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Reflective writing in math

October 19, 2014

Somehow there’s a disconnect between my writing expectations in math and what’s being turned in. For the subtracting integers portfolio which I’ve written about here and here I provided ample time, included both the criteria for success and rubric, plus discussed the reflection in class on multiple occasions. Either I’m not making my expectations clear or my students are not used to doing reflective writing that includes an analysis. I’m guessing it’s both so I created a graphic organizer to help them be more reflective and analytical.

First of all here’s the rubric the students are using:


And here’s an example that would benefit from a graphic organizer:

“In my portfolio I use multiple strategies. For the first thing we had to We had to use visual models you things certain ways to solve it. Such as -x -y or x-(-y). Then we had to use five positive numbers and negative to do this I first made a bunch of positive and negitive fives and just put it together and solve it and then I just made of a problem of -20+ positive 25 equals five.”

The student below is beginning to meet my expectations, though a revision using a graphic organizer will help provide specific evidence that’s currently missing.

“I can solve problems without giving up. What I used to help me was examples from the past worksheets that we did a while ago. … I can use math tools and explain what I use them. I used a number line as a math tool and I use this to show in understand my work. Kind of like a visual for me. I can work carefully and check my work.  I work carefully keeping my work organize and neat. I also check my work by doing the opposite of the problem and see if it kind of matches. It’s hard to explain…”

This Written reflection subtracting integers graphic organizer isn’t perfect, but I think it will help.  I’ve clarified and provided some examples of insightful analysis:


Using the graphic organizer, students now have a work space to brainstorm which of the eight math practices were utilized and how.


Image edited for space. Practices 5-8 are also listed on the graphic organizer, just not shown.

On Monday, I’ll share student work samples and ask the students to both critique and evaluate the reflections based on the rubric and graphic organizer.

I’ll update and report progress in my next post.


Formative assessments using Explain Everything

October 12, 2014

A couple of weeks ago I wrote a piece about student portfolios, specifically a subtracting integers portfolio my students were about to create. Their media of choice is Explain Everything. Now that the projects are trickling in I’m discovering conceptual errors which I might have overlooked on a traditional paper-pencil assessment. Here’s an example:


While the screenshot does capture the number line error when subtracting a positive number, I’m drawn to the student’s voice and explanation, “Since 2 is negative I move to the right.” The portfolio is no longer a thing to grade, there’s a human being behind that work and I can see and hear her. Her portfolio submission has instead become a formative assessment.

This next problem demonstrates further confusion. Here 2 is negative, but she moves to the left.


The student below, on the other hand, has mastered subtracting integers. She begins by demonstrating the number line movements, explains why she moved to the right, then proceeds to show its equivalence by adding the opposite.


When solving this equation the student admitted to a lot of guess and check, however I’m not bothered by the extra practice she got from doing the problem.


After introducing this slide she explained how she turned it into an addition problem by adding the opposite.


Then she solved.


So far my favorite student created real world problem is:

integer real life problem

Only three projects have been submitted thus far, but I’m anticipating most of my time will be spent giving feedback rather than issuing grades. I don’t feel comfortable grading something that’s a work in progress. In terms of time, the portfolios run between 6 and 7 minutes each. I really need to front load the time now so it will be easier later when the students apply the integer rules to negative fractions and decimals.


Here’s another project I looked at this morning. How much conceptual understanding does this student have? He provides no explanation and relies on adding the opposite to demonstrate the number line. Also, he only provided a screenshot of the equivalence problem with no explanation of how it was solved.


Goal: student portfolios

September 28, 2014

I haven’t seen much talk lately on twitter or in the blogosphere about portfolios. I want to deepen students’ understanding of concepts so I thought I’d steal/modify a portfolio idea that was shared at last week’s math curriculum meeting. If you can help me figure out a way to add more choice while still addressing the concepts, I’d love your input. As it stands right now choice is limited to product, and in a small way–process.

The original idea shared was too open ended for me, “Prove you have mastered the learning objective.” While I’m a free spirit in many ways I think the students need some focus so I came up with specific criteria for both the artifacts and written reflections, along with a rubric for each.

Subtracting Integers Criteria for Success


Submit three artifacts demonstrating conceptual understanding of subtracting integers. Record your thinking using Explain Everything, iMovie, or other format.

  • create visual models. Include the following scenarios:
    • x– y
    • x– y
    • x– (– y)
    • x– (– y)
  • create and solve an equation using a series of five different positive and negative integers on BOTH sides of the equation to make the statement true.
    • Example: –a – (– b) + c – d – (–e) = f + (– g) – (– h)  –  i + (– j)
  • create and solve an original, real-world integer problem where absolute value is applied.

Here’s the artifact rubric. It’s generic so it can be applied to any collection of artifacts. I didn’t include point equivalents or percentages because I want the students to focus on the “feedback”. They’ll be able to resubmit.



Submit a written reflection of your artifacts. Include:

  • an analysis of the artifacts
  • the math practices applied and how you have applied them
  • the modes of representation used
  • proper grammar, spelling, conventions

Here’s the reflection rubric. It’s also generic.


When I’m ready to grade I’ll translate the levels into percent equivalents. For example:

  • Level 4 = 100%
  • Level 3 = 90%
  • Level 2 = 70%
  • Level 1 = 60%
  • Level 0 = 50%

As the portfolios are turned in I’ll share student work and let you know how it goes.

Other bloggers sharing their goals can be found by clicking the link below.

Good, bad, ineffective feedback using Canvas

September 20, 2014

Last week I introduced integers using a James Tanton Math Without Words visual puzzle.  Below are examples of the student feedback I gave. I used our new Canvas LMS system as the work flow, but what’s important is the varying degrees of feedback I found myself giving students.

Since some students don’t bother to read feedback–especially when everyone, including me, is getting used to Canvas, I started class by asking the students to pull up the assignment and read the comments.

The student below read my feedback and asked very politely, “What am I supposed to do?” This is yet another reason why I will never earn National Board Certification.


Had I thought about it ahead of time, my feedback should have read, “If every bump is a hill and every dip is a valley, what remains?” That’s something I did for another student. See last image.

I gave this kid nothing. At best I offer hope and a growth mindset but in terms of informative and constructive feedback, it’s terrible–absolutely terrible.

Here’s another student. I acknowledge he’s made the connection to integers, but…

canvas feedback3

…I should have added this:


For students who clearly made the connection to integers, I should have been prepared to offer feedback that extended their thinking, but I wasn’t.


Here’s another example of awful feedback. I acknowledge the problems are correct, but I offer nothing.

canvas feedback2

I should have extended this student’s thinking. I could have added, “If these were numbers what would they be?”

Finally, here’s some feedback that is potentially helpful.


When I gave the feedback I didn’t have a plan. I just viewed the student work one at a time and made comments. That made my feedback inconsistent and not effective. What I should have done was examine the student work collectively, and prepare feedback based on common misunderstandings and extending student thinking.


Boring bit: vocab makeover

September 7, 2014

This 3-2-1 summary is a bit different. I’ve been focused on one thing, that was talked about by two people, more than 3 days ago.

What I’m talking about is Justin Aion’s recent post and twitter chat with Tracy Zager on vocab. The major takeaway is that a need must be created for a vocab lesson to stick.


Next week I’m starting number properties and I’d be a fool to not apply what I’ve learned from lurking and eavesdropping. Here’s what I have so far. Your help could make it all the better.

Students will be paired and will receive this Number Properties Partner Password game that has been cut in half hotdog style. Partner A begins by reading problem 1. Partner B does what his partner says in the space provided.


Only one student who took the pre-test could provide an example of any of the properties so I know nearly all my students will have difficulty coming up with the terms commutative, associative, etc. Originally my resource card for commutative referred to passenger trains that move people to and from work, but that’s beyond my kids’ vocabulary, so I resorted to hangman as the resource card.


Now it’s Partner B’s turn for round 2. The partners alternate until the game is finished.


The directions also include using precise math vocabulary. My hope is students will use sum, product, the quantity, etc along with identifying the property in their sentences. I may end up saving that for another lesson because the primary focus is on number properties.

After debriefing, and time permitting, I’ll reinforce with a number talk, maybe start small with 16 x 5 and see where that takes us. Then, when doing number talks, I can make connections to the number properties.

This activity is better than the blank page I started from several days ago, but I truly welcome suggestions.

Others blogging about their 3 – 2 – 1 summary can be found by clicking the link below.


Authors who have influenced my practice

August 24, 2014

One of the most contentious areas in middle school is work completion. When I first began teaching I was of the mindset, I need to get kids ready for high school. If their homework is one day late, the max the student could earn would be 80%, two days late: 70%, three days late: 60%; more than three days late: teacher discretion. Retakes–no way, they had their chance; they should have studied. Or when I did allow retakes the maximum grade a student could earn would be 70%.

In effect I was using grades as a punishment. Equally troublesome was the fact that this system created a tainted report card. I’m supposed to be reporting academic progress not academic progress with two scoops of behavior and a cherry on top. Now I’m not only questioning my overall grading policy I’m starting to rethink how I assess.

There wasn’t a single turning point. It was an evolutionary process. However two author/educators who caused me to reflect are Thomas Guskey and Rick Wormeli.


Several years ago Guskey came to our district and presented a talk, Developing Grading and Reporting Systems for Student Learning.  He discussed the merits of standards based grading and a narrative report card that separates behavior from learning. His book and talk nudged me to reconsider my practice. Over the next five years I continued to contemplate grading and assessment. Guskey’s book led me to Marzano’s Formative Assessment and Standards Based Grading and that is where I am today.

Wormeli has been equally influential. Chapter 8 from his book Fair is Not Always Equal is particularly compelling. Why Do We Grade, and What About Effort, Attendance, and Behavior?

He contends there are six reasons why we grade:

  • To document student and teacher progress
  • To provide feedback to the student and family, and the teacher
  • To inform instructional decisions

  • To motivate students
  • To punish students
  • To sort students

“Notice the dividing line between the top three and bottom three…The bottom three reasons cross a line. When we grade to motivate, punish, or sort students, we do three things–we dilute the grade’s accuracy; we dilute its usefulness; and we use grading to manipulate students, which may or may not be healthy” (p102).

I’m still a work in progress, but I’m getting there.

Click the link below to read other bloggers who are writing about professional development books.


First five days–subject to change, of course

August 16, 2014

Once again I’ve altered my plans from last year. I suspect before school starts on the 25th the plans will have changed again. This year I’ll need to be especially flexible and rethink learning experiences since our 7th grade will be 1:1 iPad.


We have students for 10 fewer minutes each period on the first day of school in order to issue lockers, hand out assignment notebooks and the like.  Because time is limited the kids will complete a brief cooperative group and communication challenge called You Want Me To Do WHAT? (see page 10). Students create a prototype out of miscellaneous supplies, complete a write up, then hand the directions to another group to recreate. This is my first time doing this activity but I’m looking forward to trying it out.


Classroom norms and expectations will be discussed in social studies and reinforced in math. I’ll note what went well yesterday and what didn’t. After the lecture burst, students will share a bit about themselves using an activity I found on A Sea of Math. It’s called Figure Me Out!  The students use the numbers in their life–birth year, house number, etc. and create expressions using rational numbers. I modified the directions a bit to include a criteria for success.


If there’s enough time I’ll have students swap their Figure Me Outs so they can figure out each other! If not we’ll get to it on Wednesday.


Continuing with getting to know you, the students will complete A Create Your Own Graphing Story iMovie. This is something I did last year (minus the video). I show the “How to be Interesting” book trailer, then follow it with a brief presentation on graphing stories that include several examples.



Pre-assessment on integers and wrap up the iMovie.

Students will have a four day weekend so I’ll have time to look over the pre-assessment and hit the ground running on Tuesday.

Use the link below to check out other bloggers who are sharing their first week plans.


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