I’ve been thinking a lot about the sequence of odd numbers and its relationship to accelerated motion.

1, 3, 5, 7, 9, 11, 13…

In a previous post, I came to this sequence by discussing a special kind of motion that I defined as always covering 3x the distance in the second half of a trip than in the first. You can check that the odd number sequence meets this criteria.

Moving up the sequence, each number merely represents the Δx covered in successive intervals of time. Because they are equal time intervals, they also represent the average velocity over successive intervals. The sequence shows that the average velocity is increasing by 2 chunks each second. This of course means, that the ball has a constant acceleration of “+2”

You might think that we’d have to craft a whole new sequence for different accelerated motions. But you don’t. If we want a sequence with twice as much acceleration, we just multiply each number in the sequence by two. If you want to cut the acceleration in half, just half every number in the sequence.

You can also use this sequence to describe motions not starting from rest. For example, if you want something that starts with some initial velocity and speeds up, you just start somewhere else in the sequence besides the beginning. If you want something slowing down, you just move backwards in the sequence.

I’m not saying this is anything new. I’m just saying it’s interesting to think that sequence 1, 3, 5, 7, 9… is a representation for constant acceleration, and indeed any motion involving constant acceleration.

Does this way of thinking help us to solve problems?

Let’s say you want to answer the question, what is the acceleration of a ball that rolls down a 16ft ramp in 4 seconds?

All you do is try to come up with the sequence of 4 numbers that gets you to 16 ft. This one is easy: 1 *(1 + 3 +5 +7 ) = 16 ft.  Since the series steps in increments of 2, the speed is increasing by +2 ft each second. As a result, the acceleration is 2 ft/s each second.

I personally thought it would be hard to come up with the sequence for 200 cm in 10 seconds, but it’s not. The answer is

2 * (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19) = 2 + 6 + 10 + 14 + 18 + 22 + 26 + 30 + 34 + 38. This clearly has an accelerated of 4 ft/s each second. It’s easy to check my sequence adds to 200 by adding in groups of 40, which is the insight you need to make producing these sequences so easily. Getting good at this takes some practice, and it is still terribly difficult for many situations. But the idea is simple: The numbers you add up must be the odd number sequence or a multiple of it.

OK. Where to next? One way of moving beyond writing down a million sequences for every new problem is to benefit from the fact that a sequence of odd numbers is always a square. For example,

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

In general if you are summing to the Nth odd number, then Σ (2n -1) = N² … this is of course the same as saying that accelerated motion goes like t².

Aside: You can prove this equality by showing that difference between two successive squares is always an odd number… n² – (n-1)² = n² – (n²-2n +1) = 2n-1

This tells us that our original sequence of odd numbers, which we said shows an acceleration of “+2”, goes like Σ(Δx) = n² , where n is the final place in the sequence. In other words, if you go for 10 chunks of time, you get to 100 chunks of distance.

This allows you to some really nice estimations: Let’s say you are trying to figure out what acceleration is for a ball covering 372 ft in 19 seconds (starting from rest).

Let’s use our rule of squares: 19² = 361ft, which is the distance that an object accelerating at 2 ft/s/s would get after 19 seconds. Given that we landed at 372,  the acceleration must be a little big bigger than 2ft/s/s? How much bigger? Well, 372/361 = 1.03… or three percent further than you’d expect. Thus the acceleration should also be 3% greater than expected, or  2.06 ft/s/s

If you want to write the sequence down, it’s just 1.03 * (1+3+ … + 35 + 37) = 1.03 (19)² = (2.06 /2) (19)²

I don’t think you have to. But I do think that reasoning from well-understood special cases is an important skill. The a =2 case is really compelling and interesting to think about because

The successive distances are simply the sequence of odd numbers

The successive positions are simply the sequence of square numbers

Understanding how this special case relates to other cases is also interesting

(1) Different magnitudes of acceleration are merely multiples of the odd number sequence

(2) Starting with some initial velocity is merely starting mid-sequence

(3) Slowing down is running the sequence backwards

I know I have a different way of thinking about why accelerated motion goes like time squared, and I have some news tools make it really easy to estimate accelerations, positions, etc

I’ve been blogging over the past week about how some students in my inquiry course are unhappy about my capstone-for-an-A grading policy. Here are what some students had to say in the anonymous feedback I asked for:

“I know there are some people who aren’t as invested in science as I am and are now settling to make a B simply because they’re not passionate about discovering something new, or don’t enjoy science. I feel that, in a way, it’s like punishing those who don’t like science, even if they put forth their best effort

“I’m worried that I can’t get an A in this class, and I’m an A student. It irks me that if I try my best, I’ll still just get a B”

“I find the grading policy system extremely frustrating, because why try? You won’t get an A unless you do a paper… I’m so frustrated, I am willing to just settle for the B.”

“I’m worried that I can’t get in A in this course without sacrificing A’s in the my five other courses”

“I don’t understand how you can say a 100% is the same as 83%. 100% should be an A no matter what

Mindset is a really interesting issue here. The effects of a schooled culture is a interesting issue. The real and perceived importance of grades is an interesting issue.

Thoughts?

I’m feel fairly capable fostering “thinking about phenomena” and “understanding of concepts”. I feel I can motivate it. I feel like I can engage students in it. I feel like I can even break down that such thinking and concepts into bite-size parts for student consumption. I feel like I can structure sequences of activities and questions that help students grapple with their own thinking and understanding of concepts. When I’ve done all that–when I’ve motivated tasks and students understand concepts–I feel fine teaching students procedures for solving problems and helping them to become proficient and careful in working through problems.

But I am lousy at teaching procedures for procedures sake. Granted, I don’t want to teach mindless procedures. But the truth is I have to. I am teaching a physics class where students need to become proficient at procedures that make no sense. For this purpose-of learning how to teach procedures without concepts–I need to think more like Khan Academy. I need to think more carefully how to motivate procedures, how to break down procedures into consumable parts, and how to sequence student contact with aspects of those procedures. Lastly, I need to foster up in my gut a sense that I care that students learn this (even if I don’t feel I should). I can become better at teaching procedures. It is a new goal of mine. Even if I never have to teach mindless procedures again, it will help me teach mindful procedures as well.

This pair of questions was on the multiple-choice test my students took:

A rock is thrown vertically upward, slowing as it rises until it reaches it high position, where it stops momentarily before falling back to the ground. Take the positive y-direction to be upwards.

1. Immediately after the the rock is thrown, its y-component of acceleration is _______

A. Negative

B. Zero

C. Positive

D. Not enough information to tell

2. Just before the rock hits the ground,  its y-component of acceleration is _______

A. Negative

B. Zero

C. Positive

D. Not enough information to tell

For all 250 some students,

30% of students answered negative for both (NN): To me, the answer pair might serve as a proxy for understanding acceleration as a change in velocity and an understanding of signing conventions for vector quantities.

25% answered negative on the way up and positive on the way down (NP): To me, answer pair might serve as a proxy for understanding acceleration as change in speed, with student thinking that negative signs mean slowing down and positive signs means speed up.

30% answered positive on the way up and negative on the way down (PN): This pair serves as a proxy for not having disentangled velocity (speed) and acceleration

15% for all other combinations: This serves as a collection of students who either misunderstood the question or have other confusions about concepts.

As is, each questions was graded independently. This makes it so that answering negative on either question gets you one answer correct. To me, this make no sense, because students’ answer to any one question is meaningless in terms of what they might know about acceleration. From the MC-question alone, we have evidence to support a claim that about 30% of students understand the concept fully, but 55% of them are getting some points. Additionally, we have evidence to support a claim that 25% understand the concept partially, but 55% got partial credit.

How might I grade this question?

I might grade this in terms of the pair combinations, and give more points to the student who answered NP than the student who answered PN. My reason for this would be that thinking of acceleration as denoting a change in speed is further along than someone who still has to figure out that acceleration is different from velocity (or speed).

The other option I would consider treating the two questions as one question, and only give credit for NN pair combinations, and give no credit for any other answer pairs.

Whether I did one or the other would depend on what I was trying to assess precisely about their understanding of acceleration.

What do you think?

Today, we took a break from our inquiry into light to do a mini-inquiry into motion and to inquire into childrens’ thinking about motion by watching and discussing a video-case study. We also took some time to examine and discuss some of National Standards and AAAS Benchmarks about what 2nd and 5th graders are expected to be able to do and understand about scientific inquiry. Most of that went pretty well, although we were all a bit zonked out after looking at a bunch of standards. Wednesday, we are back at light for just another day or two. I think we’ll all be happy to wrap up some of the loose ends and have a fresh start at something new.

After class, I got to read over their anonymous feedback about how class is going for them.  The range of things students bring up is quite broad, but the big three complaints are:

(i) Concerns about getting an A in my class and frustration at my Capstone A grading policy

(ii) Not getting the big picture of what we are doing and what they are expected to learn (hopefully I addressed some of this today)

(iii) Annoyance and frustration toward students who have a negative attitude about class

There were a range of other complaints including, boredom, not wanting to work in groups, wishing we were covering more topics, wanting to know the answers, complaining about the workload, being sick of light as a topic.

On Wednesday, I want to re-explain my Capstone policy and try to frame it in a more productive manner. I also want to talk about the exam that is coming up to reinforce what this class is about in the big picture. On the exam, students will be asked to do what we have been doing a lot of, namely

Explain their thinking about some physical situation, using words and diagrams to come to a prediction.

Make some observations and reconcile their prior thinking

Construct plausible alternative predictions and respond to them

Situate their explanations and ideas within emerging classroom ideas.

They’ll be assessed on what I’ve been assessing them on- clearly articulating your thinking, constructing arguments and counter-arguments, reconciling one’s ideas with new observations, etc.

Lastly, on Wednesday, I want to convey to them my (re)commitment to creating a prepared learning environment that can be fun and engaging. I do think class should be intellectually fun in ways that instill curiosity, puzzlement, wonder, and awe. I’m going to leave complaint #3 alone and just remind us all that at the beginning of the year, we all agreed that it was (as a class) we have responsibility to come to come to class with a good attitude and an open mind. I’ll work harder at making class engaging, they’ll work harder at allowing themselves to become engaged.

Brian,

I’ve been thinking a lot about my inquiry class this weekend. At some point I remembered what I actually did on the very first day of class the last time I taught an inquiry course for elementary school teachers. Instead of jumping into a month long inquiry into a topic, our class did an hour long inquiry into one physical situation and then we watched a video of children talking about the very same situation. In retrospect, this was a better decisions than I knew at the time.

First, it likely helped my students frame the science learning (I was going to ask them to do) in terms of their future careers as teachers. In particular, it sent a strong message about the potential connections that exist between their inquiries and children’s inquiries. I also, unknowingly, picked a topic and situation that adults and children tend to think the same about. Note that this isn’t always the case. Kindergarteners ideas about magnets, for example, tend to be centered around magnets being “energized”; whereas adults tend to think about magnets in terms of electric charges. Both,as we know, not how scientists think about it. But, importantly, there are many topics where children and adults tend to think the same. In the particular situation I asked them to reason about, it is likely the case that 4 or 5 ideas will always come up, and those will be the exact same 4 or 5 ideas that come up in the video. The video  shows children having these ideas, sharing these ideas, and listening and responding to each others’ ideas in a sophisticated way. Thus, what we do on the first day (and what the children do) serves as a model for what I want them to do over long periods of time during a sustained inquiry into a single science topic.

Another big advantage is that we get to talk about the fact that they were able to easily make sense of what the children were doing and saying, and that this might have been because they had just spent an hour thinking, listening, and sharing their own ideas about the situation. This way, I get to let a mini lesson and video make the case that they will better understand their children by doing science based on their own ideas and not simply memorizing canonical science understandings. That way, when we are 4 weeks deep into a muck of ideas about a topic like light and shadow, I can remind them why we are doing this. We can even take a break to inquire again into childrens’ thinking.

Next time you teach this course, come back an re-read this post. It has lots of good ideas, at least I think. Best of luck next time.

Brian

I decided to rewrite and organize my ideas about some possible standards for my inquiry course. Right now I’m only focused one strand:

A.  Creating, Supporting, and Developing My Own Ideas

1. Articulating Ideas
Generating Ideas
A1.1  I think about my own experiences in the world (looking for possible connections), as a source for ideas about how the world might work
A1.2 I ask questions about how the world works and discuss my own curiosities about the world
Expressing Ideas
A1.3 Given time to reflect and write, I can articulate my own initial thinking and ideas about phenomena in clear and specific ways
A1.4 I express my ideas and thinking through diverse representations that enhance or clarify my thoughts (e.g., drawings, sketches, diagrams, graphs)
2. Supporting Ideas

Evidence-based Support
A2.1 I consider supporting evidence or examples when making claims or presenting an idea.
A2.2 I go beyond citing evidence by detailing the rationale that explains why I think evidence supports claims and ideas.
Theory-based Support
A2.3 I can explain how different ideas that I have seem to ‘fit’ together in a logical way.
A2.4 I attend to the implications of my own ideas and articulate how those implications lead me to make certain conclusions or predictions
3. Sorting through Ideas

Theoretical Progress
A3.1 I look for inconsistencies among the various ideas that I have and I can compare and contrast competing ideas I have
A3.2 I seek to resolve inconsistencies among the ideas I have …
Empirical Progress
A3.3 I seek ways to put my ideas to the test through new careful observations and/or experimental design
A3.4 In the face of evidence that run counter to my ideas, I return to my ideas to ask questions and reconsider my thinking.

4. Maturing Ideas
Explanation
A3.1 I can provide explanations that tell a gapless story which detail exactly how specific conditions give rise to certain outcomes
A3.2 I can use diagrams to carefully work out the implications of ideas and use such diagrams to explain phenomena
Awareness
A3.3 I can explain the history of previously held ideas and the reasons for abandoning them
A 3.4 I am aware of my own level of commitment to ideas and can explain why in terms of evidence, arguments, & theory.

Chapter 1: Vermin and Ribbits

Here are two important things you need to know. There are ribbits and vermin. Ribbits tell you how much. Vermins tell you how much and what direction. Through the whole semester, you will need to think about the difference between ribbits and vermin.

Chapter 2: Yolks and Whites

Here is an equation

Y = Q/ R … if I ask you for Average Yolk, please find Q, find R, and plug those numbers in for Y. Q and R are ribbits, so Average yolks are ribbits, too. Please note that Q = Q1 + Q2 + Q3… and R = R1+R2+R3+…

Here is another equation

W = (q1-q2) / R … if I ask you for Average Whites, please find q1, q2, and R and plug those into the equation for W. (q1-q1) is a vermin, so W is also a vermin.

Don’t confuse Yolks and Whites. This would be a very easy mistake to make. Remember that yolk is a ribbit, which is always positive, while Whites are called vermin, which could be positive or negative. The positive and negative tell you the direction of the vermin.

Chapter 3: Changing Whites

There are four equations relevant to this understanding changing whites.

Equation 1: q2 = q1 + wi u + 1/2 s u²

Equation 2: wf = wi + s u

Equation 3: wf² = wi² + 2 u (q2-q1)

Equation 4: q2 = q1 + (wi +wf)/2 * u

We call “s” the snappy-juice, and wi and wf the starting and final whites. Don’t confuse starting whites and final whites with average whites. Average white is an only average, but starting whites and final whites are instantaneous. Get it? Great.

If you notice, there are six different letters above. q2, q1, wi, wf, u, and s. These are the six quantities you are going to have to always think about and list. If you notice, each of the above equations is missing at least one of these six quantities.

Equation 1 is missing wf

Equation 2 is missing all q’s

Equation 3 is missing u

Equation 4 is missing s.

This makes life really easy. Anytime I ask you for something, just list the six quantities and write next to it what I’ve told you. If I didn’t tell you something, list that letter as well, but put a question mark next to it. Here’s what you should do next: find the equation that’s missing the letter you don’t know but I haven’t asked you for. Picking that equation will ensure you will have no trouble finding what I’ve asked. Please then manipulate that equation to isolate the quantity you don’t know but I’ve told you to find, and then plug in the values I’ve told you. Listen carefully  to what I’ve told you will be really important. If you don’t, how could you possibly pick the right equation? Get it? Good.

Footnote: Listing knowns and unknowns is an expert problem-solving strategy. It’s what all experts do. Ask an expert. If you ask them what they do, they will certainly say, “I list what I know and don’t know, before I start thinking or proceeding to do anything.” This is why we are training you to do this. Even if you never have to learn physics again, you will know how experts solve problems. This will be really beneficial to you in your life.

Chapter 4: A special case of changing whites

OK. When I say a problem is about funfare, you should know that the vermin of s is b = 42. B is merely the ribbit of the vermin. B, the vermin, always points down because the flabber due to the eagle is down.  B, the ribbit of the s vermin, is always +42, and it would be wrong to think of b as -42. Rather sometimes s will = -b and sometimes +b, but b is always 42. You will decide whether s = +b or s = -b, on your choice of whether up or down is the positive direction for vermin. Get it? Good.

Now you know how to solve funfare problems: just do the same thing you did in chapter 3, except you know that the ribbit of s is 42.

Footnote: You may be tempted to think that b is not always 42. That would be wrong, all funfare problems have b = 42.

On top of a bad day of teaching last Wednesday, I also had to manage a mini-rebellion from some students in inquiry class:

Several students are unhappy about a few things, including

(1) That I use a Capstone Project for an A. In my course, a 100% for notebooks, tests, homework, and participation is the same as an 83%. It gives you a B and makes you eligible to get an A by doing an independent project.

I see it this way: You can bomb 1/6 of my course and still get an A. This puts in lots if wiggle room for mistakes, for missed assignments, struggling early on, etc.

Students see it this way: If I get an 100%, I still can’t get an A.

I see it this way:  Independent projects are optional. You can pass this course without doing one.

Students see it this way: To get an A, the teacher makes us do things that are optional.

(2) That this is an inquiry course and its different from other parallel classes. In my course, we cover three topics, so we spend a lot of time thinking, discussing, sharing, doing investigations, whiteboarding, etc. In other parallel courses, students cover a new topic each day or each week. Students simply aren’t used to classes like this and it doesn’t help my case when other classes are different. I understand that learning in a new way can be frustrating, especially when you’ve mastered the routine of (1) taking notes, (2) doing homework, (3) passing exams. There are definitely some students who are very frustrated, and I think they are frustrated for different reasons. Some are frustrated because they don’t know where we are going as a class and if we are really learning and making progress. I know that they are learning and that we are making progress, but it’s true that I don’t know exactly where we are going. Other students are frustrated because they wish they were in the other class, where it fit within their comfortable model of what a science class should be.

(3) That we are spending too much time on science In other parallel courses, instructors enrich the course by having students write up lessons plans and having students share them with each other. I am choosing to enrich my course by talking about the nature of science more and by watching videos of children doing science. I would say my course is 70% inquiry into science, 20% inquiry into the nature of science, and 10% inquiry into children’s thinking about science. Other courses are perhaps more like 60% activities related to science and 40% activities related to teaching science. Students are unhappy that the other courses are about teaching science,  where they get to collect, make, and share lesson plans; whereas my course we mostly do science.

I understand their frustrations. I also understand that not every student feels the way these students do. Many students have told me that they really enjoy the course. The Joss in my head, however, is reminding me that I didn’t spend enough time in the beginning of the course selling it–explaining why and convincing students that this way of teaching is in their best interest. So next week, it’s time to talk a little more about why I grade the way I do and why I am running the class the way I am.

What would it look like if I tried to do standards-based grading in my inquiry course? That’s a question I’ve been thinking about a lot. Here’s a quick rough draft of some more precise learning goals.

Creating Ideas

I reflect upon my own experiences and explore my own thinking as sources for possible ideas about how the world works.

I articulate my own thinking and ideas about phenomena in clear and specific ways that help others to understand them.

I ask questions about how the world works and document those questions as they arise

I express my ideas and thinking through drawings, sketches, diagrams, graphs, etc.

Developing Ideas

I seek out evidence to support claims that I make, and use evidence or examples to support or refute claims.

I go beyond just citing evidence by providing the rationale that explains why evidence either supports or refutes claims.

I attend to the implications of my own ideas and articulate how those implications lead me to make certain conclusions or predictions

In the face of evidence that run counter to my ideas, I return to my ideas to ask questions and reconsider my thinking.

I develop explanations that tell a gapless story that detail exactly how specific conditions give rise to certain outcomes.

Monitoring Ideas

I monitor how my ideas and thinking change over time and compare and contrast ideas I had at different times

I look for inconsistencies among the various ideas I have and compare and contrast competing ideas I have

I explain things that are confusing to me in ways that help others understand exactly where my confusion lies.

II. Others’ Thinking

I listen to my peers as source for ideas about how the world works.

When I don’t understand someone else’s idea, I inquire either by asking questions, trying to summarize what I thought they said, etc.

When I understand others’ ideas, I show my understanding of those ideas by writing or talking about them

When I understand another person’ ideas, I follow the logical implications of those idea (even when I disagree)

When I disagree with an idea, I provide a critical perspective toward this idea by constructing counter-arguments, citing contradictory evidence, or finding the flaw in some reasoning or premise.

I construct plausible counter-arguments by attending  to others’ ideas or thinking through their implications

I respond to counter-arguments by attending to the argument itself–perhaps by attending to some inconsistency in the reasoning or the faultiness of some premise.

III. Accountability to Community Norms

I can think and reason in ways that are consistent with our class’ foothold ideas

When my thinking departs from our class’ foothold ideas, I recognize this to be the case and point it out.

Things left out so far:

Distinguishing Observations vs. Inferences?

Designing an experiment? Tinkering?

Careful construction of diagrams?

Organization of writing?

Constructing clear definitions?

Clarity of writing?