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

Thank you so much for including my student`s work here. I love seeing how the children in my class are reacting to math this year. I`ve made a huge change in my approach to teaching math, and I see a big difference in the students as well. You`ve definitely captured this here.

Thanks!

Aviva

Aviva, you’re welcome. I am thrilled for you, to have such a positive response to your math teaching.

People say “Math is boring”; that’s obviously not true in your classroom.

There is only one proper response to a situation where kids in grade 5 are counting on their fingers to do simple arithmetic: find those responsible for this appalling act of child abuse and sue them. If legally possible, bring them up on criminal charges.

Doug, thanks for the comment.

I am appalled by the lack of basic mathematical ability that I witnessed, but I wouldn’t blame the teacher/s quite so much as you seem to.

To be fair, and what I did not admit in the post, the class I observed was a “lower ability” class. Nevertheless, like you, I feel strongly that memorization of basic facts is fundamental to children of whatever ability. Were they my students, I would make it an immediate priority to correct this gap in their learning by whatever means I had available.

I suspect that this class was not that unusual, which is even more sad.

Oh, I still have more kids than I can count on one hand in middle school who still count with their fingers. When I asked them to try not too, they tuck their hands under their desks, but their fingers kept the counting beat. I don’t know, Peter. It is sad.

I need, as always, to be careful not to come across as too judgemental. There are plenty of kids who use fingers, who will find it hard to stop if anyone asks them to. And there are some benefits from using fingers as a counting device – for a while. But I’m convinced it still is a really, really bad idea, and if parents and teachers have any influence over how their very young children/students learn basic facts, they should do all in their power to convince them to use other methods.

I tell my students that they are better allowing the use of blocks and other counters at the beginning, because after a while these artificial aids will be put away. But if a child starts using fingers, they cannot be removed later if they become a hindrance. Well, not without a lot of trouble and tears 🙂 .

I agree that seeing 5th graders counting on their fingers is a red flag. But I would never tell a student (who is at least thinking about meaning through this tool) to stop using a strategy that works for them. Instead, I’d offer them opportunities to internalize those number facts in a way that wouldn’t feel like ‘memorizing’. Playing ‘make ten’ games is one way.

I see college students counting on their fingers. (How are they going to learn algebra if their brain power is tied up with the little things?!) I tell them they shouldn’t be embarrassed (mostly, they hide it) – it’s better than just depending on a calculator. This ‘debate’ makes me think I need to also offer them ways to move forward, as I have on the multiplication facts.

Sue, I agree: handling this should be done carefully and sensitively. If I were the teacher of the grade 5 class, I would try to wean them off counting, but without crushing whatever confidence and enjoyment they experienced in math. But to me, the lack of understanding of math I witnessed was directly linked to virtually non-existent knowledge of even basic arithmetic, including addition facts.

Thanks so much for the comment!

Hi Peter,

I think you miss one additional point: Finger counting past first grade greatly increases the risk that students have a poor understanding of number positions, and how complex math problems can be broken down into simple step-by-step arithmetic.

That so many kids looks at 35 + 46 = ?? as an insurmountable problem rather than two easy arithmetic steps plus a position carry is sad. I am heavily involved with my daughter learning math (she just completed third grade), and I know many of her friends who are still finger counting. When they see the problem above, it is not visualized as something easy, but something for which they lack enough fingers (yes they are about to enter 4th grade). Over the course of the last three years, her friends who are still finger counting struggle more and more and more with math. Their end-point seems certain, yet their parents accept it. Why?

Another issue about finger counting is that it very probably inhibits understanding and memorization of multiplication tables. When my oldest daughter started memorizing the tables, she had done enough flashes with the app I wrote for her to have instant one-digit addition recall. As she worked through the ordered progression of multiplication problems, the pattern of multiplication being simply addition repeated multiple times was obvious to her. Tell me how a child who has to finger count is going to recognize the pattern of 6,12,18,24,36… 7,14,21,28,35… etc. And of course fractions become hell because there is no instant recall of the pattern by which common denominators are found. It is really like compound interest. Kids that have zero foundational, instant-recall arithmetic skills never get any leverage as the curriculum progresses.

Finally there is confidence. In the U.S. virtually everything is oriented toward saying “math is hard”. Learning is game-i-fied, teachers and students all dismiss math as “hard”, and, most importantly, so do parents (answer to my Why? question above). If a child is in fifth grade and is still finger counting, she/he will agree with all the many societal inputs she/he gets that math is hard, and she/he is was never meant to be good at it. Review the user review of MathFlashApp by Ladd Campbell for a first-hand account of how this affects kids.

Transparency: I wrote, and am now trying to sell an application that specifically addresses this issue. http://www.mathflashapp.com. It took me about two years because the sole intent was to help my daughter as she address the different stages of her grades 1-3 curriculum (e.g. I did multiplication last fall when she started 3rd grade.) Enough people encouraged me to put it up for sale that I decide to do so.

I wanted an application that was optimized for right-mind, visual learning (basis for instant recall), and which highlighted that complex problems break down into simpler solutions. Found nothing of merit, so did it myself. MathFlashApp proved everything I wanted for my daughter. It also worked Ladd’s sister, and some other friends. Bronwyn needs about 1,000 reps with MathFlashApp before she got it. It was not that she was bad at math, it was that she had never done enough of it in the right problem-solving format.

Sorry for the length. -steve

http:www.mathflashapp.com

Thanks for the long comment, Steve. I’m happy to see we are on the same page, and you have brought up several other valid points.

Hey Peter,

I’m a concerned third party to a child I’m mentoring who is in grade 5, and can’t even do 3+3 without finger counting.

In one of your other comments you said you’d use whatever material resources you had available to correct this issue.

What material resources would you recommend? I’ll be purchasing flash cards sometime this week, but I feel like I need a more concrete way of working on memorization with this child.

Andrew, thanks for the comment. Sadly, there are all too many children in higher elementary grades who don’t know the basics that would, all being well, have been learned in the earlier grades.

The main thing here is for the child to be able to “see” the number facts, and begin to visualise numbers and the relationships between them, so that over time they become well known and can be remembered. For addition facts, I would make up a pair of ten frames, and use them with counters to model all the numbers to 20, and especially the patterns in the number facts. For example, the “3 + 3” fact can be shown as two rows of 3 counters, taking up 6 of the 10 squares on the ten frame, leaving 4 uncovered. The child should learn to recognise the 6 without counting, and should therefore have a model for “knowing” 3+3 without any need for counting. I believe this would be helpful before using flash cards, which may help with the “drill” part of memorizing the learned facts.

You can go to my store for worksheets eBooks that cover each operation in detail. The books are aimed at specific grade levels, but with this child I’d use simpler worksheets. The worksheets are not marked with the target grade, so there is no stigma for the child to do “easy” worksheets. I’d try out some worksheets first as a sample before buying anything. You could start here, where there is a link to a sample eBook:

http://profpete.com/

Let me know how you go!

This is such a wonderful site and it has really made me consider further options for my students. We are fortunate at our school to have an instructional math leader incorporating many of your ideas. Not everyone is going to excel at every subject, but I think that we can get more students excited with programs that create friendly mental math competitions in our schools. In Texas, our third graders can start competing in UIL competitions and our sixth graders can start competing in Math Counts competitions. I have been fortunate enough to be involved in both programs through my students and my own son. Certainly there is a way to get students excited in lower grades.

Thank you for the encouraging comment! It sounds like you are doing a good job in Texas of focusing on student learning through engagement. Competitions are certainly a way to motivate students to try harder, just as they are in adult life.

Thanks again for the contribution.