Square Roots and Cube Roots

I’ve posted a lot about Calculus this year, and a bit about Multivariable Calculus too. But I’m not saying too much about Algebra II. Sorry. This year something is off, and the students aren’t as successful as in years past. I’m not exactly sure what to do. I’ve asked the student-led tutoring program to lead an Algebra II study group (we’ll see if anyone signs up). I also might change my teaching practice to allow more time for students to work problems in class — because I need to see more of them working and catch their errors in thinking earlier — before they go and practice the material wrongly at home.  We’ll get through less curriculum though if I do that, and that itself is a problem, since we’ve pared down the curriculum so much.

Anyway, that’s generally where I am with Algebra II.

Specifically, I just wanted advice on how you guys teach cube roots (and fourth roots and fifth roots, etc.).

My ordering usually goes:

1. Turn to your partner, and explain to them what \sqrt{5} means to someone who doesn’t know anything about square roots.

Students generally say that it’s the number that when multiplied by itself will give you 5. I then say “if the person doesn’t know anything about square roots, you might want to give them an easier example, like \sqrt{25}… and explain how that is 5. But that \sqrt{5} isn’t a perfect square so you’d get some number between 2 and 3. Yadda Yadda. I also then talk about the geometric interpretation (the side length of a square with area 5). Then I go back to the “it’s the number that when multiplied by itself will give you 5.”

I do not talk about there being two answers to “the number when multiplied by itself gives you 5” and the principal square root business. Because I want to use this to capitalize on their understanding of cube roots.

2. Then I put up \sqrt[3]{8} and say this is 2. And to think about what this funny \sqrt[3]{} symbol means. They get it. I put up a bunch more, and they usually can solve them. I put some negatives under the cube root symbol too.

3. I then ask them what \sqrt[3]{} means, and they say “the number that when multiplied by itself three times gives you the value under the cubed root sign.”

4. I then throw up a bunch of problems, and three of these include \sqrt{49} and \sqrt{-49} and -\sqrt{49}.

This is where the trouble comes in.

Some students now say \sqrt{49} is \pm 7. Because 7 and -7 are numbers when multiplied by itself which equals 49.

Here’s where I use the whole: “Don’t lose what you already know! Would you say \sqrt{49} is -7 ten minutes ago? No. You’re right, that there are two numbers which, when multiplied by themselves, give you 49. So we can tell them apart, we say \sqrt{49} is the positive one and -\sqrt{49} is the negative one. So don’t lose what you know. When you see a radical sign, it just represents a single number. If there’s a negative in front of it, it represents a negative number. If not, it’s a positive number. Just like what you’ve always known.”

Okay, now I know the idea of “principal square roots” and all that. And I honestly don’t want to have this whole discussion about principal square roots with them, because every time I do, they come out more confused.

So here’s my question.

How do you introduce cube (and higher) roots? How do you engage with this idea of principle square roots so that students don’t leave confused? I just can’t get it totally right.

And just so I am being clear, I know the properties of square roots and cube roots and all that. I’m not looking for someone to explain that to me. I want a way to teach my KIDS these without confusing them all up. And I bet crowdsourcing is a good way to get ideas for next year.

Where do I go from here?

Today in one of my two calculus classes today, we got on the topic of 0.\overline{9}.

I think it came up when we were talking about how to approximate the instantaneous rate of change in a problem: we had a function v=20\sqrt{T}, and we wanted to estimate the instantaneous rate of change when T=300.

So a student said let’s pick another point, such as T=299.99. And we found an approximate instantaneous rate of change. Of course I asked “how could I get this answer more precise?” and someone said “add more 9s!”

So we realized we could pick a closer point, such as T=299.9999999.

Of course, then we had some precocious youngster say: “why not get it super duper exact and plug in T=299.\overline{9}?”

Ah hah. Many of them thought that 299.\overline{9} was SUPER close to, but definitely not equal to, 300.

I went through the whole standard argument, which usually convinces most kids:

Let x=299.\overline{9}.  Then 10x=2999.\overline{9}.

So 10x-x=2700. Which means 9x=2700. Which means x=300.

I thought I had them. One student said I was breaking her worldview.

Ha.

But then, THEN, they asked me an awesome question.

One said, and the others jeered: “Isn’t 299.\overline{9} kinda breaking the rules of what you’ve been saying. How infinity is a concept? How this decimal goes on forever? And you said we couldn’t mix concepts with numbers. We can’t write 6(\infty) or \infty+6 because we’re mixing concepts and numbers. So  why can we talk about a number with a decimal that goes on FOREVER? Aren’t we mixing concepts and numbers? Isn’t this thing totally nonsensical?”

Okay, okay, they got me there. And they’re thinking deeply. And they’re getting me to think deeply.

So then I said: “okay, you have a point. So let’s see if we can mathematize this in a way that works with our understanding of things.” So I made a list:

299.9=299+\frac{9}{10}.

299.99=299+\frac{9}{10}+\frac{9}{100}

299.999=299+\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}

so

299.\overline{9}=299+\sum_{N=1}^{\infty} \frac{9}{10^N}

But then I noticed we are still mixing infinity (as a concept) with these numbers because we’re adding an infinite number of them… so then below it I wrote:

299.\overline{9}=299+\mathop{\lim}\limits_{M \to \infty}\sum_{N=1}^{M} \frac{9}{10^N}

Now we’ve written this decimal as what it truly is: a limit as the number of terms gets large without bound. And the limit has a definite value — the sum is getting closer and closer to 1, as close as we want to 1, infinitesimally close to 1, so we can conclude the limit of the sum is 1. So we can conclude that this sum is 300.

And then, sadly, I moved on.

The kids were interested in this conversation, and I think it could get at the heart of what we’re doing with limits (and then relating it to derivatives), and how infinity is a concept (for us) and not a number. But I don’t know what to do from here, where to go from here.

I didn’t convince everyone, and I don’t want to go too far afield with this unless someone out there can suggest a good idea. I mean, this idea of the limit, two things infinitesimally close together, is powerful [1]. So is there a way to extend this discussion meaningfully? Philosophically? Anyone have any good activities out there, any good worksheets out there, any good readings out there, any good videos out there? I’m not even sure what sort of end goal I have. Just something that acknowledges the weirdness of repeating decimals, relating them to limits, and the concept of infinity…

For context for the class, this is the non-AP calculus class, where my kids are at very different levels of understanding.

[1] From my historical understanding, both Leibniz and Newton (and their followers) were still plagued with the idea that you would be kinda-ish dividing by zero when calculating the derivative, because they didn’t have the concept of limits in their formulation of calculus. This division by zero was unsettling for a number of contemporaries. And it wasn’t until Cauchy came along with his limit concept that he was able to give derivatives a solid philosophical foundation to rest upon.

Reassessments

Tomorrow is the last day I’m allowing kids to reassess for this first quarter. For those just getting up to speed, I broke calculus into discrete skills and I’m testing kids on these skills. If they do well, great. If they don’t, they get the opportunity to show me they’ve learned what they were missing.

I told them on the first day, and now that I think about it, I have no idea why I don’t say this every week…

I want assessments to be opportunities to show off what you’ve learned, not opportunities where you are scared to make mistakes on.

So now that the first quarter is over, I am contemplating slightly switching my reassessment routine. Slightly, but just for my own sake.

Here’s the context. I have 27 students total in my calculus classes. I’ve tested them (some, a few times) on 17 skills.

My assessments tend to have straightforward and involved questions…

I expect full understanding. And my kids have risen to the challenge. For the most part, I think I can deem this a success. I have a number of rags to riches stories — where students struggled, but then were able to figure things out and saw their understanding, their confidence, and their grades all dramatically increase simultaneously. Seeing that alone is enough to justify my extra hours of work and the increased stress resulting from keeping this project afloat. (Again, it’s not like anything is hard individually, but together, it’s a lot of names, skills, grades, meetings, grading to keep together… it’s juggling a lot of balls and trying to keep them all in the air…) There are probably a few students frustrated to be at the B level, who just can’t make it to the A level. And there are a few students who have dug themselves into a calculus hole, and it’s been a hard time going. But for the most part, I feel good because I know these grades MEAN something, and that students have OWNERSHIP over them. Kids are reassessing.

Now to reassessments. This quarter I wrote 70 reassessments. I’d say all but 5 or 6 covered two skills. So let’s say each covered 2 skills. That’s 140 extra questions I’ve had to compose, set students up, and grade.

(For context, I assessed 17 skills, but a few twice, so I get that to come to 594 skills I’ve graded this quarter.)

To me, that proportion feels about right. Especially for the first quarter, where kids are adjusting, and figuring out how much work (and what kind of work) they need to put in to succeed… without me breathing down their necks.

But as I lamented above (and here)… individually the writing, setting them up, grading, and recording these assessments don’t take long. But collectively, it is overwhelming. A lot to keep track of, all the time. I have been holding onto hope that students have now figured out: this works for me in calculus class, and this doesn’t. So next quarter they’ll put some of those ideas in actions. (I had them explicitly articulate those things yesterday and I’m going to read through them this weekend.)

Luckily, my school has a nice thing set up where during lunch periods, we have a study hall and teachers can put tests in their for students to take. (As you can imagine, near the end of the quarter, recently, the study halls have been flush with calculus students.) I tell my kids they must email me by Sunday at 5pm to reassess in Tuesday study hall, and by Wednesday at 5pm to reassess in Friday study hall. So I don’t go crazy when making these, I only let them choose 2 skills to reassess.

This system has helped me A LOT — restricting things to 2 days. But I am still overwhelmed, because there’s a constant flow of things.

I’m thinking of switching things up next quarter.

1. Students can only reassess in Study Hall on Friday (we have 2 of them on Friday, both of which don’t have classes in them…)

2. Students can only reassess 3 skills per week (in this Study Hall).

3. Students have to email me by Tuesday at 5pm if they want to reassess on Friday.

Why these changes? Both personal and pedagogical.

I need a break from the constant stream. Having only one day for these will make the week more manageable. And students will have the convenience of picking which of the two times works for them — instead of having it only be their lunch period (or having to work with me to find a common free in our schedules to proctor them).

I want to reduce the number of reassessments per week, because there are a few students who are using the reassessments as a crutch still. They aren’t a safety net, but a lifeline. And by the end of the year, I hope to see them weened off of reassessments. This is one tiny step in that direction. The other steps to achieve that involve a lot of talking with students one-on-one about working smarter, not harder.

I’m not 100% sure I’m going with these changes. But I’m leaning towards it.

Hyperbolic Paraboloid Inspiration

Inspired by a link from here, I just sent my multivariable calculus kids an email.

***

Hi all,

I ran across something so neat I wanted to share it with you immediatamente! Google the “Philips Pavilion” — either in regular Google or Google Images. A stunning building.

Also, a mathematical building. HELLO HYPERBOLIC PARABOLOID!

I love this model that someone, somewhere built. I mean, it’s simple enough that one of you could build it. And the best part is… You can talk about the math and PROVE that the mesh forms a hyperbolic paraboloid. Anyway, and idea for a final project.

Last year I worked with Ms. TEACHER on a simpler problem — given this set of lines, what curve does this trace out:

In other words, what’s the equation of the blue line:

(In case you cared, we got y=\frac{(\sqrt{5x}-5)^2}{5} using some nice calculus.)

In any case, I see this as the 3D extension to that problem.

So keep it in mind for your final project.

Best,
Mr. Shah

State of the Class: Dude, This is what’s going on…

I had my calculus kids fill out a survey that I wrote — with their thoughts about what’s going on. I usually leave things like evaluations to be narrative, so students can say what they want to say without my questions leading them along. However, for this, I had some specific questions I wanted answers to. Now, before you read the survey, or start critiquing it: I know some of the questions are leading. And what the heck is the difference between “pretty interesting, often wild” and “sort of wild, sometimes interesting”? (You’ll see.) Whatever. It’s informal, and I’m cool with it.

I so far have 22 of the 27 kids in my class responding, but since it’s Sunday and I won’t have time to do this during the week, I thought I’d give the preliminary responses.

NON-Narrative Questions

This is actually pretty nice. If you consider a 3 to mean they are indifferent, then I’d say most students find this philosophy attractive.

You’d think I’d be slightly disturbed by this, but in fact, it is a relief. I feel like sometimes I teach to the lower end of the spectrum, instead of the middle or higher end. It’s that thing in me that wants to reach every kid. But I’ve been feeling like we’ve been moving slow, and it’s nice to know that others in the class feel that way too. I get to pick up the pace! Which is what I’ve been wanting to do anyway. Phew.

This is expected. I felt a few more kids would say that it was easy, but on the whole, expected.

I struggle with making class interesting, to draw them in, to capture their interest. I am just not there yet. Many students in my class love learning, but haven’t yet ever loved math. They’ve struggled. This is a class they’ve decided to challenge themselves with. To test themselves. Or for college. Or because they’ve been told they had to. But it’s hard, and it’s rigorous, and a totally different way of thinking for many of them. Procedures aren’t as available (though they are available) to be crutches for many things. Concepts are king. Wait, this is me (a) rambling and (b) making excuses. I’ll see if as the year progresses I can’t get a few more of them to find what we do wild and interesting.

This was expected. Most all of my kids do their home enjoyment (our super corny word for homework). The ones that don’t, well I think some of them should, and honestly, some of them don’t need to. Doing the home enjoyment is required, and I’ve made that clear. And they are expected to do it professionally and with integrity — whether it is graded or not. But I might soften my stance and tell students to only do what they feel is necessary to master the material. I know some other SBGers are like “duh, we already do that!” But I guess I wasn’t quite comfortable with saying that, until I saw how students responded to this.

Narrative Questions

Statement: “I really know my strengths and weaknesses better because of the new grading/feedback system in calculus. And I can now do something about the weaknesses while basking in the glory of the strengths.” Do you agree or disagree? Discuss.

I agree but I really miss having homework grades

I agree with this statement, because due to the new grading, I do actually know my strengths and weaknesses, and I can really focus on that while studying for the next assessment.

I disagree with the idea that I didn’t know my strength and weaknesses without this system, but this gives me an opportunity (and the motivation) to learn something I didn’t  understand the first time around.

Agree. Well, I don’t feel like I’ve had any major comprehension issues yet, so any mistakes I’ve made are more procedural. This means I haven’t really needed to take advantage of the system yet, but I like it a lot anyway.

Yes indeedy ! Because you have to plug in your scores onto the skill checklist you can really visualize your grades. You can see the areas you struggle in and the ones you excel in. Also if you do well in a skill and dont do so well the next time you know exactly what you need to do because it is on the skill checklist.

Because of the new grading system, I defnitely have an easier time discerning which parts of calculus are my strengths and which are my weaknesses. It helps me to isolate conceptual misunderstandings and address them accordingly.  Whether that’s re-doing homework problems or going over class worm, I feel more productive because I’m sure that I’m bettering myself as a math student.

i agree but there are certain problems where i have made really simple mistakes and because i only have two free periods its hard for me to find time to make it up.

I agree with this for the most part. I feel like my biggest weakness is retaining everything old while learning new things (I don’t think the information causes as much trouble individually) so my biggest challenge is making sure that I keep up with the past things and don’t dig myself into a hole! But I think that I understand this about myself more now with this system which allows me to really find ways to be aware of this and fix mistakes/hold on to information.

Agree. Because of the system, I can see what I can do with no problem and other skills that give me a little trouble. I know that there is one skill that is consistently bothering me and the system allows me to work on it and make it stronger.

Agree, you really have to learn the material and it’s helpful to have a second chance to prove you really know it.

Agree, the reassessing does force me to understand the things I don’t know and remember the things I do.

AGREE. My weakness is definitely learning material for a test and then forgetting it (which ironically is what I HATE in school…) and also I have a hard time making appointments to reassess and such.

I agree with this though I dont think this grading system shows any of your effort. Thats the largest flaw.

Agree

Somewhat disagree, I like that this new system lets you retest, yet at the same time I feel like this system really punishes you for making a silly mistake. There’s a massive difference between an 87 with a 3.5, and an 100 with 4, so I see it as everyone needs to do perfectly to really get a great score.

I full agree that it helps students understand what there strengths and weaknesses are, but the assessments are so often that it’s hard sometimes to fully grasp the skills. Why else would so many students be re-assessing? Also it would be nice to review every once in a while in case those old skills coming back.

ABSOLUTELY. For once I’m not totally screwed in a math class because of tests.

I totally agree with this statement, because it gives me a chance to see whether or not the mistakes I make are conceptual or careless, as if I get a grade that isn’t satisfactory, I am forced to look at it and see why I made that mistake. Also, being able to reassess proves not only a potential further understanding of the concepts, but also proves that we are willing to put a lot of effort in order to make this system work for us.

I definitely agree that the breakdown into skills shows me exactly what i need to work on and what i have totally down. However, occaisionally the system backfires; for example, if i had a 4 on a certain skill each time i was tested on it and i was having an off day when you tested me on it at the end of the quarter and got a 2, my grade wouldnt reflect the fact that i had consistantly mastered the skill, rather it would reflect one off day. Still, overall it is a good system.

I mostly agree with this. The best part is having a skill checklist on which I can keep track of my grade of every assessment, and have an idea of what skills specifically I need to work on. This way, instead of thinking “OMG IDK GRAPHZ AT ALLLLL” I think “I need to work on finding holes and explaining what they are in words.”

I agree with this statement. The new grading system really allows me to see which skill i lost points on through the grades on each assessment. Therefore, i will be able to focus on improving whatever specific skill i was weaker in.

I agree, the new grading system does help me know where my strengths and weaknesses are, so I can do something about the weaknesses.  However, as the year goes on it will be more and more difficult to retain all of the skills some strengths might become weaknesses.  Then, to help re-make them strengths, it will add more stress to the stress of learning new skills and mastering other weaknesses.

Statement: “When I have trouble in Calculus, I have found some good ways to overcome the difficulties because of the new grading/feedback system in calculus.” Do you agree or disagree? Discuss.

I don’t agree because when you’re told to review all the material, it’s a lot to review and therefore you have little slip ups which because everything is repeated can really affect your grade and we have college stuff to worry about. It’s a lot on our plate.

Due to the new grading in class, I have found new ways to overcome difficulties and study in general – instead of studying things that I got wrong, I can study the things that I genuinely don’t understand, so that I can get it right during the next quiz.

Yes, I go to my peers a lot (especially [redacted]) and get help.

I think that your comments on our tests are really good because I’ve been able to understand exactly what I did wrong, but I don’t know if that has to do with the system or not.

This is super easy with the skill system. Because each skill and numbered and includes a short explanation about the skill you know EXACTLY what you need to know. So you can take what you don’t know and turn it into what you do know/

I agree, for the reasons mentioned above.

yes i agree, being able to reass is good

I would agree, because I think I talk to people more (you and classmates) when i’m confused before or after a test.

Disagree. The new grading system is helpful in seeing what your strengths and weaknesses are but it doesn’t necessarily provide new methods to learn your stuff. You still have to develop your ways to learn the skills; the system just tells your weaknesses.

I haven’t had any real troubles in Calculus, yet.  But the little details I do leave out, I can go back and see the correct way of doing it without just racing ahead.

Agree! after I figure out my mistakes, it becomes much easier to correct them which makes the reassessments awesome!

I’m still having a hard time with that mostly because I’m really busy and I’m trying to keep everything under control. But when I have trouble I ask a friend or see Mr. Shah!

Yes
-meeting with a tutor
-meeting with a friend

Agree

The grading system hasn’t really added or subtracted in my overcoming difficulties, the idea of separating everything into “skills” is a good way to differentiate areas of concern though.

I’ve figured out that the best way to review is know what you did wrong and to explain it in class. The new grading system helps, but I can’t quantify the amount it has helped. I don’t think it’s a major factor into overcoming difficulties.

Agree. With my crazy schedule I do find it hard to schedule meetings to discuss the problems I have, but the opportunity to retest is a total life saver. So many times I don’t do well in math classes its because of a test here or there, but this time I can actually learn the content that is being tested and if needed study more and take the test again

What I have done to overcome my difficulties is retake the problems that I didn’t do well on before the next time I reassess. Also, on occasion, I’ve met with kids to discuss concepts. It isn’t difficult to do, and, even though it’s more work, it’s more convenient.

Yes i agree, the grading system shows me exactly what i am messing up which makes it much easier to target the mistakes and fix them.

I don’t really have many difficulties, but I assume that if I did having the skill lists would help me see what I need to work on and overcome the difficulties.

I think the new grading system really only helps me overcome my difficulties because i have another shot if i don’t do well the first time, which will motivate me to want to learn the material even after the test on it.

I disagree.  The new grading system doesn’t really give me any new ways to overcome difficulties.  I can still apply the same skills to overcome them with another grading system (asking for help from a friend, using the web, getting more practice problems, etc.).

The best thing about the new grading system/philosophy is…

Retesting

The best thing is the fact that you can get feedback on exactly what you did wrong, and reassess if necessary.

means you have to retain information you may have forgotten using the old system

Making a mistake isn’t that big of a deal, because you have a second chance, and also I really know that I know it.

Reassessing. Period.

It grades you on WHAT YOU KNOW and WHAT YOU DONT KNOW!  Reassessments help give students a second chance to prove just that, and it encourages learning from your mistakes.

That I can for the most part take the time to digest the information and if it doesn’t work out the first go around work on how to digest and understand it better for round two. Also, I’m less anxious at assessments because its like “ok show that I know skill blank blank and blank, and then remember the old stuff which we did great last time so it will be ok!” and that mindset helps me stay relaxed.

Independence. You’re the boss.

Retesting & Retention.

Reassessments

It makes you learn stuff! (Instead of just cramming.)

You can keep trying untill you sucseed!

How skills are divided and specific

It allows us to retest

leaning calculus.

Division of work into manageable sections for studying

How our performance in this class really is in our hands. Our effort can clearly be seen through our marks, which is something I really appreciate, and is also very rare when it comes to education of any sort..

It focuses on retention which, while difficult, is really important in a math class and will probably serve us well later in the year.

Breaking topics down into specific skills makes it much more easy to mentally manage. Doing poorly on one assessment doesn’t overwhelm me and discourage me from thinking that I don’t know a topic. I can pinpoint my weaknesses and take them on one at a time.

that improvement is rewarded

How the assessments are usually relatively short.

The hardest thing about the new grading system/philosophy is…

The hardest thing about the new grading system is retaining everything that I learned.

putting in the time to be able to retain all of the information we are expected

You have to really show that you know how to do the problem, not just give the answer, so it takes a lot longer to take tests.

The independence of it all. Most of the benefits of this system lie in the ‘proactive-ness’ of the student so it’s sometime a bit hard to be so on top of things

It’s easier to do badly because you’re graded on a 1-4 scale, as opposed to a 1-100 scale.

even though a 3/4 is knowing the information it is technically only a 75% which is not a B so i struggle deciding whether i should spend my two frees trying to reassess the 3 or not.

Making sure to stay on top of old mistakes and take care of reassessing right away instead of waiting it out/hoping it fixes itself over time with out doing something.

Keeping in track of everything. You can’t be lazy to do well. If you want to do well, the teacher isn’t going to hold your hand through it.

Having to make the effort to set up a re-test.

you would think hw would be it but it isn’t because I am used to always doing my math hw and deciding not to do it would just be stupid because then I would get extremely confused.

Making lots of appointments to reassess.

If you do a great jobb one day and get everything right then one day your brain stops working and you do poorly.

Nothing

The hardest thing is easily that small errors earn big penalties. On these tests, a quick error could get you a 3/4, or a 75 as most of us see it, which is both unsettling and discouraging. It’s nice to know that you can always reassess, but it’s also nervewracking that you could just as easily do well on a reassess, have the same problem on a new test, make another dumb error and go right back where you started from.

remembering allllllll the skills.

scheduling retests on only Tuesday and Friday

Probably how much effort it requires. We’re very busy as is, and this additional work, although beneficial, does add up to a lot.

The fact that the scale is only out of 4 points. That means that a 3.5, which reflects almost perfect mastry on the skill, is still only a B+, something that is misleading and not totally fair.

Knowing when to reassess. Say I get a 3 on a skill but a 4 on most others, I don’t know whether that means I should reassess the 3 or just move on. Some guidelines might be helpful on this.

it is much easier to lose points and have my grade go down because of mistakes on an assessment

The hardest thing about the new grading system is that as the year progresses we have to retain old skills while trying to balance learning new skills.  Through our years of high school we have learned to keep the old skills in the back of our minds (until the midterm or final) but now we are trying to change our learning habits around in 12th grade, which has been pretty difficult so far.  It is definitely a good idea but it would have been a lot easier to deal with if we learned it earlier on in our educational careers (and/or if all the classes had the same grading system and not just calculus). Also, homework and class participation should be factored into the grade so more students would do it, so the class couldn’t be slowed down by those who don’t understand the material.

There’s a lot of juice in these comments. Some initial thoughts:

What’s clear is that I still have some work to do with them, in getting them to see the new system as something APART from grades, and more about learning. My kids haven’t yet come to the place I have — and that makes sense because I’ve been thinking about this for a year, and they are just getting used to it. The fact is, many still see a 3.5 (out of 4) as an B, and a 3 (out of 4) as a C. I realize they aren’t looking at the totality of their scores holistically. I mean, on a test on linear regressions and got 4 out of 8 points on a word problem, they wouldn’t say “Oh, I got a 50% on linear regression word problems.” They have come to see these 2s and 3s as less like markers of understanding, and more like points on a test. Which makes sense. And now that I know that, I can address that. What’s clear is that my kids are very grade conscious. Heck, I was too when I was in high school. And they have good reason to be grade conscious.

The students who are finding it time consuming and difficult to come in for reassessments — well, clearly I need to get them to see how they need to work towards doing well enough on assessments that they don’t need to reassess. (Also, the absolute most anyone has reassessed is 3 times — which comes to at most 6 skills.)

About so many students reassessing… well, it turns out about 1/2 the students who are reassessing have almost all 4s, and are working to complete the scorecard for perfect mastery and retention. The other 1/2 are assiduously trying (and totally succeeding, by the way) to dig themselves out of the hole they’ve created. I don’t dictate reassessments, and I let kids make their own choices, good or bad. When students come to me asking if they should reassess a 3, I don’t answer them but just ask “What do you think? It’s your life. Your choice. You’re old enough and have earned the right to make these sorts of decisions on your own.” So I won’t ever answer that question for a student. (If a student asks me about a 1 or a 2, I generally say something similar, and then follow it with a “but if you want to show me, and also more importantly yourself, that you can do it, you ought to.”

With regards to the fear of retention, I totes get that. But I don’t think they realize (even though I mentioned this) I can only test a quarter’s worth of material — once the quarter ends, I need to enter final grades. Those skills are shut out. But what’s interesting is that when I put old skills on their assessments, I choose ones where students generally didn’t do so well. And almost universally (with a few exceptions here or there) students either keep the same score on the skill or go up (sometimes dramatically).

The one statement that shows me how much we’ve trained our kids in our schools to think about learning: “I dont think this grading system shows any of your effort. Thats the largest flaw.” Because if there is a system that actually shows your effort, and rewards for your effort, this is the one.

 

Some Downsides to SBG

SBG — or at least the way I’m implementing it — has some downsides. I see a ton of positives, but there are things to weigh against it. I am pretty sure my final verdict will come from my students, when I ask them about it.

1. I have already written 25 reassessments. (I don’t meet with kids to reassess — they take it in a “study hall” that students usually go to when they miss exams or schedule their extended time for exams.) I wish I could meet them individually and do what other teachers are doing — reassess kids by talking with them, sussing out what they know — but I can’t. I simply don’t have the time. But I lose that personal touch with the kid, that conversation after a good job or a poor job, about the process of learning. I mean, I have those when I meet with kids generally, but right now it’s just a lot of tests they’re seeing.

2. Students want to meet a lot more with me. This is good, but the downside is: I am getting way too burned out. It’s not feasible to carry on like this for the rest of the year. I hope that after the first quarter, students will be better equipped to get in front of the material, instead of lagging behind it.

3. My kids are seniors, and grades are important to them (hello, college admissions). At my school, kids almost always get As and Bs with an occasional C and even rarer D. But currently I have a few kids failing and a few kids with Ds. Right now the grades are lower than they historically have been. (They also mean something totally different. I get that. And I’ve had a number of kids who have been absent a lot.) The result, however, is that I am waiting for the other shoe to drop. I suspect I will get flak from somewhere. I hope that the trend I’ve been seeing recently continues — and the grades continue to go up as kids realize the need to (a) stay on top of the new material and (b) remediate the old material they didn’t realize they would have to stay on top of.

4. Everything is going way slower – because I’m spending more time in class assessing. I do it about once a week, for 35 or 40 minutes. But right now I am about a week behind where I was last year. I am teaching limits now and I’m getting bored with some of the dry stuff I’m emphasizing (e.g. limit laws, etc.). So are the kids. I am going to try to cut some stuff out so we can get to the good stuff — and make up for lost time. But still. I’m going slower.

5.  I haven’t changed any of my curriculum so far. I’m using the same SmartBoards (well, modified, but generally the same) from last year. So the course should feel the same. But I feel like everything is getting choppier. Kids are more focused on the skills, and it’s hard to get them to see the big picture. They latch onto the skills. It’s their security. I fear they don’t care about learning anything else. I think they have trouble zooming out and seeing the connections — and I need to be super conscientious about that. But to them, the skills are the of all and end all. Something is off, and I can’t quite articulate it well yet. This was my attempt.