A little too much estimation…

I’m a big fan of either using 10 m/s/s or 22 mph per second for g, and giving students reasonable numbers when they are doing exercises that involve projectile motion. The real world won’t be so kind when they take data, so I want to start them off where they have a fighting change of reasoning proportionally on the basis of concepts. I’m trying to fight the “iterative plug-n-chug” approach, by doing things we can think through.

But, in one of my classes, I’ll need to be using 9.8 m/s/s, because that’s what they’ll have on their exam. And I won’t have reasonable numbers always, because the problems I’m supposed to provide example are not picked by me. So, I have decided to do a lot of  “figure out a number you know is too high”, “figure out a number that is too low”, and “make a good guess”, before doing any plugging into equations.

Take for example, this kind of question I am supposed to work through as an example:

A ball is thrown vertically upward with a speed of  7.2 m/s. On the way down the ball hits a flag pole that’s 1.2 meter above from where it was thrown.

It’s a goalless problem, so I’ll have to decide something to solve for. Let’s go with, “How long will the ball take to get to its highest point?” … Here we go. Well, we know that the ball loses 9.8 m/s every second, but it only needs to lose 7.2 m/s. That must mean that the time to reach its highest point is less than 1 second. How much less? Well, it’s certainly greater than half of a second, because half a second would be  time to lose 4.9 m/s.  So, a number that is too high is 1 second, and a number that is too low is 0.5 s. I would guess it’s probably pretty close to 3/4 of second, because 7.2 seems pretty close to in between 4.9 and 9.8.

What’s the actual answer? About .73 seconds.

OK. Now on to another question. “How high will it go?” Well, the average velocity of the ball during the trip will be right in the middle of its starting velocity (7.2 m/s) and its ending velocity (0 m/s). That makes its average velocity 3.6 m/s–I can divide by two in my head. Given that I’ve concluded in the first part a travel time of 3/4 of a second, this must mean it must have traveled less than 3.6 m, because it would have taken an entire second to travel 3.6 m, and it didn’t have that much time. My reasoning is that it should go 3/4 of the way to 3.6m in 3/4 of a second. This is actually pretty easy to reason about, because 36 = 9 * 4. So, I’d say it would have gone about  2.7 meters during that time.

OK, so what’s the actual answer? 2.6 meters

OK, so how much time to hit the flag? Well, it took 3/4 of a second to go up. It would have taken another 3/4 of a second to go down, but it didn’t quite go all the way down because it hit the flag. So, How far did it get? Well, it got a little over half way down (2.7-1.2 = 1.5). Let’s just call it half for good measure. Now it might be tempting to think it took half the time to go half way down as the whole way down. That would be true if it was moving at constant velocity. But it’s moving slower earlier and faster later, because it takes time to speed up. Given that, we know that it must take MORE than half the time to cover half the distance, because that’s the slow part of the motion. By that alone we can say that it must have taken more than 3/4 + 3/8 = 9/8 of a second. So we have a lower bound of 1.1 s and an upper bound of 1.5 seconds. Let’s just guess that’s it took 0.5 second, because that’s more than half of .75 seconds.

That would mean the whole trip should be about 1.25 s = 0.75s on the way + 0.5s on the way down seconds.

The real answer? 1.2 seconds.

OK. So, some of this is a bit much. But, I didn’t have to use equations.

 

Newer than New: How to Help?

Yesterday I met a new physics teacher in our area. He has no background in teaching, beyond the two weeks he has been in class already. He has a science degree and worked in the public sector. He is teaching sections of physics and chemistry on an emergency certification. He is looking for help, suggestions, ideas and advice–whatever he can get.

I am going out to visit his physics class next week.

What do you think is the most important thing to focus on with a new teacher like this, especially with no training? I have never had to work with a teacher with absolutely no training. I can only imagine that there are lots of places where he’ll need support; but right now it’ll have to be like triage, deciding where help is needed most. You can certainly tell he is a bit frazzled, although not much more than you would expect for anyone teaching their first year. From his perspective, what he needs is ideas for what to do in the classroom–activities, lessons, things to get student engaged.

Anyone out there been in his position: What kinds of mentoring and help would you have found most helpful? What kinds of mentoring support would have been frustrating or unhelpful?

 

 

My Flipped Physics Class

Besides my inquiry course for elementary teachers, I am also teaching is a section of non-calculus-based physics. The class is highly structured, so that I am told exactly what to do each day. The basic setup is this:

(1) Students come in and take a 2-3 minute reading quiz, which consists of five MC questions that I write (so that each section has different quiz). They are supposed to have read “lecture” materials online, and quizzes are intended to have required reading but not full understanding.

(2) Students get into a group (8 groups of 4 I think) and work through a series of conceptual questions on a computer (10-15 minutes). This isn’t graded, but the questions give immediate feedback.

(3) Students then come back to listen and watch as an instructor works through a sample problem (20-30 minutes). The example problem is a goalless problem. It’s also probable that the instructor will discuss the computer exercises. I am told exactly what example problem to go over.

(4) Students then go off to work in groups on different but related goalless problems, structuring what they do, in part, on the sample problem that was just worked out.

(5) Students discuss their problem solutions with the whole class. They work on their problems in whiteboards, so that is also how discussions are structured.

(6) Students go work on a lab. Most of the labs are not graded. They are just checked. I think 2 or 3 labs are graded more closely, but students are told which ones will be ahead of time.

Everything happening in any given day of class is around one concept. Class is 2.5 hours long and is twice a week. Often for the labs, they will collect data one day and analyze it the following day. There is also a 1.5 hour lecture, where all the sections meet. The lecturer runs some Peer Instructions with clickers, but also discusses the data students should have collected. They discuss issues in graphing data, linearizing data, etc, in order to prepare them for the next day where they do analysis in the lab.

As part of the lab class, they also have to carry out an independent investigation, write a paper, and give a presentation. I still need to learn more about what this is all about.

I’ll try to keep everyone updated on the how the flipped class goes for me.

Looking at Lakes and Scattering Experiments.

Back when I was blogging using blogger, I posted a series about light and water.

First, I shared with you my intrigue over how a lake looked like a mirror in the morning and a sheet in the afternoon. To follow up on my intrique, I shared with you another pair of photos, in which another lake looked like a mirror and a lake from different vantage points (but at the same time of day). Some while later, I shared with you a rather contrived observation–a whiteboard can act like a mirror or a sheet in the same way a lake does, depending on your vantage point and how light shines on it.

This lent some support to the idea that angles are really important. One reason they are important is because shallow rays don’t penetrate  deeply, so it’s like you are seeing the surface. When the sun is low and you are low, you have a high intensity beam glancing off the surface, showing you the mirror effect off the relatively lat surface. When the sun is high, you still get some low glancing reflections, but the mirror image is overwhelmed by the scattering of rays that penetrate deeper into the lake. When you are high, glancing rays never have a change to get to you, so all you see is the sheet effect.

Today, I’m struck by how similar looking at lakes is to what physicists do when they investigate matter. Physicists shine light at all kinds of matter and look what comes out. They vary the intensity. They vary the frequencies. They vary the angles. What comes out tells physicists about the structure of the matter that light was interacting with. In that sense, looking at lakes, especially from different angles from different perspectives with different intensities, is the naturalist’s scattering experiments.

Modeling: Data, Evidence, and Science

One of the things I don’t like about most of the modeling curriculum I have experienced is the tendency to jump into taking data. In the summer workshops I attended at ASU and the several mini-workshops I have attended elsewhere, I was pretty much always asked to go take some measurements of things.

(1) Go measure some distance and time data

(2) Go measure some force and distance data

(3) Go measure some current.

(4) Go measure some frequency and length data.

And then we graph the data and look for relationships among variables. Next, we’ll often try to make a best fit curve to model the situation and interpret the meaning of the parameters in the equation.

The problem is that I always felt like I was just measuring things and looking for relationships without purpose. There were no puzzles we had identified as worthy of my epistemic curiosity. We had identified no perplexing questions that made me wonder, “what kind of data should I take to help answer that question?” What relationships am I expecting and why? How will I know if the measurements I’ve taken are good enough to either support or refute one idea or another? None of that was going on–not for me at least. And that was the thing that was puzzling to me.

Now don’t get me wrong–empirical data is important in science. But it alone is not science, at least not to me. I have been more apprenticed into starting science with something perplexing and letting that perplexing situation be the source of ideas, arguments, and explanations that need to be sorted out. In my mind, there’s now a reason to take data–that data will be EVIDENCE to support for claims. To me, the the subtleties that entangle and distinguish data and evidence are crucial for understanding the nature of science–both to do science and to be scientifically literate.

Leslie Atkins is fond of this quote that supports this view:

Observation and experiment are not the bedrock on which science is built, but rather they are the handmaidens to the rational activity of generating arguments in support of knowledge claims. (Driver, Newton, & Osborne, 2000, p.297)

Now, I’m certainly not saying that scientists never just muck around with data, or discover interesting things by looking for at relationships, or plotting data, or trying to make sense of mathematical equations. So what am I saying? I’m not exactly sure. Maybe, I’m asking someone to explain to me what I’m not understanding about the modeling curriculum.

Ruminations on Acceleration and Generative Ideas

Acceleration as Simple but Vast Idea

So take the definition of average acceleration: a = Δv /Δt. Simple right? Well, yes, but not really. In order to really understand this definition, you’re going to have to explored a vast array of ideas, including ideas like

If an object is slowing down, its acceleration is in the opposite direction of its velocity

If an object is speeding up, its acceleration is in same direction as velocity

If an object is turning, there will be (a component of) acceleration in the direction of the turn.

It takes time to speed up or slow down. Some things speed up quickly and others speed up slowly and this has to do with an object’s acceleration.

Given a constant acceleration, speeds change linearly –an object gains (or loses) the same amount of speed in any equal interval of time.

A greater change in velocity in the same amount of time indicates a greater acceleration

The same change in velocity in less amount of time indicates a greater acceleration.

A greater acceleration will result in a greater change in speed in an equal amount of time.

A greater acceleration will result in an object taking less time to change its speed.

When an object is accelerating, the distances covered are not equal for equal times; objects cover more ground during times in which its moving faster and less when it’s moving slower.

Also, if you’re human, you’re going to have to become aware of and then wary of, a variety of other possible problematic ideas, like

Faster objects have more acceleration

A greater increase in speed means more acceleration

No velocity means no acceleration

Objects moving with same acceleration move in identical ways.

And we haven’t even begun to worry about having procedures for determining or estimating velocities at particular times or procedures for subtracting two velocities. We haven’t concerned ourselves with when it might be appropriate to model a situation with constant acceleration. We haven’t concerned ourselves with the difference between average and instantaneous velocity, or with strategies for selecting convenient intervals of time for carrying out one’s work. We haven’t talked about graphs and other representations. It turns out that acceleration is a high density idea.

Some Place Generative to Start:

Changing topics a little bit, one of the questions I have been thinking about is this: Given that there are so many ideas packed into definitions such as acceleration, which ideas are most generative? That is, which ideas serve as a good starting point for generating the entire set of ideas? Given such a good starting point, are there other ideas that come along for the ride? And I don’t mean logically generative–like you could derive certain ideas from others. I mean generative from a human learning perspective. What ideas serve as productive anchors or as productive leaping off points… So, now, I’m think, “Isn’t it odd to juxtapose the words anchor and leaping off point?” Like, one implies, “keeps you grounded somewhere.” The other implies “strong base from which to leave.” Those are totally different metaphors for generative starting place.

I also think about this a lot: Does the generative starting point need to be correct? or like a baby-version of correct? If I go with the anchor analogy, then yes, the generative starting point should be correct. It’s like “home base”–the place you are tethered too so you don’t get lost. It’s a trustworthy place to ground your thinking. But if I use the leap-pad analogy, then the most generative starting point can actually be a place you never return. It’s the place that launches you to the next place, which may be quite different, and possibly even wrong. I think we tend to operate under the tacit assumption that the starting points should be “anchors.” I think we have a hard time thinking about what a generative (but incorrect) launch pad would look like. I know I do. But still, I keep returning to the idea, because it has so many implications for how we might think about teaching, learning, and assessing progress.

Assessment within an Emergent Curriculum

This year, as has been the case other times, I won’t know exactly everything about how students are going to be assessed in one of my courses. There are multiple reasons for this:

(1) On the first day of class, students will help decide how their notebooks will be assessed based on on activity where we examine several scientists notebooks and try to reach some consensus about what the purpose of notebooks are and what should be included in one. From this, I will draft a rubric for which students will have to self-assess by pointing me to various points of their notebook that show evidence for standards and criteria that are set. While I have ideas that will contribute that will almost certainly make it the rubric, there are certain criteria that are bound to emerge to be particular to this class. Students will assess their notebooks three times during the semester. I’ll use these rubrics, in addition to examining their notebooks for completeness, in order to provide “grades” for this part of the course.

(2) Students are also accountable to the people and knowledge that is developed and made public within our class–including investigations carried out by other student research groups, various models of physical phenomena as they develop–including the ones that are proposed and later discarded, evidence we collect, arguments we construct, and foothold ideas we establish along the way. Of course, a lot of this will be very closely aligned with canonical scientific understandings; but they will also be embedded in our specific classroom discourse, the particular investigations we carryout, and the arguments we construct. I know that we will make contact with ideas such as “light travels in straight lines”, “light goes out in all directions”, and rules about how light interacts at various surfaces, but I’m not just assessing them on whether or not just “know” and “understand” these rules. I am assessing them on their ability to make claims, to explain and construct arguments, and to do so in ways that are accountable not only to this knowledge, but to the ways in which our class has come to know and understand them and to talk about them. While there are specific criteria explained to the students for what it means to be accountable to knowledge and the communities that generate them, the conceptual substance of the knowledge for which they will be held emerges within the curriculum rather being imposed at the start.

(3) Participation is a part of the grade for this course, but not in an attendance sort of way. They are assessed on their participation in the community–as a participant who contributes to the development of knowledge and the activities which serve to generate it. In the class, students work both as part of a “research group” and “writing group” to which they are accountable for making contributions both as developers and critics. Students also play a role in sharing work from their research groups to the whole class. As with the notebooks, students have to self-assess and submit to me evidence of their contributions to the knowledge community. Of course, students are not expected to participate the very same way. The self-assessment rubric allows students to participate differently, but it still must be significant and substantive. One of the biggest part of the self-assessment is them telling where they could improve their participation. Once again, there are specific criteria I share with students and ask them to provide evidence, but what exactly it looks like to be a contributing member within a community varies from person to person and from class to class, just as it does from community to community within science.

It sounds like there is a lot of ambiguity. I think there is and there isn’t. So I have decided this. The goal I have for myself, during and after this semester, is to work on better articulating each part of the course such that I might be able to create more specific learning standards. The nature of the course makes some of this difficult, but I think it will still worth it in the long run.

More on sig. figs.

Andrew asks us sig-fig-haters, “How would you want students to report this measurement?

” I measured the length and width of a sheet of printer paper in centimeters. I came up with l=27.95 cm and w=21.60 cm. Each of the measurements I believed to be within ±0.05 cm. If I want to find the area, what value should I report? l×w=603.72 cm2 without regard to the number of figures being reported.”

Ignoring units for a moment, I’d be happy with any of following, plus some more:

604 ± 2
603.7  ± 1.9
“A little more than 600 cm , give or take a few”
“Most likely somewhere in between 601 and 605.”

So, beyond all of that, to me, it’s important to distinguish among three different things:

(1) The habits of mind we are trying to cultivate (e.g., here a sensitivity toward describing measurement in terms of distribution in which the actual value is likely to reside, and an understanding of why we might care to do so)

(2) A particular strategy or set of strategies we want students to feel confident using when determining those distributions (e.g., crank three times, monte carlo, calculus methods, etc.)

(3) The particular standards or conventions for reporting measurment and uncertainty when publishing for an broader audience (e.g., error bars, sig figs, confidence intervals, etc).

My thinking: If you are conflating these three, you are going to run into trouble. Of course, I want students to learn #1. But I don’t think you can learn that in the abstract. So we’ll have to talk about #1 in relation to a variety of #2’s. But doing #2’s of course doesn’t imply that they are supporting #1. Students can learn to do carry out the procedures of #2 without any idea of #1.  #3 is where I think I should be really careful. Me personally, I’d hold off on demanding particular standards until students are publishing their work for an audience greater than the teacher. Different communities, even different journals, have different standards for reporting measurements and their uncertainty. Those conventions serve a purpose, but I want students to make contact with the necessity for convention when they are communicating their research to an audience (even if just peers in class), not just completing exercises for homework or on exams.

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