Tag Archives: Curriculum

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.

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Filed under Early education, Education, Educational technology, Higher ed, Inverted classroom, Student culture, Teaching, Technology

Monday discussion thread: Business curricula

It’s been a packed last couple of weeks, leaving me picking up the pieces and trying to clear some of my grading backlog before the Thanksgiving break. Rather than leave the blog alone for another week, let’s try an open thread based around something that’s come to mind just now, namely: Business degrees and the pedagogy used in the curricula for those degrees.

The usual way business courses play out is the usual way any set of courses plays out: You have a sequence of classes on various topics, the early ones being mainly theoretical or overview courses, maybe not in the business department at all. (For example, all business majors at my school have to take Calculus, which is why I am thinking about this in the first place.) The classes get more specialized and, usually, harder as you climb the ladder. Eventually you get to the top of the degree program and have a “seminar” class that is project-based, usually involving case studies.

So, for your discussion, consider this idea: Business degrees should not be conducted in this way. Instead, EVERY course should be project-based, beginning with the first semester of the freshman year and continuing on. (For the sake of argument, restrict your attention just to courses in the business department, not outside classes like calculus.) You can have a syllabus of basic learning outcomes for business majors if you like, and maybe some way of assessing student acquisition of those outcomes prior to graduation, but EVERY business class should be predicated on project-based learning — and let students go learn the theory on their own, with professor guidance if necessary, if and only if that theory has something to say about the project they’re working on. Courses based on imparting “material” through lectures or lab assignments disconnected from the context of a specific problem would be eliminated.

I’m not saying I am in favor of this. It’s just a provocative alternative. Discuss.

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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?

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Should everyone go to college?

I’m reading through a number of books and articles related to the scholarship of teaching and learning this summer. One that I read recently was this article (PDF), “Connecting Beliefs with Research on Effective Undergraduate Education” by Ross Miller. There are lots of good points, and teaching tips, in this article. But Miller makes one assertion that doesn’t seem right. He brings up the point, under the general heading of “beliefs”, that “questions arise, both on and off campuses, about whether all students can learn at the college level and whether everyone should attend college” [Miller’s emphases]. As to the “should” part of that question, Miller says:

According to Carnevale (2000), from 1998 to 2008, 14.1 million new jobs will require a bachelor’s degree or some form of postsecondary education—more than double those requiring high school level skills or below. Given those data, it makes sense to encourage all students to continue their education past high school. Consistent high expectations for all students to take a challenging high school curriculum and prepare for college (or other postsecondary education) benefit everyone. Our current practices of holding low expectations for many students result in far too many dropouts or graduates unprepared for college, challenging technical careers, and lives as citizens in a diverse democracy.

So, Miller answers, yes — everyone should attend college. But the reasoning seems spurious for a couple of reasons.

  • How much of the increasingly common requirement of a bachelor’s degree for new jobs is the result of an existing oversupply of people with bachelor’s degrees? Miller claims that people need to have a postsecondary education because more and more jobs require it. Maybe so. But is that because the jobs themselves inherently use skills developed only through a college education? If so, we have to ask if our higher education system is consistently giving students that kind of education. If not, and if students should get a BA or BS  merely because there are so many people out there with BA’s and BS’s that you have to have one to avoid the appearance of intellectual poverty, then this encourages superficial education at the postsecondary level, and the reasoning here is more mythological than anything and needs to be repudiated.
  • As Joanne Jacobs noted back in early 2008 (quoting an article by Paul Barton) it’s not at all settled that the claims about jobs here are even valid. According to that article, only 29% of jobs in 2004 require college credentials, and the percentage is expected to rise only to 31% by 2010 — not exactly a clarion call for all students to matriculate. Also, Barton notes that the wages earned by males with college degrees have slipped, which indicates an oversupply.

College is just not the best choice for every person, and to say that it is merely sets students up for wasting four years of their lives. Some people may have a vocation into a field for which four years of college are a massively inefficient use of time and resources. If you’ve got a vocation to be an electrician, go learn how to be an electrician. If it’s to be a stay-at-home mom, then go do that. Both of these vocations can benefit from a college education if the person is inclined to get one, but neither requires a college education. If you want to go to college and then do those things, fine; but let nobody say that you should go to college, irrespective of your life situation.

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