Giving Feedback with a Highlighter

I attended John Scammell’s excellent 3-morning sessions on Formative Assessment at #TMC15. We were asked to share strategies that we may already be doing to give students feedback . I shared about how I used highlighters for this. I promised my group that I would write a short post about it, but I waited until now since I needed the school year to begin to have student samples to share.

I used highlighting to give my 6th graders feedback on their first PoW (Problem of the Week from The Math Forum).


It’s challenging, as I hope all PoWs are, and even more so when it’s the first one they get. I give no specific instructions on how they should write up their solution — nothing more than the usual “show all your work in order to receive credit.” I want to see what raw stuff I get on this first submission. We’ll worry about quality control soon enough.

I’m familiar with what I can expect with the first harvest of solution write-ups. One-fourth of the papers are pleasantly stellar, one-third show candid efforts (especially the ones with parents’ writings on them), another third make me get up and stick my head in the fridge to find a cold-and-alcoholic beverage, and the rest of the papers remind me that some of my 6th graders are still working on finessing the opening of their combination locks. The other right, sweetheart. There you go.

Years ago I taught a writing elective. I was at the beach — at the Oregon coast — because that’s where you should read and grade all writing papers. I forgot my red pen. I only had a yellow highlighter. The highlighter transformed my grading. I no longer cared so much about the writing mechanics — fuck spelling and punctuation and syntax. You got voice in your writing, kid. Your heart was wide open in this third paragraph. How did you know the rain smelled differently depending on what part of Portland you were in?

I highlighted sentences and words that spoke to me. I highlighted a brave sentence. I highlighted the weak ones too. The highlighter allowed me to interact with the kids’ writings differently. I didn’t add to or cross out anything they’d written. The highlighter didn’t judge the same way my red pen was judging.

And that’s the history of using the highlighter for me. But back to math. I have over 100 students and to write feedback for their bi-weekly PoW write-ups is all too time consuming. The different colored highlighters come to my rescue.

I’m going to continue using my binary scoring system because it worked well last year. I look through all the papers, separating them into two piles: papers that got it (full 1o points) and papers that fell short (1 point). These kids will get another week to revise their work and re-submit.

I use my yellow highlighter — just swipe it somewhere on their paper — to show that I’m having trouble understanding their work or that their work is lacking.




I use the pink highlighter to show that the answer is not clear, not specified, is partially or entirely missing.




I use another color (like green or blue) if the papers warrant another something-something that I need to address. I didn’t need to with this week’s PoW submissions.

If necessary, I will write on their papers directly. But I don’t have to do too many of these because kids’ mistakes, more often than not, are similar to one another.


When I pass the papers back, I tell students what each colored highlight means and what they need to do to revise their work, including coming in to get help from me. It’s a helluvalot faster than what I used to do.

Guess that’s it. Feels good to write in this space again.

Posted in Problem Solving, Teaching | Tagged , , , , , , | 20 Responses

A Love Letter to MTBoS (a.k.a. my #TMC15 keynote)

Thank you to Lisa Henry for asking me to talk at TMC and for believing that I could pull it off. Thank you to Baylor for the letter below that kicked me in the gut and said, “Stop whining and finish the slides.”


I looked out to the audience and began with this ad lib.

And off I went. Here are the slides for my keynote.

Thank you for being the kindest and most gracious audience.

Much love,

Posted in Teaching | Tagged , , , , , , , , | 6 Responses

A Book: Thinking Mathematically

I’m re-reading Thinking Mathematically, an assigned book from a math course I took years ago in Portland. I was teaching science at the time but signed up anyway because I’ve always loved math.

Thinking is still so good and resonates much more now that I’ve been teaching mathematics.

In the Introduction, under “How to use this book effectively!”:

Recalcitrant questions which resist resolution should not be permitted to produce disappointment. A great deal more can be learned from an unsuccessful attempt than from a question which is quickly resolved, provided you think about it earnestly, make use of techniques suggested in the book, and reflect on what you have done. Answers are irrelevant to the main purpose of this book. The important thing is to experience the process being discussed.

… our approach rests on five important assumptions:

  1. You can think mathematically
  2. Mathematical thinking can be improved by practice and reflection
  3. Mathematical thinking is provoked by contradiction, tension and surprise
  4. Mathematical thinking is supported by an atmosphere of questioning, challenging and reflecting
  5. Mathematical thinking helps in understanding yourself and the world

These assumptions need to live in our classrooms.

The problems in Thinking are mostly brief and simply stated — yet each one has the potential to make you linger a bit longer because you want to savor your own thinking. Not even productive struggle, this is sweet struggle.

How many rectangles are there on a chessboard? [Page 43]

I have just run out of envelopes. How should I make myself one? [Page 35]

A certain village in Jacobean times had all the valuables locked in a chest in the church. The chest had a number of locks on it, each with its own individual and distinct key. The aim of the village was to ensure that any three people in the village would amongst them have enough keys to open the chest, but no two people would be able to. How many locks are required, and how many keys? [Page 176]

I’m finding out that the 2nd Edition came out in 2010. Amazon does not have it in stock currently, but when it does become available, we can rent it for $54.77 or buy it new for $91.29. What??

Posted in Problem Solving, Teaching | Tagged , , , , | 8 Responses

Heavy Heart

But we would not have you ignorant, brethren, concerning them that fall asleep; that ye sorrow not, even as the rest, who have no hope. (Thessalonians 4:13).

The Rev. Sharonda Coleman-Singleton, 45

DePayne Middleton Doctor, 49

Cynthia Hurd, 54

Susie Jackson, 87

Ethel Lance, 70

The Rev. Clementa Pinckney, 41

Tywanza Sanders, 26

The Rev. Daniel Simmons Sr., 74

Myra Thompson, 59

The math task can wait. The lesson on polynomials can be postponed.  I want to talk about racism in America. Now. I think about the many interactions I have each day with my students. Each of my colleagues has as many. There are 600 students here. There are over 2,000 at the high school a few miles away. We teachers see many students all across cities and suburbs. I want today’s conversation to be about love and humanity.

I want tomorrow’s conversation to be about kindness and tolerance.

Eventually these conversations don’t need to happen every day because love, humanity, kindness, and tolerance have become part of our breathing — they are in our blood, in our black, white, yellow, purple skin. This is my prayer.

Posted in Teaching | 1 Response

Not That You Care about My Schedule

Another two weeks and I will have completed 22 years of full-time teaching. Time flies when you’re grading papers and having fun. Says no one.

I have a handful of commitments that I’d like to share — maybe you’ll be in attendance or in town and we can say hello!

Now: I’m working with Max Ray-Riek, Rafranz Davis, and Elizabeth Statmore on a writing project that if-I-told-you-I’d-have-to-kill-you-so-please-just-eat-a-cracker. I’m just honored to collaborate with these three amazing people.

June 5-6: I’ll be at NCTM Headquarters — in Reston, Virginia — to attend my first meeting as a new member of the Professional Development Services Committee (PDSC). We’ll meet again in August and November of this year.

June 25-26: Ashli Black has invited me to speak at the Oregon Math Network conference. I wonder why. :) I’m grateful that Ashli thought of me. Elizabeth will be there too! I’m planning to stay in Oregon for an extra week to visit with family. Actually that’s a lie because I really just want to hang out in all 56 breweries in Portland.

July 23-26: Twitter Math Camp! On Friday, 4:00 to 5:00, I’ll be co-presenting with Matt Vaudrey on Barbie Bungee and Desmos. (Matt is the presenter. I pass out the rubber bands.) On Saturday, 1:30 to 2:30, I’ll be giving a keynote. Lisa Henry rejected my request to have the talk at the local karaoke bar. #killjoy

July 27-29: The day after #TMC15, I will be one of the presenters at NCTM Interactive Institute in Anaheim, CA. My 3 sessions (over 3 days with the same attendees) will focus specifically on ratios and proportions. Andrew Stadel is the other presenter on the same topic, but he’ll have his own attendees in a different room. Guess the demand is higher for this topic thus they’re offering two concurrent workshops. [Update, July 10: Due to lower than expected enrollment numbers, I’d volunteered to cancel my part in this. Andrew will rock this!]

August: I need to carve out some time to fulfill my role as one of the judges for the Item Writers project.

August 6: I’ll be doing a full-day workshop in Santa Barbara. It’s the 3rd Summer Institute of Teaching Beyond Textbooks.

August 12-14: The UCSB Math Project is my favorite probably because I’ve been at it the longest (5 years?) and it’s local and the people whom I work with know how cranky I get when I don’t eat. We are putting on a leadership retreat at the beautiful La Casa de Maria.

August 21-23: I’ll be back in Virginia for the NCTM PDSC meeting.

October 22-25: Chris Hunter had invited me to present at the Northwest Math Conference in Whistler, Canada. I’m honored and excited to hang out with these familiar faces and meet new people! With any luck, I might finally get to meet Timon Piccini.

November 6-8: I had a wonderful breakfast with Brian Shay on the last day of NCTM Boston. Told him how busy life had become and that I needed to say “no” more. A week went by and I got an email from Bruce Grip asking me to speak at CMC-South. Guess who told Bruce to contact me. (I said yes only because this presentation will be a repeat from the Whistler one above.)

Have you ever driven through the intersection only to realize that you’re now stuck in the intersection because traffic is just not moving? The yellow light that you try to beat is about to turn red — which means within nanoseconds of that happening the light will turn green for the other people. Except these other people can’t go anywhere because you’re the asshole inside the car that’s blocking their passage. But a miracle happens. Someone has pulled up right behind you! Yeah. Someone who is an even bigger asshole than you thinks there’s room for their car to squeeze through the intersection before the red light gets any redder. So you’re now blocking traffic but smiling and thinking, Hey I’m not the biggest asshole after all. Everyone is now mad at the douchelord behind me. What a great day this has been. I’m going to get some sushi.

I forget the point of this story. Or the point is that lately I feel like I’ve been that first asshole. So sad.

Anyway, have a wonderful summer if you’re already there. (Translation: You suck.)

Hope to see you at the next gathering! xox

Posted in General | Tagged , , | 5 Responses

What song was it?

I miss having time to read and write. I miss my kids. Nicolai is graduating from college in two weeks. Gabriel has decided, after freshman year, that college is not for him. He thought about being a truck driver because he likes to drive. I once wanted to be a truck driver too. Contemplating the life of open road and truck-stop diners — and realizing that only one of these is appealing. Sabrina finished her sophomore year and went right into doing research this summer, I won’t see her until late August.


A few weeks ago my students took the SBAC Performance Task (PT). We had to do a classroom activity prior to them taking the computer-based PT.

The main purpose of the classroom activity is to ensure that all students have a common understanding, at a minimal level, of the contextual elements of a PT topic so they are not disadvantaged in demonstrating the skills the task intends to assess.

One of my 6th graders sounded rhetorical, “Don’t we all know what a video game is.”

I heard the unspoken agreement among her classmates. This was unfettered privilege, I thought. Then I remembered something and told them this quick story.

I was already two years out of college and teaching middle school science. Our large district offered a 3-day science workshop — retreat style at the breathtaking Silver Falls Lodge. Two deer came out as if to greet me when I pulled into a parking space. Our first meeting was an evening of social gathering in the cozy Smith Creek Meeting Hall. I knew fewer than a handful of people. The program director took the mic and welcomed us. He said we should sing a song together to begin our fun-filled days of science workshopping. As a way to bond, he added. Everyone agreed and almost immediately broke into chorus. Everyone but me. I just didn’t know the words to the song. Nor have I ever heard of the song. The singing seemed to have gone on for much too long while I stood small and insignificant. I felt like a foreigner. All over again.

One student asked, “What song was it?” I replied, “I don’t know. I didn’t know it then, so…” I ended by telling my students that the director had assumed everyone knew the song. Who we are and what we know are our privileges. Everyone in here may know what a video game is, but we shouldn’t always assume that.

Gabriel — my possible future truck driver — reminded me once that not all his friends lived in homes and apartments. His friend was living in someone’s garage.

Posted in Shallow Thoughts | Tagged , , | 1 Response

Desmos Puppy House

We don’t cover quadratic in Math 8 this year, so we really can’t do Des-man. I come up with “Puppy House” instead so my students can still create something that allows them lots of practice with writing linear equations with domain and range restrictions.

They draft the house on paper. The house needs to have a minimum of 6 slanted lines, 5 vertical lines, 5 horizontal lines, and one image of a puppy at entrance of house.




Below are the ones from my Math 8 Honors kids who learned quadratic last year. They completed these within one 55-minute period. So proud of Ashlynn for remembering how to write inequalities.




I’d love to see what amazing dog houses your students will create. (Someone create a flap door for me to play peek-a-boo with the puppy please!)

Posted in Algebra, Math 8 | Tagged , , , , , , | 3 Responses

Students Practice Scoring Short-Text SBAC Responses

A few weeks ago I attended an all-staff PD at the County Office. During the morning session we scored samples of 2-point short-text items from grades 4, 8, and high school. It was time well spent.

I wanted to duplicate that experience for my students with two goals in mind:

  1. See how well they can interpret and use a scoring rubric.
  2. For them to attend to the same thoroughness and precision in their own solution writing when it’s their turn in May.

What I had my students do:

1.  Get to know the short-text item.

They worked on the grade 4 item below. This was intentional to diminish any math anxiety and to keep our focus on the scoring of the task.

While it was good to learn of my kids’ different solutions, it was also disheartening — but not too surprising as they are the same ones who struggle mightily — to learn that 20% of my 6th graders did not get the correct solution for this grade 4 item.


2.  Go over the solution.

I collected their papers and just had a couple of kids share their strategies to the whole class. Considering 1 out of 5 kids in the room didn’t quite know how to solve the problem, this step was really for them.

3.  Get to know the item-specific rubric.

I gave the kids quiet time to read the rubric, reminding them that they would use this rubric to score 9 students’ solution responses. I told them that they could expect to return to the rubric over and over again as they scored each response.

4.  Score the responses.

I gave them quiet time to fill out Score 1 column of this handoutI reminded them that this was one of the main goals of the task — to score the sample responses fairly and accurately using the item-specific rubric.


Why are you giving this response a 1? What is it missing to not get a 2? What does it have to earn a 1 and not a 0? Keep referring to the rubric! Does spelling matter? What does your rubric say about spelling errors?

After everyone was done filling in Score 1 column, I asked them to talk to their neighbor/s and only fill in Score 2 column if they changed their mind. (They were not to erase any score in Score 1 column.) This also made it easy for me to see how many scores they’d changed their mind on.

My favorite thing in the whole wide world is to listen in on their conversations about math.

5.  Reveal the actual scores.

[The actual scores are on 2nd page of handout above.]

If the whole class agreed with the actual score for a particular student response, then we moved on. But if anyone disagreed, then I had that student tell the class why. Then I had another student who agreed with the answer to share his/her reason.

Out of 66 students, 24 students scored 9 of 9 correctly, 19 students scored 8 of 9 correctly, and 6 students scored 7 of 9 correctly.

That meant 65% of my 6th graders did this scoring-using-a-rubric better than I did. Whatever.

I also asked the kids to write a couple of sentences about what they got out of doing this. Most of their responses echoed these:

This was helpful to me because now I know I need to be much more thorough with my work and explain why I might of did something.

This was helpful to do because it let us see how these problems are graded. Even though the problem was for 4th grade, I think the grading scale of conclusion and math will be similar or the same for all problems like this.

I believe this was helpful because when I take the test, I will be more aware of the questions and what is expected of me. I will make sure to always back up my answers with evidence.

Posted in Teaching | Tagged , , , , | 6 Responses

Grade 6 Rocks Visual Patterns

I’m always happy to hear how teachers use with their students.

Michael Fenton shares how he uses the patterns with Desmos. And this.

Alex Overwijk’s students use the big whiteboards.


Bridget Dunbar removes some figures, and kids need to draw them in.


Kristin uses the patterns with 5th graders.


I do patterns with my students on Mondays as part of our warm-up routine. I’ve already shared 28 pattern talks (and 28 number talks) on, but I’d like to share a couple more here because my 6th graders have made incredible gains in seeing a pattern in different ways and in articulating an equation to go with each visual.

This is pattern #153. I’m sharing this one because I meant to only use it with my 8th graders, but my printer was acting up and failed to print a different one for my 6th graders, so I just used the same one. Fun challenge!


Student 1:

I see these 5 spokes coming out. Each one has n number of hexagons. In between these 5 are Gauss. So, the equation is… five times n, plus five Gausses.

153 marked

 Hexagons = 5n + 5(1+n)(n/2)

 Over time, my students have come to recognize Gauss addition very quickly. They have used Gauss as a verb and a noun, as in, I Gaussed it or I saw two Gausses in the pattern.

Student 2

Each step adds another ring of hexagons on the outside. Looking at the outer most ring, I see three groups of (n+2), plus a leftover. The leftovers are odd numbers. So, the outer ring alone is 3(n+2) + 2n-1.

153 marked 2

And the rings add like Gauss!

Together we write the equation carefully, talking through each step.

Gauss means adding the first and last steps together, then multiply by the pairs of steps. The last step is the outer ring, the first step is the inner ring, which is always 10. So, 10 plus the outer ring, then multiply this by the number of pairs [of rings], which n/2.

Hexagons = [10+3(n+2) + 2n-1](n/2)

We were confident we had the correct answer when both equations simplified to the same equation.

Hexagons = [(5n^2)+15n]/2


This is pattern #147. I’m sharing this one because of the many different ways kids tried to see the pattern. Normally, when I randomly call on a kid to share and someone had already shared their same way of seeing, then they just have to come up with a different way.


147 marked

Ducks = (n^2) + (2n+1) + n

147 marked 2

Ducks = (n+1) + (3+2n+1)(n/2)

147 marked 3

Ducks = n(n+2) + (n+1)

147 marked 4

Ducks = 2(1+n)(n/2) + (n+1) + n

147 marked 5

Ducks = (n+1)(n+2) – 1

147 marked 6

Ducks = (n+1)^2 + n

I very intentionally do not have kids fill in a table of values for visual patterns. I’m afraid it becomes a starting point for them every time instead of just looking at the pattern itself. For our 8th graders using the CPM curriculum, which I like a lot, there are plenty of opportunities in the textbook to tie all the different representations (table, graph, rule, sketch). These are my 6th graders who are writing quadratic equations without all the fuss right now.

Please continue to share the site. What I love most is learning that the patterns also get used in elementary and high school classrooms.

Posted in Algebra, Course 1 (6th Grade Math), Math 8 | Tagged , , , | 10 Responses

Reversing the Question

Don Steward posted this on Sunday. Like Don, I really like this task and also think it has a certain Malcolm Swan je ne sais quoi about it.

I showed only the top part to my 6th graders, and I gave them 2 minutes to write down what they notice.

In addition to noticing the given information, the students also mentioned:

It takes a lot of grams to make a sponge cake.

A kilogram must have a lot of grams in it.

The unit of measurement is changed in the cake and in the big bag.

This problem doesn’t have a question.

The sponge cake has no price.

There is frosting on the cake.

You need to change the measurements first (kg – g)

The cake weighs more.

The cake is really small and the flour is really big.

Not much information and there’s no question.

There is no question. The weight of the bag is in kg but the flour it takes for the cake is in g.

You have to convert 24 kg to grams.

It uses only very little of the flour.

I gave them another 2 minutes to write down what they wonder.

How much the sponge cake costs.

How many sponge cakes can you make.

How big is the sponge cake.

How to convert from kg to g.

If the sponge cake is good.

24 kg is <, =, or > 150 g.

What we are going to have to solve.

Is the question going to be about if there’s enough flour or is it going to make us change it from kg to g.

How much 24 kilograms is in pounds.

How many grams are in a kilogram?

How many krumkakes can you make with the bag of flour.

How long will it take for the cake to be ready.

What a sponge cake is and what the recipe is for the cake.

How much flour is left.

What flavor is the cake. (Sorry. I’m hungry!)

What you’re wondering that I’m wondering about.

What the question is, and if there are more ingredients.

If we will have to find the price for the sponge cake, or maybe we have to find out how much half as much flour is worth.

Then we moved on to next part of providing questions that would go with the calculations. This was so very tough for my kids. While 21 of 31 kids could come up with the correct question for part (a), they were lost with what to write for the other three parts.

reverse the question 2

Because we didn’t care what the numerical answers were, the kids didn’t do any calculations, instead they were supposed to focus on the operation(s) in each problem and decide on the question that would prompt a specific operation.

The most common question for part (b) was, “How much does 1 kg of flour cost?” (I’d swapped out £ for $.)

Then my next step would be for them to go ahead and use a calculator to get the numerical answers. They will see that for part (b), 24 divided by 21.50 equals approximately 1.12. And if 1.12 were the cost for 1 kg of flour, then 24 kg of flour should cost more than $24. But, wait. We already know that the bag of flour costs $21.50. Hopefully they’ll arrive at this contradiction on their own, and re-think their question.

We’ll then attach the units to the numbers accordingly and let the conversations continue. When kids tell me that they will calculate two numbers using operation w, I always follow up with, What does your answer mean? What unit or units does it carry?

Too often kids have trouble with word problems. Too often they don’t know what to do with two numbers let alone a bunch of numbers. They guess at division when one number is big and one is small. They add when they see two fractions. They multiply because that was how they solved the last word problem.

I will also do this with my 8th graders because I suspect they will have trouble too. And this is exactly the kind of trouble we need to get into. Now rather than later. This task gets them thinking about ratios — which is like the most important math thing in all of the math things.

Steward is exactly right about this task. And I’m thankful he shared.

It was used as a fine example of how reversing the question can often lead to a more challenging task.

Posted in Course 1 (6th Grade Math), Math 8 | Tagged , , , , , , | 8 Responses