Teaching Confidence

In this 2008 Science Daily article, Dr. Nadya Fouad tells us that her research (into girls’ interest in math) has produced a startling result.

“The relationship between confidence and interest is close,” says Fouad. “If they feel they can do it, it feeds their interest.”


It’s important to focus on making topics interesting, but some students have a fundamental belief that they can’t “do math,” and convictions like that can be a roadblock to any lasting interest.  In my class, I had students who would finally, after days of practicing, be able to perform a new skill – and go right back to saying they can’t do math.  It doesn’t help when their classmates all seem to learn every skill way faster, and their teacher has a degree in math and is always showing off at the front of the room.  Not that I make a big deal out of estimating square roots with absurd speed and accuracy, or anything.  Bee tee dubs, I am world champion at square-root estimating.  And at confidence.

How can we teach confidence in our math classes?  As I see it, there are two main routes to take:

1: Make it clear that “not knowing” is not “failure.” (yet)

  1. Ask for wrong answers first.  “What’s a number that wouldn’t solve this equation?  What number is way too big to be the answer here?”
  2. Model being wrong in a healthy way.   When you make a mistake, say, “whoops, that was a mistake.”  Don’t try to hide it or make some excuse – even jokingly.  If you find it hard to make a mistake because you’re such a math whiz, you might want to ask students about things they’re expert in.  That’s a good idea anyway, and you can show yourself learning and not understanding immediately.
  3. Something like this post about Painting with Functions lets students play without anything to “know” in the first place.  There’s no way to be wrong.
  4. Be on the look out for risky behavior – a guess, an idea, anything a student offers that he or she is not sure about – and then praise that behavior.
  5. Never lock a student in to a bad grade.  If you have a student with five 0% in a given topic, adjust your grading system so that it won’t take five 100%s just to get to a high F.  Some students think your bad grade means you think they’re bad at math, and if you aren’t willing to change that opinion, they’ll give up.  (Self-promotional link: ActiveGrade can help you manage an active grading system).

2: Work in contexts in which every student is already fluent.

  1. Start with something all of the students know – throwing a ball, walking, measuring distance – and ask a question about it. I write a lot about this in my “Bringing the Problem to Reality” series.  When students understand the problem enough that they can play with its parameters and watch the outcome, they’ll feel confident experimenting.  If they’re using something they don’t know anything about – a new equation, or a new notation – and they don’t even know what they could change, they’ll feel stuck and insecure.
  2. Ask questions they already know how to answer.  It’s OK to spend time on things they already know, even if they don’t “need” to review.  You can keep it lively with a game or two.  If a student who is usually insecure says he wants to move on because, gosh Riley, he gets it already, make sure to point out his high skill level before you continue.

Teaching young people to be confident and creative is an important part of being an adult.  I hope this gives you some specific ideas about how to do it!  Leave a comment with your experiences and techniques!

The Pseudocontext in The Belfry

By the united aid of medals, manuscripts, and inscriptions, I am enabled to say, positively, that the borough of Vondervotteimittiss has existed, from its origin, in precisely the same condition which it at present preserves.  Of the date of this origin, however, I grieve that I can only speak with that species of indefinite definiteness which mathematicians are, at times, forced to put up with in certain algebraic formulæ.  The date, I may thus say, in regard to the remoteness of its antiquity, cannot be less than any assignable quantity whatsoever.

Edgar Allan Poe, The Devil in The Belfry

Math problems abound in this great story.  Read it if you’ve got kids in geometry or algebra or calculus – or if you’re looking for a good laugh!

Things are really coming together

Beta release estimates for ActiveGrade (free spots still available).  The top line is 95% certainty, the middle line is 50%, and the bottom line is 5%.

Sorry to keep posting these graphs – I think the dimensions are so interesting!

Notice that I am now only 5% sure that we’ll be ready to launch on November 1.  On September 15th I was 95% sure we’d be ready to launch BEFORE November 1!

I will say that studying these graphs has brought my average estimation accuracy from 400% (I think a 4-hour task will only take me 1 hour) to 180%.  Keeping data can be good for you!

Active grading: “weights.”

[edit: throughout this post, I am talking about weighting assessments within a standard, not about weighting different standards]

Sarah asked if ActiveGrade will support weighting different assessments different amounts.  A this-assessment-is-worth-twice-as-much-as-this-one kind of thing.


Here’s my thing: I already feel pretty weird when I give an assessment and say, “I can tell from this that you understand about 80% of how to find the roots of a quadratic polynomial.”  Right?  We have to accept that the most we can say is, “on this problem you did about 80% of what the solution required.”  We can’t know what our students know – all we get is evidence about what they know.  Descartes & Plato, right?  I know:  deep.

So when I’m already admitting that my assessment is only a grave-rubbing of my students’ actual understanding, and that I can’t know both my student’s skill level and his velocity, I feel a little sheepish saying that one of my assessments is more accurate than another.  When I give an assessment, I’ve got to be willing to stand behind it as a valid measurement.  Calling assessment B twice as valid feels like calling assessment A half as valid.

On the other hand, shouldn’t final exams be worth more?  Big projects?  It’s not too hard to find ways that a carefully thought-out paper might be better evidence than a pop quiz, but maybe that just shows that we shouldn’t give pop quizzes.  Of course, you can also show why a pop quiz might be better evidence than a paper.

So, I’m conflicted.  Those of you who weight different assessments differently: how do you decide what the weights are?  Those of you who don’t: what problems do you run in to?

Announcing ActiveGrade!

Hey Y’all,

This year I’ve been working with a partner on developing a piece of grade-reporting software that we’re calling ActiveGrade. Here’s why:

  • There’s a lot more that software can do to help make grades understandable and meaningful.  I think grades should be the start of a conversation between students and teachers, and so one of ActiveGrade’s focuses is helping teachers communicate exactly what they mean when they report a score.
  • ActiveGrade is also a reaction against grading software that treats grades like the final product of education.  I feel strongly that damage is being done to students who are motivated by grades (ugh), so I’ve been striving to create something that functions mostly as a feedback tool which students can respond to.  Grades should not be an end.
  • Support for standards-based grading in other software is weak.  ActiveGrade is strictly SBG, and it will provide tools for teachers and students to really participate in active SBG.  You may have noticed: I liked the term “active SBG” so much I named my software after it. Well, I should say that we liked it – Dan really came up with the term. Anyway!

ActiveGrade will be ready for beta testing next month!  A big goal in the testing is to figure out what kinds of tools are most important to teachers using active SBG.  If you want the chance to make ActiveGrade what you really want it to be, I hope you’ll sign up!

If you’re interested in helping, sign up at ActiveGrade.com!


Which Active Grading Scheme is Best?

Active SBG means:

  1. Emphasizing the learning that grades represent, and trying to avoid holding grades as the final product of education.
  2. Allowing students to react to their grades. Grades are the beginning of a conversation, not the end.
  3. Helping students to understand their grades by organizing them into topics (vanilla SBG).
  4. Actively keeping students informed by assessing their skills often and giving them feedback as soon as possible.

These ideals are great, but we have to be really careful about how we implement them – there are some major traps to avoid. For example, if we implement #2 by throwing out old grades, do we inadvertently overemphasize tests and contradict #1? If we break our whole course down into discrete topics to implement #3, do we run the risk of trivializing the broader connections _between_ concepts in our curriculum?

I’m creating grading software, and trying to decide what the default grading scheme should be.  Here’s my current favorite.  What do you think?

Rubric for each assessment:

  • 0. Has not demonstrated any skill or understanding
  • 1. Has demonstrated the beginnings of understanding, but still makes conceptual errors
  • 2. Has demonstrated understanding of the standard, but still makes mechanical errors
  • 3. Has demonstrated basic mastery of the skill
  • 4. Has demonstrated mastery of the skill, can connect this skill to other skills, and can creatively apply it to new situations.

It may not be possible for a single assessment to accurately measure a 1 and a 4, here, since any assessment that can measure a 4 would probably overwhelm a student who is at a 1 level.  An assessment that could measure a 4 would probably include many different standards simultaneously – its score could be entered in multiple standards at once.

Reassessment policy:

Students may ask for a reassessment on a given standard whenever they like.  Reassessments might take the form of problems, conversations, a project, etc.  If the student wants the chance to earn a 4, he should ask ahead of time, since these assessments take time to create.

Method of calculating a standard score:

A decaying average that counts the most recent score as 60% of the total grade, recursively. Whenever a student gets a new score for a standard, that single score is worth 60%, and the combination of all the old scores is worth 40%.  Examples:

First assessment Second assessment Third assessment Final score The math
2 3 4 3.4 0.6\cdot 4 + 0.4(0.6\cdot 3 + 0.4(2))
4 3 2 2.6 0.6\cdot 2 + 0.4(0.6\cdot 3 + 0.4(4))


First assessment Second assessment Third assessment Fourth assessment Final score The math
2 3 4 4 3.8 0.6\cdot 4 + 0.4(0.6\cdot 4 + 0.4(0.6\cdot 3+0.4(2)))

Method of Calculating the Overall Grade

To find a student’s overall grade, I evaluate the list below from the top down.  The highest qualifying grade is assigned.

An average of at least 3.6 with a minimum score of at least 3: A

An average of at least 3.3 with a minimum score of at least 2.5: B

An average of at least 3.0 with a minimum score of at least 2:  C

An average of at least 2.7 with a minimum score of at least 1.5: D


Examples, with 6 standards in the course:

S1 S2 S3 S4 S5 S6 Average Minimum Final Grade
Student 1 3 4 4 4 3 4 3.7 3 A (on average, shows high-level understanding.  Has no gaping areas of weakness)
Student 2 3 4 4 3 3 4 3.5 3 B (on average, shows somewhere between mechanical and conceptual mastery)
Student 3 2 4 4 3 3 4 3.3 2 C (shows a mix of low- and high-level understanding, is good with the mechanics, but still can’t really get standard 1)
Student 4 3 3 3 3 3 3 3 3 C (always shows mechanical mastery and has never shown higher-level understanding. Has no areas of bigger weakness)
Student 5 4 4 1 4 4 4 3.5 1 F (usually shows a very high level of understanding, but really doesn’t understand standard 3.  Note that a single assessment with a score of 3 or higher on standard 3 would launch this student to a B)

This answers a lot of my worries, as long as I can really keep up with the demand for level 4 assessments in a way that connects multiple standards.  Students must show a lot of high-level understanding to get an A, and they must be able to connect concepts.  Students who just want to pass the course can easily get a C by learning the rote, independent skills of the course.  This seems about right to me.

Worries I still have:

  • Under this method, if I give a final exam and a student with straight 4s gets a 1 on one of the covered topics, is it really right to drop that student to a C?  His grades would be something like 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2.2 4 4 4 4.  A C?  Maybe it’s not the minimum that I should be testing, but some measure of deviation?  Maybe I need to leave 3 days for emergency reassessment after any “final” exam?
  • This system is complex.   Is it so complex that students would feel completely confused about their grade?
  • All standards are “weighted” equally here.  Do I need to reserve some topics as more important?

What do you think?  I’m trying to make this software for anyone who wants to use Active SBG in their class.  What do you need to effect your favorite grading system?

Strategies for keeping active SBG cohesive

Does breaking the grade into 10 or 20 different topics help? or does it foster a reductionist attitude toward learning—that everything is discrete and independent of everything else?

Does allowing lots of reassessment help? Or does it focus kids on point chasing?

GSWP left these poignant questions about SBG for us, and today Sam Shah said that so far, the answers for his classes are bad.  To keep the punches coming, Alfie Kohn points us to “When Good Teaching Leads to Bad Results: The Disasters of ‘Well Taught’ Math Classes,” which concludes that “Despite gaining proficiency at certain kinds of procedures, the students gained at best a fragmented sense of the subject matter and understood few if any of the connections that tie together the procedures that they had studied.”

A scary and sad morning for people just starting to use active SBG!

Sam’s last point is that practice of active SBG needs to include some kind of protection against the choppiness that splitting your curriculum into discrete chunks brings.  If we assume that students will see their grades as the final result of our classes (a depressing but realistic assumption), what can we do with our grades to include learning and connections in them?

My Strategies

Here are two strategies I used to hold my courses together and fight tendencies of reductionism.

  1. I worked hard to reward students in ways that didn’t include grades, and heavily rewarded higher-level, holistic thinking.
    • Praise: “Great realization, John – that’s the real root (hah!) of the connection between factors and intercepts here.  This isn’t going to be on a quiz, but it’s one of the most beautiful parts of this stuff.”
    • Recognition: “And this is another case of what Sarah was describing before!”
    • Interest: “Whoa, how did you think of that?” followed by a 90-second conversation.  (Requires students to do something interesting, of course).

    This stuff felt cheesy at first, but I realized that if I only used it when it was really genuine, then, well, it would be really genuine.  Unfortunately, it tends to reward some students more than others, but we also give some kids As and some kids Cs, so I’ll leave that debate out of it.

  2. I reserved my highest grade for students who showed more complete understanding. At my school the highest grade was “honors,” but you could use “A” or “A+” or whatever.  If a student earned 100% on every skill in the whole course, he or she still wouldn’t earn the highest grade without completing a few projects that brought together a larger scale of knowledge.  This is, I admit, very un-SBG.  The effect was that students who just learned each concept individually and minimally could never get an A.  I made that very clear from the beginning, so that students wouldn’t be surprised, and I spaced the projects so that I introduced about one per month, keeping the emphasis on holistic learning throughout the year.I didn’t like requiring more work for the highest grade, but I did really like requiring a different magnitude of understanding.

SBG is no guard against point-chasing, and even active SBG has a lot of loopholes kids can exploit.  And, I mean, of course.  Teaching is freaking hard and teaching 80 kids at a time is even harder.  We’re set up to use grades as a reward by the system, and have to fight to keep them low in importance.  When grades are the reward, how can we really expect learning to be most important to all kids?

Strategy 2 above is the most practical – an administrator could just drop that into your class without caring if you’re a total grump with your kids.  But strategy 1 is really the most important, I think.  You know how I feel about personally connecting to our students already – I think it’s our only tool that actually does elevate learning over grades.  And with that our only tool, every point-chasing kid looks like a nail!

Active SBG

SBG is all about description and specificity, but “SBG” doesn’t describe what I’m talking about when I say “SBG.”  Here’s the problem:

“Standards-based” means “organized into topics,” but we’re doing more than that.  When we talk about letting students improve and show their improvement, we’re not just talking about organizing information.  When we talk about breaking kids of their addiction to points, we’re not talking about adding 20 columns to a gradebook.  When I talk about SBG, I’m talking about a philosophy of empowered students who have control of their education and their grades.

We implement this philosophy by organizing our feedback into helpful topics, making sure that our students can understand our feedback, and allowing students to react to that feedback.  You (not you, of course, but one) can implement SBG without any fundamental changes to your philosophy, and students in an SBGed course may still chase points, so “SBG” is not enough.  I think we need a new term.

“Active SBG” means:

  • In conversation with students, emphasizing the learning that grades represent, and trying to avoid holding grades as the final product of education.
  • Allowing students to react to their grades.  Grades are the beginning of a conversation, not the end.
  • Helping students to understand their grades by organizing them into topics (vanilla SBG).
  • Actively keeping students informed by assessing their skills often and giving them feedback as soon as possible.

After reading this list, can you see why “standards-based” wasn’t cutting it?  Joshg wonders “whether SBG really means anything without a slight philosophical shift,” and countless others blog about the “philosophy” of SBG.  “Active” is a great word to sum up the extensions that SBG needs to really shine – active student involvement, active feedback, reactive grades.

Most importantly, active SBG means that grades are used as one of the catalysts for learning in a class – that even though all that’s going on your report card is a single letter, the 100+ hours of imagination, concentration, and sweat are the real prize.

Man. There’s a Lot of Math to do.

Ironically, as a math teacher I hardly needed to use any math at all.  Now that I’m starting a business, I’m keeping track of hours, money, and stock shares & options, and using all that boring stuff I hated teaching.

It’s straight out of a precalc text book, actually:

  • “How long can Riley afford to work on his own if he eats $4.5 worth of food every day, pays $60 a month in utilities, $2000/year in taxes, and $4/day in drinks?”
  • “How many hours per day does Riley need to work to finish by December if Riley estimates he has 100 hours left, doesn’t want to work on Sundays, and usually underestimates how much work there is to do by 50-70%?”

These make really boring textbook problems, but I tell you what, they get more interesting when you might not eat if you’re wrong.  Talk about intrinsic motivation.  I’m lucky enough to have learned to do these kinds of calculation in school (a lot of them only after I started teaching, btw), and I simply couldn’t start a business without being able to do them.  It feels like running in the dark even when I can grapple with pretty complicated models.

I always felt cheesy as shit saying things like, “you might need this if you ever want to xyz,” and avoided using that kind of motivation whenever I could.  But what about the kids who will want to xyz?

Smart Criticism of SBG

One of my new hobbies is collecting criticism of SBG.  SBG has a lot of great properties, but the enthusiasm for it sometimes glosses over some of its problems.  One writer examining SBG pretty closely with an open mind is the author of Gas Station Without Pumps.  He/she writes about a lot of things, but if you like my blog you’ll like his/her posts on SBG.

The thing I worry about most with SBG is that we might get complacent – that our students can learn to play SBG just like they’ve been playing the rest of school their whole lives, and GSWP hits that nerve hard with Just scoring points.