Rubrics, Feedback, and Learning

We have found a formative assessment plan for our Algebra I learners that is working for us.  We have always struggled with the idea of the generic 4-point rubric, because it didn’t give our learners enough feedback.  If you want to see some examples, you can search on formative assessment in the blog and find several. 

Our team has now begun to think about using the 4-point rubric to help our learners understand our expectations with regard to effort, listening, questioning, collaboration, homework completion and other non-graded skills that we want them have.  We have used the assessment below twice, once in February and then again in March.  Our learners level themselves and then write about their growth and areas where improvement can/should occur.

Only now have we started the process of using the document above to calibrate our perception with our student’s perception of these important topics.  Even without the calibration, my learners seem to use our rubric to help self-reflect and create action steps. 

The following is a reflection written by RM about her understanding in Algebra I on February 9, 2011.

“This semester I have really improved on helping others in the classroom. I feel like I now go around more to try and assist others who don’t have the right answer and don’t understand what’s going on. This makes me feel like I am really helping and participating in the classroom. I have started to ask more questions to Ms. Gough when I don’t completely understand what we are talking about and I ask my table mates first to see what they think. This is another way of better learning!! I feel like this is a great way to start off the semester.”

Then, a reflection written by RM about her understanding in Algebra I on March 17, 2011:

“During this semester I have learned new things and tried my best to comprehend them and apply them to everything that we do in class. During the course of this semester what has really helped me is during class after we have finished a problem we get up and help other people that don’t get it; this method has really helped me to find other peoples mistakes and repeat my knowledge of the lesson we are learning. Also during class when we finish problems we check with other people who think they are right but we did not get the same answer which helps you to either find what you have done wrong or find what the other person has done wrong and that is your chance to explain to them why they got something wrong or why that concept makes sense, both thing I have done this semester that really makes me more confident about what I know. During Office Hours when my friends and I do our homework we pull out our phones do a problem and then check it together, then we do what we do in class which is helping a friend if they didn’t get it right or they don’t understand.  The story above has really helped me when it comes to test day or as we sometimes call it in class “the championship game” which is when we show what we have learned that unit and apply it on the test. The things we do in class I feel makes me very confident and helps when we get to the test because I know that I know what was going on in class and that really helps me.”

I am thrilled to witness RM’s growth in knowledge and confidence.  The skill of comparing thinking to find a solution is incredibly powerful and important.  Wouldn’t it be nice if our country’s leaders would take the time to sit down with each other and hear what the other side thinks and learn from each other? One of RM’s greatest strengths is that she is willing to listen and consider another solution.  She is definitely a lead-learner in Algebra I, and she demonstrates this by her willingness to teach and to be taught by others. 

The following is a reflection written by SJ about his understanding in Algebra I in February.

“I think that my performance in math class since January 4, 2011 is pretty good, but could be better. I consistently focus on tasks, but I require occasional encouragement from Ms. Gough to do my work. I usually participate when working in groups. I ask questions most of the time when I need help, but I don’t really help others learn as much. I listen well in class and I think my questions keep myself and others engaged during class. I feel that most recently, I have hardly forgotten any homework and I usually complete deep practice. I believe that for formative assessments, I don’t prepare as much as I should. I am at a level 2 on every aspect except on the part that says “I do not get serious about my learning…”

And then, SJ’s reflection in March.

“I believe that I am constantly improving in math class. I have fun while getting better in your class. I struggle with doing well on the first chance tests. While I usually nail the second chance tests, I usually need the extra practice before the first chance tests to do better on those as well. For example, I got a really disappointing grade on the first chance test on factoring. I really went over that test and improved. I believe that I can be an honors level math student if I put my mind to it. I got a 92 on our last test. That is one of many examples of my mathematical abilities. I hope that I never stop learning and that from here I can only move forward.”

Isn’t it great to see how SJ has self-corrected since the middle of February?  In February, SJ wrote about how his performance could improve.  It has!  I am most impressed with SJ’s work on the polynomials test.  I loved his approach.  He worked in class to make most of his corrections.  The next morning he arrived before school with more questions.  In fact, he had highlighted notes that said ask “Ms. Gough about this.”  He was at level 2 in February and is now a solid level 3. 

The following is a reflection written by WAM about her understanding in Algebra I.

“So far in Algebra I this semester, I have learned a lot, but have also been challenged. I am almost always focused in class, and trying to finish the task at hand. Sometimes, my table team and I get distracted by each other, and get off task. Mrs. Gough encourages us to stay on topic, so we can learn more. In our class, we have many group activities, where we have to work as a team. I participate in these assignments, and help others if they do not understand. I have learned from these group activities that when my team is there to help me along the way, I learn the material better, and it is clearer. When I do not understand something, I ask my table members for help. If my team doesn’t know, I ask my other classmates, or Mrs. Gough. I know that when I ask questions, it helps me have a much better understanding. Also, to practice, and to learn, I do my homework. I do my homework every night, and come to Office Hours regularly to complete it. Sometimes, I forget to do deep practice, which is an opportunity that I should take. I know that doing deep practice helps to learn from your mistakes so you don’t make them again.  I always study and put my best effort into the first chance tests, and the formative assessments. I know that putting a lot of effort into them makes me understand what areas I need to work on. I like the fact that we can have a second chance test, because it makes you learn the concepts that you didn’t know when you took the first test. Over all, I have put in a lot of effort, and try my best to improve.”

And then her reflection in March:

“Since the last interim, I feel like I have improved a lot in Algebra.  I have started to take math more seriously, and focus on my studying. Before, I made good grades in math, but I think I was learning the material, but it wasn’t sticking with me. Now, I am trying to really understand the material, so that sticks with me.  Since then, I have tried to study my hardest to prepare for the first chance test, rather than using the second chance test as a comfort. When I go into the First Chance test with a mindset of thinking that my best and final work has to be now, I do better. If I try my best on the first chance test, then the second chance test is only used to make my understanding better, rather than a re-take of the test. In the first Interim, my first chance tests weren’t so great. The more answers I got incorrect on the first test, the more problems I had to redo on the second. The more problems I have to redo, the more likely I am to get some problems wrong. Lesson learned, put all your effort into the first time. Also, since the first interim, I have put more effort into my homework. Before, I rush through my homework, just so I could finish it.  I would sometimes do deep practice, but would frequently forget. I think that before I was being lazy with my homework. Now, I do my homework with more effort. I do each problem, then check the answers, and if I get it wrong, I keep trying it until I understand how.  Overall, I am putting more effort into the class, and more effort into my learning.”

I confirm that she is living what she has written.  While I think the algebra that we are learning is important, the lessons of effort and work ethic that WAM expresses above are much more valuable skills to learn.  I am very proud of her learning and her dedication to the learning of others. 

Once our expectations were clear, our learners begin to work to meet them. 

Isn’t it great that they talk about their learning?

Tearing Down Walls

We live in an increasingly connected world. Yet barriers to connection continue to operate in schools. Kathy Boles at Harvard has described school as the egg-crate culture. With some exceptions, teaching can be an isolated and isolating profession, unless teachers and administrators work to be connected to other learners. It is far too easy to go into one’s classroom and teach…relatively alone…siloed. Classes right next door to each other, much less across a building or campus, often have no idea what is going on outside the four walls in which they are contained. And departmentalization makes for an efficient way to deliver content in neat, organized packages, but departmentalization is not the best parrot of the real, inter-connected, messy-problem world.

What can we do to step closer to modeling and experiencing real, inter-connected problem-addressing?  How do we communicate with each other when we are assigned classrooms where we can be siloed?  What could greater connectivity look like for learners of all ages?

Recently, learning partners Jill Gough and Bo Adams submitted a roughly made prototype of a three-minute video to apply for a speakers spot at TEDxSFED. It’s about “Tearing Down Walls.” It’s about experiments in learning by doing. It’s about learning.

Create a Guiding Coalition of Students – The Way It Should Be – Examples of Good Work

If you read my blog, then you know I have students that are hiding in plain sight.  Yesterday afternoon during Office Hours, my learners made an appointment with me for today’s study hall. I said yes and was pleased with the request. However, today my advisees are knitting for charity, and I wanted to be there too. I cannot be in two places at once and there was other adult supervision, so I went to study hall.

Eight learners were waiting on me as I arrived, all with papers in hand and questions to ask. It was GREAT. It was a mix of my two sections of learners. They took turns; they helped each other. They were patient and considerate; they were motivated and demonstrated self-discipline.

Twenty minutes into the event, I paused and said “this is the way class should be; why can’t our class be like this? I want our class to work like this. I want you to be successful; I want to help you learn. I can only do this if you ask me questions and work together. I want you to ask questions.”

In that moment, I think a guiding coalition was born.

My class was much more about learning and asking questions.

Today’s lesson was to formatively assess our progress on solving quadratic equations. In my class that hides in plain sight we worked one level at a time, collected data using the TI-Navigator, and decoded individual problems. It was GREAT! (I’m so sorry that I don’t have screen shots to show, but I was in the moment and did not think to save. Boo. I’ll do better in the future.)

Level 1 was about substitution and order of operations. You can’t do the quadratic formula successfully if you can’t do this. Together, we learned that all but two of us can do this. In community we discussed what errors might have been made by these two learners. LM said “I know how 41 became the wrong answer, because I almost made that mistake.” LM went on to explain what another learner must have been thinking and what to look for to avoid this type of error.

Level 2 was also about substitution and order of operations, but the expression was much more complicated. Again, you won’t be successful with the quadratic formula if you can’t do this. We learned that only half of us could do this. The biggest error was with integers and substitution. Now we had half learners and half lead-learners. Everyone engaged in the process of getting all of us to success. Differentiation, decoding of errors, and individualized instruction was high.

Much more important: There was success and confidence. When we got to level 3, our target level, I began to hear comments like “this is much easier than I thought.” “This makes sense to me, now.” I watched QB slide his chair up next to DG and decode DG’s error. QB said “Oh no, man, you’ve got this. You just need parentheses right here.” “Ohhhh…” said DB, “I get it now.”

Shouldn’t it always be like this?

Can it always be like this?

Is it that my hiding-in-plain-sight learners just did not know how they should function? Had they lost or forgotten what concentrated group learning requires and needs? Have they ever known? Did I finally communicate to them what good work looks like in my classroom, a classroom that calls for collaborative learning?

Which learners form your guiding coalition?  Do they have examples of good work?

A Kind of Paradise – Multiple Representations

Have you watched this TED talk?

Chimamanda Adichie: The danger of a single story

As is our habit early on Saturday mornings, AS and I watched HBO’s Happily Ever After: Fairy Tales for Every Child.  This morning we watched Cinderella.  I have always loved HBO’s versions of fairy tales.  How many of us “see” Walt Disney’s version of Cinderella when we think about this fairy tale?  How many versions of Cinderella’s story are there?  If you haven’t seen HBO’s version, you can view the first 9-ish minutes of Cinderella:

From Chimamanda Adichie:

“So that is how to create a single story, show a people as one thing, as only one thing, over and over again, and that is what they become.” 

All of this makes me worry about the unintentional assignment of labels and stereotyping that happens at school.  I worry that we are telling our young learners a single story when we classify them as recommended-for-honors or not-recommended-for-honors.  I worry that we do the same with our adult learners.  Does the single story become the definitive story?

More from Chimamanda Adichie:

“It is impossible to talk about the single story without talking about power.”  

“Power is the ability not just to tell the story of another person, but to make it the definitive story of that person.”

How do we celebrate the strengths and contributions of each individual?  How do we show that we are not a single story, but a collection of stories that create the anthology of who we are now?  How do we convey that the story is not complete, that it is a work in progress?  That there are many choices and crossroads ahead? That we have control of the choices and pace?

Again from Chimamanda Adichie:

“The single story creates stereotypes. And the problem with stereotypes is not that they are untrue, but that they are incomplete. They make one story become the only story.”

How often do our learners overcome their stereotype, self-imposed or otherwise?  Do we just accept the label that we carry?  Do we teach our learners how to overcome a stereotype that they don’t want or accept?

More from Chimamanda Adichie:

“The consequence of the single story is this: It robs people of dignity. It makes our recognition of our equal humanity difficult. It emphasizes how we are different rather than how we are similar.”

The target of our assessment plan is to indicate to our learners what is essential to learn, point them to where they are now, and show them how to reach and exceed the target.  With each formative assessment, learners stand at a crossroads and choose to work (or not) for the target.  While we have set the proficiency target at level three, some choose to strive for more.  It is exciting and motivating.  However, some buy into the stereotype and that is discouraging.

Chimamanda Adichie concludes her talk with:

“I would like to end with this thought: That when we reject the single story, when we realize that there is never a single story about any place, we regain a kind of paradise.”

We strive for our learners to have multiple representations of ideas and concepts; do we also help them (and us) have multiple representations of who they are and can become?

Turnpikes, Toll Roads, Express Lanes

Atlanta:  Traffic, traffic, and more traffic…

Coming Soon! Peach Pass available in Spring 2011. 
I-85 Express Lanes in Atlanta open in Summer 2011. 

View the Peach Pass video to see lots of accessible math connected to a real community issue. Learn more about the I-85 Express Lanes.

  • What’s the difference between an express lane, a toll road, and a turnpike?
  • Are you charged by the mile or by the minute?
  • Why is the target speed 45 miles per hour?  What is the target speed for other express lanes?
  • How will the Peach Pass know when I should pay (because I have less than 3 people in my vehicle) and when I can ride toll free?
  • What is the mathematical model that determines the toll?  We know it is positively correlated.  Will the model be linear, exponential, or some other type of function?
  • What will the revenue generated by the Peach Pass be used for and who controls these monies?
  • Are the Peach Pass and other E-ZPass-type cards cost effective or just convenient?
  • How do the other locations listed in the video charge for the use of their express lanes?  How do other states collect this money?  Utah, for example, uses an ExpressPass.

The Pennsylvania Turnpike is the oldest turnpike in our country.  Beginning in January, 2011 there was a rate increase; cash tolls increased 10% while E-ZPass tolls increased by 3%.  Is there a savings to use the E-ZPass, or is it just for convenience?  Since there is a Pennsylvania Toll/Mileage calculator, we can investigate the cost to drive on the PA turnpike.  Would this help indicate a reasonable rate for driving on any toll road or express lane?

To see if there is a pattern to the cost, I chose to collect data entering the Pennsylvania Turnpike at Interchange 57-Pittsburgh and then vary the exiting interchange for a class 1 vehicle with 2 axels.  I wonder what the toll rate for an 18-wheeler would be compared to my passenger vehicle.  My learners have many choices.  They may choose to start at any entry point on the turnpike and vary their exiting interchanges.  I suppose they could vary both the enter and exit interchanges.  They could also change the type of vehicle to investigate the charges and the rates for different size vehicles.

Is there a pattern to the data?  It the relationship linear, exponential, logistic?

To see the relationship between the data, we graph.

Cash toll charged vs. miles driven on the PA Turnpike:

E-ZPass toll charged vs. miles driven on the PA Turnpike:

To compare the two data sets, graph on the same grid.

More questions:

  1. What are the mathematical models that could represent these data sets?
  2. What are the meaning of the slopes of these lines?
  3. Is it cost efficient to purchase the E-ZPass?
  4. Is there a relationship between the E-ZPass toll charged and the Cash toll charged?
  5. What is the mathematical model that could represent these data?
  6. What is the meaning of the slope of this line?

Which leads to more questions:

  • How does the rate charged by the PA Turnpike compare to the rates of other turnpikes?
  • How does the rate charged by a turnpike compare with the charge on a toll road or express lane?
  • From $0.60 to $6.00 is a pretty big swing in cost to use the 16 miles of the I-85 express lane in Atlanta.  How will traffic volume be determined since tolls go up when traffic volume increases and the toll is lowered when traffic volume decreases?
  • How do the toll roads, turnpikes, and express lanes in other countries compare to our toll roads, turnpikes, and express lanes?  How do they compare in cost, in speed, and in access?

Can our learners aquire the needed content through a problem or project based approach?  Will they find the content more interesting and engaging?

As we learn more about problem-based learning and project-based learning, would this be type of lesson help learners see the application of content? … the blending of content? … the relevance of content?

I think so.  Are we willing to experiment?… to learn by doing?

s=v*t + 0.5a*t^2 ~ Wanneer heeft u vorige bezoek?

If you teach a foreign language, what is the most important or essential for your learners? 

  • Is it that they are grammatically correct? 
  • Is it that they can read a book or watch a video in that language and understand the message? 
  • Is it that they can communicate with others when in a country where this language is the primary language?
  • Is it that they can solve problems that arise while in a country where this language is the primary language?

While being grammatically correct and reading are both very important, I’m pretty sure that we would agree that being able to communicate with others and solve problems in another language would be more important.  If I’m sick, I need to find a doctor or the hospital.  Right?

I made several new friends at last week.  Culturally, we are very different.  For example this came across my Twitter timeline from @mvanast: 

Laatste ontbijtje hier. Gek dat ze deken dat je ijsthee wilt, als je thee bestelt.” 

Roughly translated it says “Last breakfast here.  Crazy that they bring you iced tea when you order tea.”  It is shocking to expect hot tea and be served iced tea.  Michel speaks (and tweets) in English as well as Dutch.  I, on the other hand, speak (and tweet) in English only.  I can use the Live Search translator to read Michel’s messages.  He does not need a translator to read my tweets. 

Our colleagues, the teachers of foreign languages, want their learners to be prepared when they visit another country.  We want our learners to understand the culture, the climate, the traditions and the customs as well as the language itself.  Is this true for me, a teacher of mathematics, the language of science? I dare to say that most, if not all of my colleagues teaching foreign language have been to a country where the language they instruct is spoken.  Have we, the teachers of mathematics? 

Are we teaching a language when
we have never visited the lands where the language is spoken?

Has it been so long since we have visited these lands that
we have forgotten about the culture and the traditions that are important?

Photo by @fnoshese, 2010 (Cross River, NY)


What are the components of the culture, climate, traditions and customs? What are the conventions and must-knows for the lands our learners use our language to survive, function, and thrive?  What serves as the Live Search translator for our learners when they are immersed in one of these lands?  How can we, teacher-learners, develop opportunities for “foreign exchange programs” and visit these lands to experience the culture and practice our language?

“If we teach today as we taught yesterday,
we rob our children of tomorrow”
~ John Dewey