## Number Facts Resources That Students Love

Do students enjoy learning times tables?

You’re kidding, right?

If you remember learning times tables by reciting them over and over and over, you are probably skeptical that students could actually enjoy the exercise.

But if you provide students the strategies they need to make sense of the numbers, they do in fact enjoy it. And what really motivates them is to be able to improve on their times day by day.

Developing Number Fluency worksheets eBook series

• Aligned to curriculum documents for students in US, UK and Australia
• Strategy-based approach so students develop understanding as they memorize facts
• Daily worksheets, including fortnightly assessment sheets
• Comprehensive system to teach every fact for every operation, in 10 minutes a day
• Integrated homework sheets so parents can support classroom teaching
• Duplication license allows for unlimited copying by the original purchaser: no more to pay
• 30-day money-back guarantee, no questions asked
• Year-level bundles for enough worksheets for entire year cost less than US\$70

Sample worksheets to try at home or in class:

The “Developing Number Fluency” eBooks Series

Visit the Professor Pete’s Store site.

## Why Did 74% of Facebook Users Get This Wrong?

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?

## A Tale of Two Classrooms: Finger Counting in Grade 5, Visualising in Grade 1

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.

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?

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

## So, You Will be Teaching my Grandchildren? Job Interview Questions

I’m going to be a grandfather, this coming August.

This momentous event is not unexpected, since our daughter has been married to her wonderful husband for 3.5 years. Nevertheless, it has made me think again about small children I have a particular interest in, and what happens to them at school. This post is just a touch indulgent, but I invite you to join me in thinking about the qualifications for teaching children.

What do children need from their math teachers? How important is it? If you really were going to be my grandkids’ teacher, these are the questions I’d be keen for you to answer:

Math Teacher Job Interview: Questions for Candidates

• How did you do at math when you were at school?

It’s not important if you were an “A” or a “F” student; you could become a brilliant teacher of math from either starting point. Now, on balance, I’d rather that you got great marks at math, since you obviously “get it” and will be able to understand math more deeply than most. But either way, I’d like an answer to the second part of this question:

• Do you understand why lots of people find math difficult, scary or downright horrible?

To teach students who really don’t understand math, and therefore most likely hate doing math, you need to be able to empathize with them. Did you ever feel that learning math was pointless, illogical, impossible or just plain frustrating? If so, great! Can you put that knowledge to work as you help students who feel that way? If not, are the ways for you to learn what it feels like to students who hate math?

• Do you love math? If not, are you at least enthusiastic about math?

Not everyone loves math. Even math geeks get that. But teachers of math have a responsibility to encourage students who do love it, who will go on to do amazing, important math later in their lives. I have decided to never discourage a student who has, or is developing, a passion for something, anything almost. If a student loves dancing, or BMX bike riding, or astronomy, or writing poetry, I will do everything in my power to support, encourage and promote success for that student in that endeavour. How can you do that in math?

• Which is most important to you: understanding, or correct answers?

We should never pretend that correct answers aren’t important. But just as giving a poor person a fish is a lousy substitute to teaching that person how to catch fish, focusing on correct answers should come after we are sure the student understands the math, never before. Great math teachers have the attitude that correct answers are the product of correct thinking, and in some cases aren’t even that important, providing the student understood the process.

• Do you teach “tricks and shortcuts” to math processes?

This is really the corollary of the previous point. A shortcut (eg, to convert a decimal to a percentage, move the decimal point two places to the right) will lead a non-understanding student to the correct answer, providing (and this is the deal-breaker right here) the student can remember the correct trick to use. But the shortcut or trick will let the student down in situations in which the math is not so obvious, and thinking has to be applied to solve a problem. Good mathematicians do use shortcuts, but that happens way, way, way after they understood why the shortcuts worked.

• How important is memorization of times tables and number facts to you?

Since calculators first became a consumer item, teachers have had a challenge with this question. Do we really need to spend the time that some of us remember from our own school days on learning times tables, or can we now let calculators handle that process? Answer: unless students have fast, accurate, confident recall of basic number facts and times tables, they will be hampered and slowed down by the need to reach for a calculator every time they need one of those facts. If you have to know “13 x 24″, by all means use a calculator – it will be quicker. But if the question is “6 x 7″, recalling the answer quickly will save huge amounts of time and cognitive load.

By the way, to see a series of classroom-tested, systematic workbooks for teaching all the basic facts and times tables in 10 minutes a day, visit our store.

• What math teaching blogs, forums, wikis or Facebook pages do you read?

In most places where teaching is a profession, a certain amount of professional development is required every year. Where once that PD came in the form of visiting academics to run workshops and sessions after school, nowadays it should also include use of online communities and other sites, where other teachers and educators meet to exchange ideas and encourage each other. Some examples from hundreds of great sites:

I already wrote about the necessity of math students knowing their number facts. The way some non-math people talk, this is an “either-or” situation: either you learn times tables by rote, or you use technology. My view: the true answer belongs in the middle somewhere. Yes, students need to learn number facts. No, they don’ t need to use unthinking rote methods (see here for more on this). Yes, they should use technology. A great math classroom will have lots of appropriate technology: calculators, iPads, iPods, laptops, data projector and interactive whiteboard, internet connection (duh!), etc., etc. The technology opens up unlimited possibilities for connecting classroom math to real life math, for using microchip tech to make more complicated math possible, for practising skills using online interactive widgets, and so on and so on.

• Do you assign math homework? What sorts?

Homework is a huge topic which I don’t have space to address adequately here. But for the purposes of this post, let’s agree that homework is a great way to connect your math lessons with parents and other family members. By sending home suitable activities to apply and practise the math being learned in class, you make space for child and parent/grandparent/carer to have conversations about math. My advice: make the homework something every child can succeed on, so that it becomes an enjoyable part of family life, not one of the biggest frustrations. Build in activities for children to talk about math at home, for the benefit of child and adult.

• Do you integrate math into other subjects’ lessons? What about integrating other subjects into math lessons?

Math can seem a pretty lonely subject in some classrooms. It can be the only one which lots of students don’t like, the only one done solely using textbook or worksheets, the subject most likely to lead to the question “When are we ever going to use this?”. You can help to change these perceptions by showing students how useful math really is, by allowing math to serve other subjects and help solve their particular problems. For example: in social studies or geography classes, use math to examine statistics and graph the data on how people behave, how the environment is being affected by pollution, etc. The reverse is also worth doing: include real-life examples from other subjects when preparing problems and investigations for math class.

How did you go? Do you agree with my views? Would you be happy to teach my grandchildren? Leave a comment below and let me know…

Photo Credits:

• Grandfather and boy:  © iStockphoto.com/Pavel Losevsky

[related-posts]

## Ignorance of Number Facts “No Barrier to Success”?

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?

 Traditional View Progressive View Learning by rote (repetition) Learning through understanding Facts learned in isolation Facts learned as connected to other facts & topics Facts believed to be essential for proficiency Facts believed to be less important than understanding Forgotten facts difficult to retrieve Facts not known may be derived by thinking Memorization of number facts regarded as essential for all students The ability to work out facts from understood principles regarded as essential

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.