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.

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

Best Math Blog Posts: May, 2012

If  you are a regular visitor to this site,  you will have seen a previous post in which I highlighted a number of outstanding blog posts written by teachers who really “get it” when it comes to engaging learners in mathematics lessons.

This week, I do a round-up of recent great posts from Tom Whitby, Larry Cuban, Fawn Nguyen, Daniel Schneider & Matt Vaudrey. Let me know in the comments what you think, or if you have other suggestions for my reading list.

Tom Whitby [My Island View] > “We Don’t Need No Stink’n Textbooks” #Beyondthetextbook

  • Tom Whitby discusses the history of textbooks, particularly in American math education, and argues for a fresh look at how students are provided with the information they need to learn mathematics.
  • ‘A decade into the new century we have a new way to deliver content. The internet not only delivers text, but allows it to be manipulated, transformed, evaluated, analyzed, merged with video and audio, created, and published.’

Manifesto for dumping textbooks for C21st tech from @tomwhitby “We Don’t Need No Stink’n Textbooks”http://ow.ly/afoTI

I came across this post a little while ago and tagged it for a future blog round-up. Checking again today, I’m in good company in thinking this is a great post – it has been reblogged or mentioned in blogs at least 18 times!

It’s fair to say that Tom is a veteran in the mathematics education field, and his writing is always eloquently expressed, thoughtful and thought-provoking. This piece asks questions that ought to be considered by every teacher, every school board, every body that decides how to spend money on classroom resources. The idea that schools should spend millions on dead-tree resources in order that students can receive what is considered the best material from which to learn math is fast becoming an embarrassing anachronism.

Larry Cuban [School Reform and Classroom Practice] > Student “Learning Styles” Theory Is Bunk (Daniel Willingham)

  • Larry Cuban takes on a theory that is practically a religion in some circles. I often hear beginning teachers say that such-and-such an teaching strategy is used to cater for “visual learners” or “kinaesthetic learners”, without any foundation for the idea that learners need instruction tailored to their particular way of thinking.
  • The author points out that teachers have much to do already catering for individual differences in their students, and following this theory is not only difficult to fit into what teachers already do, according to the research evidence it’s a waste of time.
  • ‘When you think about it, the theory of learning styles doesn’t really celebrate the differences among children: On the contrary, the point is to categorize kids.’

I totally agree with this one: “Student ‘Learning Styles’ Theory Is Bunk” http://ow.ly/aint7 #edchat

This article, written by Daniel Willingham in 2009, is still relevant today, and I am grateful to Larry Cuban for reposting it. I think that the “learning styles’ theory is attractive to many educators, in the way it presents a simple way of thinking about the different ways that students learn. However, there are two clear problems with the idea: (i) the research data simply doesn’t support it, after studies going back to the 1940s have looked at various ideas about different ways of learning; and (ii) the main result of adopting the theory is to pigeon-hole students, into “visual”, “auditory” and “kinaesthetic” learners.

Great teachers cater for individual differences in their classes every day using as many methods as they can come up with. The other strategy I think is really powerful here is to use as many different resources that exemplify as many media as possible. Provide students with pictures, auditory stimuli, videos, interactive computer apps, text, teacher instruction, physical out-of-seat activities, etc, etc, in an attempt to find the best ways for every  student to connect with the topic.

Fawn Nguyen [Teaching Math in Middle School] > Staircases and Steepness, Continued

  • Fawn follows up her post from the day before, when she asked grade 6s how to judge which of a set of drawn staircases was the steepest, and to work out a method for justifying their decisions
  • In part II, Fawn taught about slope, following up on the students’ discoveries and discussions
  • ‘By Friday morning the kids who did “base times height” learned that these numbers didn’t match up with the steepness ranking. They said, “That just gives you area.”‘

The payoff in @fawnpnguyen ‘s lesson on slope: Don’t jump to “rise over run” too fast! #mathchat

I have told Fawn that I am a fully paid-up member of her fan club: she is simply the best teacher blogger I have yet come across sharing lessons for teaching mathematics. She has an adept touch in gently prodding and leading her students to think mathematically, to actually do the math, without once feeding them a formula or driving them to distraction through endless symbolic routines.

Fawn Nguyen

The other bonus when you read Fawn’s work is she is just so witty and caring, and you get a clear sense of her love for both math and her students.

The highlight for me in this particular lesson was Fawn’s comment ‘I finally said the word slope, but I never said “rise over run”‘. In my opinion, this finesses the learning for students: they learn about the idea of slope, and they learn how to calculate a measure that allows you to compare any two slopes for steepness, but they discovered it for themselves! I think it is clear that these students will remember what slope is and how you work it out long after students who learned to use “rise over run” have forgotten all about it.

Daniel Schneider [Mathy McMatherson] > The Wall of Remediation (Or: My Low-Tech Version of Khan Academy)

  • Daniel shares a method he uses to differentiate instruction and remediate students’ difficulties in a remarkable, low-tech way: he provides students with a structured array of worksheets for them to practice the skills they’re missing
  • The strategy is evidently producing impressive results, with students choosing worksheets to brush up on tricky math skills without being told to
  • ‘Having this board available lets me quickly walk up and grab the particular skill they need to work on (integer operations, algebra, etc) and get them started on it. It also lets students be self motivated and do the exact same thing for themselves!’

Low-tech, high-impact differentiation of math teaching: Brilliant stuff from @MathyMcMatherso http://ow.ly/aRyVr #mathchat

This post is a brilliant example of how an excellent teacher can find ways to truly engage students in learning without cutting-edge technology, without some fancy new resource or fancy idea about teaching. It seems to me that the students are learning and all being kept on track because their teacher has made it easy for this to happen, and because his focus is on learning, not completing exercises.

Daniel Schneider

Daniel’s resources are not flashy at all; I asked him about the content of the worksheets, and he confirmed that they are just plain exercises to give targeted practice.  The magic is in what he does with them.

Matt Vaudrey [Mr. V’s Class] > Mullets: The Only Lesson They’ll Remember

  • Mr V goes above and beyond what he was taught at university about teaching math, and gives himself a mullet (really), then teaches an inspired lesson on proportion using measures of “mulletude”
  • ‘I gave myself a mullet. It was totally worth it; every student came into class with a smile, already curious.’

Using ratios to measure Mulletude: brilliantly memorable math lesson from @MrVaudrey http://ow.ly/aOAYt#mathchat

This is another recent post that has generated a huge interest among math teachers, with 26 responses to date. Matt Vaudrey has taken a reasonably dry topic, ratios and proportion, and injected life and fun into it in a way that his students are unlikely to forget, perhaps for the rest of their lives. This lesson really is that good.

Not every teacher could wear a mullet even for the cause of a good math lesson, of course, but Matt pulls it off with flair and panache.

Matt Vaudrey

The important take-aways for me are the way he weaves real mathematics into a lesson that revolves around questions about the length of a person’s hair at the back and at the front, then uses that premise to generate lots of mathematical thinking. If you want to teach memorable math lessons with impact, there is a lot to pick up here, no matter what the topic and what the age group.


Powers of Ten > Based on the file by Charles and Ray Eames

  • This interactive site shows images at all scales from 10-18 to 1026 metres from the now-classic 1977 movie, Powers of Ten.

More mind stretching via @mathforlove Powers of Ten. Based on 70s film by Eames. http://ow.ly/av412#mathchat http://ow.ly/i/AmNt

Powers of Ten

This is not a blog as such, though there is one on the site. But I just couldn’t resist including the site in this round-up. If you teach mathematics, and you want ways to help your students grasp the different sizes represented by powers of 10, you have to look at the site. The movie itself, unsurprisingly, is now available for free on YouTube. The music soundtrack and narration are now dated, but the images blow me away every time I watch it.

Image credits:

  • Red typewriter:  alexkerhead at Flickr
  • Slope diagrams: Fawn Nguyen
  • Bulletin board: Daniel Schneider
  • Mullet: Matt Vaudrey
  • 10 Meters squared: Powers of Ten

Without These 3 Components, Your Primary/Elementary Mathematics Lessons Won’t Work

As a primary or elementary teacher, you are probably not a specialist in mathematics. As the old adage goes, “high school teachers teach subjects; primary and elementary teachers teach students”.

As a result, knowing how to teach students mathematics really well may be something you find a challenge. If so, this post may help you.

1. Focus First on the Mathematics

This is the biggest, most important factor in getting mathematics teaching right. If you don’t nail the mathematics and put it at the center of all your math lessons, it will be very difficult to capture students’ interest, and impossible for them to really understand the topic. This step is needed right at the beginning, at the planning stage.

For example, if the topic is “symmetry in flat figures”, you would teach the terminology of symmetry, the two different types of symmetry, correct terminology and mathematical ways to analyse symmetry (such as the angle of rotational symmetry).

If you lack the content knowledge in a topic, I would go to Wikipedia for a quick brush-up on what the basic math is. As an encylopedia, rather than a blog or social media site, Wikipedia will have succinct summaries of all the topics you are likely to teach, and is highly likely to have the facts correct.

2. Work Out What Mental Processes Students Will Need

The second major step in planning a great math lesson is to consider the student. What processes will they need to practise for this topic?

For example, if you are teaching number facts, the standard we are aiming for is instant recall of every fact. So the mental processes needed are memorizing the facts in the first place, then recalling them from memory.

Note that with this approach, there is no argument about whether students need to remember all those facts or can “invent” ways to come up with them when they are needed. The bottom line is that number facts are needed for just about every math topic, and using up precious brain power (and time) to work them out when needed is just too inefficient.

3. Use the Best Methods to Connect Math to Mental Processes

Lastly, you need to find the best methods you can to truly connect students and their thinking to the mathematics. This has a couple of important components:

  • Make the mathematics the primary focus of the lesson. In other words, it’s not about games, exercises, routines, or any other activity or behavioural focus.
  • Don’t focus on making math “fun”; true mathematical activity requires attention to detail, discipline and following rules. Students will experience satisfaction from understanding the mathematics and the  processes needed to reach solutions and correct answers, but “fun” is the wrong focus.
  • Doing mathematics involves a good deal of mental effort and mental processes. It is essential that your students are engaged in thinking for themselves, using a variety of mental processes including memorization, visualization, mental computation, exploring options, testing hypotheses, following logical connections, holding pieces of information in the working memory while applying mathematical processes such as operations, etc.

I have decided that I will no longer help teachers to inject fun, simplicity, tricks or cute pseudo-math processes in an attempt to buy students’ affection. (As an aside, I am working on a new product about fractions. We brainstormed titles for the series, and rejected “Fun Fractions”, “Fantastic Fractions”, “Spectacular Fractions” and the like because they send entirely the wrong message.)

What do you think? If you are a primary/elementary teacher, does this article help? Does the above advice match your own practices? Please leave a comment below!

Graphic Credits:

  • Primary student group:  © iStockphoto.com/Chris Schmidt
  • 5x Table:  © iStockphoto.com/Dougall Photography

Teach the 9x Nine Times Tables

This podcast video is from my Free Math Worksheets series, which you can access here.

If the video does not display, watch on YouTube.

Download Free Worksheets: 9x Tables

Multiplying by 9 raises some really simple and interesting patterns, which you can use to help children to learn this set of number facts or times tables.

Teaching the 9x Nine Times Tables

9x Tables: Start With 10

As explained in the video, if you start with the equivalent multiple of 10, you can then compensate to quickly find the 9x answer. For example, think about 8×9:

  • 8 x 9 = ?
  • 8 x 10 = 80
  • Subtract 8 ones (one for each set of 10): 80 – 8 = 72
  • 8 x 9 = 72

9x Tables: Sum of Digits Equals 9

Look at the multiples of 9 up to 90, and check out the sum of the tens digit and the ones digit in each multiple:

  • 09  :  0 + 9 = 9
  • 18  :  1 + 8 = 9
  • 27  :  2 + 7 = 9
  • 36  :  3 + 6 = 9
  • 45  :  4 + 5 = 9
  • 54  :  5 + 4 = 9
  • 63  :  6 + 3 = 9
  • 72  :  7 + 2 = 9
  • 81  :  8 + 1 = 9
  • 90  :  9 + 0 = 9

Students can use a two-step process which takes advantage of the above pattern:

  • 6 x 9 = ?
  • There must be 5 tens (because we know it is a bit less than 60)
  • So, 6 x 9 = 5?
  • If the digits in the answer add up to 9, what is the other digit; ie, 5 + ? = 9. The other digit must be ‘4’
  • 6 x 9 = 54

9x Tables: Finger Trick

Put both hands up in front of you, palms facing away. Imagine that each finger is numbered, from left to right, from 1 to 10.

To find a multiple of 9:

  • 3 x 9 = ?
  • Hold up your hands, thumbs together
  • Put down the finger that corresponds to the number multiplying the 9: the third finger
  • Count the fingers to the left of the finger that is down: 2 – this is the number of tens
  • Count the fingers to the right of the finger that is down: 7 – this is the number of ones
  • 3 x 9 = 27

Math Teachers Survey 2012

Do you teach mathematics?

Are you a teacher who teaches mathematics?

In 2011 I asked my readers what they were looking for to help them to teach mathematics, and their responseswere very helpful.

This year, my list has grown, and the number of visitors is significantly up also. I’d really like to know what you think is important in your classroom, and how I can best help you with your mathematics teaching.

Click to enter the survey: http://www.surveymonkey.com/s/W62S5NJ

Click to go to 2012 survey

The survey is completely anonymous, and should only take you a few minutes. Your time would be most appreciated – thanks in advance! So you know I take note of what my followers tell me, in a few weeks I will put up a follow-up post to let everyone know the results.

More Information About This Survey

In case you wondered why I’m putting up this survey, I’m happy to tell you that I am planning to start showing my products at trade shows, in Australia, the UK and the US. (When I do, you’ll be sure to know if you come to this site or are on one of my mailing lists).

The information provided in the survey by classroom teachers is straight-forward market research; basically, I need to know what teachers think, what they are looking for, and how decisions are made about purchasing resources to use in the classroom. Producing quality resources is really expensive, especially in time, and I don’t want to guess what teachers are actually looking for and miss the mark.

How You Can Help

I would appreciate it if you could let your network of fellow teachers know about the survey – the more teachers who respond, the better prepared I will be to take the next step. Feel totally free to share this page on Facebook or Twitter, or by email.

If you care to add more information right here, please leave a comment below, and thank you!

Would Khan Academy Work for Elementary Math?

Last week I wrote about Khan Academy’s apparent moves to play a more active part in thousands of classrooms, and my concerns that there was a hidden agenda of trying to make the curriculum “teacher-proof”. This new train of thought was triggered by two recent posts by Dan Meyer (I recommend you check him out; his blog is outstanding).

Khan Academy: Only for High School Math?

While the Khan Academy has videos for all levels from kindergarten/preschool up to university level, the discussion I’ve read about his philosophy of teaching, his vision for education, and the uses made of his material by teachers has almost all been in the context of high school math education. And the discussion has been, shall we say, pretty heated. High school math teachers especially, it seems, are critical of Khan’s methods of teaching, possibly the unfair influence applied by the Gates Foundation’s funding of the academy, and the suggestion that the Khan approach could be used to “fix” what ails math education in the developed world of the 21st century.

Elementary school class with teacher outside

So, what I’m wondering is, have elementary teachers of math had the same discussions around the staff lounge, or in their blogs, or has this highly contentious debate passed them by? And what would the discussion look like if we suggested that perhaps the Khan approach could fix the problems in earlier math education, before kids get to high school with a bad attitude and poor understanding of math?

I’d like to propose some basic points about this situation. And remember, Khan is just the most obvious example of an approach that was probably inevitable, given the expansion of the internet, delivering teaching episodes via online videos. So what we’re discussing is not really the Khan Academy per se, but the idea of replacing a teacher with a recorded lesson prepared by an “expert teacher”.

  1. This is not a new idea, that expert teaching could be captured and recorded, and delivered to students in a “perfect” form, bypassing the teacher, who of course is flawed and makes mistakes. Back in the day, lessons were packaged into slide shows or filmstrips, with audio recordings and flash cards. I remember having a set of these things in my classroom, and being amazed that some syllabus publisher thought I needed a script to make sure I taught everything correctly. The “teacher-proof curriculum” has been an attractive idea to governments and various commentators who don’t understand classroom teaching, and think the real problem with education is the teachers.
  2. Let’s admit that Khan’s output is nothing short of astonishing. The guy is clearly a workaholic, and has a vision for helping students with their math, science, and many other subjects which is attractive in many ways. I am sure that lots of teachers could find ways to use Khan videos to help students learn, to support the other activities that go on in the classroom.
  3. Given that students need to understand what they are learning in order to make sense of it and apply it in their “real” lives outside math class, both now and when they are grown, videos are going to be extremely limited in the ways that they can effectively produce that sort of learning.
  4. Yes, Khan’s videos can supply revision of once-learned, now-forgotten material, they can help explain and demonstrate algorithms and processes for approaching set problem types. But they can’t possibly engage a student as a real live teacher can, in conversation about the topic, to connect to students’ learning.

Elementary Math Teaching and “Teacher-Proof” Videos

It’s probably fair to say that many elementary teachers are not as confident with mathematics content as the average high school math teacher. This is understandable, given the wider range of subjects which teachers of younger students have to manage, and the different preparation they had at university. Does this difference mean that the Khan Academy videos are more attractive to elementary or primary teachers (do tell me your thoughts!)? In fact, would heavy adoption of KA materials be a good thing in elementary classes, as a way of “shoring up” the teacher’s lack of confidence and depth in mathematics content?

In a word, in my opinion, NO. Teachers of elementary students have a significantly different role to play in the education of the next generation: not only are they expected to teach the content knowledge and skills of each subject. They also have a responsibility to develop:

  • students’ attitudes to learning
  • their self images
  • their views of life and the parts they will play in it
  • their confidence
  • etc.
  • etc.

In mathematics specifically, elementary teachers ought to be (and many are) inspiring their students to construct a robust, flexible, deep understanding of what mathematics is about, how it makes sense, and how it may be applied in real life. To suggest that the teacher should hand over this job to a “video teacher” is ludicrous.

Your thoughts, as always, are invited – leave a comment below if you’d like to add to the discussion.

Photo References:
  • Elementary Teacher with Students:  © iStockphoto.com/Catherine Yeulet
  • Bored Child with Computer:  © iStockphoto.com/zhang bo

Best Math Blog Posts: March, 2012

I thought I’d write a different sort of post this time: a round-up of some of the best recent math teaching blog posts.

I share these posts via my Twitter and Facebook accounts already, but those posts soon move off the top of the stream, and are lost pretty much for ever. But by sharing here on the blog, these links can remain accessible for much longer.

By the way, I’d love to hear what you think; I am planning to write similar posts in future weeks, if they are useful and interesting to you, my dear reader.

Math Blog Post Roundup

Fawn Nguyen [Teaching Math in Middle School] > Always Sometimes Never

  • Students sort mathematical statements into three piles: those that are always true, those sometimes true, and those that are never true.

Brilliant math activity for any grade from @fawnpnguyen: Always Sometimes Never ow.ly/9PnSB#mathchat

I linked to this article on March 24, 2012 via Twitter, and it was far and away my most clicked tweet all week. I guess other teachers agree that this is a wonderful article, explaining a simple activity that any math teacher could use with their class. Fawn’s students worked on statements which included “p + 12 = s + 12″ and “If you divide 12 by a number, the answer will be less than 12”. I love these statements, and the activity, for several reasons:

  • the statements themselves are easy to understand at first, which will help develop “buy-in” by students.
  • I would expect just about every student to be happy to get started.
  • None of the statements can be answered immediately via some standard routine procedure.
  • Each one requires a level of intuition, investigation, even lateral or creative thinking.
  • For many statements, there is an obvious answer; and like many obvious answers, it isn’t always true.

David Ginsburg [Coach G’s Teaching Tips] > There Are No Stupid Questions, But…

  • How teachers respond to students’ questions may have a big impact on how likely students are to ask questions in the future.

Another great post from Coach G: There Are No Stupid Questions, But… ow.ly/9PmVZ

David is another blogger whose writing I admire. He manages to get right to the heart of an issue for teachers, grab your attention, and then get the reader to honestly think about his or her own teaching. This particular post is a great one if you care about the impact your comments have on students’ feelings of well being and self esteem.

I have to say, this isn’t really “Questioning 101”, but more like “Questioning 404”, for teachers or preservice teachers who have understood the basics of eliciting students’ responses, but realize that there is a higher standard to aim for. Key statement by David: “[students will] never feel such freedom unless we as educators value their input rather than just evaluate it”. Amen!

Malke [The Map is Not the Territory] > All in Good Time

  • The author’s 6-year-old daughter asks to be taught to play the penny whistle.

Music, math, reading, kids develop at different rates > All in Good Time @mathinyourfeetow.ly/9A9bb #edchat

Malke always writes interesting posts, illustrated with lovely photos of her daughter and the activities they share. This post caught my attention because of the focus on playing music, and also learning math, two of my loves. The key point: given the freedom to choose when to learn something, children will often reveal when they are ready, “all in good time”.

Ms Cookie [Math Teacher Mambo] > Shadows on Planet Earth….

  • Ms Cookie finds out that her students’ difficulties with similar triangle questions had less to do with the math, and more with fundamental misconceptions about shadows.

Not understanding shadows interferes with learning similar triangles… from Ms Cookie http://ow.ly/9Pwmn #mathchat #scichat

I love science as well as math, and so this post was a fascinating one to me. Students really do struggle with scientific concepts quite often, but even so I was surprised at their ignorance about shadows and how they are caused. How often do we think difficulties in math problem solving are caused by lack of math knowledge, when they might come from misunderstandings of the question itself.

Dr Mike Hartley [Math Games for Kids] > Is Math The Primum Movens?

  • Could mathematics explain the existence of the universe?

Is Math The Primum Movens? | Philosophical post > Does math alone explain existence? ow.ly/9Pp33 #mathchat

If you enjoy philosophical discussions and the study of mathematics, you’ll like this post. Dr Mike poses some really tricky questions about ultimate reality and First Causes (Primum Movens, in Latin). Is God the Cosmic Mathematician? You’ll have to decide.

That’s all I have space for this week. I hope you’ll follow the links to read other bloggers’ posts, and let me know below your own thoughts. If you have a favorite blog or two, let me know and I’ll check it out.

Photo References:

Khan Academy: “Teacher-Proof” Curriculum?

I follow Dan Meyer’s blog quite closely, and find the discussions over there really stretch my thinking sometimes about how we teach math, and the best ways to engage students in thinking.

Dan Meyer on the Khan Academy

I first encountered Salman Khan on his TED video, perhaps like lot of others. (Incidentally, that’s also how I first heard of Dan Meyer, watching his TED talk.) I found Sal Khan’s methods surprising and challenging, and incidentally, his business practices pretty remarkable also. If you look at his site, it’s hard not to be impressed by the sheer volume of material he has there, with a huge list of videos, all free for watching.

Recently Dan has posted a couple of articles about the Khan Academy:

Dan points out several really important points about the Khan academy’s approach, including an apparent shift in emphasis from supporting the work of teachers via flipped lessons to supplying an entire curriculum for students. Crucially, Dan comments that students actually find watching the Khan videos quite boring, which surely is a critical flaw in the program.

“Flipped Classes” – a Solution to Bad Teaching?

To summarise, in case you haven’t been keeping up with this debate, the idea put forward by Khan at the TED conference which has captured the attention of many educators, is “flipped classes”. In this model, instead of the teacher teaching in class and then assigning practice work for homework, students watch the teaching at home via Khan’s videos online, then in class the teacher gets to follow up the video presentation, offer one-on-one tutoring help, and generally support and troubleshoot students’ learning, freed from having to spend hours planning and teaching didactic lessons.

What’s the philosophical idea behind Khan’s approach? Note the low-tech quality of the videos: it can’t be able visual engagement, hooking students with exciting music, animations or the like. No, what Khan is attempting, without really admitting it, is to produce a set of perfect teaching videos. If you like (and I doubt you do), a teacher-proof syllabus. How does that strike you? I find it insulting: why does Mr Khan feel that a disembodied voice track and a screen showing the teacher’s written notes for a math process is better than what real teachers do in a real, physical classroom, with students who are present in the same space?

The only way to accept KA as a replacement for what teachers in general do in classrooms is if you subscribe to the idea that most teachers suck at teaching math. If that premise is accepted, then the idea that a single source of “expert instruction”, delivered uniformly to all students, could supply all the teaching might look pretty attractive.

However, critics point out, often with some heat and passion, that there are several problems with this scenario:

  • lecturing to students is not the best pedagogical approach to teaching
  • video recordings lock every student into a single lesson for each topic
  • there is no opportunity for students to ask questions of the video teacher, to have something explained again, other than replaying that part of the video

What do you think?

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:

  • Do you use technology to help your students learn math? How?

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


Are Wind Farms The Solution? Do the Math!

It’s Agreed: Teachers Should Teach Environmental Responsibility

Teachers around the world are expected to teacher their students to be responsible citizens, to reduce their impact on the environment and to support sustainability. For one example, see the new Australian Curriculum’s Cross-Curriculum Priority: Sustainability.

Great, good stuff. In fact, it would be difficult to imagine a teacher anywhere who did not believe in protecting the environment for future generations, and teaching today’s students to be responsible adults in the future.

But does mean that we should agree to promote every green cause or environmental suggestion, “just because”. Some ideas are really dumb, including those with a green label applied. How can we tell them apart? Do we just let others decide for us, or do we think for ourselves? I reckon we should use math to check out the facts – and teach our students to do the same.

Will Wind Farms Fix the Energy Crisis?

A lot of environmental activists and politicians are currently promoting wind energy as the solution to global warming and the world’s reliance on dirty fossil-fuel-generated energy. Are they correct?

The answers aren’t all in yet, but it is important to do some basic research, rather than just believing the promoters. There are some great sites out there with lots of information on outputs, days of output, daily averages, hourly averages, etc.,  which give a much better idea of what is actually happening.

Will wind energy on its own replace other forms of electricity generation?

Other considerations for developing an informed basis for developing an opinion could be factors that are unique to wind farming. Such as:

This is the final podcast episode that was recorded on our trip in Europe. Future episodes will be videoed in the classroom.

Podcast Location:

South Lanarkshire wind farm. You’ll notice that the Google maps photo doesn’t show the wind turbines; the photo must have been taken before they were constructed. If you check the Street View, you will see them.
View Lochhead (Wind) Farm, South Lanarkshire in a larger map

So Where’s the Math?

Teaching about environmental issues is a great opportunity to promote socially responsible outcomes in your students’ lives. It is also a wonderful opportunity to put those mathematics skills to really practical use. Environmental projects always produce controversy, because they always involve costs to someone or other, along with the benefits. In order to reach sensible, informed conclusions, we should return to the data, and ask “What are the facts?”. Your students are ready to answer questions of concern to them, that are matched to their level of maturity. Use math to help them decide!

With your class you could do some activities such as actually planning a family’s daily needs based on the output from turbines on different days. How is industry affected? What is the daily consumption, or even the hourly consumption of power of a city? Are its needs steady or fluctuating? How does this affect the power industry? With these ideas it would be possible to follow through on an idea and make a great math project where the application of math is almost limitless.

The Wind Farm Performance site has lots of real-time stats and graphs on Wind Farms:

What do you teach in environmental studies that could use some reality in the form of data? I’d love to hear your ideas – please leave a comment below.

Related Videos

Explanation of the components of a wind turbine:

Turning the Weather Into Power (article + Flash animation) – click image for link:

Danish wind turbine fails in storm:

Wind energy in west Texas, Wind Turbines:

Wind Farms in the Media This Week:

Photo Credit

Children Recycling: © iStockphoto.com/Randy Plett