Notes on Math Fluency - Part One

 How should children acquire math fluency? There is a rising volume of debate on this
topic, world-wide, and a number of conflicting points of view.

What is our historical response to controversy? … We ask * What do the experts say? *What actually works?      * What are the needs of the 21st Century?  *How do we increase equity?  *What does it feel like to be a child learning math? 

First, try this exercise in math learning, using an (imaginary) new number system, which will give you a trip back to kindergarten/first grade. All in the name of developing empathy with our young learners!

How did that go? Not easy, was it? Some of us might even be feeling tremors of the frustration or math anxiety we felt as young students – especially those of us who needed more time to learn math.

Let’s investigate how we might make learning math less stressful, more authentic, and more enjoyable – and create foundational math understanding in the long run.

The goal of math fluency has changed over the last few decades. Partly, of course, due to technology. Who wants to use a pencil when they’ve got Siri in their pocket?

But more importantly,  the expectations of an educated adult have changed in the 21st Century. Therefore, although calculation is still one goal of math education, it is no longer the main one.

The recognized educational priorities for the 21st century include critical thinking, communication and collaboration skills, creativity, and technology skills.  We still acknowledge the value of math

It is our firm belief that one of the best places to build these vital 21st Century skills is in the math classroom.  More specifically, in a thinking  math classroom.

fluency – a competent mastery of the “math facts” – but that fluency should be built on a strong visual/conceptual foundation of number sense.

Fluency is simply the last stage of the development of number sense, when strategies, visualization, derived facts and base-ten understanding become strong and fast.

Number sense on steroids, so to speak. 

How to teach fluency more authentically

Teach the way children learn: Concrete -> Pictorial -> Abstract (C>P>A). Although this approach is often associated with Singapore, it actually is based originally on the writing of prominent education researchers of the 20th Century in Europe and North America.   If you’re interested, here’s a video of Yeap Ban Har (arguably the world’s leading expert in Singapore Math) as he describes some of the main ideas from these writers.

  • Jean Piaget (Swiss, 1896 – 1980)
  • Maria Montessori (1870 – 1952)
  • John Dewey (US 1859 – 1952)
  • Richard Skemp (UK, 1919-95)


  • Zoltan Dienes (Hungary, 1916 – 2014)
  • Lev Vigotsky (Russia 1896 – 1934)
  • Jerome Bruner (USA 1915 – 2016)

Many of these writers point out that the C>P>A approach – because it is based on how children learn – is also the most equitable approach to math teaching. If we use manipulatives, strategies, discussion, and good questions in mathematics, and give children the respective time they need,  we can grow a generation of math students that

  • Know how they best learn math
  • Enjoy the process of learning math
  • Develop more confidence in their ability to learn (and without tears!)


Here are two examples of how C>P>A develops number sense in (a) addition and (b) multiplication.


(a)  9+6

If students have had plenty of experience with 10-frames, and recognize the 10 as a “Quantity I Do Not Have to Count”, we can then ask them to use the ten-frames above,  and  leave them to figure out 9 + 6  on their own.  Most children will physically move one block from the 6 to the 9, making frames of 10 and 5  …. so 15.

Why is this strategy more useful than flash cards?

  • With enough practice at this concrete level, students will have internalized the movement of the blocks. They soon can look at an empty 20 frame and visualize the relationship between 9 and 10 and the resulting sum. This is true for all students, but especially those who traditionally would have struggled with math facts because they don’t memorize well.
  • If students who need more time at the concrete level don’t get that time, they will  become discouraged and confused, and say “Just tell me what to memorize and I’ll do that.” Unfortunately, this is a weak and ephemeral solution to their long-term learning. 

But if the 6+9 above visually makes sense to the child, then soon, so do the following examples.

Can you use the above (sliding)  strategy to solve each  of these easily?

  • 79 + 6
  • 79 + 16
  • 70 + 60

(b)  6 x 14

Iditarod dogs each need 14 pounds of food a day! How much food is that per day, for six dogs?

We see that 14 for each dog means

there are 6 rows of one ten and 4 ones.  

  • The 6 rows of 4 can be thought of as 3 groups of 8 ->
  • Or 2 double-columns of twelve  ->
  • How else could you see it?


The total is  6 tens and 6 fours, therefore 60 + 24 (six 10s and 2 tens and 4) ,   so 84. Given time and blocks, all students can calculate this mentally. They don’t need a pencil!

In middle school, this makes the distributive law easier. We can ask “What is 6 rows of “10 plus something?” and write it 6 (10 + x).  They see it as 6 tens and 6 rows of x, so 60 + 6x.

Many of the 8th graders at my school say that “Algebra is just arithmetic with letters instead of numbers.”

How long do children need manipulatives?

The short answer is Much longer that we currently use them. That being said, the time varies from child to child.

One child might have to add 9 + 6 with blocks 13 times, while for another, twice is enough. Not to worry –  they can all build number sense and – given time – fluency.  S L O W   D O W N !

Here are 2 examples of how we use manipulatives with different children.

“The Racer”

Our first grade Ronnie often finishes first. He memorizes easily, races through work (sometimes with accompanying errors from rushing), and has trouble explaining his thought processes. “My brain just figured it out” is a common explanation. We try to give Ronnie problems that use strategic, flexible thinking, not bigger numbers.

  • This example comes from “”. We give the students 10 blocks when we read them the problem, and then leave them to work it  out. Sometimes there’s a delay before they reach for the blocks, but eventually, almost all of them do.

There are 10 eggs and 3 baskets. The BROWN  basket has 1 more egg than the red basket. The GREEN basket has 3 more eggs than the red basket. How many eggs are there in each basket?

  • And here’s an example we made up.

Mork says “When I mean TEN, I just draw a     .”

Mindy says   “Me too!  And when I mean 9, I write  .”

Mork says “I actually prefer to write 9 as        .”

Mindy answers , “Okay, but my favorite number is twenty! I write it like this:      .”

How would Mork and Mindy write these numbers?   (a) 11 ____    (b) 18 _______               

(c) Write out this whole  number sentence in Mork and Mindy’s symbols: “11 – 5 = ?

(d) Can you write out this whole number sentence in 2 different ways  (Mork’s way and Mindy’s way) : “2 fours plus 2 nines = ??”     (and yes, Ronnie figured this one out!)

Here is a list of links we use to find challenge questions

“The Builder”

Our first grade Sam needs longer at the concrete level. He might need to add 9 + 6 with blocks 13 times, while for Ronnie, twice was enough. Not a big deal – Sam is building a solid foundation that will follow him through years and years of schooling. The best way to give Sam enough practice at the concrete? GAMES!

Here  is a list of games that meet our 3 criteria:

  • They use manipulatives
  • The games are not speed based
  • They should encourage strategic thinking, along with a dose of luck.

We find that games are attractive to all students , regardless of learning speed, and the luck component evens the playing field for a group.

Our final weigh-in on the “Fluency” debate:


  • Fluency is fragile when it is built on memorization. Facts can desert a child under stress, and the facts do not transfer to new number combinations or topics.
  • In Singapore, teachers have found that flexible number sense is highly preferable to memorization, and no longer drill students in timed math facts. All this with no long-term detrimental effects on mathematical achievement.
  • Studies (here , here) indicate that visualization skills show a strong correlation to success in mathematics. Visualization and model drawing help clarify and simplify novel and complex questions.  (and here is a great online course, if you like James Tanton as much as we do!)


Building fluency with a C>P>A approach is worth the time it takes.

  • Students who struggle with math learn with much greater long-term success.
  • Students who progress more quickly can be challenged with questions that offer more depth.

Stay tuned for “Fluency Part 2”

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