Every time our math committee meets we are energized by the progress we’ve made but at the same time we feel deflated because there is so much more work ahead of us. Last week we were again reminded that our math units need bolstering.

On Wedesday we once again analyzed our progress using the Tri-state EQuIP Rubric designed by Achieve. For the 7th grade rational numbers unit there are four areas we need work on:

Linking Mathematical Practices to learning opportunities and assessment items.

We hadn’t formally identified the applicable MPs to each learning sheet and activity so we went back and did that. Examining our assessments was next. My colleague and I experienced a giant YIKES on the decimals assessment because hardly any items could be linked to a MP. Since we ran out of time we need to revise those items plus look at the other assessments in the unit.

Explicit writing, speaking, and listening opportunities

We made some progress here by modifying directions on some activities. For example our subtracting integers portfolio was modified to include two peer reviews. The next time I assign this task students will create a draft of their video using Explain Everything then two students will watch, listen, and provide written feedback before creating the final video.

Instructional supports

We have some intervention and enrichment resources but we need find more and better organize the ones we currently have. The rubric also identifies the need for a performance task for the unit. I think we inadvertently listed it in the instructional supports section.

Scoring rubrics and pre-assessment reflection plans

Formally identifying the characteristics of partial credit, high partial credit, and full credit are on the drawing board. We also need to create a pre-assessment reflection plan for students to identify their strengths and what they need to review. This also includes helping them design an action plan before they take the assessment.

Before we jump into adding and subtracting negative decimals I thought we should spend some time defining what estimation is and is not. That way students would be able to determine if their answer was reasonable. It took about a period and a half to clear up misconceptions but I think it was time well spent.

The basic structure of the lesson was 15 minutes of group work using giant whiteboards, followed by a 10 minute a gallery walk for feedback, then the remainder of the period to refine their definitions and examples.

I posted the following directions: Create your “bestest” definition of estimation and provide one example and non-example.

Here are the first drafts of student work from one of my classes:

Some groups were on the right track, but their definitions and examples could be more explicit. When it came time for the gallery walk the groups examined each others’ work, commenting on two things: one thing they like about the work and one thing that needs clarification. Groups struggling with the definition or example/non-example really benefited from seeing others’ work. The post it feedback also helped them to refine thinking.

At the end of the period I asked students to use their iPads to take a photo of their work and the next day we debriefed. The definition of estimation being an educated guess was lacking for me, so I added the phrase, “by using a strategy.” I think that condition to the definition will allow us to continue our estimation conversation when we talk about scaling up or down with proportional reasoning.

Having students create their own version helped them participate in the final draft. We talked about where rounding occurs in estimation so we could identify the characteristics. One student thought it helpful to include arrows to clarify what is being rounded, and I threw in my caveat about using a strategy as part of the definition.

While it took a lot of time the students ended up owning the definition more than if I had told them what to write in their notes. The collaboration and post it feedback also helped those students who may have had some trouble getting started on their own.

I am taking baby steps towards converting to a math workshop model with the help from reading Minds on Mathematics. There is much more to the model than breaking the class time into chunks: 1) roughly 10% of class time on an opening task, 2) 20% devoted to the mini lesson, 3) 50% work time in which the teacher confers with the students, and 4) 20% sharing and reflection. It’s about creating a culture of learning, where students…

engage with worthy, minds-on tasks and important mathematical ideas; enjoy time to think, work, and communicate; construct mathematical understanding for themselves; and notice their own development as mathematicians.

A lot of planning and preparation goes into this approach, but once class begins those doing the most thinking are doing the most learning. Over break I’ve been doing a lot of thinking about my “practice” and I have learned that I am not pleased with myself. In terms of planning and preparation to create this culture of learning, I’d rate myself as mediocre–which is a scoch to the right of basic on the Danielson model.

I’d be fool to say I’m going to fully implement a workshop model as a New Year’s resolution but there are some measures I can take along the way. Originally I thought the “right” thing to do would be to commit to one change then slowly implement another. For example if I don’t like the way I begin class (which I don’t) I’ll work on the opening task. But since the opening task leads into the mini-lesson, which leads into work time, etc. I need to rethink the entire class period for that day.

Wendy Ward Hoffer, the author of Minds on Mathematics, suggests an opening task be: a new problem, a problem from previous homework, recent class work, or a concept central to the unit of study. The task should also be a springboard into that day’s lesson and it should have multiple entry points so all students have access to the problem.

When we return from break the seventh graders will be multiplying and dividing fractions. Based on their pre-assessment they don’t have a deep understanding of the concepts. So I’m working on a series of brief, scaffolded tasks that will lead into two mini-lessons.

The recipe problem is heavily scaffolded, suggesting repeated addition as a visual representation, and using multiplication to represent the problem. At this point I’ll conduct a skinny, mini lesson on multiplying fractions by whole numbers.

The next problem is less scaffolded.

At this point the lasagna problem will launch a mini-lesson on multiplying fractions by fractions. I created a video for students who will be absent.

Plus the students will have this handout to take notes during the mini-lesson. I’ll do some white board work to check for understanding the visual model. Now here’s where I’ve run out of gas for the time being. I need to find or create some differentiated resources to use following the mini lesson. Some students will catch on immediately and I may want them to explore simplifying fractions by examining their common factors:

For students who struggle, I’ll need to have at the ready some additional visual model problems for in class practice.

Additionally, I may have the students work on this MARS yogurt problem while I’m conferring.

Sharing and reflection will be students presenting a variety of problems they’ve been working on during their work time.

I’ve spent hours thinking about this ONE lesson and I’m still not fully prepared. The sheer fact of thinking about it however will make it a thousand times better.

Next up will be dividing fractions by a whole number then dividing fractions by fractions. I haven’t sketched out the whole number mini lesson yet, but here’s the direction I’m headed with dividing fractions by fractions.

Dividing fractions by fractions

I know I’m getting ahead of myself but the dividing fractions by fractions lesson will look similarly. I haven’t yet created a video or handout but it will be based on these problems.

I’m troubled by section 4. It’s too scaffolded and not enough discovery. I need to work on that. Suggestions???

Again, I’m troubled by section 4.

The visual models you see in this post are based on a handout I found while scouring the internet. I liked the overall idea, but tweaked it to be a bit less paint by numbers. This is still a work in progress.

Earlier in the week I posted on my 180 math post-its blog that students were pre-assessed on fractions. I was disappointed to see that only two pre-tested out of many of the concepts. Given the caliber of this class there should have been at least five more ready for enrichment on the skill of adding and subtracting fractions.

I probably should have started the class immediately by differentiating for the two students, but I didn’t. I had enrichment ready for them, yet I launched into a whole class whiteboard/quick check activity of adding and subtracting fractions. I admit I short changed those two girls, but I’m glad I did the whiteboard work because within 10 minutes five more students revealed to me that their pre-assessment was a signal that they simply forgot.

It troubles me that students do not retain their learning, but that’s for another post. In this instance I truly believe those five kids had, as they say, a brain fart.

For the rest of the block they worked at two tables on challenge problems. Which leads to…

Lesson 2: Students can be completely engaged with non-real world problems.

The kids loved the challenge. I know there is little context with these problems, yet they were engaged for the entire block. You can’t describe them as puzzles, but for 7th graders the problems were puzzling and they wanted to solve them.

By the way this same class is enjoying the NCTM palette of problems I’ve been using for determining importance.

Recently I’ve been presenting them at the start of class. Students work independently then we share solutions, discuss their “arguments” etc. Some students want me to do this everyday.

My SMART goal to raise self efficacy was to implement four tasks in the first quarter. I only had two. My standard students experienced the Money Munchers and Variety Show tasks. My pre-algebra class had Molly’s Locker Combo task and the Boomerang task. While I didn’t reach my goal, I am making a wee bit of progress with one student’s self efficacy. I have to admit, it’s not me. Her teacher last year had her in math lab nearly every day.

On Wednesday, she was elated that she was part of a group that was allowed to work in the hall on an extension activity while I retaught a concept. It was the first time she ever experienced such a thing. “I’ve never been out in the hall before! I feel so smart!” Since the start of the year she’s only had to reassess one time. Attending math lab has paid off.

The extra time we devote to our students may or may not pay off this year, but it will in the future.

My students do not have much experience with non-routine problems. On occasion they’ve completed complex tasks, but certainly not enough, or a variety, for students to build self-efficacy. I’m trying to change that by living a SMART goal I created for myself. Last Thursday, day 8, I presented a formative lesson designed by the Mathematical Assessment Project.

The problem began: Emily doesn’t trust banks with her money. She has stored $24,000 in one dollar bills under her mattress. The rest of the problem can be seen below. This student’s attempt was typical.

After they worked on the task independently for fifteen minutes, I collected their work. I found myself providing the same feedback over and over. (My handwriting is atrocious. I know. )

As I reviewed their work, I came upon this response:

One or two students thought to estimate the size of a mattress, but their dimensions weren’t reasonable. For them my feedback was, “How tall are you? What do you think the dimensions of your bed are?”

For estimating a stack of one dollar bills, nearly every student has this feedback: “How could a book help you? How could the pages in it help you estimate a stack of money?”

When I see the students tomorrow, they’ll take that feedback and work independently for about 10 minutes. They’ll then work in their groups to share their progress and come to a solution.

One thing I will add is criteria for success. Your work is: a reasonable estimate that is organized and clearly labeled.

I love this lesson. I get to see individual thinking. The feedback offers progress while not enabling. Plus, when they collaborate every student should be able to contribute to the solution.

Like Fawn, I’ve been thinking about how to assess the 8 mathematical practices. Eventually I will want to “grade” them, but at the start the students will need opportunities to practice the practices. In addition I can’t just let them practice without giving them feedback. So I’m going to use this ASCD resource to drive my formative assessments, It provides a great framework and walks through the entire process using the Boomerangs lesson, a high school MARS task.

As you can see the feedback is mostly through written questions. When I first read the section on suggested questions I was thinking, “They sound a lot like the 8 mathematical practices.” For example the issue of difficulty in getting started could fall under the math practice of making sense of problems.

These questions and my comments will hopefully provide the necessary feedback for students to improve on the mathematical practices, but I’ll have to do it on a regular basis for the feedback to be effective. Plus I’ll have to provide them with a rubric. Our district created this rubric for students and teachers.

In the fall I’m going to focus on only two or three practices to start with, then add other practices. I’m stealing that idea from language arts teachers who use Six Traits of Writing. From what I understand middle school language arts teachers only focus on a few traits at a time.

What I’m also planning on doing is using this recording tool for anecdotal, informal observations. Sometimes I feel like a mad scientist walking around with a clipboard and jotting down my observations but I’ll have to get over it. The tool and rubric were created last month so I have not had a chance to take them for a test drive.

Assessing the math practices will be new to me. If you have experience or see some red flags feel free to chime in.

Update

I’m beginning to get incredible feedback. Your comments deserve attention so I’m placing them within the post. Please continue to share your thoughts.

Assessing them is not about “can they” do a specific SMP, but “do they” and can they eventually do so habitually?–Just one of Jessica’s suggestions.

Reading their reflections not only gave me great insight into how the students believed they were using the practices but also how they were beginning to think about solving problems in general–Jennifer shared her blogged about journaling.

Daniel Schnieder and I are on a similar journey. He recently wrote a thoughtful piece in which he discovered the domino effect while implementing Standards Based Grading. His post focused on three dominoes—standards, assessments, and grading—and how his move to SBG has resulted in a reevaluation of every aspect of teaching and learning. He concludes by suggesting there are at least 50 more dominoes in the chain.

I think one of those fifty dominoes combine differentiation with formative assessment.

I thought I knew differentiation—content, process, product, readiness, etc. But today I realized how much there is to learn. I watched portions of a Carol Ann Tomlinson webinar, Common Core State Standards: Where Does Differentiation Fit? and learned how the MARS project is transforming differentiation by using formative assessment to facilitate student thinking and problem solving.

In the past I used formative assessment to differentiate instruction by groupings or as a reactionary tool for when students didn’t get it. If you read the previous sentence closely, I’m differentiating before or after a lesson or task. What’s different is how Tomlinson suggests using differentiation and formative assessment during a mathematical task.

Here’s an example. In this MARS task students work independently for 15-20 minutes to solve and graph an inequalities problem. After the allotted time, the work is collected and analyzed based on the 8 Mathematical Practices (Not all practices may be present in the task).

Is the student having trouble getting started, or misinterpreting the problem (Make Sense of Problems)? Is there an organization of thought, or is work haphazard (Construct a Viable Argument/Attend to Precision)? Did the student present a variety of mathematical models (Reason Abstractly and Quantitatively/Model with Mathematics)?

The differentiated feedback is given to the student in the form of questions. Can you organize your numbers in a certain way? How would someone unfamiliar with your solution understand your work?

When the students resume the task the next class period they are given 10-15 minutes to review the feedback and answer questions. They are then paired or organized into triads to collaborate—combining ideas and creating a new and improved solution.

This particular task is designed for high school, but MARS has a variety of problem solving and concept development tasks for middle and high school. You can take your pick of Classroom Challenges.

What I’ve been doing

The mathematical tasks our math department created this year use questioning to differentiate and guide problem solving but not in the same manner. Those tasks are much more group worthy in nature and the questions are presented to the groups as resource cards to allow for multiple entry points.

They both serve a purpose but the MARS example allows for differentiation based on a formative assessment, not on anticipated misconceptions.

In the fall, I had planned on introducing the students to the mathematical practices and collaborative work through a series of short group worthy tasks. It will serve them better if I differentiate by using the approach suggested by Tomlinson and the MARS classroom challenges.

Julie’s prompt this week is…How do you help students in your class that are behind in math? What a great question.

Here are some reasons I thought of as to why students fall behind:

Students:

lack of investment in their own learning

insufficient self-advocacy skills

inability to set goals and monitor their own progress

inattention, social-emotional, or learning disability

absences

Teachers:

infrequent formative assessments

pacing is too fast for the student

lack of differentiation during class or homework

insufficient monitored and/or independent practice

BEFORE they fall too far behind

Let me share what happened on Friday. I returned an equivalent ratios formative assessment to one of my classes and six students absolutely bombed it. Fifteen students earned a 2.5 or better (3 is the target, 4 is exceeds the target in my standards based grading grade book). The six who struggled demonstrated a level of proficiency of 1.5 or less. So as a class, they are either “getting it” or they “are not getting it at all”. There is no middle.

This was the first formative assessment so the group hasn’t fallen too far behind. But if I don’t do my job they’ll fall even farther behind.

After we discussed the assessment, I regrouped the class so I could meet with the six students. I assigned a Thinking Blocks online ratio activity to most of the class while I met with the struggling group. We had a conversation and this is what I learned…we were both at fault.

Let’s start with me.

It turns out that my pace was too fast and I didn’t give them enough monitored and independent practice. The short of it is I should have better differentiated this lesson to accommodate their needs. It also turns out that I need to help them develop self-advocacy skills. Students already set goals and monitor their progress (Student Goal Setting 6 RP1), but they need to take it more seriously.

Now them.

I want students to tell me to slow down. I also want them to ask questions. A few are not as invested in their learning as much as I am. I get it; they’re kids. But that also means they’re not old enough to choose not to learn. I intervene by insisting they either attend math lab or schedule a date with me after school.

WHEN they’ve fallen behind

When I pull students for math lab (fourth period) I reteach and monitor their practice problems. If it’s not busy, a student can receive one-to-one attention. Other times it’s a bit busier where students drop in for help with one problem. Or it can be chaotic where I’m re-teaching two different classes at the same time while students drop in with a quick question.

Math lab fills a need but it is a horrible substitute for when a student misses class. On Thursday, two students who were previously absent thought they could make up an 84 minute lesson on one step equations in 30 minutes. It was disastrous.

In those situations perhaps I should insist students come before or after school. It adds to my day however it also forces the student to have “skin in the game”. I’m well aware there is life beyond school. Kids have after school activities and responsibilities. Those are important yet so is learning. It’s a dilemma.

Providing extra monitored practice, or some re-teaching to get a student caught up is one thing. I can handle that in class, in math lab, or after school. What I haven’t been able to do well is getting students caught up when they’ve been absent. That requires a heck of a lot more dexterity, juggling, and time.

I hope I’m not the only one out there who is having trouble differentiating math homework during this Common Core transition period, especially when it comes to the content.

New content + closing gaps + a wide range of student abilities = Yikes! I need to differentiate, but how?

Even if your middle school tracks math like mine does (in sixth grade we have five levels) it is the at-grade level math class where student ability has the widest range. In my standard (at-grade level) classes I have students ranging from the 25 to the 90 percentile in MAP scores.

B.C. (before the Common Core)–It wasn’t perfect but I was handling it

In the past I daily differentiated homework by offering students a choice between two tiers of practice problems. Lower ability students chose A or B level worksheets (provided by the textbook publisher) and the rest chose B or C level worksheets. I admit, I did not offer the high ability students a choice; they were assigned the Level C problems.

Reviewing the homework the following class period was tricky. Students formed groups according to their assignment, discussed the problems, and checked their answers with the answer keys. But my classroom was far from perfect. If we reviewed any problems as a whole class I was certain to lose some students because that problem was not on their homework. Another related challenge: some groups took longer than others and I should have been prepared with extensions or additional practice for the groups who were waiting.

A.D. (during this Common Core transition)–Holey Moley

This year I’m not differentiating as well as I’d like. Our geometry unit was created late last year and over the summer so we were able to gather adequate resources to differentiate. But our other units are being crafted as we go. We’ve been busy pulling resources just to address the standards.

If you are familiar with the Common Core you know the emphasis is on the eight mathematical practices. Our textbook only supports the computational processes, not the deep levels of understanding. And from what I understand the textbook publishers have only been able to “align” (frankly I think that means rearrange) their current problems. It takes time to create rigorous tasks that address the practices, so we’re waiting for the publishers to truly offer a Common Core textbook with resources.

And those textbook differentiated worksheets? I use them but less often. I’d be fool to think I was truly differentiating to meet the Common Core. I’m just doing the best I can with what I’ve got.

I know this is a transition period. I hope when the dust settles I’ll be able to do a better job differentiating.