, 2016

Filed under: General Math — Focus on Math @ 6:51 pm
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Screen Shot 2016-08-25 at 6.36.45 PMIt is nearly the beginning of the school year here in BC, Canada. New beginnings are always special with untold opportunities and challenges before us. As a teacher, educational assistant, or parent who is home schooling a child, it is important to keep one question at the forefront and ask it of yourself daily: “Why am I doing this?”

I am particularly interested in this question in reference to math lessons because I think the intent behind a lesson matters greatly. For most of us, in our past experiences in school, the main point of doing anything in math was to get the answer, and usually with the added hope of getting it quickly and efficiently. Full stop. The answer was what was important. Indeed, it was the only thing that was important.

But is producing an answer quickly really enough? Consider some other intentions you might have for a lesson:

  • I want my students to be able to explain how they arrived at the answer.
  • I want my students to be able to explain why the answer they got makes sense.
  • I want my students to see how this particular aspect of mathematics fits into the bigger mathematical picture.
  • I want my students to understand this math concept and be able to explain the concept with concrete materials and/or visual representations.
  • I want my students to understand how this mathematical concept relates to the real world.
  • I want to make sure that all of my students have an opportunity to link this new knowledge with previous math knowledge.

I hope your answer to the question is not, “This is the next lesson in the book,” or “The curriculum mandates that I teach this.” Both of those things may be true, but I urge you to rethink your lesson and focus on an intention that will make a difference for the learning of your students.

Mathematically yours,