The trouble with most so-called “educational software” for teaching math is that is was designed by a programmer or a business owner, not an educator. As a result, it was designed to connect questions with answers really fast, in the mistaken belief that if you could just automate the testing of students on matching question and answer, they would learn faster and better.
Yeah, so what?
Trouble is, most of the math adults use in real problem solving involves figuring out what math is needed, finding the best tools to use, then applying them and evaluating the result. If the chosen tools do not produce a solution, then alternative pathways are tried, until the solution is reached.
Very little actual use of mathematics involves memorizing and single fact in response to a simple question, which is exactly what is targeted by most math software.
A New Pedagogy: Provide Tools to Support Thinking
Let me introduce a new set of interactive, cross-platform teaching and learning tools, which we call Professor Pete’s Gadgets.
Professor Pete’s Gadgets are designed to interactively support students’ development of mathematical thinking, by presenting them with symbols, pictures and words which they can manipulate to understand how they relate to each other. It is a fantastic teaching tool that, combined with well sequenced lessons and worksheets, allow students to understand the math concept.
For example, they can see how common fractions, percentages, decimal fractions and ratios relate to each other, what they look like in a pictorial form and where they sit on a number line between 0 and 1:
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:
Process whatever is in brackets first (applying later rules if necessary)
Apply “other” operations such as indices/powers or square roots.
Apply multiplication and division, in order as they appear from left to right.
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?
This week, my wife and I are in South Korea to visit family.
I have noticed when visiting museums that I am often able to use math to make sense of information that is presented in Korean, a language I don’t speak.
For example, we visited Gyeongbokgung, the Korean royal palace from the 14th Century, in Seoul. In the museum next to the palace grounds was a large wall chart in Korean, showing the Royal family tree up to 1910, when Japan annexed Korea and the royal family was dissolved. Look at this section (click for a larger image):
Korean Royal Family Tree 1790-1863
Unless you read Korean script, at first glance this will appear almost unintelligible. But with a little deduction, there is actually a lot of information which may be gleaned from the chart.
Click the photo above for a larger image. Each box has a pair of dates under it, noting the years of birth and death of that person. What can you learn about the Korean royal family in the 19th Century? The detail that is available to us using quite elementary math is surprising.
The study of history gives an excellent opportunity to put to use mathematics skills to learn more about the people being studied. Here are just a few pieces of information deduced merely from the dates provided and the graphical layout of the information:
Looking down the left-hand vertical column, birth dates are 1790 (the King), 1789, 1785, 1793, 1788. These people were thus of the same generation, since their birth dates were only a few years apart.
The six people in the left-hand column are in pairs, and each pair is linked by a line to an individual in a column to the right – obviously a couple’s child.
The person represented by the dark brown box was born when his parents were 19 and 20 years old, married someone approximately a year older, and they had a child when they were 18 and 19 years old.
The child became the 24th king at age 7, and reigned until the age of 22. In fact, the throne had skipped a generation, and the 24th king took the throne in the place of his grandfather, his father having already passed away 4 years earlier. His mother, on the other hand, outlived the king by 41 years. His grandmother also outlived him, and was 45 years old when he took the throne.
I wrote previously about using math in a visit to a graveyard to learn about the history of the people interred there. It is quite amazing how much one can learn using math in a similar context, even when the rest of the text is in a foreign language!
If you would like to use other photos from this museum to give students an experience of finding information using math, download the accompanying zip file containing 5 photos:
How do you use math in history classes? Feel free to leave a comment below.
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):
At first glance it is a fairly typical childish drawing of a teacher and a student, drawn by a girl who loves her teacher.
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.
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 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 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.
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 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’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.
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.
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.
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.
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!
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.
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!
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.
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”.
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.
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.
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.
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
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.