# Emphasize Learning

Should students be rewarded for being friendly, prepared, compliant, a good school citizen, well organized and hard-working? Or should good grades represent exclusively a student’s mastery of the material?

The article that quotation comes from is spot on, but in a scary way.  The author, Peg Tyre, nails some of the primary benefits of standards-based grading, like the emphasis of knowledge and skill over classroom participation and direction-following.  She describes the improvement in parent-teacher-student communication after a school in Minnesota switched to SBG.  As you may know, these are benefits I believe in.

But the whole article is framed in the context of alignment with standardized tests.  The idea of standardized testing is not inherently repulsive to me as long as we can find some way to keep the focus of education on learning and away from scores.  To have this great article about knowledge and communication framed in terms of test scores makes me worry that our ActiveGrade software, and even SBG in general, will be turned against us as another method of points-grubbing that is simply more specific about where to grub for points.

It’s up to the teachers and parents and kids to keep the focus on learning.  No grading system can help much.  ActiveGrade and other programs that report SBG grades can help by organizing the grades into topics, and emphasizing those topics over any final grade, but they all must ultimately rely on summaries of learning that can be interpreted as a score.

Here’s how I tried to keep the focus on learning when I was teaching:

• I made sure the class was interesting to me – if a lesson from last year felt boring, I tried to remake it.  I let my interest show in class with enthusiasm and engagement.  One day a young woman asked me if you could really crush a can with boiling water (like Bill Nye does), and I’d never done it – the next day I brought in the materials so we could try it!  Clearly the can experiment doesn’t give me a pay raise or anything – it’s just interesting to me and I showed my students that I was interested.
• I tried to model learning for its own sake by letting students see me learn things I had no expertise in.  As the math teacher, they didn’t relate to me when we were learning math – even if it WAS something new to me, I learned it faster and easier than they did because, I don’t know, I know the properties of multiplication about a thousand times better.   I joined soccer practice for a semester, in which several of my math students were at least a thousand times better than me, and I asked them for help on small parts of my technique.  They saw me falling down, getting muddy, going up for a goal and completely missing the (stationary!) ball (it’s funny now, but it felt really bad then), but staying with it and getting better.On a smaller scale, I did this in class by basing scenarios around things that the students knew a lot about (e.g. fashion or video games or music or juggling) but I didn’t, and just taking 30 or 45 seconds to let one of them share their interest.
• I went out of my way to hold students up as authorities in a subject.  If a student had shared about juggling earlier, and so we were discussing something about it, I referred to him when another student had questions like, “how many times does a club flip over?”  I did a lot of group work, with each group studying something independently, and I would refer the group studying vertex form to the group finding roots if it was appropriate.  Giving students the responsibility of being authoritative in a subject gave them a new kind of reward for learning.

How do you emphasize learning in your classroom?  One thing I never did well was including parents – I met with them once a year, if they requested it, for 10 minutes.  I regret not talking about this stuff more explicitly with them, and also with the students.

I just realized that I’ve started to forget how much there is to keep track of while you’re teaching and planning and grading and reporting – writing about my regrets brings back all the stress of feeling like I’m not doing ANYTHING.  I’m turning this into a run-on post, now, but please remember that you are doing a LOT of work for a LOT of other people.  I hope you’re proud of yourself – you have my admiration, respect, and gratitude!  Sure, SBG is great, and we can all improve at everything, but don’t sweat it: love those kids, first, and show them the beauty all around them!

# Math Teachers At Play: A Ballad

Welcome the blog carnival for math teachers at play.  Try not to think of Gilligan as you read it today.

Respect children and be sincere.  This is what they need.  Kimberly Moore points out that there is no call for speed.

How many different ways are there to cut up a square box?  Here’s a seven-year old girl who sees no paradox.

Kate looked at a dollar bill, said “let’s see what we see.”  On looking further with her class found the square root of three.  And Illuminati.

Ever want to make a giant geodesic dome, but only have materials you find around your home?  Jen has two adorable and curious young girls who show you how to make one without pawning off the pearls.

Showing every single step of arithmetic work can be a pain and make you think your math teacher’s a jerk.  A student of Tom Kendall’s knows a superior way of writing what you’re gonna do next to the thing you say.

A billion is a number that can capture students’ minds, and Ryan describes his success engaging them with times.

Our students may not always know exactly what they mean;  When we say that x equals y our meaning can’t be seen.  It could be so much clearer with a good taxonomy – I recommend the one you’ll find at SG Without P.

Patrick Vennebush gives us some jokes not so funny.  It turns out you’ll more likely laugh if you’re in company.  That finds it all funny.

There are lots of math puzzles designed for their devotees, but Caroline has five math games for those who feel unease.

Alexander extends a math problem to the reals, and finds that there are more solutions than the world has seals.

John Cook us’lly writes about some esoteric stuff; divisibility by seven should not be enough.

But the last digit….  Times two…

Subtracted from… the front end…

Shows whether, the orig’nal was divisiblehere on Gilligan’s ISLE!

# The Indistinction Between Summative and Formative

Confusing half-truths:

• You change your teaching based on formative assessment but not summative
• Summative assessments are formal but formative assessments are informal

There’s a confusion that causes stress when grades are involved.  Grades are supposed to:

1. Tell students where they stand in the class.

For me, and I think ideally for most of us, “where they stand in the class” is as close as possible to “how much of the subject they understand, and to what degree.”  So, “they understand a lot and can really extend their knowledge to new areas” is encoded to an “A,” while “they’re getting it!” is encoded to a “C.”

Hopefully they’re not at a D, which means something like “I guess he has been in class.  Most days.”

Great!  The summative and formative (non)dichotomy is not causing any pain yet – here’s a summary of your knowledge, let’s respond to it.  Boom.  Active SBGers out there are thinking, “yeah, and since I’ve got it broken down into specific areas it’s way easier to get to that response part.”

Grades are sometimes also supposed to:

2. Show how well your student can manage assignments or big projects
3. other performance- but not understanding-based stuff

We make them mean those things by counting participation for 10% or homework completion for 20%. And it’s not unreasonable. We feel like it’s our job to a) teach math and b) raise responsible kids, and we have to cram a full report on both of those objectives into a single letter. There’s also c) get them all high scores on the Tests and sometimes d) keep them safe or e) provide a different view than they get at home and f) prepare them for the unknowable challenges of the 21st century-entury-entury.

When students are thinking, “does this count?” they aren’t thinking about any of your objectives.  When parents are looking at a GPA (a smashing of all of these already overloaded encodings into a single number, if you can believe it!), they aren’t understanding if a) their kid knows math, OR b) their kid is behaving in class and doing his work.

And, ultimately, I think this is why we sometimes grab on to “formative assessment” as a way to get around the horrible restrictions and meaninglessness of most grades.  We say, “it’s ok, it doesn’t count, class,” and if we’re lucky our students can relax enough to just enjoy learning something.  “This is just formative assessment – we’re just exploring to see what works,” as if there’s any other way of learning.

This is why active SBG is so great.  It lets us give grades that mean something – “hi John, here are your grades,” and John can understand them and react to them and now you’re communicating with John about his learning.  Don’t be fooled – these can still be grades that you record, that “count” – but you’re still helping John learn.  My favorite part of teaching was being with the kids in class, because that’s where the learning actually happens.  Grades are just administrative bullshit compared to that – unless they are also part of the learning process.

I think of active SBG as reclaiming grades as feedback, and de-emphasizing the evil baggage that can lead our students away from learning.  For me, the difference between formative and summative isn’t very important.  Instead, we’ve got to focus on the distinction between meaningful and meaningless.

# Math Teachers at Play Blog Carnival: November 19th

The next Math Teachers at Play blog carnival will be hosted here on November 19th.  It’s a fun way to share “tips, tidbits, games, and activities for students and teachers of preschool to high school mathematics.”  You’ve still got a week to either write a new article or pick one of your favorites that hasn’t appeared in the carnival yet to submit.  Let’s see what you’ve got!

And PS: you http://mastersinwhateverdegreeyouwant.com people can stop sending me submissions now – you’re not getting in! (Think dozens of “top 50” lists, emailed to me at regular intervals.  Argh!)

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.

I think a lot of us like the ideas of active grading because we care more about helping our students learn than about

1. their transcript or
2. comparing them with each other.

We give feedback as a way of helping students learn.

But we also want to give feedback in the form of numbers.  Numbers have all these great properties that meaningful feedback doesn’t have – you can average numbers but not comments, and you can compare numbers “objectively” but not comments.  It’s faster to read numbers than comments, and I can scan a transcript with a GPA to decide whether to let someone into my college much faster than I can read 30 pages of writing.

So, we condense our knowledge about students’ learning down to numbers (or, more extremely, a single number!).

Once we record a grade as a number, we’ve lost information.  I gave Mike an 85% last year in calculus, but that doesn’t tell you that he just couldn’t get his head around the idea of a differential equation.  You gave Sandeep a 75% because he aced every test but never handed in a piece of homework and skipped every other class.

But now I’m a college, Mike has a 3.5 GPA, Sandeep has a 2.5 GPA.  I can clearly see from these marks that Mike is a better student – by twenty-five percent of the scale.  Maybe Sandeep has some extracurriculars or something, but he’s got some major catching up to do!

When we do this we’re acting like recording grades as numbers adds information to them!  We can’t sort comments  in order of academic achievement (automatically), but it’s no sweat sorting GPAs – even from different teachers in different schools, each with his or her own idea about what the grade levels even mean!  This is inappropriate.  The numbers are not orderable.

I’ve been thinking about this a lot in the last couple of months, as I build ActiveGrade.  How do we use numbers to represent grades where it’s appropriate – to get the power of the best-fit line, or the correlation – without giving those numbers too much power, like the power to rank (which is nonsense with different definitions) or average (a 0% F averaged with a 100% A is: a 50% F.  Talk about effed up!).  I think I’ve hit upon a few great ideas – more on that soon.  What do you do?

# I’m Just Saying

This is something SnapGrades chose to show in their demo video.

Cool how it automatically translates your message to Spanish, eh?  Now even the exclusively Spanish-reading parents can tell their kids to “study more!”

How much value does “Good improvement!” hold for Laura or her parents?

# More Geogebra Tools: Programming Will Fix _Everything!_

Everyone’s talking about programming in math, and I’m here to tell you: I AGREE!  Let’s do more programming in math!

In order to successfully program a computer, students must understand the math they are doing.  Computers give students instantaneous feedback, but the feedback is 100% neutral – it does not say “hey, you’re right” or “try again.”  It says, “This is what you did,” and Johnny must decide for him- or herself if it was right.  This is what we’re talking about, right?

The annoying part is that learning to program a computer takes a long time, even in simple cases.  The best thing I’ve found is Geogebra, which has a language very similar to normal, standard, algebraic notation – you can feel OK about spending 20 minutes practicing it with kids in your math class.

A little example (search my blog for others!): Newton’s method with tools in geogebra!  If you just want fun, interactive applets to use in your class, scroll to the bottom.  Double-click one of the applets to open up your own window, from which you can save to your desktop or whatever you want.

Type f(x)=x^2-2 to create the function f.

### 2. Create your “initial guess.”

Choose the “New Point” tool from the toolbar – the one with the blue circle and the ‘A.’  Click on the function you made to create an initial guess.  I’ll choose a deliberately poor one for demonstrative purposes.

### 3: Create the line through that point, tangent to your function

Here’s something your students will need to be taught – takes about 5 minutes: x(A) means “the x-coordinate of A” and y(A) means “the y-coordinate of A”.

With that, we can make the equation of the tangent line here.

y-y(A)=f'(x(A))*(x-x(A)) – good old point-slope form!

At this point you can choose the pointer tool (the first one in the toolbar) and drag the point A around – note that the line updates automatically.  This is more feedback for your students that they have to understand and analyze.

### 4. Find the zero of the line.

There are a bunch of ways to do this, and your students can decide.  The algebra:

• $y - A_y = f'(A_x)(x-A_x)$
• $0 - A_y = f'(A_x)\cdot x - f'(A_x)\cdot A_x)$
• $f'(A_x) \cdot x = f'(A_x) \cdot A_x - A_y$

And finally,

• $x = \frac {f'(A_x) \cdot A_x - A_y}{f'(A_x)}$

The geogebra:

• x_1 = (f'(x(A)) * x(A) – y(A))/(f'(x(A)))

Now, x_1 is the zero of the tangent line.  If you want to keep this conceptual and not get bogged down in the algebra, you could use geogebra’s “root” command as well.  But if you’re going to use root, watch out for questions like, “why don’t we just use root?”

### 5. Find the actual value of the function at your guess… in point form, for visualness.

Type (x_1, f(x_1))

Point B is created, as your next guess for the root!  This is one complete cycle of Newton’s method, programmed into Geogebra.  When your students move A, B moves accordingly.  Try it out for yourself below – it’s an actual geogebra file, not just a picture!

Sorry, the GeoGebra Applet could not be started. If you're looking at this in a reader, try going to the actual post.

### 6. Create a Tool

This is the coolest part, and the part that your students will love the most, the part that will give them the most feedback and the greatest sense of power, and the part that will take you at least 20 minutes to teach them to do.  Geogebra can create “tools,” which encapsulate all of the work we just did, and can apply the same steps repeatedly.  I won’t go too far into it in this post since it’s explained for a different situation here.  Basically, the function f and the point A are the inputs to your tool, and the line and the point B are your outputs.

Creating the tool and applying it repeatedly has very neat effects – you can get an arbitrary number of steps in your Newton’s method calculator.  I’m telling you that kids who can do this

1: Really understand Newton’s method

2: Will start thinking about math in a new way

3: Will freaking love your class.

Just look how fun and interesting the tool is when applied three times.  Drag the point A around.

Sorry, the GeoGebra Applet could not be started. If you're looking at this in a reader, try going to the actual post.

And here’s an example of a function (provided by Shawn) that resists the method.  Imagine students trying to classify which points will work and which won’t!

Sorry, the GeoGebra Applet could not be started. If you're looking at this in a reader, try going to the actual post.