kitchen table math, the sequel: the new, new thing

Wednesday, January 18, 2012

the new, new thing

When was it school personnel used to talk about what students should "know and be able to do?"

The 90s?

The 80s?

For the last few years, whenever I've thought of it, I've checked my district's documents for the words "knowledge" and/or "know." So far, I haven't found them.

What I have found is lots of references to understanding. Students in my district will understand this, that, and the other.

Occasionally, too, a district document will mention 21st century skills.

Which sets me to exactly does one acquire a 21st century skill?

Can you memorize a 21st century skill? (And if you can, would that be kosher?)

Can you practice a 21st century skill until you get really good at it?

And does aptitude for 21st century skills fall on a bell curve?

Can you be gifted in 21st century skills?


Below average?

Are there specific 21st century skill learning disabilities?

This could be big.


Catherine Johnson said...

In fairness, the UBD website does mention "know" and "do":

Catherine Johnson said...

I suppose there is nothing to be gleaned re: 21st century skills from the Iowa Test of Basic Skills.

Anonymous said...

Now you are on the right track.

And the know was added later.

Because what we want you to do sounded bad.

As it should for anyone not completely permeated with 21st century skills.

And yes their meaning of understanding is more emotional than cognitive.

Should there really be a difference in our social justice ed world?

SteveH said...

"Understanding by Design (UbD) is a framework for improving student achievement. Emphasizing the teacher's critical role as a designer of student learning, UbD works within the standards-driven curriculum to help teachers clarify learning goals, devise revealing assessments of student understanding, and craft effective and engaging learning activities."

Apparently, textbook authors can't do the job. That must be because it's too small of a market for UbD.

"Students reveal their understanding most effectively when they are provided with complex, authentic opportunities to explain, interpret, apply, shift perspective, empathize, and self-assess. When applied to complex tasks, these "six facets" provide a conceptual lens through which teachers can better assess student understanding."


What about the skills of selecting the correct governing equation and practicing how to apply it in many different situations?

If students can do a complex task, that really only means that they can do that one problem, no matter how much understanding is applied. If they can't, it's very difficult to find the gap or problem in understanding. It reminds me of unit testing versus system testing in programing. Testing doesn't start at the "authentic" system level. You might define tests that work, but you will never properly exercise the code.

For any complex system, you have to validate the parts before you put them together to test and validate the whole. For education, you have to validate that students have mastered basic skills before letting them loose on complex "authentic" or real world problems. Like Duh! I prefer a prototype approach where you start with a little bit of core knowledge and skill and then add on little pieces at a time, verifying and validating as you go.

If you do ensure content and skills as you go along, then how does understanding fit into all of that? Part of proper mastery involves understanding. How is that understanding different than the kind of understanding that allows you to tackle new problems? There is understanding and mastery of the tools and then there is understanding an mastery of how you can apply those skills. A top-down or real world approach to higher levels of understanding is just not possible.

"The potential of UbD for curricular improvement has struck a chord in American education. Over 250,000 educators own the book."

["Understanding by Design (Wiggins & McTighe, 1998, $26.95).

Let's see, 250,000 * $26.95 = I'm in the wrong business.

It all sounds like really bad systems analysis.

SteveH said...

"how exactly does one acquire a 21st century skill?"

With drill and kill.

My son had lots of practice with the internet, "keyboarding", word processors, spreadsheets, and blogs (in 6th grade, they worked on one at school and the teacher encouraged them to set up their own). There is no discussion of quality content or understanding. I think he spent more time on these skills at school than on learning the times table and other traditional skills. Apparently, if something is a 21st century skill, then drilling is OK.

SteveH said...

I listened to Grant Wiggins podcast on math and problem solving, and how schools should build a math curriculum by working backwards from non-routine problems. I also read other comments he makes on problem solving.

He makes a case on how "exercises" dominate math and how there is little effort placed on solving non-routine problems. He talks about having a balance between exercises and problem solving, but never defines exactly where that should be. However, he does take the position that problem solving should drive the need for exercises. The issue is that one never learns how a curriculum (or even one class like pre-algebra) should be built this way. It's sooo easy to argue with generalities.

Is "traditional" math bad because it is done poorly, or is it bad by definition? Does "traditional" math fail because it does not keep going into problem solving after the exercises are done, or should exercises be driven top-down from problem solving?

This is where all educator discussions on math break down due to lack of detail. I assume that I would get more details if I buy the book or I take one of their courses.

Most of these discussions tend to throw exercises and problem solving into two distinctly different categories. Problems are often defined as non-routine ones with no one answer. Wiggins' discussion of problem solving doesn't talk about how the basic math skills are applied to problem solving. The implication is that all skill development should be driven by problem solving. But what set of non-routine problems defines the breadth and depth of a course like pre-algebra? Pre-algebra is defined by a set of topics and skills.

One, suposedly non-routine, problem he offers seems quite traditional. It is:

"A train is leaving in 11 minutes and you are one mile from the station. Assuming you can walk at 4 mph and run at 8 mph, how much time can you afford to walk before you must begin to run in order to catch the train?"

No explanation is given about how much time a curriculum should spend on simple D=RT "exercises" before one is asked to solve this one. If you have a proper introduction with D=RT concepts, why would this be a problem and not an exercise? Problems and exercises don't fall into neat buckets.

So what does understanding and problem solving mean for D=RT problems? How do you figure out when to use that governing equation? Problems like this are common in traditional math, so why the complaint?

That problem was taken from a math class at Phillips Exeter Academy, which he holds up as a shining light.

"Math class at Exeter is entirely problem based. Students are given these problem sets each week, and homework consists in being prepared to offer your approach and solutions (or difficulties) in class the next day. In short, Exeter (arguably one of the best schools in the United States) takes it as a given that the point of math class is to learn to solve problems. Content lessons often follow upon the attempts to solve them rather than always preceding them."

Did he evaluate the expectations placed on these students compared to regular kids? Is it a top school because of the curriculum or the students? Did he compare the curriculum with Phillips Andover?

With UbD, is the content of a class like pre-algebra changed? Do they use different textbooks? Do they cover the same amount of exercises plus add in non-routine problem solving? What are the trade-offs?

Jen said...

"No explanation is given about how much time a curriculum should spend on simple D=RT "exercises" before one is asked to solve this one. If you have a proper introduction with D=RT concepts, why would this be a problem and not an exercise? Problems and exercises don't fall into neat buckets."

This is just so stupid. Not to mince words. :-p As you point out, this is truly just another math problem, same as it ever was. A word problem, yes, but not one that is materially different from any other textbook problem. Or is he saying that this problem is what high school students should do rather than more complex math?

Presenting this as some exemplar of problem-solving problems makes it even odder. If this was truly a problem I had to solve, my answer would be simple: I would run the whole mile to the station and use the extra few minutes to catch my breath, walk through the station and get myself by the right track.


Of course, even that assumes that my 8 mph pace is in clothes I'd wear on a train and possibly carrying a bag or two, not in running clothes and shorts. Given that, I might barely have time to catch my breath before needing to hop on that train.

Ohhh, how my teachers would love me, no?

MagisterGreen said...

"The potential of UbD for curricular improvement has struck a chord in American education. Over 250,000 educators own the book."

["Understanding by Design (Wiggins & McTighe, 1998, $26.95).

Let's see, 250,000 * $26.95 = I'm in the wrong business.

It all sounds like really bad systems analysis.

Yeah, 250,000 educators own the book. Some of us because our school bought it for us and told us to read it.

Just sayin'

Allison said...

The false dichotomy is so annoying. "We want students to Solve routine problems quickly, But we want them to solve non routine problems, too." But it's Not "But"! It's AND!
Problem solving is not Zero Sum. Being good at solving routine problems doesn't make you worse at non routine ones, not at all. Being good at routine ones is a pre requisite for solving non routine ones. And the best way to solve non routine ones? Have solved so many that most aspects Are routine, so you don't get troubled by various aspects of the problem, but can focus on what is novel.

Allison said...

Someone should ask what they think of Singapore's math program since it is explicitly built around problem solving. there they have an actual example of pedagogy, content, and results. Do they approve or not? those kids don't waste a night solving that problem of getting to the train station.

SteveH said...

"Being good at routine ones is a pre requisite for solving non routine ones."

Yes. Prerequisite. The UdD people play fast and loose with this. They talk out of both sides of their mouths. They won't deny the need for prior basic skills, but they seem to think that the learning process must change quickly to top-down problem solving, which apparently does not or cannot be seen as an "exercise." Problem solving seems to come down to generalized thinking and organizing principles, not skills like applying specific governing equations for which you've spent many hours mastering the different variations. The goal of math is to make most things exercises; to minimize non-routine probems.

SteveH said...

Wiggins bases his "problem" analysis on "Deitz' distinction":

"An exercise is a question that tests the student’s mastery of a narrowly focused technique, usually one that was recently ‘covered’. Exercises may be hard or easy but they are never puzzling...the path toward the solution is always apparent. In contrast, a problem is a question that cannot be answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving."

– Paul Zeitz, The Art and Craft of Problem Solving p. ix

Wiggins uses this distinction to define the following types of problems.

"A puzzle: not all needed information is provided explicitly. Some “investigation”, inference, logic, and filling in of what is implicit is required. Lots of prior knowledge is tapped and tested."

"e.g. What is the area of a square whose perimeter is 16?"

It's not explained where and when these prior knowledge and skills are mastered.

This problem is the bread and butter of traditional math. These questions are used to "exercise" one's mastery of basic skills and understandings. Obviously, this sort of problem does not explain the raisons d'etre of UbD. However, this defines the vast majority of math problems. Math gives you skills and understandings that can be applied to all sorts of problems.

"Paradoxical: on the surface, it seems unsolvable, self-contradictory, i.e. involves leading seemingly impossible conclusions, logic, or assumptions. More broadly it must be a “non-routine” problem."

"e.g. “Take the number 4, and using it five times, write a mathematical expression that will give you the number 55.”[1]"

44 + (44/4) = 55

Don't you just love these "math" problems? Do they define this as part of a class of math problems with a theory behind it? Do they explain and master these techniques? No. They just pose these problems and expect you to think. It sounds like Professor Hill and his "Think Method".

That is not math.

SteveH said...

The next type of problem is:

"Open-ended: once into it, the work may require us to re-frame the problem statement, consider what counts as an answer, and consider that the answer may end up being: “well, it depends...” "

"e.g. How big a warehouse is needed to store a week’s worth of newsprint for use in printing presses by a major daily newspaper in a big city?"

This reminds me of having kids offer their opinions on current events even though they have little to no background knowledge. Will their analysis skills improve this way? No, they will just learn how to argue. Is this the sort of active learning they are looking for?

This is of "no one right answer" type of problem. Another classic problem of this type is to give the student a fixed amount of money to spend on a birthday party for 10 kids with a menu of party supplies and their prices.

Is this math? Very little. It's background knowledge and judgment. If you are talking about developing a cost model, then there are skills one can learn to make that process easier. One can learn about weighting functions and risk factors. One can learn how to quantify judgment into a merit function. Those skills can be applied to various real world problems once the specific background knowledge is developed. Math is about developing skills, not thinking and guessing about knowledge and judgment.

Wiggins tries to claim the high ground by not dismissing pure math problems.

"In short, we will get more problem solvers the more secondary math courses are framed around real problems, pure and applied; ..."

He is making a case for a problem driven curriculum, but there are no details. He does not address the curriculum issue of separating students and tracking that usually starts in 7th grade. He does not address whether STEM career doors remain open or not. He does not address the needs of kids once they get past algebra I or geometry in high school. He does not address the problems of keeping STEM doors open in K-6. It's another case of arguing with generalities, but glossing over the critical details.

Does this problem solving approach ensure that basic skills are mastered? What are the basic skills and how does UbD ensure them in K-6. Are they approached as a separate mastery goal, or does it assume that mastery will be developed in the context of doing problems?

SteveH said...

Here is another problem cherry picked from Phillips Exeter's 9th grade math curriculum.

"Pick any number. Add 4 to it and then double your answer. Now subtract 6 from that result and divide your new answer by 2. Write down your answer. Repeat these steps with another number. Continue with a few more numbers, comparing your final answer with your original number. Is there a pattern to your answers? Can you prove it?"

Prove it? With repeated tries to see the pattern? Just ask them to write the equation and simplify it. This is an exercise, not a problem. Few "problems" remain with enough skill and mastery. Math makes this an exercise. Math is not some vague thinking skill.

Phillips Exeter's philosophy is again cited:

"The resulting curriculum is problem-centered rather than topic-centered. The purpose of this format is to have students continually encounter mathematics set in meaningful contexts, enabling them to draw, and then verify, their own conclusions...The goal is that the students, not the teacher or a textbook, be the source of mathematical knowledge."

The students are the source of mathematical knowledge? Where are my hip boots? Remember that Exeter is the home of the Harkness Table.

This loops back to discussions of discovery and the Socratic Method. In a perfect (and driven, like Exeter) world, I can see this working ... maybe, but few schools use this approach. What I see in most K-12 examples, however, is poor implementation and low expectations. To have any chance of success, you have to increase the expectations and expect students to do more skill mastery on their own.

You don't get something for nothing.

SteveH said...

From Wikipedia:

"Exeter does not teach math with traditional textbooks. Instead, math teachers assign problems from workbooks that have been written collectively by the Academy's math department. From these custom workbooks, students are assigned word problems as homework. In class, students then present their solutions at the blackboard. This means that in math class at Exeter, students are not given theorems, model problems, or principles beforehand. Instead, theorems and principles emerge more organically, as students work through the word problems."

At Phillips Andover, the math is much more traditional. Both schools produce good results, so something else is going on here besides a problem-centered approach.

At best, I can see this approach working in high school, but how does it translate to the lower grades? Like the Harkness Table, a general concept is crammed into all levels and types of education. Why not use the Harkness table in orchestra classes? Students can offer their own interpretation of the music rather than study music history and theory.

A traditional lecture approach does not preclude student interaction or problem solving at any level, but their solution is not to work harder on that area. Their goal is to use it as an excuse to change the process of teaching to match their own pedantic sensibilities - student centered with the teacher as the guide-on-the-side. Criticism of problem solving is only an excuse to get to that point. That it can work in theory is all that they need.

SteveH said...

The issue of not enough problem solving is mixed up with the idea that the material has to be driven by problem solving rather than skills or content. It is also mixed up with issues of teacher-centered versus student centered classes. Since student-centered classes are less efficient, where and when is the rest of the work done? Also, adding more problem solving to the curriculum takes time away from mastery of basic skills. Clearly, Exeter students fill in the gaps on their own.

One can have a directly taught, problem-centered approach that would be a better use of class time. Students could be grouped to do discovery as homework, especially at a boarding school. Students could be expected to do individual discovery.

Their primary objective is not better problem solving, but student-centered discovery in class.

Anonymous said...

Phillips Exeter is modeling the 21st century learning approach being pushed by Common Core where students construct their own knowledge and problem solving is on deliberately unscripted, indeterminate situations.

It has the effect of teaching students that all knowledge is relative and transient and certainly not universal. There is little emphasis on the transmission of inherited cultural knowledge because the assumption is that belongs to a previous social, economic, and political system that needs to be replaced. The heavy emphasis on problem solving encourages students to look for political applications.

Back to work as I finish documenting what is really going on.

Student of History

Catherine Johnson said...

My son had lots of practice with the internet, "keyboarding", word processors, spreadsheets, and blogs

OK, I have to (re)-tell my Powerpoint story.

Chris spent YEARS 'learning' Powerpoint in "technology" class. Everybody at the middle school had to take "technology."

One day I was trying to create a Powerpoint I thought might help teach Jimmy (who is autistic) to read. Because Jimmy seems to see motion much better than he sees stationary objects, I was trying to create a Powerpoint where the letter would slide in from the left of the frame and stop in the middle, while the sound of the letter would play.

I couldn't figure out how to do it the way I wanted it, so I asked Chris.

Chris couldn't do it, either.

"I'm not very good at Powerpoint," he said.

Catherine Johnson said...

With 21st century skills ... it's interesting.

Probably most of you know that 21st century skills seem to have originated with the Partnership for 21st Century Skills, which is a coalition of tech companies, the NEA (NOT the AFT), and some govt departments. (Department of Ed, probably)

Tech companies want to make money selling 21st century doodads; the NEA presumably wants to fight accountability.

Not surprisingly, there's been no real push to measure or assess 21st century skills, or to declare schools in need of improvement because their kids don't have them.

On the other hand...I've also seen members of the education establishment push the idea that we should spend more time assessing 21st century skills and less time assessing 19th century skills ...

It'll be interesting to see whether a special-ed emerges that's specific to 21st century skills.

I'm going to guess 'no,' but we'll see.

Catherine Johnson said...

I'm going to guess that the fundamental impetus behind 21st century skills is to induce more spending without more accountability.

Catherine Johnson said...

Magister Green - if you're around & feel like commenting - what did your school want you to do with UbD? (Or learn - ?)

MagisterGreen said...

I can't really say what our little venture into UbD was supposed to do since I'm not entirely sure where we stand with it at the moment. We only cottoned onto UbD maybe 5 years ago after a huge round of curricular mapping for our accreditation. So we had these maps that took forever to develop and then we had to put them into UbD format.

But last year we got a new favorite thing ever, "Strands". So now we're putting everything into "Strands" which we'll then examine to see where our curriculum has gaps or repetitions or what-have-you.

We'll also be checking our "Strands" against our "Core Curricular Values" that were plucked from the ether by a committee last year.

So...UbD has rather vanished and no one really talks about it anymore. As such I'm afraid I have no idea what we were supposed to do with it or learn from it.

SteveH said...

So now we're putting everything into "Strands" .."

Do you have any web links to explain what "strands" are? I tried searching, but didn't pull up anything definitive.

Anonymous said...

Strands are the various activities and tasks used to interact with and maybe reenforce a given concept. Spiraling has a bad rep so the same idea is now called a learning progression.

It is a theory.

We should not be restructuring education based on theories but Catherine is right upthread. It's quite lucrative for the designated companies and organizations and it's OPM. Unless you are the one paying all those taxes that create the OPM or are on the hook for the debt.

NEA goes along because it's a means of dictating what goes on in every classroom and every school. A long time dream. Teachers closing the door and doing what they thinks works best will go away. Or the teacher will be deemed not effective and be out of a job.

palisadesk said...

The term “strands” has different meanings. When referring to the K-8 mathematics curriculum, “strands” are the components that are taught and reported on discretely: my district has 5 strands: number sense and operations, geometry, algebra, measurement, data analysis and probability. Our curriculum does not “spiral” very much – of course each successive grade has the same strands, but the ones at the next grade level take off where the previous one stopped. There is no “trust the spiral” concept.

For some reason, Blogger won't accept the hyperlinks. It seems to be cutting the URL's off. But this Wikipedia article : Principles_and_Standards_for_School_Mathematics

explains the curriculum usage of the term “strands.”

On the other hand, strands seem to refer also to other components of mathematics at the theoretical level : see strands.htm

(you may have to take the spaces out that I put in so Blogger didn't cut off the URL. Something isn't working right today.)

Grant Wiggins and Understanding By Design have not made much of an impact here. At first I thought Catherine was referring to Universal Design for Learning (UDL), which is something quite different. Right now we are much more involved in implementing Elmore’s initiatives and some of those recommended by Robert Marzano.