Category: Times Tables

I recently posted the following graphic on my Facebook page:

Having taught many classes of primary/elementary students, I have taught my fair share of lessons on the Order of Operations. To me, the rules for applying operations in the correct order are not that difficult:

1. Process whatever is in brackets first (applying later rules if necessary)
2. Apply “other” operations such as indices/powers or square roots.
3. Apply multiplication and division, in order as they appear from left to right.
4. Apply addition and subtraction, in order as they appear from left to right.

The reason for having these rules, of course, is so that we can all agree on the value equal to an expression with multiple operations. Otherwise, we would have ambiguous situations, which would be highly inconvenient, to say the least. For example, how should we evaluate the following:

2 + 5 x 7 = ___

• If we work in order from left to right, we get 49: 2 + 5 = 7; 7 x 7 = 49
• If we apply multiplication first, then addition, we get 37: 5 x7 = 35; 2 + 35 = 37. This is the correct answer, but only because we collectively agree that we should carry out multiplication before addition.

My intention with the Facebook post was to generate interest, attract people to “Like” the page, start conversations, and so on. My Facebook page at that time had around 150 people who had “liked” it, and I was used to seeing around 100 interactions a day on the page. I was unprepared for what happened in the following 2 weeks:

• over 70,000 people in all saw the post
• over 6000 people left comments
• more than 140 shared it with their followers on Facebook
• over 400 liked the post
• more than 15,000 interacted with the post

Apparently, this question caught the interest of a lot of people on Facebook, and many felt the need to respond (which was the whole idea, of course). I guess almost all adults learned to answer questions like this at school, and over 6000 who saw it were confident enough in their abilities to answer it in public. I presume they believed they were correct with their answer, and in fact several backed up their numerical response with comments emphasizing that they believed with great confidence that they were correct:

• “8 use BODMAS”
• “… and in arithmetic you apply the multipliers and dividers first, then the additions/subtractions afterwards, so the answer is 8”
• “7-(1×0) + (3/3) = 7-0+1 = 8”
• “YOU GUYS ARE SO DUMB YOU HAVE TO USE FREAKIN ORDER OF OPERATIONS SO ITS 8, im seventh grader and i got that right and adults cant WOW”

Most amusingly (or worryingly), even those who were incorrect often tried to justify their responses:

• “it’s 4 you idiots”
• “It is 6 if you apply principle of BoDMAS”
• “6 do the BODMAS rule! Brackets first then in order, pOwers, Division, Multiplication, Addition, Subtraction!”
• “1 because 7-1=6×0=0+3=3/3=1 SIMPLES”
• “PEMDAS 1*0= 0 / 3=0 +3=3-7= -4”
• “Think you all need to go back to school. Anything X 0 is 0…!!!!!!!”

Seeing the huge number of responses I thought it would be interesting to analyze them to see just how good today’s Facebook users are at primary / elementary level arithmetic. The results were, to say the least, disappointing:

All up, out of the sample of 865 responses which were analyzed, just 25.8% of responses were correct. Of the three-quarters of incorrect responses:

• 43% of respondents apparently applied the operations strictly in order from left to right: 7 – 1 = 6; 6 x 0 = 0; 0 + 3 = 3; 3 ÷ 3 = 1
• 16% of responses were 3. This may have been the result of ignoring the “multiplying by 0”, getting (6 + 3) ÷ 3 = 3
• 6% of answers were 6, perhaps because the respondent made a mistake in calculating 3 ÷ 3
• 4% of respondents gave the answer 0. A number of people explained that “anything multiplied by zero equals zero”, evidently applying that to the entire expression

I have some ideas about why this little experiment found just 26% of Facebook users who saw and responded to the question could answer it correctly. What do you think? Are we teaching the order of operations badly? Should we even bother to teach it, since it seems not to be successful?

Leave a comment below to say what you think:

How well do your students cope with numbers? I mean, can they picture them? Do they visualise number combinations and number facts in their minds with ease? Or do they count on their fingers?

Two Classrooms. Two Stories. One Day.

I have been struck today by two contrasting stories of children working out addition facts. One I observed when I visited a Grade 5 classroom, the other was shared by Aviva Dunsiger via Twitter (@Grade1), relating what her Grade 1 student did the other day.

The contrast between the behaviour of these two children is so striking, and says so much about how the two children were coping with math, I couldn’t resist sharing.

Grade 5: Finger Counting

I was visiting a Grade 5 classroom today, and saw a lesson on calculating area. The teacher had three really cool activities for the kids to engage with the idea of area, and working out ways to calculate it using 1-cm base ten blocks, the formula for area of a rectangle, and a transparent 1×1-cm grid.

What I saw when I looked closer at the children’s working was that they couldn’t calculate areas simply because, in the main, they couldn’t recall simple addition number facts (let alone multiplication facts). A child wanting to know “8+4” in the middle of a multiplication algorithm counted on her fingers. In fact, as I looked around the room for a moment, I saw several children all counting on their fingers.

It was clear that these students had learned at some earlier time that finger counting was a legitimate method for working out unknown number facts. It seems likely that this happened a few years ago, since children would normally be learning these addition facts in around Grade 1 or 2. So for at least 3 years, it would appear that the students have been relying on their finger counting for finding out the answers to addition questions.

What do you think? Am I getting old and out of touch to expect that 10-year olds could recall addition and multiplication facts mentally, instantly?

Grade 1: My Teacher Loves Math. I Can Picture Addition in My Head.

Contrast the above with Aviva Dunsiger’s student’s picture, which Aviva shared a few days ago (and which I noticed today):

It’s posts like this that just make you happy to be a teacher: missd.commons.hwdsb.on.ca/2012/05/11/sch… Happy Friday everyone!

At first glance it is a fairly typical childish drawing of a teacher and a student, drawn by a girl who loves her teacher.

Aviva Dunsiger

Look closer, and you can see that Ms Dunsiger has written some math on the board, and the child is thinking of that math in her head. In fact, the words the child put to these thoughts were “I think that 4+4 = 8 / 10 = 6+4”. The other nice touch is Ms Dunsiger’s own thought bubble, which contains the thought “I love math!”

Isn’t that cute? And what a confirmation of the positive attitude toward math that Aviva is sharing with her students.

But what really thrills me about the picture is the indication it gives that the child is truly able to visualise the addition facts she is learning, and can “see” them in her mind. Am I imagining too much? That’s what I see in the drawing.

Should Children Recall Number Facts?

I think the answer to this question is obvious. On one hand, Grade 5 students in the class I visited relied on finger counting in order to complete a multiplication algorithm, in order to work out the area of a rectangle. Along the way they made so many mistakes that I seriously doubt they understood much at all of what they were learning about area.

On the other hand, a Grade 1 student is able to recall simple addition facts, and moreover by choice draws a picture to express her enjoyment of math, like her teacher. We can hope that she continues to be inspired to love mathematics as she grows older, and that in 4 years she is able to calculate area easily and quickly, using her knowledge of number facts.

How would your students fare in this comparison?

Image credit:

• Finger counting by ckmck – CC license at Flickr
• School & Miss D – Francesca, via Aviva Dunsiger

This week, a flurry of news and blog articles appeared, proclaiming that children don’t actually need to learn their number facts off by heart, as not knowing number facts doesn’t stop them from being good at maths. Is this really true?

The articles in question include these:

• BBC News: “Sums tables ‘not needed for maths success’”
• The Guardian’s Teacher Network Blog: “Children don’t need to know all their number facts to succeed at maths”
•  Daily Mail’s Mail Online: “Scandal of the primary pupils  who can get full marks in maths without even knowing their times tables”
• Independent Education Today: “Primary school children succeed at Maths without knowing their tables”
  
• Yahoo UK: “Children don’t need to know number facts to be good at maths”
  

My attention was caught by a tweet by @wanstad73 curated on my daily paper over at paper.li on September 13, linking to the Guardian article. Having taught number facts myself as a classroom teacher, and now teaching preservice teachers the importance of laying a good foundation in number facts in all 4 operations, I have a strong interest in the topic.

Let’s just say I was alarmed by a claim that memorization of number facts is unnecessary for success in primary or elementary maths. Surely, my thinking goes, without knowing number facts by heart, children will be unable to tackle later maths, not just in computation, but also in geometry, measurement, probability, algebra; pretty much all mathematics topics.

“The Development and Importance of Proficiency in Basic Calculation”

I had a look at the original article by Professor Richard Cowan at the Department of Psychology and Human Development, Institute of Education in the University of London. Surely the study’s author himself hadn’t said number facts were not necessary, as these secondary reports were saying?

It shouldn’t be a surprise that media outlets have picked on the idea that number facts are not really important. The English National Curriculum requires all addition facts to 20 to be memorized by the end of Year 3. So the idea that a research study has proven not only that Year 3 students aren’t learning all their facts, but furthermore those facts aren’t really that important could be expected to catch the interest of journalists whose bosses want to sell more advertising. But is that really what the research showed?

To summarize, Prof Cowan and his fellow authors say the following:

• proficiency in basic addition and subtraction to 20 is a key indicator of general mathematical ability, which later leads to adult proficiency
• students in Years 3 and 4 in the study showed above-average mathematical achievement, yet none out of 259 knew all their number facts
• only 10% of the children themselves reported that they were recalling number facts to answer most of the questions

The report’s authors describe the differences between a traditional view of learning number facts and a progressive view. According to them, the traditional view, in vogue in the 1920s and 1930s, favours rote memorization of number facts, whereas the progressive view focuses on children ‘learning numerical principles and patterns and knowing how to use them efficiently and accurately’ (Cowan 2011, p. 4).

Rote Learning vs Developing Understanding of Numbers

A comparison is thus set up between rote learning of facts and developing understanding of mathematical principles. What can we learn from this comparison?

Which view is better? And which one is favoured by the report authors?

Contrary to the tabloid headlines, Prof Cowan and his co-authors believe that being able to carry out basic addition and subtraction quickly (the standard used in the study was 3 seconds for a correct response) is vital for developing a wider mathematical proficiency to lay the foundations for adult mathematical skills. The authors certainly did not say that number fact shouldn’t be rapidly accessible to all students. In fact, what Prof Cowan did state (according to the Mail Online) was ‘We are not saying that fact knowledge is irrelevant’, and ‘Facts help children grasp principles, and applying principles helps children learn facts’.

Conclusions

1. Children do need to know their number facts, either via memorization or via developing conceptual knowledge.
2. While children are learning the number facts, it is quite acceptable for students to use a strategy based on conceptual knowledge to quickly work out the answer.
3. Big media is wrong to imply that number facts aren’t important after all. Children need understanding of numbers first, and then need to memorize number facts. A more accurate headline than those chosen by editors would be “Children need to understand basic number concepts to succeed at mathematics”.
4. Most primary age students will use a combination of strategies based on understanding and memorized facts, as they develop greater speed and proficiency. Not having the complete set memorized is not a significant flaw, provided the child has a set of tools to derive those facts that have not yet been committed to memory.

 Links

Cowan, R 2011, The Development and Importance of Proficiency in Basic Calculation, Institute of Education, London, http://www.ioe.ac.uk/Study_Departments/PHD_dev_basic_calculation.pdf [accessed 13th September 2011].

If you are interested in resources to teach number fact recall via an organised system of worksheets based around classroom-proven strategies, check out our eBook “10 Minutes a Day: Times Tables Worksheets“. If you would like to trial part I of the system, sign up to receive it; all we ask is for your email address so we can keep you updated on new resources and content as they become available. If you are ready to buy the complete eBook, visit our store where you can purchase it.