Posts Tagged ‘math’


What’s your history of classroom tech?

The racket of IBM Selectrics in typing class? The square holes of data cards? (I can’t even type phrase that without hearing it in my Indian college professor’s voice.) Did you – or if you’re old enough, your parents – collect grocery receipts for Apple IIe computers? Mario Typing? Channel One?

It’s a history of promises and utopia, isn’t it? Promises of:

  • Individualized, self-paced learning
  • Global connections and awareness
  • Classroom and system efficiency
  • Parent-school cooperation
  • Financial savings
  • Preparedness for the workplace and the modern world in general.

How much has been written on these issues? Millions of words. I don’t want to add too much to that. What I have to say is particularly directed at Catholic education, which, in this country, has – not surprisingly – jumped right on the Tech Train, seeking, as it does it so much else, to do nothing more than ape public education.

The lack of critical, counter-cultural thinking in Catholic education is not surprising, but still continually disappointing nonetheless.

What I have to say today is pretty simple, and I’m going to say it mostly by quoting from others. I’d encourage you to follow the links and read more.

Bottom line:

The push for screens and internet-based learning to replace books and paper is sold to us as an inevitability that is, of course, best for students.

I invite you to never, ever, accept that premise, and to question it, from top to bottom every time it’s presented to you. 

Because the push for screens and internet-based learning is not about students. It’s about profit and data. 

Let’s go back to the dark ages – the 1990’s, when Channel One entered classrooms. It’s an instructive example because it was so controversial at the time, and what’s happening now with computer-based learning, particularly Google Classroom, raises similar issues, but does not seem to be raising the same kinds of questions.

The deal was this: Whittle Communications provided classroom televisions and satellite receivers to schools in exchange for schools having their students watch a daily news show provided by the company – called Channel One.  I taught in a school that took this deal, and yes, every day after opening prayer and the Pledge of Allegiance, we had to watch this – what, maybe 10-15 minute news show with advertisements (that was the controversial part).

As I said – it was controversial – accusations of schools selling out students, forcing them to watch advertisements and whatever editorial slant Channel One offered in its programming.

And those are questions that should have been raised. It was certainly problematic – not to speak of being a pain and an intrusion. But hey! We got free televisions!

What’s happening now is no different – well, it is different – because it’s worse. What’s happening now in so many systems is an unquestioning, eager acceptance of faulty premises about what’s best for students, allowing tech companies to simply take over education, set the standards, and dominate every aspect of the process from pre-assessment, to instruction, to testing to information infrastructure.

And all the while scraping your kids’ lives for data.

Tomorrow I’m going to take the issue on from a more personal perspective, ranting about sharing various experiences my own kids have had with this in their classrooms – amy-welbornsince I haven’t taught myself since the advent of intensive, intrusive classroom tech – in my day  – it was a big deal to get one classroom computer, period.

Today, I’m just going to leave you with some citations from other writers. I don’t agree with everything every one of these writers have to say about every issue – some of them tend to tilt definitely more leftward than I do, and many are hard-core opposed to charter and private schools – but on these matters, I’m indebted to their passion.

Here’s where I stand – before I get to the links:

  • Education – even up through high school – should be as screen-free as possible. I really don’t see any reason at all for elementary students to use computers or screens. Their brains need the holistic connection between mental and physical activity that comes with reading real books and writing on paper and using concrete manipulatives.
  • Everything I have read indicates that reading retention is stronger from reading from printed paper pages than it is reading from a screen. There is an aspect to spatial awareness that assists in retention. I know this is true for me because I can often, when trying to remember something I read, can retrieve it by thinking about where I read it on the page – top, bottom, middle. I don’t have that with a screen, which is why the only books I read on an e-reader are out-of-print books I can’t find anywhere else, for the most part. For sure, if I am reading non-fiction – serious history or theology – I must read a book – I must be able to have that experience of holding something physical in my hands, flipping back and forth, physically highlighting.
  • Anecdotal evidence suggests that students tend to prefer printed material, as well: “real books.” There are serious questions, as well, about the physical impact of all-day screen immersion, not only on brain chemistry and attention, but on other aspects of our physical health.
  • As the links below will emphasize, this is mostly about perceived financial savings (by schools and systems) and financial gain (by corporations). It is disappointing, as I said, to see Catholic schools buy into this.
  • Classroom tech does not improve efficiency. At all. It takes time to learn, there are bugs and disruptions, the Internet goes out, the power goes out, it presents distractions.
  • While there are great teachers out there, teachers as a group are not noble saints immune from human weakness. As I said, great teachers still work out there – my children are sitting in the classrooms of some excellent teachers as I write this. But – again – teachers are not saints or superhuman or uniformly excellent. There are lazy, inefficient, ignorant teachers whose worst habits are encouraged by classroom tech. I mean – who among us hasn’t encountered the teacher who does nothing more than hand out worksheets? Now he/she has a classroom full of kids who can be told to work in their Chromebooks and call it a day.

So yeah – basically, for me, it comes down to : The tech needed in a classroom is going to vary: kids studying AP Physics might need to use it more than they do in English class (or maybe not – I sort of have my doubts on that score, too). But the presumption should be: less tech is better. 

Now for the links:

Dear teachers: Don’t be good soldiers for the tech industry

There is an entire parasitic industry making billions of dollars selling us things we don’t need – standardized testsCommon Core workbook drivel, software test prep THIS, and computer test crap THAT.

We didn’t decide to use it. We didn’t buy it. But who is it who actually introduces most of this garbage in the classroom?

That’s right. US.

We do it. Often willingly.

We need to stop.

And before someone calls me a luddite, let me explain. I’m not saying technology is bad. It’s a tool like anything else. There are plenty of ways to use it to advance student learning. But the things we’re being asked to do… You know in your heart that they aren’t in the best interests of children.

I know. Some of you have no choice. You live in a state or district where teacher autonomy is a pathetic joke. There are ways to fight that, but they’re probably not in the classroom.

It’s not you who I’m talking to. I’m addressing everyone else. I’m talking to all the teachers out there who DO have some modicum of control over their own classrooms and who are told by their administrators to do things that they honestly disagree with – but they do it anyway.

We’ve got to stop doing it.

Corporations want to replace us with software packages. They want to create a world where kids sit in front of computers or iPads or some other devices for hours at a time doing endless test prep. You know it’s true because your administrator probably is telling you to proctor such rubbish in your own classroom so many hours a week. I know MINE is.


The EdTech shell game is not about improving student learning. It’s a commercial coup, not a progressive renaissance.

Think about it.

They call this trash “personalized learning.” How can it really be personalized if kids do the same exercises just at different rates? How is it personalized if it’s standardized? How is it personalized if it omits the presence of actual people in the education process?

It’s teach-by-numbers, correspondence school guano with graphics and a high speed Internet connection.

Personalized Learning Without People – An Education Scam from the 1980s Returns

This is seen as a way to save money by teaching without teachers. Sure, you still need a certified educator in the class room (for now) but you can stuff even more children into the seats when the teacher is only a proctor and not responsible for actually presenting the material.

The teacher becomes more of a policeman. It’s his job to make sure students are dutifully pressing buttons, paying attention and not falling asleep.

Moreover, this is sold as a way to boost test scores and meet the requirements of the Common Core. You can easily point to exactly which standards are being assessed on a given day and then extrapolate to how much that will increase struggling students’ scores on the federally mandated standardized test when they take it later in the year.

In fact, students’ answers on these programs are kept and recorded. They are, in effect, stealth assessments that can be used to judge and sort students into remediation classes or academic tracks.

Co-opting the Language of Authentic Education: The Competency Based Education Cuckoo

That’s what the whole program consists of – forcing children to sit in front of computers all day at school to take unending high stakes mini-tests. And somehow this is being sold as a reduction in testing when it’s exactly the opposite.

This new initiative is seen by many corporate school reformers as the brave new world of education policy. The public has soundly rejected standardized tests and Common Core. So this is the corporate response, a scheme they privately call stealth assessments. Students will take high stakes tests without even knowing they are doing it. They’ll be asked the same kinds of multiple-choice nonsense you’d find on state mandated standardized assessments but programmers will make it look like a game. The results will still be used to label schools “failing” regardless of how under-resourced they are or how students are suffering the effects of poverty. Mountains of data will still be collected on your children and sold to commercial interests to better market their products.

On “Competency-Based Education” ….and B.F. Skinner:

Parents, here’s the moral of the story: if you want your child “constantly interacting” with whatever corporate testing company your state has contracted with, and if you trust that company to be your child’s teacher, then by all means, CBE is for you.

And then a general rant from a couple of years ago – with good links. 

Dr. Kentaro Toyama, an associate professor at the University of Michigan’s School of Information, once believed that technology in the classroom could solve the problems of modern urban education. No Luddite, he had received his Ph.D. in computer science from Yale and had moved to India in 2004 to help found a new research lab for Microsoft; while there, he became interested in how computers, mobile phones and other technologies could help educate India’s billion-plus population.

Rather than finding a digital educational cure, he came to understand what he calls technology’s “Law of Amplification”: technology could help education where it’s already doing well, but it does little for mediocre educational systems. Worse, in dysfunctional schools, it “can cause outright harm.” He added: “Unfortunately, there is no technological fix…more technology only magnifies socioeconomic disparities, and the only way to avoid that is non-technological.”

From Ed Week:

What this discussion boils down to is a concern about student learning and a skepticism regarding the idea that technology is always necessary or appropriate. New tech tools might promote engagement, but students might also enjoy colorful pens and giant pieces of chart paper as a change of pace in environments that are proudly, and rigidly, paperless. Virtual discussion boards might be crucial for drawing out introverted students; they might also give students permission to sit back and type canned responses.

In his 2003 book The Flickering Mind, author Todd Oppenheimer argued that education technology had failed in its promise to transform education and that it may paradoxically impede learning. Oppenheimer, a journalist who visited a range of schools and institutions in the United States to examine how technology was shaping education, found that educators often conflated sleek but content-thin presentations with evidence of deep learning.

Educators also erroneously assumed that the use of tools like PowerPoint counted as relevant skill-building for the workplace. Oppenheimer suggests in the book that students are more likely to prosper if they develop “strong values and work habits,” and master “the art of discussion.”


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I am finally getting myself together here for Melanie Bettinelli’s linkup on “Learning Notes.”  Melanie is a homeschooling mother of many who writes a very fine blog. Check out the series she’s doing on Shakespeare with kids, for example.   There are innumerable ways and styles of homeschooling, and if you are curious about this growing phenomenon and want to understand its appeal to families, I think Melanie’s blog is a great place to start.  Conversations, creating, exploring with people who love you? …the best kind of education, for sure.

My first foray into this linkup isn’t going to be a day-by-day account this time because I’ve got math on the mind these days.

For a non-mathematician, I think a lot about math, and this blog post has finally spurred me to put down some thoughts on it.


I come from an academic, humanties-centered household.  There was no mathiness or science or business-type activity to speak of in my parents’ lives or in their parents’ lives. My mother  joked about her wildly contrasting verbal and math scores on the GRE. I did fine in math in high school, took the minimum I needed to in college, and that was it.  I had no opinion of it one way or the other.  I certainly had to work hard and think things through in the higher math (the highest I got was what they called “Advanced Math” in the day – maybe there was a bit of Pre-Calculus in it, and a little trig, but I never even attempted calculus.  I don’t think the school offered it, come to think of it.), but I often had the weird experience of hitting a wall at night when I was doing my homework, then waking up the next morning, saying “Ah-ha!” – my brain evidently having worked it all out when I was sleeping.

As a parent, I’ve had one older kid who needed help in math, but the other two breezed through on their own, doing very well. My second son never studied in high school and made straight A’s, even in Calculus.  Daughter had to study, but still did well, and liked it – “Math is like a puzzle to me, and I love puzzles” is what she’s always said.  And now she’s studying for the LSAT which she was emboldened to do, not just because she took a Civil Rights/Liberties class and really enjoyed doing case analyses, but also because she looked into what the LSAT is and joyfully discovered, “It’s LOGIC!” So.

And then, for the others…. it was time to homeschool.

Math is something that some non-mathy homeschooling parents dread, but I never have, mostly because I picked a program that I found easy and even interesting to work with – from The Art of Problem Solving.  I’ve written about this program before, so I won’t repeat myself.  I’ll just say that Joseph worked through the Pre-Algebra text last year and is making “A’s” in Algebra in 8th grade right now.  He never minded it too much, and neither did I – in fact, in many ways, I found it illuminating.  Plus I love the videos.  There, I admitted it.

Now, I have a theory about teaching.  I actually think that people who are a “natural” at a subject don’t necessarily make the best teachers of that subject.  Think about it – if you have an intuitive grasp of a topic or skill, it might be a challenge for you to communicate the process to someone who doesn’t have a clue.  On the other hand, if you’ve had to work through a process step-by-step and have actually struggled with various aspects of it…you might just be a really effective teacher to the equally clueless.

All that is to not to say that I’m a fabulous math teacher.  But it is to say that I’m not a bad one – at least to my own children –  and I think it’s because I understand their lack of understanding.

Anyway, math is not only on my mind these days, it’s on the mind of many because of Common Core-related issues.  I’ll say straight up that I’m (not surprisingly) opposed to Common Core simply because I’m opposed to all federal standards in educational content, period, without exception and also because I believe that the push for Common Core is primarily profit-driven.  As I’ve said before, no one makes money when teachers are using five-year old textbooks using methods they’re familiar with.  People make money when new textbooks must be written and printed, when workshops on new pedagogies must be paid for, when consultants must be consulted and when – above all – children must be tested.

But what has gotten folks riled up above all is the content of the standards, especially in math.  I saw a bit of this in the text Joseph was using in his old school, and which we used in the first year of homeschooling (because at that point we weren’t sure if he would be returning to school after our fall in Europe…just in case he was, he needed to be on track.) I rather liked the text because it invited the student to look at problems in a number of different ways and introduced various problem-solving strategies, but I could see how it could be confusing.

(My problem, though, with how this is shaking out in schools is this: I think the various strategies should be introduced.  What I don’t think is right is then tying “success” of a child – and by extension, a teacher and a school – to that child’s mastery of all of the strategies.  It’s terribly confusing and really confounds the purpose of introducing various strategies, doesn’t it?)

So now, to the present. With the Art of Problem Solving and the curriculum which my younger son is using from the same group, Beast Academywe are encountering “new” strategies. That is, they are new to us, all of us having been taught more or less “traditional” math, even if it has been 40-45 years apart.

And here’s the thing.

They’re so much better. 

They make sense.  They are, as far as I can tell from my limited perspective, truer reflections of what is going on with the numbers with more explanatory power than anything I was taught, which was mostly about learning rules and formulas and plugging in the numbers and doing the computations, period.

I’m going to start with a simple example.

(Caveat – I’m only going to say this once, but it applies to every example.  You may have learned this stuff during math.  Maybe I was taught it, too.  But I don’t think I was, and if I was, it didn’t stick.)

When my older son started PreAlgebra with AOPS, he re-learned a lot about basic arithmetic operations.  It seemed, at first glance, kind of silly, but it wasn’t because, as we soon discovered, it really helps to understand exactly what these basic operations are.  So take division.  What is division?  Well, division is a few things, I suppose, but one of the things division is is simply multiplying by the reciprocal of a number.   So…20 divided by five is also 20 times one-fifth.  Right? So there’s your definition of division:  Multiplying by the reciprocal.

Now. Flash back to..I don’t know.  Fourth, fifth grade math.  When you were taught how to multiply and divide fractions.  Multiplying: easy.   Just multiply straight across.  But dividing?  Ooooh…tricky.  You had to remember that weird thing you had to do – you had to flip the divisor and multiply by the resulting reciprocal.  I don’t know about you, but I never understood why you did that.  Why do we have to do that?  Who knows? It’s a rule!

But hey….isn’t that what division is? Isn’t that the definition?

"beast academy"So when you have to divide fractions, you multiply by the reciprocal…because that’s what division is.

My point is – this was taught to me as a rule with no theoretical foundation.  I probably would have had an easier time remembering it if I’d been taught the reasoning behind it in this really very simple way.

Properties were another thing.  Every year we’d be taught those blasted properties, and never did any of them except the Commutative (because that’s easy) make sense to me.  I had to relearn it every year, and barely did so, because the properties were presented as one little section in one chapter and then essentially neglected, probably until Algebra.

In these AOPS books, students are taught the properties early on, and they use them..constantly.  Multiplication and Division are taught within the framework of the Distributive property – basically, they are taught to break the numbers apart in order to both more easily mentally compute, but also to understand, once again, the operations from the ground up.  And really, this is something a lot of us do anyway, right?  I know I do, and always have – if I have to compute, say, 78 times 6 in my head, I do so by breaking it up into 70 times 6 plus 8 times 6.  It’s just that I never knew what I was doing.

SO.  Finally.  Back to this blog post – in which the author says that the way kids are taught to do multiplication  – the algorithm (or system) – undercuts their understanding of place value. 

If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.

But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.

Ding-ding-ding.  As a non-mathematician, I am in total agreement.

So now back to my issue.  Have you ever tried to explain 2 and 3 (and more) digit multiplication to a kid?  And what “carrying the ones” means?  And keep any sense of place value?  I mean..try it.  Right now.   Explain why you do those things to an imaginary (or real) nine year old.

It works, sure. You get the right answer.  And there’s a reason it works.  But now….let me tell you about how Michael is learning multiplication of 2, 3 and more digit numbers..  It may not be new or radical to you…perhaps it’s being incorporated in some of the other new math materials out there.   But it’s new to me, and I’ll admit when he first started, I got nervous.  I was thinking, “Wait. This isn’t the way I was taught.  I mean, I don’t really understand the way I was taught..but this is different! I don’t think it’s what they’re doing in regular school.  WILL HE BE AN OUTCAST?”

Well, not really on that last part. So let’s go to the photos:

"amy welborn"

That’s how I learned it.  You, too, probably.  Again, imagine explaining to a kid why you carry the 1 and then the 3 and why you put a zero in the units place on that second line. Try.

Now here how Michael’s learning.

"amy welborn"

Do you see? It’s the Distributive property, in action.

In case you  don’t – it’s (3X5) + (3X40) + (70 X 5) + (70 X 40). It’s an accurate, clearly laid-out expression of what is happening in the act of “multiplying” these numbers.

The beauty of it is that if you can do part of it your head –  if you know that 45 X 3 is 135 right off the bat – feel free to just put it down that way. Doesn’t mess anything up.

"amy welborn"

This makes so much more sense.  To me, a non-math person. Yes, it takes up a bit more space on the paper, but it preserves the sense of what the numbers are and what is going on in the act of multiplication.  In other words, it’s not just a “rule” but a clear process.

Has this been the dullest blog post ever on my blog(s)?  Probably.  But at least I got it out of my system.

The examples of “Common Core” math that I have seen do, indeed seem unnecessarily complicated and frankly convoluted.  I think the intention is to encourage a deeper “number sense,” but they end up confusing instead.  My point is that what I have encountered in the AOPS programs has certainly been new to me, but as not-mathy person I haven’t found them confusing at all,but rather illuminating and quite interesting.  There is a way of teaching a way of doing math that is a more accurate expression of what is going on and which doesn’t seem so random, especially to the non-mathy person. The tragedy is that a worthy end is being massively screwed up and, as a consequence, raising suspicions against any attempt to develop better ways to teach our children math, better ways that are out there and that are not crazy or needlessly confusing – in fact, are the opposite.

Below are some of 9-year old Michael’s math pages from last week. You’ll probably have to click on them to get a better view.

"amy welborn"This was the first workbook page on which he had to work with this new algorithm.  Robots optional.

DSCN4447On this page, he was given just a few numbers of each problem and had to work out the rest.  So, for example in #144, he would have to work out what do you multiply 6 by that gives you a number with 8 in the units digit..well, it could be 3 or it could be 8..so you have to go from there and figure it out.  We left the last one to do as review later.

"amy welborn"

He started exponents late last week.  On this page he had to work out where to put parentheses so the equation would work.  If it worked without parentheses, circle it.  (Obviously it was also an exercise in understanding Order of Operations.)

The way that Beast Academy is planned (they haven’t finished all the books yet…) the student will be ready for Pre-Algebra after competing level 5 (this is 4C, with one more to go in the 4th level) – I had my doubts when I heard that, but as we go on…I can see it.  Michael is going to have a completely different, deeper understanding of math than any of his siblings..and it will be better, I have no doubt.

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