How much should study guides mimic an assessment?

Last week during parent-teacher conferences an 8th grade parent stated their student saw the pattern between the study guide problems and the assessments. Mom didn’t say whether it was good or bad, it was just a statement of fact. Obviously we don’t want students to be surprised on an assessment, but are our study guides overly transparent?

This has been on my mind since Bryan Meyer tweeted images from @ilana_horn‘s book Motivated. Most of our grade 6-8 math study guides mirror problems that are on the assessment. Are we distorting grades by doing so? Could we include a few study guide problems as opportunities to deepen learning while “protecting the integrity” of the assessment? What would that look like?

When studying 6th grade ratios, the overarching goal of the standard is for students to use ratio reasoning to solve problems. For example if students use a double number line to solve for a missing quantity our typical study guide would replicate that problem using different numbers and ingredients. Instead of mirroring the problem, what if the study guide showed a double number line and the students were asked to pose a question then answer it? Would that develop deeper learning?


I tried this idea on two students during math support. At first they were stumped. It never occurred to them to think this way. They were used to “problem solving”, not “problem posing”. Once they understood what I was asking they were able to pose a question, but their questions lacked clarity. Then, as you can see from the photo, their questions were much more clear. As an aside, every question they posed, while accurate, resulted in an answer that was found in the second part of double number line.

A problem posing question is not on our assessment, however I could see this type of problem being valuable on a study guide as long as students have had some practice posing questions.

In seventh grade the end of unit assessment is on proportionality. One problem on the assessment asks students to examine an off the grid line graph where they are to determine all true answers about a proportional relationship when given a point on the line. Another problem from the assessment asks students to examine a different off the grid line graph with a point whose coordinates represents the unit rate.  They are then asked to write the equation for the proportional relationship.

My idea for the study guide is to pose this problem–something I modified from Khan Academy. It’s still a work in progress.


Writing the equation for this relationship may be challenging since the constant of proportionality isn’t identified by the ordered pair of (1, 40)  But there are enough clues for students create a table. Doing so might indirectly address identifying  “true” statements about the relationships.

I would love to hear others’ thoughts on study guides. The conversation I am having in my head is also spurred by the piloting of the Illustrative Math resource. Study guides are currently not a part of this resource. If IM is in the process of creating study guides, I wonder if the problems mirror the assessment, or are they different?




Clarifying the learning journey; revising IM’s learning targets


As we continue our Illustrative Math pilot and prepare for the next unit, the sixth grade teachers recently examined the learning targets identified in Grade 6, Unit 2, Introducing Ratios. We took a deep dive using a document created by @normabgordon  in which she captured all the (@openupresources ) IM learning targets in Unit 2.

We appreciate IM’s work, but to clarify our students’ learning journey, we determined many targets needed revision. Some edits were minor, others required more tweaking. Here is our interpretation of Grade 6, Unit 2. Below is a selection of revised targets based on our district’s brief list of “look fors”.

Using student friendly language

The IM targets were student friendly, however we wanted to make them more concise. For example in Lesson 2:

Learning target student friendly

…and in Lesson 9:

learning targets brevity.JPG

We felt these changes still capture the essentials without jeopardizing the learning intentions.

Selecting “observable” verbs to describe Bloom’s Taxonomy or Depth of Knowledge level

IM did a good job of selecting “observable” verbs for the learning targets, however the verb “know” was used in six of the 17 lessons in the unit. Since it’s difficult to observe “know”, we chose different verbs that are observable–such as identify, or explain. For example in Lesson 11:

learning target verbs
Note: we also combined the second and third learning targets and made the targets more brief.


“Understand” is also unobservable so we revised this learning target as well.

learning target verbs2

Omitting context

There’s some debate on whether to include the context in which the outcome will be observed. When creating learning targets, we’ve been advised to omit the context as it relates to the learning activity. For example in Lesson 4 the targets included  color mixtures which we omitted:

learning targets activity

I appreciate the heavy lifting IM has done in writing learning targets for each lesson. It is always easier to revise than to start from scratch. But the other sixth grade teachers and I felt it necessary to clarify the learning journey a bit more. Tackling the targets was not a solo effort. I work with a great group of teachers both in the building and at our “sister” middle school and they deserve a shoutout: @CathyDicksonD95@MichelleRuotolo ,  @Mr__Manahan, and  @MrsOlsenLZ.








Friends of the Library Book Sale task

The other day a friend of mine, who is a Friends of the Library board member, presented me this real life dilemma which might make for an interesting  group task.

Hey, Mary!

We’re getting ready for our Book Sale and we’ll be unpacking hardcover books stored in Hinckley & Schmidt water boxes. We want to display the books spine up on 8ft x 30in tables. I don’t know the dimensions of the box but I do know each box holds six 1-gallon plastic water bottles. The boxes are packed with most books laying flat, stacked on top of each other, with a few others standing upright in the remaining space.

My questions are:

  1. About how many books can we display spine up on each 8ft x 30in table
  2. Approximately how many boxes should I assign to each table for unpacking so I can fill the 8ft x 30in table


The Friends of the Library

In a way it reminds me of MAP’s Money Munchers task I did with my students some time ago. Various book sizes and thicknesses impact the number of books that can be placed in the box, just as various mattress sizes impact the number of dollar bills covering the area in the Money Munchers task.

When giving this task I plan to have on hand an empty 1 gallon container, a Hinckley & Schmidt box, and a sample hardcover book for students who need a visual aid. Rulers will be available should students ask for them.

What can I do to improve this task before I present it to the students? I’d love your thoughts and comments.

The endless chase for the perfect lesson

Sometimes I think we teachers are so invested in our work that we exhaust ourselves and flame out before spring break. What fatigues me the most is the endless chase for the perfect lesson. I feel like I’ve been on a treadmill logging miles, getting tons of exercise, but in the end I’m right back where I started. I can’t tell you how many hours I’ve spent searching 3-Act Tasks, Desmos, Shell, #MTBOS and other resources for that perfect lesson. What caused this endless search and what am I truly looking for?

Shortly after the Common Core was adopted, our math department took on the task of breaking down the standards to write our own curriculum. At the time, you may recall, textbook publishers were simply slapping the Common Core label on their lessons.

expectOur math department knew there should be something more, but we lacked the time and curriculum development expertise to write a coherent and cohesive scope and sequence. Following a structure and level of rigor based on our previous textbook adoption wasn’t enough. Thus the endless search for the perfect lesson.

Since we’ll be piloting the 6-8 IM curriculum this year, I won’t be searching for the perfect lesson. That’s not to say every IM lesson will be perfect, but we need to stay true to the pilot if we want to evaluate the curriculum with fidelity. This pilot will allow me to hop off that treadmill.

But the question remains, what am I truly looking for in a lesson? If I am to trust that each lesson has clear learning goals and is sequenced appropriately, my job then becomes to ensure continuous, embedded, formative assessment so I can offer effective feedback to advance learning. That’s a mouthful; it may be a cliche; but it’s true.

This year I’ll be spending my time and energy on the least dazzling but most essential part of a lesson.


Homework and practice

The 6-8 math teachers received Illustrative Mathematics training this summer and one key question we posted on our “parking lot” is the purpose of homework and daily practice. We will be meeting as a department next week to discuss homework and its role, if at all, in daily instruction. Should we spend class time reviewing each problem? Select problems? Don’t go over it at all? Should we collect and review student homework as a formative assessment tool? Do we scan for a completion grade based on following the criteria for success? Use it as an opportunity to teach responsibility? Do students learn from the worked solutions we provide? Should the practice be blocked or interleaved? There are a myriad of questions we face when evaluating the purpose and effectiveness of homework.

None of the teachers I collaborate with assign “too much” homework. But there’s potential for any homework assignment to slide down the slippery slope from independent practice which can be successfully be completed by the student, to dependent practice involving the parent, tutor, etc. to downright no practice where the student simply copies the worked solutions.

In grades 6-8, students follow a homework criteria for success process of GCS–Grade, Correct, Submit. Grade by marking C, PC or NY; correct each PC or NY using the worked solution, then submit. While a worked solution is a helpful learning tool, I have two issues with using worked solutions: 1) it locks the student into showing work using only one method, and 2) some students simply copy the key and do not use it as a learning tool.

When we have our homework discussion, I would like to hear my colleagues’ opinions on assigning fewer problems from the IM practice, providing worked solutions to similar problems and continuing to have students self correct their work outside of class.

For example if we ask students to complete problem #5 from this grade 6 IM lesson, one worked solution would be provided and the students are then asked to show two other ways.

worked solution

Here, the advantage is students can reference the worked solution in order to solve it other ways.

This is merely a suggestion to jump start the conversation. And to be sure, this should not be considered a homework policy that every teacher must adhere to. How a teacher uses homework in their instructional decisions is up to them.



Tackling Tanton’s Pinwheel Area Problem

While our middle school math summer success program has ended I wanted to share how the students grappled with James Tanton’s Pinwheel Area problem. I was only working with seven students that day because the rest of the class was on an EL field trip.

I gave each student a copy of the problem and 10 minutes of quiet think time before solving as a group.

pinwheel area

After 7-8 minutes none of the students were able to get started so I asked, “What shapes do you see?” Their first response was the pinwheel and squares. Then I asked, “What other shapes to you see?” There as a very long pause. I sensed the students eyes were solely focused on the pinwheel and grid so I asked them to look at the entire image. Another long pause.

“Do you see any other shapes besides the pinwheel and squares?”

A few moments passed and a student responded, “I see triangles.”

I should have anticipated this task better. I was expecting to scaffold, but as we moved through the task, the students needed additional support.

“What triangles do you see?” My eyes naturally saw the larger triangles, but his eyes didn’t.  I asked the student to point to the triangles.


At this point I recognized the grid was a distractor–which is great in terms of adding rigor to the problem while providing another scaffolding opportunity.

Let’s draw this on the board.



Now we were able to discuss a strategy. It took some time, but we agreed that if we were able to subtract the non-shaded area from the 5×5 grid, we would find the area of the pinwheel.


Students enjoy working on the whiteboard and wanted to shade the triangles.


Students recalled how to find the area of a square or rectangle, but could not remember how to find the area of a triangle. I drew a rectangle, and labeled the length and width as 3×2. “What’s the area?”


I cut the rectangle in half horizontally and asked, “What’s the area of one section?”


I erased the horizontal line and replaced it by cutting the rectangle in half vertically. “What’s the area of this section?”


I erased the vertical line and cut the rectangle in half diagonally. “What’s the area of this section?”


So if we multiply the base and height of a triangle and cut it in half (divide by 2) we can find the area. We revisited the printout of the problem to find the measurements and began calculating.


Now we were able to put it all together.


To achieve productive struggle this group required more scaffolding than I originally anticipated. I need to reread Five Practices.

Students Evaluate Summer School Math

Yesterday I asked students to evaluate our four week Middle School Math Summer Success program which I previously wrote about here and here. One of the biggest challenges of teaching a multi-grade level class is, as you can imagine, meeting the needs of the students.


The makeup of this grade 6-8 class was quite diverse. It included 1) students recommended for summer school by their math teacher, 2) students enrolled in the summer EL program to receive math support, and 3) students who were enrolled by their parents to keep skills fresh.

When a teacher recommends a student for summer school, certain expectations are presumed. When parents enrolls their student, they have certain expectations as well. Given the make-up of the class, I needed to reach rising seventh, eighth, and ninth graders whose mathematical prowess ranged from developing to proficient.

I structured each 90 minute day to include number talks, individual and group tasks, number sense activities, and grade level review. I wanted student feedback on each of these areas plus one recommendation for improvement. Here are the results.

Students’ Evaluation

For the four focus areas students simply placed an “x” on a helpfulness continuum. I gathered the data and compiled it into four separate graphs. I also debriefed with each student individually using the feedback they provided.

number talk
Students who did not find number talks helpful already had strong number sense.
Individual group tasks
The one student who rated tasks less helpful was the most “skilled” student in the group. This student could also be considered the lead learner.
Here again, students who had strong number sense did not find these activities helpful.
grade level
One student, a rising 9th grader, felt the grade level practice was extremely helpful. A rising EL 7th grader gave the lowest rating.

It really didn’t surprise me how much students value the grade level practice. For several students in the class, this was exactly what they needed. Also, when working with intervention students, some do not recognize the value of foundational skills because, “I need to know how to do [insert grade level concept] that’s due tomorrow.”

Students’ Recommendations

I love the honest feedback the students provided.

In addition to the EL teacher and me, most days we had 1-2 high school student volunteers working amongst the students.
This feedback is from a rising 8th grader, the lead learner in the class, referring to the thumbs’ up sign when giving wait time during number talks.
Referring to number talks–from a rising 7th grader.
From a rising 9th grader.

I appreciate the students’ suggestions and will take them to heart. It would be lovely to have even more high school volunteer in the classroom facilitate the learning. I had one volunteer, “Annie”, who was exceptional. She knew how to give hints without “giving it away” and gave students a growth mindset pep talk–“I was where you were in middle school and through hard work I’m now in Calc III.”

Overall I’m pleased the students found the class valuable. When I debriefed, I asked the lead learner, “Bradley”, why he enrolled in the class. To paraphrase his response, “I was recommended for reading and that was the first time slot. My mom didn’t want to pick me up right away so she signed me up for the this.”