There is confusion over the aim for number fluency and the means by which is it achieved.

Let me explain.

Many people learned their “times tables” by rote, sometimes under threat. This led to people know the right answer, but not necessarily knowing how to apply them. For example if they were asked what is five times three they could say 15, but given $150 to split up between five people they would not know how to proceed. (Having said that, if it’s money involved people often do better.)

I have written about this previously in How to help your child with her multiplication facts and Fluency in maths. “Knowing your tables” is a common concern.

This time I asked a bunch of mainly middle school teachers about when they think learners should be fluent with their times tables and what they do to develop it. Though it was a very short survey of a convenience sample of 68 teachers there is still value in the findings.

Over two thirds of the respondents indicated that learners should be automatic with their multiplication tables in Years 5 and 6 (equivalent to Grade 4 and 5 for the US).

Teachers were asked, “How important do you think fluency with multiplication tables is for future mathematics learning?” on a scale of 1 Not important to 5 Extremely important. Two thirds of the respondents said 5 – Extremely important, and all but two gave a 4 or 5.

Clearly there is a consensus among teachers that fluency with multiplication tables is extremely important for future mathematics learning. This was also indicated in an earlier survey of high school teachers who rated multiplication facts as second in importance after an understanding of place value, though in an open question, multiplication facts were the most common response.

Comments included:

“Instant knowledge of those times tables makes teaching fractions so much quicker – I say fractions because I am taught improper to mixed to my bottom set year 9s today. The ones who instantly knew how many times 3 went into 13 got the concept so much quicker than those who had to subtract over and over. It really slowed them down so much”

“Helps with so much with real life situations, games and means the student can concentrate on the concepts etc without having to spend half the time working out the multiplication facts”

“Without the foundations of knowing you basics and tables, this can be a barrier for student when learning new concepts that rely on this knowledge.”

In the original survey of middle school teachers the question was worded “At what year level do you think students should be automatic or close to automatic with multiplication facts?” with the choice of Before Year 5, Year 5 or 6, Year 7 or 8, Year 9 and Year 10. No one gave Year 9 or Year 10 as a response. It is unanimous that learners need to be fluency with multiplication before they get to high school.

Here’s a thing – many are not. Maybe it is because I have been working with adults who need basic maths skills or maybe it is because I provide individual tuition, but my experience is that many students are not fluent. A quick straw poll of some high school maths teachers in New Zealand gave the following percentages of Year 9s who are fluent in multiplication facts: 3 out of 27, 70%, 80%, 10%, not many, 55% from previous survey, 50%, less than half, definitely less than half, 10%, 20%, 40%.

Another issue discussed in my straw poll was order of operations which is unfortunately called by its acronym. Sounds like another post.

I was interested to know how teachers worked towards this fluency. The bar graph shows the frequency of the different responses. Teachers use more than one method and nearly all use games.

Nearly half the respondents included “Frequent Timed Testing” in their methods. This was a surprise to me as there has been a push from some quarters to reduce the timed aspect of fluency. Fluency Without Fear is compelling reading based on research.

Two teachers commented that they did not teach tables as they taught high school. I find that interesting.

Fluency with multiplication facts is needed for later learning in mathematics. As the curriculum is reviewed, maybe this can be given higher priority. We need to be careful though, as learners who struggle to memorise tables may see themselves as poor at mathematics. I would suggest having a tables chart available to help avoid working memory overload and encourage an understanding based on structure.

In the meantime I am doing my bit by creating innovative resources for teachers to use to develop fluency along with conceptual understanding. You can see them here:

A free sample is available to download at this link. Full copies are available on the Creative Maths shop.

I would love to hear what else people are doing and in what ways you think fluency is important.

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## 2 Comments

It was very late in life when I first heard PEMDAS It was taught to me, and I to my students as

“Please Excuse My Dear Aunt Sally. Was surprised when I first heard it that none of my adult learners were aware of PEMDAS but most knew about Aunt Sally. Teaching adult learners Electronics Technology I would simply assume most had either math weaknesses or some level

of math anxiety. Solution use any and all tricks I knew to help them over come what ever problems

they may have or thought they had. Most thought that the required programmable calculator was

there to assist with math (and it did) but the real reason for the programable calculator in addition to helping those who were weak in math was to provide all members of first year electronics with experience in programming concepts, such as stacks, registers, subroutines, jumps and calls, program counters etc. The machine best suited for this task happened to be Hewlett Packard brand RPN calculators. My timeframe for doing this was from around 1980 through 1999. In the 70’s we taught the slide rule operation. Electronic calculators were invented and most who used to teach the slide rule operation no longer saw a need to teach calculators. Who wrong they were and for the most part still are! The world is much poorer with the unfortunate choice of Reverse Polish Notation and math teacher insistence on algebraic calculators that worked like the math book examples. The HP machines were much closer to the way we think and calculate mentally while the algebraic front end only hides the machine nature of machine math.

Jan Lucasiewicz, the Polish has been poorly served by the algebraic community that has ignored or is not aware of his contribution to mathematics. His parenthesis free notation actually enforced order of operations and with computer applications listed operations first followed by the data….reading his list backwards reversed the order of his “Polish Notation” ….resulting in Reverse Polish Notation. In the USA the first five letters of his name became Lucas pronounced

“lew-cas”. In Poland the first two syllables would be “wu-caz”.

My husband is a great fan of reverse Polish. There are many reasons for people to struggle in maths. Thanks for your comments.