#1 of 101 Bedtime Math Snuggles

Mother daughter snuggle_sEveryone knows that reading to and with your children helps them learn to read.  Those nighttime stories help welcome children into the world of literature and create a love of reading that lasts their whole lives.

Well you can do the same for mathematics – create a love of mathematics that lasts their whole lives.  Vary up your nighttime routine and cuddle up with a math problem.  Make it cozy and have a little fun.  This is not about homework or about doing better in school; this is about loving math. The point of bedtime snuggles is to lead your child into the wonderful world of mathematics.  A little adventure into mathematics might hook you as well. I will provide you with some ideas for bedtime math snuggles on this blog.  Here is my first…

Bedtime Math Snuggle #1

You might want to grab 15 objects before you start.  Anything will work: 15 Beany Babies, 15 cars, 15 quarters.  Choose something your child likes.


Then ask the question:

How many ways can you add two numbers to make 15?

Help your child make two piles of your chosen objects. Then count each pile out loud with your child.  For example


Point to the four objects and say together ‘four’… ‘plus’ then point to the pile of eleven objects and say ‘eleven.’  As you say ‘equals’ push the objects back together and say ‘fifteen.’

Ask your child to find another.

If your child is a little older, ask them what happens if you add a negative numbers, or a fraction, or a decimal…

Credits for the math problem:

Davis, B., & Renert, M. (2014). The math teachers know: Profound understanding of emergent mathematics. New York, NY: Routledge.


OECD Results

Boy Science_m
Canada’s performance on the OECD’s PISA exams is a hot topic right now.  In the ranking list, we appear to be moving down the list.

The results have fuelled fear-mongering and calls for a return back to traditional methods for teaching. A teaching focus on rote memorization of mathematics facts and algorithms will NOT improve students’ mathematics aptitude or attitudes. Yes, knowing mathematics facts is great, but it is NOT enough.  Knowing what the facts are good for and how and when to use them is also very important for mathematical thinking.

According to the PISA exam, Canadian students are already good at regurgitating math facts and plugging in numbers into formulas and algorithms. With 96.4% performing at or above Level 1, Canada already does very well on with “answering questions with familiar conditions… (and) “carry(ing) out routine procedures according to direct instructions and explicit situations” (Brochu, Duessing, Houme, & Chuy, 2013, p. 24).  To have more students move into the higher levels, stronger mathematical problem solving skills are required.

The PISA test instrument is NOT about measuring rote understandings. Level 6 on PISA is about “demonstrate(ing) advanced mathematical thinking and reasoning and apply(ing) this insight and understanding” (p. 24). In other words, it measures how well students can connect mathematical ideas and apply ideas to solve novel problems.

Canada is well above average on OECD’s PISA exams: only 9 out of 65 countries scored statistically better in mathematics than Canada.  We are doing well. Can we do better? Yes, there is certainly room for improvement.  Rote-teaching methods without problem solving is definitely not the path for improvement.


Brochu, P., Duessing, M.-A., Houme, K., & Chuy, M. (2013). Measuring up: Canadian results of the OECD PISA study. Toronto, ON: Council of Ministers of Education, Canada. Retrieved from http://www.cmec.ca/Publications/Lists/Publications/Attachments/318/PISA2012_CanadianReport_EN_Web.pdf




Back to the basics – not again

In response to Moira McDonald’s National Post article called Frustrated professors convince elementary schools to step back from ‘new math’ and go ‘back to the basics’ 


 mathformulas_sI was disappointed to learn that Manitoba revised its curriculum to have a ‘back to the basics’ agenda? Why is the Manitoba government so quick to throw out more than 40 years worth of research about how children learn mathematics? Rote teaching and learning is NOT associated with high achievement.  A ‘back to the basics’ curriculum is definitely a step backwards from problem solving approaches to teaching mathematics.

Besides, since when is problem solving not a ‘basic’ mathematical skill?  Excusing children from developing deeper understandings of how calculations work underestimates and underutilizes children’s innate capabilities.  What is the point of memorizing multiplication facts if there is no understanding of what multiplication means?

Supported by concerned parents, WISE Math’s lobbying is credited for Manitoba’s step backwards. Why do these concerned parents, who are unable to help their own children with homework, not want more for their children’s children? The parents themselves are a product of the same rote methods to which they are now advocates of.  Their inability to help their children is likely the direct result of how they learned math. Memorization is the lowest-order of cognitive thinking on Blooms taxonomy. Facts and algorithms that are memorized are easily forgotten.

Below is an example that illustrates the difference between rote and problem solving approaches.

With a rote method the teacher shows and demonstrates that the area of a triangle is (base x height) ÷ 2. Then to help students memorize the formula, students practice (and practice) plugging numbers into the formula with worksheets or problems at the end of the book.  When marked, the worksheets assess how accurately the student multiplied and divided: not necessarily their understanding of what the area of a triangle means.

A problem solving approach challenges and deepens understanding.  For example, rather than demonstrate, a teacher asks students to apply the property of the area of a triangle to the following problem:

Draw a straight line (in the image below) that divides A from B so the area of A and B remains the same.  Hint: Triangles with the same base and height have the same area.

Conserving Area Problem

Then students find their own solution to the problem. Rather than demonstrating a solution, the teacher provides hints and extensions as needed.  The problem’s solution requires applying an understanding of what the formula for the area of a triangle means.  ‘Apply’ moves cognitive thinking two scales up on the Bloom’s taxonomy scale.

A solution to this problem can be found by watching a Grade 8 Japanese mathematics lesson at http://timssvideo.com/67.  Notice how happy the children are working on such a tough problem.  On average, non-routine challenging problems constitute 25% of the mathematics problems children in Japanese classrooms. Japanese students are amongst the top achievers in international tests.  It is not a coincidence.

Mathematics is complicated, complex and challenging.  There is beauty in the challenge. Denying children the opportunity to embrace challenging mathematics problems, denies them the opportunity to develop their problem solving ability. I am grateful to work in Alberta’s educational system where developing children’s critical thinking and problem-solving ability is a priority.

Back to School

Back to School_sSeptember is always an exciting time of year for children and their parents.  Going back to school is always a fresh start to a new year. I always love the possibilities each new school year can bring.

Every parent wants their children to succeed in all subjects, but math is the one subject that seems to cause the most anxiety. How do you help your child succeed without adding undo pressure? The adage practice makes perfect sends us off buying math workbooks and software.  I am certainly not against practicing math facts but enforcing a drill and practice regiment can create anxiety for everyone.

How about a different approach?  Many parents snuggle up at night to read a story.  Why not add a math problem to your routine?  For instance, try drawing the impossible cube together. See my earlier post. As you are drawing, discuss why the cube is impossible.  Drawing and discussing the cube can be a fun way to help your child develop their spatial reasoning.  Spatial reasoning is important for success in mathematics.

Or you could combine both.  The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger is a great book.  Don’t just read the book. Whenever the Number Devil asks the young boy, Robert, a question, stop and explore the question with your child.  Make sure you have paper, pencils, a calculator handy to help your exploration.

Creating a warm, intimate invitation for mathematics exploration can be a wonderful bonding experience.  You will show your children that mathematics can bring as much enjoyment as a favourite story.

New Directions

After starting as the IOSTEM Director at the University of Calgary, I placed my Number Explosion program on hold. More than a year has passed since I stopped the classes and my blog posts.  I didn’t have the heart to pull down my site.  I loved the design and the stories of children engaging in mathematics.

In the past year, this blog started to get more followers, which did not make sense.  All the ‘how to blog guides’ state how important it is to blog regularly. Why am I getting more followers with an inactive blog than an active blog? How can I possibly pull down this blog now?  What can I contribute to this blog now that I do not have the Number Explosion classes for children?  Can I still encourage mathematical play?

Recently, there was an Escher exhibit at the Glenbow Museum.  At first glance his images looked perfect.  Then something would register in my mind that something was not quite right.  Inspired with Escher’s play with shapes and perspective, I found instructions for how to draw an impossible cube.  Below is my attempt. You should try it.

My Impossible not quite a cube

School is out

Our Spring term is over and we are now accepting registrations for the fall term.  Click here   to register.  The program expands this the fall to include Grades 3, 4, 5 & 6.

Number Explosion provides an exciting adventure into mathematics and encourages thinking skills through challenging problems.  Children love to be challenged to figure out tough problems.  The exhilaration from solving rigorous problems becomes addicting.  Such exhilarating experiences bring joy and happiness.

There is tons of systemic pressure to do well on exams to have success in schools.  Such pressure leads to teaching to the test. Teaching to the test leads to teaching rote skills.  Perhaps children find short-term exam writing success with drill and trill methods.  However, once the exam is done memorized skills are easily forgotten.  Don’t forget the tedium that comes from drill and trill.  A few children have success with drill, trill and test.  Other children are left feeling like they cannot keep up.  The cycle keeps repeating making many unhappy.  Teaching to the test does not help mathematical understanding.  Number Explosion is a departure from the cycle and leads into a deeper understanding of mathematics.

When John Lennon was in kindergarten his teacher asked him what John wanted to be when he grew up.  John replied that he wanted to be happy.  John’s teacher responded that John did not understand the question.  John told the teacher that she did not understand life.  At a young age John had incredible wisdom.

Number Explosion is about enjoying mathematics and relishing in the happiness found in great concepts and problems. Joy and success are not mutually exclusive pursuits.  A little joy goes a long way to foster long-term success in mathematics.

Register your child now!  Spaces are limited.  Your child will love math at Number Explosion.

I will not be blogging until classes resume September 17.   Have a wonderful summer.


Family Game Night

I am frequently asked, “What can I do to help my child do better in mathematics?”  My response is always, “play games.”  Besides being great fun, games teach so many skills including:

  • Understanding and learning to apply rules
  • Strategic and logical thinking
  • Reasoning
  • Representation
  • Arithmetic in keeping score

Children have enough pressure in school.  Schools can be stressful places.  Teachers rarely have enough time to play games in class. If you play games with your children, you will be likely doing something entirely different from school. Besides, playing games with your children can release stress and give you opportunities to enjoy each other.

I was just at a workshop with a group of PhD mathematicians.  Their passion for puzzles and games was infectious.  Dr. Andy Liu described his huge collection of games and puzzles.   He complained how he had to move to a larger apartment to house his growing collection.  Drs. Andy Liu, Ted Lewis, and Tiina Hohn’s mandate through Math Fair  is to encourage problem solving and fun with mathematics.  I was awestruck by how these incredibly intelligent people are delighted to play games.   If mathematicians can have fun and not be so serious about mathematics, so can the rest of us.

I played lots of card games with my family of origin when I was a child.  Our favourite game was Hearts. The competition was fierce and none of us were gracious when we did not win.  My grandfather’s favourite game was Crib. He insisted we play for money.   In my undergraduate university days, I spent many evenings losing quarters to my grandfather. When I lost all my Laundromat money, I did my laundry at their house. Grandpa’s favourite story was about how his youngest daughter was having trouble with her addition facts.  She was around 7 or 8 years old at the time.  He taught her to play crib and she never had trouble with mathematics again.

With my own family, we also play lots of games.  Like Andy, I have a problem storing my growing collection.  My youngest likes Cribbage so we play with her frequently. Besides Crib, a few of my favourite games are:

Sentry Box has a great selection of games.  The staff are really helpful and can usually make great recommendations.

I also like puzzles.  The puzzle games are individual.  You will need to spend time teaching your children how to play them.  Of course, my children had to remind me that it was their turn.  I try to play one puzzle everyday.  My favourites include:

Puzzle games are easier to find.  Lots of toy stores carry them.

Play GamesI know it sounds counter intuitive to play games to help your children do better in mathematics.  Remember that mathematics is about thinking and you need to think to win a game.  A board game is often cheaper than a night out at the movies.  A weekly family game night is great fun and your children will enjoy themselves learning and thinking.

49 out 50

Multiplication, subtraction, division addition“I have something to tell you,” J announced on her way in the door.  I got 49 out of 50 on my multiplication mad minutes.  “Really, what do you usually get?” I responded.  “Somewhere in the twenties,” J replied.  “Great job. What are you doing differently?”  J said, “For every question I think to myself how many times the groups of something will equal?”

J figured out a way to visualize multiplication in numbers of groups.  She associated an organized quantity in her mind.  For instance, 3 groups of 4 equals 12.  Instead of a random equation of numbers to compute, J now appeared to have meaningful understanding of multiplication.

I wondered if our work in Number Explosion contributed to J’s new understandings of multiplication. My first thought was that we never perform math fact drills.  There really was no memory work.  How could Number Explosion affect J’s rote multiplication score? I never promised that children would perform better on tests.  My primary goal for Number Explosion is for children to love mathematics.  To make children love mathematics I share the problems I love and concepts that I find fascinating.

Reflecting back on past Number Explosion classes, multiplication has frequently been a component for solving a problem. J was in the first session and has continued into the second. Multiplication and other operations were never the goal, but could be used (or alluded to) in a solution method.  For instance, we slayed the Dragon on the very first day slashing tails and heads by twos.  “How many times did you slash two tails?” relates to multiplication.  M changed the Irritating Things addition problem to multiplication.  When we played Lost Cities, we  multiplied the score by negative three if player used an investment card.  Counting and writing numbers with different bases involved the use of multiplication.  305 translated to base 10 equals 3 x 51 = 15. In this session, we focused on proportions and fractions.  Multiplication managed to make an appearance in every class.  While I did not think I was ‘doing’ multiplication with the children, in reality we were ‘doing’ lots of multiplication.  Instead of ‘doing’ multiplication in the traditional rote drill sense, we pulled multiplication out of our toolbox when we needed it. Yes, J, maybe Number Explosion is helping you understand and remember your math facts.

We embarked on testing Da Vinci’s claims in the  Vitruvian Man.  There are eleven items on the list, but we only worked on five.  We tested (1) whether the girls arm span equaled their height (only one was close), (2) whether the roots of the hair to the bottom of the chin is a tenth of their height (all no’s), (3) from the bottom of the chin to the top of the head is one eight of their height (closer, but still ‘no’), (8) the foot is a seventh of girl (all yes), and (11) the space between the legs is an equilateral triangle (not quite).  The girls’ strategy for testing each hypothesis was —- multiplication.


We finished the class with a game of Qwirkle.  We had so much fun that we are going to start next class with a rematch. There is even a little multiplication scoring in Qwirkle.


Making Progress

Finally, we are making progress. Several weeks ago C was not convinced that division and fractions are related.  At the beginning of class, she was still not convinced.

The definition of a fraction is:  , where a and b can be any integer (b cannot be zero).  The symbol for division is .

What is similar between the symbol for division and a fraction?

Our notation gives a hint there is a relationship between fractions and division.

Next, I drew a square and tasked the girls with dividing the squares into four equal parts.  I drew the first division.

How many different ways can you divide the square into four equal parts?

Dividing squares

I asked, “If you were to give me one piece of one of the squares, how much would you give me?”  With a little hesitation, C ventured “1/4.”  I replied, “yes, you divided one square by four.  One of the divided sections of the square is 1/4.  Can you see that there is a connection with fractions and division?”  C nodded.

“If you gave me 2 sections, how much of the square are you giving me?”  J replied “Two quarters”  “Exactly, how much of the square are two quarters?”  J said “one half”  “Are two quarters the same as one half?”  They all nodded.

I handed out four cards with the fractions 1/4, 1/3, 1/5 and 2/3 written. Place these fractions in order from smallest to largest.  With lots of conversation and disagreement, they placed them as follows:

Nothing was ordered accurately.  “Is 2/3 smaller than 1/3?”  C responded, “Yes.”  I drew a cookie and divided it in thirds.  “I’ll give you one third.  Who got the bigger piece?”  C responded you did.  “Is 2/3 smaller than 1/3?”  In unison, the girls replied ‘Ohhhhh”

“Is 1/3 smaller than 1/4?”  J said, “Yes because 3 is smaller than 4.”

There you have all the misunderstandings about fractions in one little statement.  Fractions throw everything one thought they knew about numbers out the window.

I have two cookies.  I am going to give C a third and you a quarter.  Who got the bigger piece?

They all said, “C.”  “Is 1/4 bigger than 1/3?”  Again, “Ohhhh.”  “Are your fraction cards in the right order from smallest to largest?”  They rearranged the cards in the correct order.  They transferred their understanding of the magnitude of 1/3 and 1/4 to 1/5 and 1/4.  We were making progress in understanding fractions.

“If we were to put these on the number line, where would you place them?”  My question was received with blank stares.  I pulled out Guy’s fraction circle from a few weeks ago.  We constructed the number line with his circle, one row of fractions at a time.  We also referred back to the fraction strips we created.  We started with one-half.

1/2 on the number line

Next thirds

Thirds on the number line

Adding quarters

And fifths

Now the number line was beginning to make sense.  The children could see how the fraction boxes lined up on the number line.  They could tell the magnitude of the fraction with cookies, fraction cards and the number line.

In the past four weeks we have explored three questions: What is a fraction?Which fraction is bigger?  Where do the fractions fall on the number line?  These are not easy questions and have taken us time to explore.  We are not ‘done.’  While I know the children are beginning to understand, these concepts will need to be revisited over and over again. To quote the Carpenters, “We have only just begun

To make it fun, I gave them one more representation of dividing a square into four equal parts: The Greek Cross and Square.

I gave them each four of the following shapes and asked them to join them together to make 1 square.

We ended well and I am feeling relieved that fractions are starting to make sense.  The girls have taught me so much about the misunderstandings about fractions.  Each misunderstanding required me to dig deeper.  Their misunderstandings took us on great detours and conversations. With deliberate interjections and cycling back and forth between representations, I can see the light bulbs turning on.  We are really close to broaching the subject of equivalent fractions.  I can hear M already, ‘what?”