Design Problem 2: Brainstorming Options

The goal this time around is to brainstorm ideas that seems promising.  We’re not going to worry yet if they fail one of the design criteria from before; we’re more focused right now on thinking of ideas that meet at least one criterion than throwing out ideas that don’t meet all of them. 

To organize the brainstorming, let’s categorize efforts by age group.  Based on class readings like Whatever it Takes, the Paul Tough book about the Harlem Children’s Zone all the TFs read earlier this year, by the time a student in a bad district is in high school, on average they’re already several grade levels behind.  This suggests we target broad catch-everyone programs at younger ages where that achievement gap is narrower and hasn’t had a chance to fester, and focus our initial efforts at the high school level on the few odds-beaters who have shown promise and achievement in spite of their environment.  The goal should be to build a good base of participants at the younger age group and follow them as they get older. 

We’ll start with some ideas (bullet-pointed) for program-based earlier interventions.  Based on our limited resources, a good approach may be to come up with existing program infrastructures where we can lend technical expertise.  Ideally this setup allows us to limit planning efforts and other start-up costs and get the ball rolling immediately in a short timeframe.

1. FIRST Robotics
   • FIRST is a program that exposes kids to building robots to meet a design challenge.  The challenge is different each year.  http://www.usfirst.org/.
   • Pros:
     • Direct exposure to hands-on technical activities
       • Design, build, test ties in especially well to engineering
       • Opportunities to enrich hands-on work and designs with basic math to solve small sub-problems – relevant application of math
     • Comes bracketed into different age-appropriate segments
       • Can start targeted to younger segments and expand into later years with same students
   • Cons:
     • Expensive
2. MathCounts
   • MathCounts is a middle school extracurricular program that revolves around preparation for a spring multi-school competition in mathematics.  Schools can field multiple “teams” of 4;  judging is based on individual and team problem-solving.    Local individual and team winners can proceed to state and national level competitions. 
   • Pros:
     • Junior high age group, more receptive
     • Teaches high-school level math in a one-off problem format, emphasizing creativity over formulaic skills
     • Can be treated as a contest or game as motivation, but without adverse consequences for failure
   • Cons:
     • Probably better geared to undergrads (or even high school students?) as mentors than PhD’s (maybe a PhD as overall admin to lead the undergrad / HS student effort?)

Self-started lower age-gr0up options: I’m not coming up with any good ones here yet.  Opinions?  Leave a comment.

High School Options:

4.   Semester-long Seminar Electives (Seminars)

• Science electives on topics of interest to grad students who will serve as primary teachers
  • plasma physics
  • biomedical topics – artificial joints, etc.
• courses open to top-performing, interested seniors and juniors (by application?)
• classes taught by the grad student as a full GSI-supported position
• Classes heavily focused on labs and demos
• Include a design component to incorporate what the students learn
• Pros:
  • great opportunity to continue with lessons learned from current TFs
  • Focuses on likely future UM applicants
• Cons
  • Very heavily dependent on the skills and emphasis of the PhD who is teaching.  Probably best to do a team-teach scenario to share the load, have a partner to keep each other motivated / prevent being discouraged.
  • Likely teacher backlash for “stealing” their best students.  Wah.  This is why it needs to be marketed as an elective. 

5.   Magnet Science Courses with Heavy Math Emphasis

• As opposed to the above “advanced elective”, this is an equivalent course for, say, earth science, bio, chem, phys, the basics
• Still heavily demo and lab-based.  Bring in a seismometer, learn logs while learning the Richter Scale.  Do Brinell hardness testing with real MSE equipment while learning rock types.  Do DNA resequencing in Bio I with sweet equipment from a lab here.  Bring cutting edge to the class, and then present it well.
• Still require top performing and application for admission 
• Also GSI-supported
• Pros:
  • Earth science was boring.  Bio was a lot of rote memorization.  Chemistry was a little better, physics better still.  I think not coincidentally, that’s also about the order of increasing hands-on learning in those classes. 
  • Including labs takes expertise, and that’s what UM can provide. 
• Cons:
  • This is a stopgap measure to prevent losing the odds-beaters who are still engaged in 9th-10th grade before the college application process kicks in.  This is also a task that would likely require 2 PhD’s per class.
  • Teacher backlash, teacher backlash, teacher backlash.  See above, except this time it’s not an elective, it’s their turf. 
  • You’re going to need a full-time teacher present, to address legal liability / trained teacher present issues.  You’re also going to need to make this teacher the backup, secondary to primary instruction by the PhD’s with demos.  Depending on the teacher, this may be a very tough (pride issues) or very easy (less stress teaching) issues.
  • You’re also going to need that teacher doing the grading and administrative to keep the PhD’s under whatever half-appointment tuition time commitment level you’re looking at.
  • The quality of the PhD is again critical here.  It’s better not to do it at all than to do it poorly. 

Other ideas:  This one doesn’t fit into an age bracket, because it’s not an in-school idea.  It’s one that can be done safely from anywhere you’re in front of a computer, though it would likely benefit from insights gained from the TF program. 

1. Data Analysis for Student, Teacher Performance Evaluation Metrics
   • Fact is, UM involvement doesn’t necessarily mean we need to be on the front lines with the kids.  A heckuva lot of PhD and professorial types are terrible with kids, but great with numbers.  Leverage that – create better tools to keep track of the numbers.
   • Get a couple of PhD’s (maybe School of Ed. plus an EECS coding monkey) to partner on writing an open-source software total monitoring system that can keep a teacher abreast of student issues at a glance, and keep the student appraised of their performance as well.
   • Aggressively support and direct these students in applying for external fellowship funding so they don’t have to cut into your $40k
   • Pros
     • Schools pay good money to get all their grades, attendance, data taken care of.  Dead tree grade books are out.  But those systems kind of suck, and I’d bet a dime to a dollar that the data isn’t effectively crunched.  Build the system, give it away and you have built-in access to the (anonymized) data.  It’s the Google model.
     • Your product has immediate applicability anywhere, not just in Podunk School District.  Broad impact.
     • If you can monetize the product without subjecting your customer (the school districts) to high deployment costs, you have the potential to spin this off as an enterprise that can fund itself and expand.  The issue here is figuring out a revenue model that, if we follow the Google example and guess ads are involved, remains tasteful and refrains from projecting a crass commercialistic veneer over a fundamentally beneficial endeavor.
       • You don’t want text messages automatically sent out to failing students advertising fly-by-night overpriced tutoring services.  This is the nightmare scenario.
   • Cons
     • This requires substantial creativity, vision and focus, and it does not offer any of the warm, fuzzy feel-good aspect that direct involvement with the students does.
     • The district can always tell you to get lost, they don’t want your system
     • You need really good intel on the school’s desires to gear your tools to be appealing to them.  You also need good insight into what they ACTUALLY need, not their own perceptions of same, to add value to the product.

For some good info on why better performance metrics are a really good idea, see these links:

• http://www.gatesfoundation.org/united-states/Documents/empowering-effective-teachers-empowering-strategy.pdf
• http://www.gatesfoundation.org/united-states/Documents/empowering-effective-teachers-readiness-for-reform.pdf

The two reports share an identical intro section, but they are different once you get about 7-8 pages in. More on this in a subsequent post. 

Outlining the Design Space (or, what can $40,000 buy you?)

Your budget is $40,000 / year.  You have access to a wealth of scientific personnel from a major research university – undergraduates, graduate students and faculty.  You’d like to turn that money and access into a reinvigorated mathematics, science and engineering education at the K-12 level.  Let’s treat this as a design problem, starting by making a concrete list of our resources, constraints and objectives:

Resources:

• $40,000 / yr
• Access to Tier I research university students and faculty
• Access to cutting-edge technical research
• Access to current education “best-practices”
• Access to a local K-12 school district

Constraints:

• Quantifiable results must be achieved within 2 years to retain funding
• School district is failing, student achievement is poor, and district resources are nil.
• Student population is predominantly minority.  University population is predominantly white. 

This is the lay of the land.  Like many design tasks, here we have a poorly posed objective.  Knowing how designs without clear goals tend to wander, let’s try to tighten up that objective statement a bit:

Objective:

• Leverage program access to university technical resources and personnel to increase K-12 student STEM interest and exposure.
• Measure student STEM exposure as number of students exposed to hands-on work with university research-related topics for some minimum rate of exposure and minimum duration, i.e., number of hours/week and weeks/year.  Select a minimum criterion and justify it.
• Demonstrate retention of participating students year over year.
• Develop other metrics of success which can be quantified and evaluated.  Justify the relevance of these metrics. 

This is still somewhat poorly posed, but it’s a little better.  For example, the measure of exposure is somewhat arbitrarily chosen by me.  Basically, you need to cut down your design space somehow, and since I think the chance of raising interest correlates with increased exposure, I linked them in that first objective.  Then I said that I want a program that grabs kids for a certain amount of time, does it consistently every week and over a long span of weeks, and potentially keeps them involved for multiple years.

That last objective is a catch-all to recognize that these objective statements still need some improvement.  Luckily, design optimization is an iterative process, so we’ll be revisiting these objectives.  But first, in my next post I’ll start brainstorming solutions and see what I can come up with. 

Algorithms: The first stage in learning or the final stage in understanding?

Consider the above as a prompt, a thought experiment.  Is it better to use algorithms as the first step in a learning process, or as the culmination?  This really hinges on that word better, as in, better for what purpose?  Better for learning what?  As we all know, the question is rarely asked, “Is our students learning?”  -- but I don’t know that we’re all on the same page about what we’re trying to teach!

First things first, let’s have a definition of terms for the lay reader.  An algorithm is a series of explicit steps to accomplish some task.  Here’s an example taken from my class’ algebra textbook of an algorithm to add two numbers:

Rules of Addition:

• To add two numbers of the same sign,
  1. Add their absolute values
  2. Attach the common sign
• To add two numbers with opposite signs
  1. Subtract the smaller absolute value from the larger one
  2. Attach the sign of the number with larger absolute value

I don’t want to be polemic, but reading that I get the mental image of that screeching sound of a sudden halt that you hear on TV.  It hits me right around the  point where I get to Subtract the smaller absolute value from the larger one… what?

At the risk of sounding arrogant, I feel like if I read something in an introductory math book and I don’t get it the first time around, despite just about having my PhD in Applied Physics, something is amiss.  It’s not a foolproof test, but it does have a good success rate.  Still, I value the input of you, my dear (imaginary?) readers, to tell me if I’m off-base here.

Frankly, I think math is one of those cases where you have to be very careful about the balance between the general and the specific.  Normally, I’m all about teaching things in the context of a general framework.  My calculus class, for example – they are great at applying things like the product or quotient rule, provided you give them two functions named f and g and tell them to find the derivative of fg or f/g.  But, give the problem in the book a different spin so that it doesn’t match up verbatim with what they’ve seen before, and… disaster.  They’ve just memorized specific ways to do specific problems, without having a framework to put it all in.

Nevertheless, I want to draw a distinction between a conceptual framework and an algorithm.  Let’s take an example from algebra: finding the equation for a line.  As it happens, there are really two fundamental ways to do this, and students are required to learn both.  They are:

1. Slope-intercept form, y = m*x+b
2. Point-slope form, (y-y1) = m*(x-x1)

Now, an algorithm for slope-intercept form would say:

• If you are given two points (x1,y1) and (x2,y2):
  1. Find the slope of the line (m) between the two points as m = rise / run =  (y2-y1) / (x2-x1)
  2. Choose one of the two points; we’ll assume you chose (x1,y1).
  3. Find the y-intercept (b) by solving the equation y1 = m*x1 + b for the value of b
  4. Write your final answer in the form y = m*x + b, where x and y are variables and not the specific values for either point
• If you have a point (x1,y1) and a slope m:
  1. You already have the slope m, so just do steps 3 and 4 from above.
• If you have a point (x1,y1) and another line in the form y = m*x + b that you are told is parallel to the line through the point (x1,y1)
  1. Parallel lines must have the same slope, so you know that the slope of your line is the same as the slope m of the parallel line, so that’s your m too.  Do steps 3 and 4 from above.
• If you have a point (x1,y1) and another line in the form y = m*x + b that you are told is perpendicular to the line through the point (x1,y1)
  1. Perpendicular lines have negative reciprocal slopes, so you know that your slope is –1/m where m is the slope of the other line, i.e., if it was y = 4*x+5 then your slope would be –1/4.  Do steps 3 and 4 from above.
  2. Don’t get confused by using m’s in two places here.  The point is that both m’s are slopes of different lines, but  (your slope) = –1 / (their slope).

I don’t think that’s even a complete algorithm!  They could give the other line in point-slope instead of slope-intercept form, they could tell you the point by specifying the intersection of two other, completely different lines that your third line has to pass through… the point is that the problem can be arbitrarily complicated, and thus so can your algorithm.  We haven’t even touched on point-slope yet!

Trying to extend an addition algorithm including absolute values may be good computer programming practice, but it’s not good pedagogy.  Here’s my counterexample of a decent conceptual framework for the finding the linear equation:

• Start by looking at a line on a graph.  Ask yourself, how can we distinguish this line from any other line we might draw?  What makes it unique, one of a kind?  How could I make any line look like any other if I could stretch it and push it and pull it and move it around?
• It turns out there are only two ways to change it, two things that are important and that make a line a line.  One is how steep it is, which we call a slope.  A hill is really steep if it goes up really far without going very far horizontally, so for example a handicap ramp would not be very steep, so it would have a small slope, while a staircase might have a large slope and an elevator would just have a super-enormous slope.  So, to decide how steep a line is, I need two pieces of information:  how much does it rise, and how much does it go horizontally, which we call how far the graph runs from left to right.  The slope is just the rise divided by the run, so slope = rise/run.
• Remember that there’s another thing we can change – I could shift a line up and down or left and right.  For example, a handicap ramp has the same slope whether it’s on the first or second story, or in this room or the next, but those are all different ramps.  In math, we just talk about it moving up and down, because our lines go on forever so a shift up and down can look the same as a shift left or right (Something about pictures and kilo-words comes to mind here, alas).
• So, we need two pieces of information to talk about lines: a slope, and how far up or down it should be shifted.  If we have a slope and a point the line has to go through, we can pin that line down and know exactly where to draw it.  But sometimes, we can be tricky about how we give our two pieces of information.  For example,
  • we could give two points instead of a slope and the line – then we could figure out from the second point how much the line would rise and how much it would run from the first point, so the second point would hold the key to finding the slope.
  • we could give one point, and then tell you the slope of a different line that we said was parallel to the first.  Parallel lines are like handicap ramps on another story or in another room – they have exactly the same slope, just a different shift up, down, left or right.  So you’d have a point and you’d know your slope was really the same as the other line’s slope!
  • kind of like above, we could give one point and then tell you the slope of another line that was perpendicular -- (discussion of the –1/m bit would require a picture).
• Of course, the upshot is, you always need two bits of information: a point to pin down your line in one spot and a slope to decide how steep to draw it.  It just happens to turn out that there are a lot of ways to represent that second bit of information, the slope, with other facts like the location of a second point, or the slopes of parallel and perpendicular lines.

If you actually read all that, bless your heart.  Brevity, I ain’t got it.  I could go on to talk about how we could pick a special spot, the place where the line crosses the vertical (y) axis as our point that we’ll always use, but I think the dead horse, she is beaten. 

To return from our super-long and in-depth example, I feel like having a discussion about that framework is crucial – I don’t know that any of my kids really understood that the difference between slope-intercept and point-slope form of a line is just that point-slope is a generalization to a fixed point that doesn’t have to be on the y-axis.  They couldn’t conceive of it as a generalization, because they didn’t know what made us pick that specific point in the first place!  It was as if God came down and said, “Thou shalt use the y-intercept”, not like we discussed it and talked about why it made things simple.

So, now let me see if I had a point in my longwinded-ness here…. ah yes, found it, the balance between generality and specificity in math teaching.  I distinguish between generality as an overarching conceptual framework, to give you a roadmap when you’re learning something new so you can figure out how it fits with other stuff you know, and generality as in an algorithm that allows you to handle any arbitrary set of inputs correctly.

I think the right order to teach those things should be something like, start with a discussion of the framework, so you get a preview of what we’re going to do and why, then do all the specific cases and drill the hell out of them, constantly referring back to where we are in our framework at every step, then at the end you make your students write the algorithm so you expose any remaining flaws or gaps in their thinking with devious, pathological algorithm-breaking test cases.  Discuss. 

Vignette #1: Your brain is a muscle

Writing on a blog regularly is something that doesn’t come naturally to me – I like to write in big, full-featured posts that include links to pictures and slideshows and background articles, and that takes a while and in a busy schedule it often ends up being the lowest priority on the totem pole.  So let’s forget the extras and ramblings and focus on a single story, a vignette.

Note that now, as always, names have been changed to protect the innocent.

Alice is a freshman in Mr. MacGregor’s algebra class, and also is part of his math support class, where students doing poorly get an extra hour of math each day.  On a couple of successive visits, Alice pulled out a romance novel and sat quietly reading during the lecture.  Now, to set the scene, Alice sits in the very front of the class, closest to the teacher’s desk – there was nothing subtle about it.  As per usual, the teacher told her to stop reading and pay attention, she ignored him, and he was too busy to really follow up much. 

So, Mr. McDonald to the rescue!  I walked over, sat down quietly next to her and started talking to her.  As it turns out, Alice has been reading since she was really little, maybe around first grade or so, and she likes to read, at least a heck of a lot more than she likes listening in math class.  As in many cases, rolling with the “disruptive” behavior for a minute without condemning it tends to produce results:

Mike: So, you’re pretty good at reading, huh?
Alice: Yeah, I guess.
M: Are you better at it now than you were when you first started?
A: Yeah, probably.
M: Why do you suppose that is?
A: I don’t know, I just got better.
M: How about math, do you like that?
A: No.
M: Are you any good at it?”
A: No, I suck at math.
M: Is that why you read instead of paying attention to what’s going on in class?
A: None of what he’s saying makes sense.  It doesn’t matter if I pay attention or not, I don’t get it.

At this point, I’ve talked to her about something she’s good at, reading, and worked in a bit about why she’s not doing what I wanted her to.  Now I want to make a connection about why her behavior isn’t very productive.

M: That’s a pretty long book you’re reading there.  Did you start out reading books that long?
A: No, when I started they were little.
M: Oh, like picture books, and not a lot of words back then, right?
A: Yeah, I guess.
M: So then after a while you just read enough books that you could kind of make sense of things, read better stories?
A: Yeah.

Now I’ve laid the groundwork that there was something she learned, practiced, improved at and now enjoys.  Let’s make that explicit:

M: You ever do any sports or anything?
A: No.
M: How about dance?  Do you like to dance?
A: Yeah…
M: What happened the first time you tried to dance, do you remember?  Did you have any good moves?
A: Nah, it was pretty bad.
M: But you’re better now, right?  I mean, you’re not still bad, are you?
A: Yeah, now I like it.
M: So because you like it, you do it more, and because you do it more, you get good at it and you like it… you see where I’m going with this?
A: Yeah … I bet you want me to put my book away and pay attention in class.

So she knew all along I wanted her to stop reading the book.  I don’t think she knew exactly when the other shoe was going to drop yet, but she knew pretty well where it was coming down.  Still, the tone of the conversation is, well, it’s conversational, not confrontational.  Confrontation doesn’t work, especially when as in my case you have absolutely zero authority to mete out any sort of punishment anyway (I often wonder how well a time-out corner would work for 9th graders!).  So, since she’s brought it up, let’s go in for the kill:

M: Well, think about it like this: When you first danced, your body didn’t know how to use all the muscles you have to make you look good when you tried to move to the music.  Some of those muscles were weak, and others you just weren’t used to using that way, so I bet you looked pretty funny.
A: (Laughs) Yeah…
M: But then after a while those muscles got stronger, and you learned to use them better, and you got better at dancing because you kept at it, you know, like practicing.
A: (Quietly) Yeah, I guess…
M: Now here’s the really crazy part – your brain is just like another muscle, and it can get stronger too.  When you learn something new, like how to read, it’s like your brain is learning a new way to dance.  At first, you didn’t have a clue what all those letters on the page meant, and now you’re reading whole books full of little symbols that make words!
A: Ok.
M: Guess what?  Your brain has a math muscle, too.  When you don’t understand what’s going on in here, it’s like you’re trying to read a whole big book but you haven’t practiced enough on the smaller books yet.  You’ve gotta train your math muscles, and  Mr. MacGregor, he’s like your coach, and he’s showing you exercises to get those muscles stronger.
A: Ok, so what am I supposed to do?
M: Well, you’ve already practiced a lot with your reading muscles… maybe you should try using the math muscles a bit.  After all, you can read at home, but Mr. MacGregor isn’t going to follow you home to help train your brain.  The work during class is like exercises – even if they’re hard, struggle through them and eventually you get stronger and better.  Or you can keep reading, but… it’ll be like your first time on the dance floor, every time you try to do math.  Not pretty.

So this conversation all happened in the span of about 5-6 minutes, for all it’s a thousand lines typed out.  And, as I expect most teachers have experienced, for all the effort I expended most of Alice’s replies were only about 2 words -- not very encouraging.  But after we were done, she put her book away and paid attention for a while without getting too distracted.  In return, I tried to come back frequently during the hour to see if  she had questions.

Over the next few weeks, I didn’t see the book ever come back out.  Now, this was no panacea – Alice passed first semester, but like most of the students passed by the skin of her teeth and likely would have failed if not for Ypsi’s cook-the-books grading scheme.  But when she gets off-topic now, she is far more receptive to me when I urge her back on, and she is also far more open to asking me questions now that I’ve spent some effort having a conversation with her that consists of more than “Put your book away, or it’s a referral.”