“The course is a short intervention designed to change students’ relationships with math. I have taught this intervention successfully in the past (in classrooms); it caused students to re-engage successfully with math, taking a new approach to the subject and their learning. In the 2013-2014 school year the course will be offered to learners of math but in July of 2013 I will release a version of the course designed for teachers and other helpers of math learners, such as parents…” [emphasis is original]

I’ve been disheartened this year to realize how limited my toolset is for convincing students to broaden their thinking about the meaning of math.  Every year, I tangle with students’ ingrained humiliation in the face of their mistakes and sense of worthlessness with respect to mathematical reasoning. I model, give carefully crafted feedback, and try to create low-stakes ways for them to practice analyzing mistakes, understanding why math in physics gives us only “evidence in support of a model” — not “the right answer”, and noticing the necessity for switching representations.  This is not working nearly as well as it needs to for students to make the progress they need and that I believe they are capable of.

I hope this course will give me some new ideas to think about and try, so I’ve signed up.  I’m especially interested in the ways Boaler is linking these ideas to Carol Dweck’s ideas about “mindset,” and proposing concrete ideas for helping students develop a growth mindset.

Anyone else interested?

This question is at the heart of a lot of my classroom’s success and also anxiety.  When I ask good questions, students are more likely to evaluate evidence thoroughly, seek contradictions, resolve those contradictions, hold each other and themselves accountable to what we know so far, and generate significant new questions for our next round of research.

A poorly-chosen question reveals itself when students don’t have enough information or skill to make sense of the information they find, or can’t think of ways to find information at all, or don’t care about the answer, or can’t see how it’s related to their goal of becoming an electronics tech.

I’m intrigued by what’s going on in this video, a clip of a TED talk featuring Bobby McFerrin (of “Don’t Worry Be Happy” fame, but also a brilliant performer of many genres).  For maximum benefit, try singing along.

The question McFerrin asks himself seems to be, “what musically honest question can I ask this audience?”

The question he poses to the audience is, “What’s the next note?”

This question works because he was able to

• anticipate the ideas participants are likely to have about the topic (the pentatonic scale is surprisingly cross-cultural)
• anticipate which ideas are difficult to learn, and which ones are not (he avoids certain scale degrees and uses a tune that’s going to be structurally familiar to an American audience)
• choose a question that’s simple enough for people to make sense of using the tools they already have
• make the task interesting (and the big picture “audible”) by doing the more complicated work himself.

I’m getting much better at anticipating common initial ideas and eliciting my students’ thinking.  I’m still not great at choosing the question, or choosing the right moments to suggest the question.

I’m not sure what I make of this.  In the video, the participants are not exactly learning something new.  They are realizing something they didn’t realize that they already knew.  This doesn’t give me much insight into tackling the topics that are difficult to learn.  But I keep thinking about it.

Student 1: “Is the charge time always the same as the discharge time?”

Me: “According to this model, it is, if the resistance and capacitance haven’t changed.”

Student 2: “I’ve got data where the charge time was short and the discharge time was long.”

Me: “Why would a reasonable teacher say something that contradicts your data?”

Student 3, excitedly: “What circuit was it?  Was there anything else in the circuit?”

Student 1: “I can’t remember what it was called — it had a resistor, a capacitor, and a diode.”

Student 2: “That’s it then!  The diode — it’s changing its resistance!”

Student 1: “Yes — it goes from acting like a short to acting like an open.  Thanks for bringing that up [Classmate's Name] — I just answered a HUGE question from that lab!”

Student services counsellor who sat in for a day

“You’re challenging my whole idea about science.”

While exploring why capacitors act like more and more resistance as they charge

“Maybe the negative side of the cap is filling up with electrons, which means less capacitance.  According to the ‘tau model’, charge time = 5 * R * C.  So if the charge time never changes, and the capacitance is going down, then the resistance must be going up.”

[I'm excited about this because, although it shows a misunderstanding of the definition of capacitance, the student is tying together a lot of new ideas.  They are also using proportional reasoning and making sense of the story behind a formula. I need a better way to help students feel proud of things like this...]

Student critique of a Wikipedia page

“There’s some great begging the question, right there!”

Student analyzing the mistake in their thinking about a resistor-diode circuit

“I didn’t think of current not flowing at all during the negative alternation of the source.  This would mean that the direction of current through the resistor does not technically change.  I thought that if current was flowing through the resistor, it would change direction even if there is a very small amount of current flowing.  I did do a good job about thinking of the electrons already in the wires.”

One student’s feedback on another student’s paper

“I understand fully what you are trying to explain!”

On figuring out why a diode works

“If you make the connection to a wire, it’s like how copper atoms…”

“If it wasn’t doped, wouldn’t current flow in both directions?”

Students discussing a shake-to-charge flashlight they are designing

“In our rechargeable flashlight, if you put the switch in parallel with the diode, when it’s closed it will just short it out…”

Student who gave a recruiting presentation at a high school

“The day was a great step up for me that I never ever thought possible.  To be able to go back to the high school where I am pretty sure most had given up hope on me and see and hear them tell me how proud they are of me for where I am today is a feeling I will never forget.”

• What is absolutely non-negotiable that my students should be able to do or understand when they graduate?
• How to I make sure they get the greatest opportunity to learn those things?
• How do I make sure no one graduates without those things?  And most frustratingly,
• How do I reconcile the student-directedness of inquiry learning with the requirements of my diploma?

Some people might disagree that some of these points are worth worrying about.  If you don’t teach in a trade school, these questions may be irrelevant or downright harmful.  K-12 education should not be a trade school.  Universities do not necessarily need to be trade schools (although arguably, the professional schools like medicine and law really are, and ought to be).  However, I DO teach in a trade school, so these are the questions that matter to me.

Training that intends to help you get a job is only once kind of learning, but it is a valid and important kind of learning for those who choose it.  It requires as much rigour and critical thinking as anything else, which becomes clear when we consider the faith we place in the electronics technicians who service elevators and aircraft. If my students are inadequately prepared in their basic skills, they (or someone else, or many other people) may be injured or die. Therefore, I will have no truck with the intellectual gentrification that thinks “vocational” is a dirty word. Whether students are prepared for their jobs is a question of the highest importance to me.

In that light, my questions about job-readiness have reached the point of obsession.  Being a technician is to inquire.  It is to search, to question, to notice inconsistencies, to distinguish between conditions that can and cannot possibly be the cause of particular faults.  However, teaching my students to inquire means they must inquire.  I can’t force it to happen at a particular speed (although I can cut it short, or offer fewer opportunities, etc.).  At the same time, I have given my word that if they give me two years of their time, they will have skills X, Y, and Z that are required to be ready for their jobs.  I haven’t found the balance yet.

I’ll probably write more about this as I try to figure it out.  In the meantime, Grant Wiggins is writing about a recent study that found a dramatic difference between high-school teachers’ assessment of students’ college readiness, and college profs’ assessment of the same thing.  Wiggins directs an interesting challenge to teachers: accurately assess whether students are ready for what’s next, by calibrating our judgement against the judgement of “whatever’s next.”  In other words, high school teachers should be able to predict what fraction of their students are adequately prepared for college, and that number should agree reasonably well with the number given by college profs who are asked the same question.  In my case, I should be able to predict how well prepared my students are for their jobs, and my assessment should match reasonably the judgement of their first employer.

In many ways I’m lucky: we have a Program Advisory Group made up of employer representatives who meet to let us know what they need. My colleagues and I have all worked between 15 and 25 years in our field. I send all my students on 5-week unpaid work terms.  During and after the work terms, I meet with the student and the employer, and get a chance to calibrate my judgement.  There’s no question that this is a coarse metric; the reviews are influenced by how well the student is suited to the culture of a particular employer, and their level of readiness in the telecom field might be much higher than if they worked on motor controls.  Sometimes employers’ expectations are unreasonably high (like expecting electronics techs to also be mechanics).  There are some things employers may or may not expect that I am adamant about (for example, that students have the confidence and skill to respond to sexist or racist comments).    But overall, it’s a really useful experience.

Still, I continue to wonder about the accuracy of my judgement.  I also wonder about how to open this conversation with my colleagues.  It seems like something it would be useful to work on together.  Or would it?  The comments on Wiggins’ post are almost as interesting as the post itself.

It seems relevant that most commenters are responding to the problem of students’ preparedness for college, while Wiggins is writing about a separate problem: teachers’ unfounded level of confidence about students’ preparedness for college.

The question isn’t, “why aren’t students prepared for college.”  It’s also not “are college profs’ expectations reasonable.”  It’s “why are we so mistaken about what college instructors expect?“

My students, too, often miss this kind of subtle distinction.  It seems that our students aren’t the only ones who suffer from difficulty with close reading (especially when stressed and overwhelmed).

Wiggins calls on teachers to be more accurate in our assessment, and to calibrate our assessment of college-readiness against actual college requirements. I think these are fair expectations.  Unfortunately, assessment of students’ college-readiness (or job-readiness) is at least partly an assessment of ourselves and our teaching.

A similar problem is reported about college instructors.  The study was conducted by the Foundation for Critical Thinking with both education faculty and subject-matter faculty who instruct teacher candidates. They write that many profs are certain that their students are leaving with critical thinking skills, but that most of those same profs could not clearly explain what they meant by critical thinking, or give concrete examples of how they taught it.

Self-assessment is surprisingly intractable; it can be uncomfortable and can elicit self-doubt and anxiety.  My students, when I expect them to assess their work against specific criteria, exhibit all the same anger, defensiveness, and desire to change the subject as seen in the comments.  Most of them literally can’t do it at first.  It takes several drafts and lots of trust that they will not be “punished” for admitting to imperfection.  Carol Dweck’s work on “growth mindset” comes to mind here… is our collective fear of admitting that we have room to grow a consequence of “fixed mindset”?  If so, what is contributing to it? In that light, the punitive aspects of NCLB (in the US) or similar systemic teacher blaming, isolation, and lack of integrated professional development may in fact be contributing to the mis-assessment reported in the study, simply by creating lots of fear and few “sandboxes” of opportunity for development and low-risk failure.  As for the question of whether education schools are providing enough access to those experiences, it’s worth taking a look at David Labaree’s “The Trouble with Ed School.”

One way to increase our resilience during self-assessment is to do it with the support of a trusted community — something many teachers don’t have.  For those of us who don’t, let’s brainstorm about how we can get it, or what else might help.  Inaccurate self-assessment is understandable but not something we can afford to give up trying to improve.

I’m interested in commenter I Hodge’s point about the survey questions.  The reading comprehension question allowed teachers to respond that “about half,” “more than half,” or “all, or nearly all” of their students had an adequate level of reading comprehension.  In contrast, the college-readiness question seems to have required a teacher to select whether their students were “well,” “very well,” “poorly,” or “very poorly” prepared.  This question has no reasonable answer, even if teachers are only considering the fraction of students who actually do make it to college.  I wonder why they posed those two questions so differently?

Last but not least, I was surprised that some people blamed college admissions departments for the admission of underprepared students.  Maybe it’s different in the US, but my experience here in Canada is that admission is based on having graduated high school, or having gotten a particular score in certain high school courses.  Whether under-prepared students got those scores because teachers under-estimated the level of preparation needed for college, or because of rigid standards or standardized tests or other systemic problems, I don’t see how colleges can fix, other than by administering an entrance test.  Maybe that’s more common than I know, but neither the school at which I teach nor the well-reputed university that I (briefly) attended had one.  Maybe using a high school diploma as the entrance exam for college/university puts conflicting requirements on the K-12 system?  I really don’t know the answer to this.

Wiggins recommends regularly bringing together high-school and college faculty to discuss these issues.  I know I’d be all for it.  There is surely some skill-sharing that could go back and forth, as well as discussions of what would help students succeed in college.  Are we ready for this?

In that light, I’m thinking back on my post and the comments.  For example:

Me:

I talk about the difference between “who/what you are” (the definition of you) and “what caused you” (a meeting of sperm and egg).  In the systems of belief that my students tend to have, people are not thought to “just happen” or “cause themselves.”  It can help open the conversation.  However, even when I do this, they are surprisingly unlikely to transfer that concept to atomic particles.

James:

“Purpose is a REAL facet in all of nature because everything has a natural function e.g., the role of mitochondria in eukaryotic cells is ATP production, or that the nature of negatively charged electrons is to attract and repel + and – charged particles respectively, etc.”

Luke:

But I think it’s the same mistake to presume that they really *mean* that the electron has desires and wants, which is a slippery slope to thinking they *can’t* access or feel the need to explore the deeper causal relationships.

I’m noticing that there are ideas I expect students to extend from humans to particles (forces can act on us), and ideas I expect them to find not-extensible (desire).  These examples are the easy ones; “purpose” is harder to place clearly in one category or the other, and “cause” probably belongs in both categories but means something different in each.  I need to think more clearly about which ones are which and why, and how to help students develop their own skills for distinguishing.

I’m trying to stop assuming that when students talk about electrons’ “desires,” that they are referring to a deeper story; I also need to avoid assuming that they are not, or that they don’t want to/aren’t drawn to.

I’m on a personal “fast” of discussing electrons’ purposes and desires, at least while I’m in earshot of my students.  It’s hard to break those habits, exactly because they are so helpful.  However, it has the useful result that all the ideas about purpose and desires that are getting thrown around in class come from the students.  The students seem more willing to question them than when the ideas come from me.  Unfortunately they are having a really hard time understanding each other’s metaphors (even though the metaphors are not particularly far-fetched, by my reckoning), and I’m having a really hard time facilitating the conversation to help them see each other’s point of view.  But that still seems better than before, when the metaphors were not getting questioned at all, and maybe not even noticed as metaphors.

Michael distinguishes between practice and hard practice, and wonders whether blogging belongs on his priority list:

“Hard practice makes you better quickly. Practice lets you, essentially, plateau. …

Put it like this: do you feel like you’re a 1st year teacher when you blog? Does your brain hurt? Do you feel as if you’re lost, unsure how to proceed, confused?

If not, you’re not engaged in hard practice.”

Ooof.  On one hand, it made me face my desire to avoid hard practice; I feel like I’ve spent the last 8 months trying to decrease how much I feel like that.  I’ve tried to create classroom procedures that are more reuseable and systematic, especially for labs, whiteboarding sessions, class discussions, and model presentations.

It’s a good idea to periodically take a hard look at that avoidance, and decide whether I’m happy with where I stand.  In this case, I am.  I don’t think the goal is to “feel like a first year teacher” 100% of the time; it’s not sustainable and not generative.  But it reminds me that I want to know which activities make me feel like that, and consciously choose some to seek out.

Michael makes this promise to himself:

It’s time to redouble my efforts. I’m half way through my third year, and this would be a great time for me to ease into a comfortable routine of expanding my repertoire without improving my skills.

I’m going to commit to finding things that are intellectually taxing that are central to my teaching.

It made me think about what my promises are to myself.

Be a Beginner

Do something every summer that I don’t know anything about and document the process.  Pay special attention to how I treat others when I am insecure, what I say to myself about my skills and abilities, and what exactly I do to fight back against the fixed-mindset that threatens to overwhelm me.  Use this to develop some insight into what exactly I am asking from my students, and to expand the techniques I can share with them for dealing with it.

Last summer I floored my downstairs.  The summer before that I learned to swim — you know, with an actual recognizable stroke.  In both cases, I am proud of what I accomplished.  In the process, I was amazed to notice how much concentration it took not to be a jerk to myself and others.

Learn More About Causal Thinking

I find myself being really sad about the ways my students think about causality.  On one hand, I think my recent dissections of the topic are a prime example of “misconceptions listening” — looking for the deficit.  I’m pretty sure my students have knowledge and intuition about cause that I can’t see, because I’m so focused on noticing what’s going wrong.  In other words, my way of noticing students’ misconceptions is itself a misconception.  I’d rather be listening to their ideas fully, doing a better job of figuring out what’s generative in their thinking.

What to do about this? If I believe that my students need to engage with their misconceptions and work through them, then that’s probably what I need too. There’s no point in my students squashing their misconceptions in favour of “right answers”; similarly, there’s no point in me squashing my sadness and replacing it with some half-hearted “correct pedagogy.”

Maybe I’m supposed to be whole-heartedly happy to “meet my students where they are,” but if I said I was, I’d be lying. (That phrase has been used so often to dismiss my anger at the educational malpractice my students have endured that I can’t even hear it without bristling).  I need to midwife myself through this narrow way of thinking by engaging with it.  Like my students, I expect to hold myself accountable to my observations, to good-quality reasoning, to the ontology of learning and thinking, and to whatever data and peer feedback I can get my hands on.

My students’ struggle with causality is the puzzle from which my desire for explanation emerged; it is the source of the perplexity that makes me unwilling to give up. I hope that pursuing it honestly will help me think better about what it’s like when I ask my students to do the same.

Interact with New Teachers

Talking with beginning teachers is better than almost anything else I’ve tried for forcing me to get honest about what I think and what I do.  There’s a new teacher in our program, and talking things through with him has been a big help in crystallizing my thoughts (mutually useful, I think).  I will continue doing this and documenting it.  I also put on a seminar on peer assessment for first-year teachers last summer; it was one of the more challenging lesson plans I’ve ever written.  If I have another chance to do this, I will.

Work for Systemic Change

I’m not interested in strictly personal solutions to systemic problems.  I won’t have fun, or meet my potential as a teacher, if I limit myself to improving me.  I want to help my institution and my community improve, and that means creating conditions and communities that foster change in collective ways.  For two years, I tried to do a bit of this via my campus PD committee; for various reasons, that avenue turned out not to lead in the directions I’m interested in going.  I’ve had more success pressing for awareness and implementation of the Workplace Violence Prevention regulations that are part of my local jurisdiction’s Occupational Health and Safety Act.

I’m not sure what the next project will be, but I attended an interesting seminar a few months ago about our organization’s plans for change.  I was intrigued by the conversations happening about improving our internal communication.  I’ve also had some interesting conversations recently with others who want to push past the “corporate diversity” model toward a less ahistorical model of social justice or cultural competence.  I’ll continue to explore those to find out which ones have some potential for constructive change.

I’m also planning to go on deferred leave for four months starting next January.  After that, I’ve made it a priority to find a way to work half-time.   The kind of “intellectually taxing” enrichment that I need, in order for teaching to be satisfying, takes more time than is reasonable on top of a full-time job.  I’m not willing to permanently sacrifice my ability to do community volunteer work, spend time with my loved ones, and get regular exercise. That’s more of a medium-term goal, but I’m working a few leads already.

Anyone have any suggestions about what I should do with 4 months of unscheduled time starting January 2014?

My Interpretation

They’re more confident in their time management, their organization, and their control over their learning.  I’m doing a better job of anticipating their thinking, and when I fail, a better job of not being visibly dismayed! They’ve made major improvements in their ability to articulate their ideas, especially their disagreements, clearly and respectfully.

Their Words

Letting myself make mistakes is how I learn the most.  Being able to reassess is allowing me to do this.

It seems there is more  material to cover compared to semester 1 — not sure if something could be moved to level out the material.

Fast-paced but able to keep up

Extensions help

Material is interesting — never boring or stale.

Students are contributing more in conversation — I see a noticeable improvement

Real-life situations — big improvement!

Hard to soak all the information in

Quit job or at least ask for time off

We are helping each other out more now than before.  It helps when others are stuck and have classmates to give a hand.

You do a great job being supportive

Teaching is great.  Having [conversations] at the end of labs really helps dig up the “funny,” also makes it easier to grasp important details that might get missed otherwise.

Things sometimes seem overwhelming but always manageable.

More people are showing up on time, prepared.

I think you have improved a lot with the understanding and being patient.

Horseplay in the lab is distracting — students should manage their time better instead of complaining about workload

Being able to book a meeting makes skills easy to get signed off, get to have 1:1 time with teach and ask questions, figure out problems.

There are a few catchphrases that damage my calm, and this is one of them.  I was discussing it with my partner’s daughter, who’s a senior in high school, and often able to provide insight into my students’ thinking.  I was complaining that students seem to treat the model (of circuit behaviour knowledge we’ve acquired so far) like their baby, fiercely defending it against all “threats,” and that I was trying to convince them to have some distance, to allow for the possibility that we might have to change the model based on new information, and not to take it so personally.  She had a better idea: that they should indeed continue to treat the model like a baby — a baby who will grow and change and isn’t achieving its maximum potential with helicopter parents hovering around preventing it from trying anything new.

The next time I heard the offending phrase, I was ready with “How do you expect a baby model to grow up into a big strong model, unless you feed it lots of nutritious new experiences?“

It worked.  The students laughed and relaxed a bit.  They also started extending their existing knowledge.  And I relaxed too — secure in the knowledge that I was ready for the next opportunity to talk about “growth mindset for the model.”

“At first I didn’t understand why we had to learn these complicated methods when we could just do it the simple way you showed us last semester.  But when you get to these complicated circuits, it makes it so much easier.  I do math every night now, even if I don’t have any for homework, because you have to exercise all the time or you lose it.”

On graphical waveform addition

“I got off to a bad start with this, I had the wrong answers for everything, and I really didn’t know how to do it.  I won’t lie.  But now after taking all these measurements, I’m starting to understand.  And I did really bad on that first quiz — I didn’t even know what DC offset was. But I made up some practice problems that are a little bit different from the quiz, and I can do them now.”

On AC voltage, sinusoidal signals, and what the time domain really means

“I just realized that the word ‘electronics’ has the word ‘electron’ in it. ” (x2) (After a conversation about how a sinusoidal signal represents a voltage or current that changes over time)

“Is this why we need DC voltage for electronics — so it doesn’t turn off all the time?”

“In an AC circuit, how to the electrons get their energy back after they’ve lost it?” (I love the insight in this question — the synthesis of ideas, the demand for a coherent cause)

While presenting some routine lab measurements

“How does an electron know how much voltage to drop in each component?” (7 months later, students are suddenly gobsmacked by the totally weird implications of Kirchoff’s Voltage Law)

During a one-on-one discussion of the group’s interpersonal dynamics

“I find no one in this program is looking for someone to give them the answers.  We might text all night long about homework but it’s never ‘Can you send me X,’ it’s always ‘How can I figure out X?’ “

While whiteboarding some AC circuit data

“I don’t like saying that KVL applies to instantaneous voltages, because it applies everywhere.”

“But if you say instantaneous, it applies in a general sense. Have you ever seen an AC circuit where the component voltages didn’t add up to the supply?”

Another whiteboarding session

“Make sure you’re talking about electrons, otherwise it’s not a cause!”

“And that’s supported by the model, because…”

While designing an experiment

“Do you have a 1uF capacitor?” “No, I guess we can use 100uF and scale it…”  (Students making big gains in proportional reasoning)

After discussing how a capacitor’s voltage approaches an asymptote

“I never noticed before how much math relates to life — like the idea that sometimes the closer you get to something, the harder it is to get there.  I guess it’s not surprising — because math comes from life.  Math is everything.”

The challenge now will be helping them get in the habit of doing this consistently.  I’m thinking of requiring this on reassessment applications.  The downside would be a lot more applications being returned for a second draft, since most students don’t seem able to do this kind of analysis in a single draft.

Understand What’s Strong

• “I thought it was a parallel circuit, and my answer would have been right if that was true.”

• “I got this question wrong but I used the idea from the model that more resistance causes less current and less current causes less power to be dissipated by the light bulbs.”

• “The process of elimination was a good choice to eliminate circuits that didn’t work.”

• “A good thing about my answer is that I was thinking if the circuit was in series, the current would be the same throughout the circuit.”

Diagnose What’s Wrong

• “The line between two components makes this circuit look like a parallel circuit.”

• “What I don’t know is, why don’t electrons take the shorter way to the most positive side of the circuit?”

• “I made the mistake that removing parallel branches would increase the remaining branches’ voltage.”

• “What I didn’t realize was that in circuit 2, C is the only element in the circuit so the voltage across the light bulb will be the battery voltage, just like light bulb A.”

• “I looked at the current in the circuit as if the resistor would decrease the current from that point on.”

• “I think I was thinking of the A bulb as being able to move along the wire and then it would be in parallel too.”

• “What I missed was that this circuit is a series-parallel with the B bulb in parallel with a wire, effectively shorting it out.”

• “What I did not realize at first about Circuit C was that it was a complete circuit because the base of the light bulb is in fact metal.”

• “I thought there would need to be a wire from the centre of the bulb to be a complete circuit.”

• “I wasn’t recognizing that in Branch 2, each electron only goes through one resistor or the other.  In Branch 1, electrons must flow through each resistor.”

• “I was comparing the resistance of the wire and not realizing the amount of distance electrons flowed doesn’t matter because wire has such low resistance either way.”

• “My problem was I wasn’t seeing myself as the electrons passing through the circuit from negative to positive.”

Improve

• “In this circuit, lightbulb B is shorted so now all the voltage is across light bulb A.”

• “When there is an increase in resistance, and as long as the voltage stays constant, the current flowing through the entire circuit decreases.”

• “After looking into the answer, I can see that the electrons can make their way from the bottom of the battery to the middle of the bulb, then through the filament, and back to the battery, because of metal conducting electrons.”

• “To improve my answer, I could explain why they are in parallel, and also why the other circuits are not parallel.”

• “I can generalize this by saying in series circuits, the current will stay the same, but in parallel circuits, the current may differ.”

• “From our model, less resistance causes more current to flow.  This is a general idea that will work for all circuits.”