Helping the community with educational technology

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Many people associated with educational technology are driven by a passion for helping students learn using technology in a classroom setting. But I wonder if many ed tech people — either researchers or rank-and-file teachers who teach with technology — ever consider a slightly different role, voiced here by Seymour Papert:

Many education reforms failed because parents did not understand or could not accept what their children were doing. Remember the New Math? This time there will be many who have not had the personal experience necessary to appreciate fully the multiple ways in which digital media can augment intellectual productivity. The people who do can make a major contribution to the success of the new initiative by helping others in their communities understand the potential. And being helpful will do much more than improve the uses of the computers. The computers could be a catalyst for turning our communities into “learning communities.”

So true. So much of education falls to the immediate family, and yet often there are technological innovations in the classroom which fail to be supported at home for the simple reason that parents and other family members don’t understand the technology. Ed tech people can make a real impact by simply turning their talents toward this issue.

Question for you all in the comments: How? It seems that the ways that ed tech people use to communicate their thoughts are exactly the ones off the radar screen of the people who need the  most help — Twitter, blogs, conference talks, YouTube videos, etc. You would need to get on the level with the parent trying to help their kid in a medium that they, the parents, understand. How is that best done? Newsletters? Phone hotlines? Take-home fact and instruction sheets? Give me some ideas here.

(h/t The Daily Papert)

A binary notion of “understanding”

Another great insight from Seymour Papert, via The Daily Papert blog. I put it up on my Posterous blog this morning but I thought it could go here too:

Many children who have trouble understanding mathematics also have a hopelessly deficient model of what mathematical understanding is like. Particularly bad are models which expect understanding to come in a flash, all at once, ready made. This binary model is expressed by the fact that the child will admit the existence of only two states of knowledge often expressed by “I get it” and “I don’t get it.” They lack—and even resist—a model of understanding something through a process of additions, refinements, debugging and so on. These children’s way of thinking about learning is clearly disastrously antithetical to learning any concept that cannot be acquired in one bite.

(Papert, S. (1971) Teaching Children Thinking. In Contemporary Issues in Technology and Teacher Education, 5(3/4), 353-365.)

And on the higher education end of the spectrum, all of the things that really matter are those things that take patience, time, and persistence to acquire. But these are the very things excluded by this binary notion of understanding in which many children are immersed and to which most college freshmen are completely habituated.

This also makes a good argument for insisting that students — particularly those in the STEM disciplines, but I would argue anybody — should learn computer programming as part of their studies. You cannot learn to program without engaging in the non-binary notion of understanding Papert is describing. Papert knows a thing or two about that subject.

(By the way, I must give Gary Stager many thanks for running The Daily Papert. It is a great resource. The month of March is ridiculously busy for me but once it’s over I am going on a massive Papert reading spree.)

Computers, the Internet, and the Human Touch

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This article first appeared earlier this week on the group blog Education Debate at OnlineSchools.org. I’m one of the guest bloggers over there now and will be contributing articles 1–2 times a month. I’ll be cross-posting those articles a couple of days after they appear. You’d enjoy going to Education Debate for a lively and diverse group of bloggers covering all kinds of educational issues.

It used to be that in order to educate more than a handful of people at the same time, schools had to herd them into large lecture halls and utilize the skills of lecturers to transmit information to them. Education and school became synonymous in this way. Lectures, syllabi, assessments, and other instruments of education were the tightly-held property of the universities.

But that’s changing. Thanks to advancements in media and internet technology over the past decade, it is simpler than ever today to package and publish the raw informational content of a course to the internet, making the Web in effect a lecture hall for the world. We now have projects such as MIT OpenCourseWare, Khan Academy, and countless initiatives for online education at US colleges and universities providing high-quality materials online, for free, to whomever wants them. It brings up a sometimes-disturbing question among educators: If students can get all this stuff online for free, what are classrooms and instructors for?

Tech author Randall Stross attempts to examine this question in his New York Times article “Online Courses, Still Lacking that Third Dimension”. In the article, Stross mentions “hybrid” courses — courses with both online and in-person components — but focuses mainly on self-contained courses done entirely online with no live human interaction. He correctly points out that learning is an inherently human activity, and technologically-enhanced coursework is successful insofar as it retains that “human touch”.

However, Stross casts the relationship between computer-enabled courses and traditional courses as a kind of zero-sum game, wherein an increased computer presence results in a decreased human presence. He refers to universities “adopting the technology that renders human instructors obsolete.” But it’s not the technology itself that makes instructors obsolete; it’s the adoption of practices of using that technology that does. There are numerous instances of traditional college courses using computing and internet tools to affect positive change in the learning culture of the institution. There are also plenty of cases, as Stross points out, where technology has replaced human instructors. The difference is an administrative one, not a technological one.

Nor is the supposed obsolescence of the instructor all technology’s fault. If universities and individual professors continue to hold on to a conception of “teaching” that equates to “mass communication” — using the classroom only to lecture and transmit information and nothing else — then both university and instructor are obsolete already, no technology necessary. They are obsolete because the college graduate of the 21st century does not need more information in his or her head to solve the problems that will press upon them in the next five or ten years. Instead, they need creativity, problem-solving experience, and high-order cognitive processing skills. A college experience based on sitting through lectures and working homework does not deliver on this point. The college classroom cannot, any longer, be about lecturing if it is to remain relevant.

And notice that an entirely self-contained online course can be as “traditional” as the driest traditional lecture course attended in person if it’s only a YouTube playlist of lectures. What matters regarding the effectiveness of a course isn’t the technology that is or is not being used. Instead it’s the assumptions about teaching and learning held by the colleges and instructors that matter, and their choices in translating those assumptions to an actual class that students pay for.

What we should be doing instead of choosing sides between computers and humans is finding ways to leverage the power of computers and the internet to enhance the human element in learning. There are several places where this is already happening:

• Livemocha is a website that combines quality multimedia content with social networking to help people learn languages. Users can watch and listen to language content that would normally find its place in a classroom lecture and then interact with native speakers from around the world to get feedback on their performance.
• Socrait, a system proposed by Maria Andersen, would provide personalized Socratic questions keyed to specific content areas by way of a “Learn This” button appended to existing web content, much like the “Like This” button for sharing content on Facebook. Clicking the button would bring the user to an interface to help the user learn the content, and the system contains social components such as identifying friends who also chose to learn the topic.
• I would offer my own experiments with the inverted classroom model of instruction as an imperfect but promising example as well. Research suggests this model can provide in significant gains in student learning versus the traditional approach to teaching by simply switching the contexts of lecture and activity, with lecture being delivered via video podcasts accessed outside of class and class time spent on problem-based learning activities in teams.

Rather than view college course structure as a pie divided into a computer piece and a human piece, and fret about the human piece becoming too small, let’s examine ways to use computers to enhance human learning. If we keep thinking of computers as a threat rather than an aid to human interaction, computer-assisted instruction will continue to lack the human touch, the human touch will continue to lack the power and resources of computers and the internet, and student learning will suffer. But if we get creative, the college learning experience could be in for a renaissance.

Another thought from Papert

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Like I said yesterday, I’m reading through Seymour Papert’s Mindstorms: Children, Computers, and Powerful Ideas right now. It is full of potent ideas about education that are reverberating in my brain as I read it. Here’s another quote from the chapter titled “Mathophobia: The Fear of Learning”:

Our children grow up in a culture permeated with the idea that there are “smart people” and “dumb people.” The social construction of the individual is as a bundle of aptitudes. There are people who are “good at math” and people who “can’t do math.” Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. As a result, children perceive failure as relegating them either to the group of “dumb people” or, more often, to a group of people “dumb at x” (where, as we have pointed, x often equals mathematics). Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated and reinforced throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable.

Haven’t all of us who teach seen this among the people in our classes? The culture in which our students grow up unnaturally, and incorrectly, breaks people into “good at math” or “bad at math”, and students who don’t have consistent, lifelong success will put themselves in the second camp, never to break out unless some “exceptional event” takes place. Surely each person has real limitations — I, for example, will never be on the roster of an NFL team, no matter how much I believe in myself — but when you see what students are capable of doing when put into a rich intellectual environment that provides them with challenges and support to meet them, you can’t help but wonder how many of those “limitations” are self-inflicted and therefore illusory.

It seems to me that we teachers are in the business of crafting and delivering “exceptional events” in Papert’s sense.

Conrad Wolfram’s vision for mathematics education

A partial answer to the questions I brought up in the last post about what authentic mathematics consists of, and how we get students to learn it genuinely, might be found in this TED talk by Conrad Wolfram called “Teaching kids real math with computers”. It’s 17 minutes long, but take some time to watch the whole thing:

Profound stuff. Are we looking at the future of mathematics education in utero here?

Questions about an enVisionMATH worksheet (part 2)

Here’s another question about the same enVisionMATH worksheet we first met yesterday. Take a look at this section, and think about the mental processes you’d use to answer each of these problems:

Got it? Now, let me zoom out a little and show you a part of the worksheet you didn’t see before:

If you’re late to the party and don’t know what’s meant by “near doubles” and the arithmetic rules that enVisionMATH attaches to near doubles, read this post first. Questions:

• Now that you know that these are supposed to be exercises about near doubles, does that change the mental processes you selected earlier for working the problems?
• Should it?

Questions about an enVisionMATH worksheet (part 1)

The 6-year old had Fall Break last week, so no homework and no enVisionMATH-blogging for me. Tonight, however, she brought home a new worksheet for her weekly homework, and a couple of things caught my eye. I thought I’d throw those out there to you all, along with a question or two, as a two-part blog post.

For the first post, take a look at this (click to enlarge):

Questions:

• In your own words, preferably those that a smart 6-year old could understand, what is the basic principle that this page is trying to get across?
• What technique does this worksheet want kids to use when doing the Algebra problems?
• What’s your opinion about the principle/technique you think the worksheet is trying to communciate? Reasonable? Natural? Likely to be useful, or used frequently later on?

 

More enVisionMATH: Adding “near doubles”

The last post about enVisionMATH and how I, as a math person and dad, go about trying to make sense of what my 6-year old brings home from first grade seems to have struck a chord among parents. The comments have been outstanding and there seems to be a real need for this kind of conversation. So I have a few more such posts coming up soon, starting with this one.

The 6-year old brought this home on Monday. Click to enlarge:

It’s about adding “near doubles”, like 3 + 4 or 2 + 3. In case you can’t read the top part or can’t enlarge the photo, here are the steps — yes, there are steps, and that’s kind of the point of this post — for adding near doubles:

1. “You can use a double to add a near double.” It gives: 4 + 5 and shows four blue balls and five green balls.
2. “First double the 4”. It shows 4 + 4 = 8, and the four blue balls, and four of the green balls with the extra green ball sort of falling to the ground.
3. Then it says: “4 + 5 is 4 + 4 and 1 more.” At this point you really have to look at the worksheet itself, because it’s hard to put into words what is going on:

And from there, in the fourth frame, one of the girls in the earlier frame concludes that 4 + 5 must be 9 because 8 and 1 more is 9.

The Guided Practice section has the kids doing four near-double sums. Clearly, the way the worksheet wants kids to learn how to do this is not simply to add 2 + 3, but (1) to recognize that 3 is 2 plus 1 more, (2) add 2 + 2, and (3) then add 1 to the result of 2 + 2:

There’s a thing at the bottom asking kids to explain the process and then a bunch of near-double sums to practice — presumably kids are supposed to use the method described above, but there’s nothing forcing them to do so — and some “algebra” questions with blanks in the place of variables.

I’m not sure exactly how my brain goes about adding near doubles — whether it just somehow does the addition in ways that are almost automatic thanks to 35 years of repetition, or whether there are little tricks it employs — but I am absolutely certain that  I don’t do it the way enVisionMATH is telling kids how to do it. I tried it. When I read the worksheet, I thought about near-doubles that aren’t so easy, like 121 and 122. Quick! Add those together. Did you think, “122 is 121 plus one more; 121 + 121 = 242; 242 and 1 more is 243” ? I didn’t — not by a long shot. I just added the numbers together. No methods, no tricks; just old-school addition. There may be some tricks that my brain invokes to “just add the numbers” — for example, I tend to visualize the two terms of the sum stacked atop each other in the classic vertical arrangement for adding, and then visually add the digits — but I am most definitely not going through the four-step process on this worksheet.

In fact, the four-step process complicates matters so much that it’s inexplicable why they are even bringing it up. Most kids at this stage can add 2 + 3 or 5 + 6 in one step. But by introducing this method, there are four operations: comparison (find the larger of the two near-doubles), subtraction (take 1 from the larger number), addition (add the two duplicates), and another addition (add 1 to the result). Technically there is a fifth operation kids have to perform, namely recognize that the two numbers they are adding are near-doubles in the first place.

One might argue that doubling a number (in the third step) is easier than adding it to itself — kids just recognize that doubling 5 gives 10, for instance — and subtracting 1 is a very easy special case of subtraction in general that nearly everybody at this age can do without thinking, similarly for adding 1 at the end. That may be so, but it can’t be so much easier that adding in steps 1, 2 and 4 results in a net reduction in complexity or a net gain in conceptual understanding.

But what about kids who can’t add two one-digit numbers together in one step? There are some of those out there, including probably a few in my daughter’s class. This method doesn’t help those kids. Again, we may argue that adding 4 + 5 is considerably harder than the combined process of comparison, subtracting 1 from 5, doubling 4, then adding 1. But I don’t think so. A four-step process is no less cognitively demanding than a single-step process, even if the four steps are easy. And besides, life does not throw near-doubles at you to add. How is a kid going to learn to add 2 + 5, or 2/5 + 7/8, or 123.38 and 99.99 this way?

If there is some research that suggests that people really do add near doubles this way, I would love to see it. Otherwise it’s hard for me to believe that any more than a tiny fraction of the human population actually does it this way. Is there going to be some mind-blowingly cool way to do complicated arithmetic in one’s head farther down the road that uses this idea, like multiplying numbers that are near-squares or something? Perhaps I should be more patient. But for the time being, I told the 6-year old just to add the numbers together like she already knows how to do, the old-fashioned way.

In the trenches with enVisionMATH

It’s been back-to-school time for everybody in our household (hence an excuse for the light posting). We started classes at the college today, and last week the 4.5-year old went back to preschool full-time and the 6.5-year old started first grade. (The 1.5-year old is rocking the local daycare.) One of the biggest changes for the kids is for our first-grader, Lucy, since she has real homework for the first time. It’s not much; the expectation is about 20 minutes a night, Monday through Thursday. Some of that homework is math, which I was very excited about — but then that excitement turned to alert caution when I learned my daughter’s class was using enVisionMATH.

I wrote this post on enVisionMATH almost three years ago, basically laughing it off the blogosphere for its happy-clappy, uncritical acceptance of unproven digital nativist frameworks and for going way over the top with smartboards. Little did I know that my own offspring would be in the middle of it just three years later. So, in an effort to process what she’s doing (for me, for her, and for anybody else who cares), this is the first of what might be many posts about the specifics of enVisionMATH, as viewed by a parent whose kid happens to be learning from that curriculum, and who also happens to be a mathematician and math teacher.

I’ll start with the worksheet Lucy brought home this evening, called “Making 8”:

I’ve never had a kid in first grade before, nor can I remember how I did this stuff in first grade, nor have I recently worked with a kid in first grade. So I’m going to share my thoughts, but realize I have no reference for what’s “normal” pedagogy for 6-year olds and what’s not.

This worksheet is really about subtraction, although it never comes out and says so. The first two exercises are attempting to build a sense about subtraction by getting kids to think about how parts fit together to form a particular quantity. enVisionMATH appears to be really big on getting kids to recognize numbers visually rather than by counting. I’ll need to blog about this in a later post, but Lucy’s had some other exercises that, for example, stress the ability to recognize this:

…as the number 6, just by looking at it and without counting the dots, almost to the point of telling kids that they shouldn’t be counting anything but rather arranging things into patterns. Again, that’s for another post.

So, back to the worksheet, kids are supposed to look at the first collection of balloons and, knowing that there are eight of them, see — and only “see” — that 8 splits into 2 plus 6, and then 4 plus 4. I did a few more of these with Lucy using coins (no balloons on hand, sadly). Biggest challenge here: Keeping Lucy from just counting the black balloons and then counting the white balloons. And the only reason this was a challenge was because, as a math person, I knew what the worksheet was getting at: recognizing quantities through visual patterns rather than counting, so the unwritten rule is for kids not to count the balloons. But other parents probably didn’t know this, and their kids just counted. I don’t think this is necessarily wrong, but it doesn’t necessarily help in the next sections either.

The next section is rather startlingly labelled “Algebra”. Remember: This is a worksheet for a first grade class. Why we are bringing up the word “algebra” at this point is anybody’s guess. I suspect this is more to make parents, school boards, and accreditors happy than it is to start getting kids to feel comfortable with the word “algebra”.  But anyway, as you can see, the two problems are just the first two problems in reverse.

Lucy had a hard time with this. First of all, she didn’t understand what “the whole” meant. This is not the first time Lucy’s struggled not with the math but with relatively esoteric vocabulary in her math lessons. Last week she had a worksheet where she was to arrange three integers “in order from greatest to least” and “from least to greatest” and we had to take a moment to figure out what all of that meant. Maybe other people’s kids don’t struggle with that, but on the other hand it’s been verified that Lucy is reading at a third or fourth grade level right now, so I wonder if it’s just her.

We had to work these out using manipulatives. We started with fingers because that’s the first thing I thought of. So, I said, if the whole is 8:

…and one part is 3:

…then what was the other part?

Lucy was able to get the answer of “5” with no problem. But… I don’t think she got it the right way. Because when we moved to the next problem and the “one part” was 1, for her, the other part was still 5! This was because when I held up one finger on my left hand this time, there were of course five fingers on the right hand. I tried holding up eight and wiggling one finger instead of putting the “one part” on one hand, but that just confused her. So, we went back to coins and built a “balloon diagram” like in the first two problems, and she got them just fine (and without counting).

I don’t think exercises 3 and 4 are bad problems necessarily, but I do think they came in here way too early. Perhaps I’m missing the context of the actual classroom interaction between Lucy and her teacher, but it would seem like a better idea to do as many exercises like 1 and 2 as possible before moving on to the “algebra”. After all, if you stick to positive integers, there are only seven ways to fill in the blanks __ + __ = 8. (And doing all seven might help kids discover the commutative property early on, which seems like a much more important thing to bring up than “algebra” in first grade.)

And then, it’s not clear to me that doing “algebra” is a better idea here than just doing straight-up subtraction.  What’s to be gained by saying “the whole is 8; one part is 3; the other part is ____” versus “What is 8 minus 3?” Again, maybe I’m out of touch, but subtraction is a fundamental skill that algebra builds upon; doing algebra before subtraction seems a little backwards to say the least. A kid who is comfortable with subtraction will be able to do these whole/part problems in a snap by using subtraction. A kid doing these “algebra” problems basically has to invent subtraction in order to do them, or else draw pictures of balloons and start counting. It feels like the curriculum is trying to be intentionally nontraditional here, just for the sake of doing things differently rather than because it works better.

Then we come to the “Journal” question, which is downright sophisticated: “The whole is 8. One part is 8. What is the other part?” Here we reach serious abstraction: You can’t draw balloons like in exercises 1 and 2, and in fact resorting to physical props is tricky.As Derek Bruff mentioned in a tweet about this earlier this evening, the use of the word “part” in conjunction with the quantity 0 is already sort of questionable. What does it even mean to say the “part” is 0? What “part”? I don’t see a “part”. The natural way of interpreting what a “part” is, is as a bunch of objects. If there are no objects present, then there really isn’t a “part”.

We had to resort to thinking not about objects but containers that hold the objects. I took two books sitting nearby. I took my eight coins and said: The whole is 8. One part is 2 — and put 2 coins on one of the books. What is the other part? — and put the remaining coins on the other book. Lucy got the right answer quickly, and she did so by looking back at exercise 1 with the balloons and noticing it was the same problem with different objects, which I thought was pretty smart. I’ll make an algebraist out of her yet! Then I repeated with one part being 1. Then I did it with one part being 6; then 7. Then I said, “The whole is 8; one part is eight.” — putting all eight coins on one book. “What’s the other part?” — showing her my empty hands and an empty book. “Zero,” she said right away.

For her, and maybe not just her, “zero” represents not a size of a part but a state of emptiness of a container — or perhaps the size of a set. It’s how much you see when nothing is there. To map the “zero” concept onto a concept of “part” that presupposes something is there just doesn’t make sense. If this sounds like the New Math, I think we’re barking up the right tree.

The “Tell how you know” was especially tough because it involves getting Lucy to talk about what she did, even though she’s doing it at a sort of visceral level, and then spell the words she needs to use — which is the other type of homework she has. I got her to say out loud what she was thinking, and then I had her say it back to me and then helped her spell the words.

So we made it through the worksheet, but there are a lot of questions in my mind about the pedagogical design of this stuff. And how in the world does this sort of thing work in a household where the parents don’t have the time, patience, interest, fluency, or comfort level in mathematics to sit down and work all this out with the kid?

Indiana teacher licensing changes now official

The sweeping set of teacher licensing changes for Indiana, which I first blogged about here last July, has officially been signed into law. Frankly, I’m surprised, on two levels.

First, although this proposal flew mainly under the radar in Indiana, it was quite polarizing. The public, especially parents of school-aged kids, seemed mainly to be in favor of the bill; while teachers, teacher unions, and university education professors were quite vocally against it. Usually something this divisive doesn’t make it to being signed into law, or else it gets gutted and compromised first. But I can’t find any changes that were made between the bill and the law. It looks like what we saw is what we will get.

Second, it was pretty clear if you scratched the surface of this bill that one of its reasons for being was to put Indiana in a position to get Race to the Top money from the Federal government. Once Indiana was declared out of the running for that money, I figured the bill would get dropped, or else gutted/compromised. But apparently not so.

There will be winners and losers as these changes are implemented. As I said back in July, probably the biggest losers will be the education departments at large universities, which are constructed for the sole purpose of preparing preservice teachers to fulfill the outgoing licensing requirements. Now that the pedagogy coursework requirements for education majors will be drastically reduced, so will the workloads of many of the profs in those departments, and one wonders what happens next. The smaller colleges, like mine, will be fine. Our education faculty are generalists by necessity, and most of our secondary education degrees — which will no longer exist — are just one or two courses shy of a content major anyway. The big winners in this are going to be:

• People who want to become teachers but lack the time, resources, or willpower to follow the traditional — and highly regimented and lengthy — coursework for an education degree. Many of these are students who come to my college wanting to get a degree in math or science and eventually find their way into teaching, and who walk away disappointed that preparing to become a teacher is an all-or-nothing proposition — you can’t just “pick up a teaching license” in a content area. You either choose to invest dozens of credit hours in education courses or you stay out of teaching. I will be very happy to tell all of my highly talented math and engineering students that as of today, if you want to become a teacher, you can.
• Indiana college students, who now have more career options open to them. College students who trained to become teachers but who later want to leave the profession for something else will have a content degree to fall back upon. Those with, or who are working on, content degrees won’t have to make the all-or-nothing choice I mentioned above; if they decide later in their degree program to become teachers, they can.
• Indiana school kids, especially high school kids who are now guaranteed to have teachers who will now be just as proficient in their subject areas as a beginning practitioner of the discipline working in business, industry, or government or going to graduate school. We all realize that content competence (if not mastery) is not a sufficient condition for good teaching; but it is a necessary condition, and far too often that condition is not met. No longer!

This is a big net win for Indiana.