Adapting Scope and Sequence for Remediation

Like many schools, we have students in every grade who start the grade with unfinished learning/misconceptions/gaps in standards that fall under previous grades. We operate with a general policy of trying to remediate through grade level standards—as an eighth grade teacher I am teaching primarily eighth grade content and always using that as the starting point/end goal of any reteaching of prior-grade-level content. However, we have not done very much work to make sure that this happens in a systematic way. I was really excited when the Standards Institute addressed exactly how necessary and challenging this work was and provided a framework for thinking about how to do this type of Tier 1 intervention.

This slide from the Standards Institute (Day 3 powerpoint for all of the math groups) illustrates how students will fall further behind and never be able to catch up if we spend a significant amount of time filling gaps in isolation from the current grade-level content.

As an alternative, they offered these ideas about where and what to remediate. The idea is to make a strategic plan instead of trying to review everything or only reacting after the fact to what you have learned about students’ foundational knowledge.

While this generally matches how I have been trying to remediate, I really appreciate how clearly and intentionally this process is laid out. As someone who is in a department where we don’t use a curriculum and primarily write our own sequencing and pacing, I think that this work is really our next step in building vertical and horizontal coherence. I am extremely excited to take this framework to my team for when we are doing our big-picture planning.

The Standards Institute also introduced me to two amazing tools that will be essential to this work:

1) Zimba Wiring Diagram: One writer of the common core’s interpretation of how the standards connect to each other both between- and within- grades.

2) Coherence Map: This is an interactive map that allows you to “zoom in” on the wiring map. You can choose a standard to start from and then follow it to a connection of your choice.
3) Content Emphases by Cluster: This designates which clusters are major, supporting, or additional standards. Shockingly, I had never seen this before.

Rigor vs. Cognitive Demand

Between work on creating my school’s instructional vision and attendance at the Standards Institute this week I have been doing a lot of thinking around what makes good instruction at different grain size—lesson, unit, curriculum, sequencing, and overall educational experience. Two concepts, which I would once have used interchangeably, have come up frequently.

Cognitive Demand:

Stein, Smith, Henningsen, and Silver define cognitive demand as “the kind and level of thinking required of students in order to successfully engage with and solve the task.” I have analyzed tasks with their four levels of cognitive demand in many settings. The biggest question that comes up for me is what is the ideal placement and balance between low- and high-cognitive demand tasks? One of the professors in my teacher prep program always suggested  “a steady diet of high cognitive demand tasks” which I found both wise and frustratingly vague.

Rigor:

The idea of rigor as a concept separate from cognitive demand is a newer one for me. In the past nine months, though, it has come up for me in two places—as part of my district’s curriculum review tools and at the Standards Institute. At corestandards.org, they define rigor as “deep, authentic command of mathematical concepts” and clarify that it is “not making math harder or introducing topics at earlier grades.”

Here are the elements of rigor. I’ve also included the definitions from the five strands of mathematical proficiency, because I think they better describe the three elements.

	Corestandards.org	Adding it Up Chapt. 4
Conceptual Understanding	The standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.	Comprehension of mathematical concepts, operations, and relations.
Procedural skills and fluency	The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others.	Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Application	The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency.	Ability to formulate, represent, and solve mathematical problems. (This is called strategic competence, not application, here)

The language of the shift calls for students to pursue the three elements with “equal intensity.” However, this leaves me with questions about how the three elements interact.  One of the tools that my district used in evaluating curriculum starts to further illustrate exactly what balance of rigor looks like. 

The authors of Principles to Actions state “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” With this in mind, we were really looking to see whether and how procedural fluency was explicitly built off of conceptual understanding.

And then there’s the question of applications. In my experiences with both the Standards Institute and on the curriculum review team, applications were generally defined as any problem with a “real world” context. I find this interpretation problematic for three reasons.

1. Instead of applications acting as a third leg in the three-legged stool of rigor, all conceptual understanding and procedural fluency problems can be categorized as applications or not
2. There wasn’t a larger conversation around how well the real-world context was executed (See Dan Meyer here on one of my biggest pet peeves).
3. There’s a reason that the phrase “real-world and mathematical problems” occurs in the common core 31 times rather than just “real-world problems.” I think we need to better define what it means to have an application that is in a purely mathematical context, but I don't think we should just drop it.

I am not yet ready to fully develop my ideas around how to better define application, but I think that the strategic competence description in Adding it Up would be a good place to start.

Relationship

Based on these definitions, it is clear to me that I can’t use the terms “rigor” and “cognitive demand” interchangeably. But these are both words that I hear thrown around a lot when trying to evaluate teaching and curricular materials. So what relationship do they have to each other?

In looking at the text of a problem and the execution by both students and teacher, I think it makes sense to identify the foremost element of rigor and the level of cognitive demand separately. However, I would imagine that conceptual and application problems have a higher likelihood of being high cognitive demand, and that procedural fluency problems have a higher likelihood of being low cognitive demand.

I also think that it’s worth trying to push a little bit beyond the concept of “a steady diet”—which I think can be applied both to high cognitive demand tasks as well as tasks of each type of rigor.  In the standards institute we started to look at arcs of the elements of rigor over a particular topic or unit. In general, we seemed to see the sequence of conceptual understanding --> procedural fluency --> application, though of course it was not quite that clear cut. I would also question the placement of application primarily at the end because I think it underestimates students and impedes their learning to say that procedural fluency is required before being able to apply a concept.

This makes me wonder if there is a similar general flow that makes sense for cognitive demand. I would certainly argue against low cognitive demand being a prerequisite for high cognitive demand. I think that there’s a lot of value in starting with high cognitive demand tasks because they are an ideal place to surface student reasoning and build upon the foundation that students already have. But I wonder if it is oversimplified to suggest high cognitive demand --> low cognitive demand --> high cognitive demand as an arc for a unit. It might make more sense to have a unit that is composed of a series of shorter cognitive demand arcs of that nature.

Practice/Review Games

There were two goals for my classroom re-set this January—to make our class more productive and more joyful. An element of this was to make my lessons feel different and more fun than the regular structure I use in about 5 out of the 8 periods a week.  This structure is guided notes that involve a mini-lesson, practice/application, and then discussion. However, the thought of trying to change each lesson so it felt different and fun was overwhelming. Instead, I decided I wanted to have a couple of new practice structures that would meet my goals, but that I could just insert into my regular lessons.

In choosing practice structures, here were the four criteria that were important to me:

a) Easy to Prep

b) Individual Accountability

c) Versatility of Use

c) Didn’t Require a Whole Period

It’s also important to note that I chose these structures  to build fluency with procedures. I don't think they are a good fit to introduce new ideas or to build conceptual understanding.

Practice Games

Climb the Ladder

Description: I first heard of this from my mentor teacher. It involves a series of problems, each with increasing difficulty. Students get each problem checked when they finish (I had three student leaders with answer keys who did the checking) and then they get to “climb the ladder” to the next problem. The worksheet, with directions, is here.

Analysis:

Easy to Prep: It took me a little while to get the formatting to show the ladder on the page, but I can now just modify that sheet whenever. Very easy prep.

Individual Accountability: Students each had their own paper and needed their own answers checked, but they were helping each other out. In general, motivation to get to the next problem was high.

Versatility of Use: I have used it for solving equations and identifying equivalent equations so far, but it could be used for anything. It is especially good if students can build on what they did in the previous rung (ex: Rung 1 is solving x +4 = 10, Rung 2 is solving 2x+4=10)

Didn’t Require a Whole Period: 20 minutes seemed about the ideal amount of time, with about 8 problems. Everyone finished the first ladder (4 problems), and fast finishers got to all of the problems.

Other Notes: I find it is extra fun if you excitedly say "Climb the ladder" every time a student gets a problem correct.

Solve-Crumple-Toss

Description: Students get a half sheet with a problem on it. It is checked for accuracy (I had three students with answer keys to do this in my bigger classes, I personally checked all of them in my smaller class). If it is incorrect, the student revises their work. If it is correct, the student crumples the paper and gets to toss it into a basket to earn their team a point. The directions I projected for students is here. Also read Kate Nowak’s description here.

Analysis:

Easy to Prep: I had to cut up all of the half sheets, but they were extremely easy to make. In the future, I will make fewer of the later sheets because not all students get to all of the sheets.

Individual Accountability: Students each had to put work on their own paper, but were generally helping each other. I saw a handful of students more engaged than I have seen them in a long time because they really wanted to toss their paper. This desire also led to several students just copying their friends’ work so they could throw sooner.

Versatility of Use: Could be used for any type of problem! The problems should be a little bit meaty so that the time spent doing the problems vs. crumpling/tossing is heavily weighted towards doing the problems.

Didn’t Require a Whole Period: I think a minimum of 15 minutes, maximum of 40 is probably appropriate here. I want to make sure that students who are taking their time/checking their notes/explaining to a partner still get the reward of getting to crumple and toss their paper.

Other Notes: I was worried about too many people finishing at the same time and the checking/tossing part being overwhelming for me. Having 3 students checking, each of whom was in charge of a group of about 9, was really helpful. And they didn’t finish as quickly as I anticipated—the line to toss the paper was never over 4 students in a class of 27. I also had one student who was pretty distressed about crumpling her papers up when she finished. I had decided that this was ok because they already had a sample in their notes that they could look back at and I just wanted them to get more practice.

Speed Dating

Description: Students each get their own problem that they have some time to work on individually and become experts at. Then they pair up and trade problems, each person helping their partner if they need it. Every 5 minutes or so, the partners shift. There are more detailed descriptions from  Kate Nowak, Elizabeth Stratmore, Meg Craig, MaryAnn Moore, and Amy Gruen.

Analysis:

Easy to Prep: Pretty easy to prep. You need a different problem for each student (though when I do this with my larger classes, I will split them into 3 groups and then only have 9 different problems rather than 27). Also, whiteboards and dry erase markers are helpful to have for each student (though they could just do all of the problems on a separate piece of paper). This has been the limiting factor for me, so I have only done this with my class of 11 so far.

Individual Accountability: Speed dating is excellent for this! Students get exposed to a ton of problems where they have their own personal helper. They also have to be able to explain really well one particular problem. It’s also good for differentiation in the difficulty level of the problem each student initially gets.

Versatility of Use: Could be used for any type of problem!

Didn’t Require a Whole Period: This one is a little bit longer. 5 to 10 minutes to become an expert at your own problem. Probably about 2 minutes to get set up into the rotating columns. Then 3-7 minutes of work time for each pairing.

Other Notes: I loved how uninvolved I was as students were doing the speed dating. It was amazing to see the amount of math talk going on and how they were really explaining to each other.

Content Auction

Description: Teams get $1000 and an “auction catalog” (read: worksheet with practice problems). They are told what they want to buy (in my case equations with one solution) and given 10 minutes to work on the worksheet in order to plan what they will bid on.  Then the auction begins! Teams bid on the problems, and the problem goes to the highest bidder. After each item is won, we go over the problem to decide whether it was a wise buy. Here's the powerpoint that I used. Further description here and here from Sarah Hagan.

Analysis:

Easy to Prep: Extremely easy to prep. You need a worksheet with some practice problems on it. I also made bidding paddles (a notecard with the table number stuck to a popsicle stick) and a powerpoint with the problems, but those were extras and not required.

Individual Accountability: There wasn’t great individual accountability when teams were working on deciding what to bid on. I did, however, require students to copy down the answer as we went over each problem.

Versatility of Use: There needs to be some way of distinguishing which problems students need to buy and this distinguishing element is where the doing the practicing the math comes in. For example students might want to only buy true true statements (as opposed to false) or functions (as opposed to non-functions).

Didn’t Require a Whole Period: You definitely need a whole period for this one.

Other Notes: I felt like this one was the least successful out of all of the games that I tried because there was a high ratio of unproductive chaos to productive math practice. Students didn’t do that many problems before the auction started and then once the auction started the bidding was a lot of time without math thinking. Then reigning the kids back in to go over the problems after they had been bought was hard. However, I think some tweaks in the future would help. I would probably start with 5-10 minutes of individual work time. I would then have the teams split up their $1000 dollars however they wanted over all of the problems. I would collect this info some way (electronically?). Then we would have volunteers to explain each problem and then reveal which team had the highest bid and thus won it. I think this would be more focused on doing the math, but still have the fun bidding and competition elements.

Student Responses

After we had done Climb the Ladder, Solve-Crumple-Toss, and the Content Auction at least once, I asked my two larger classes for feedback on which game was their favorite and why. Here were the votes:

• 1^st Place: Climb the Ladder (15 votes)
• 2^nd Place: Auction (9 votes)
• 3^rd Place: Solve-Crumple-Toss: (8 votes)

There were enough votes for each one that I let the class know we would be doing all of them again.

Here were some of the best reasons that kids gave for their top choice:

 

Other Practice/Review Games

These are games that I haven’t tried yet because I didn’t want to introduce too many new structures at once and/or they didn’t seem as good a fit for my specific criteria. They may however be useful for others!

War: Descriptions from Denise Gaskins and Kate Nowak

I Have, Who Has: Sarah Hagan and Sarah Hagan again

Ghosts in The Graveyard: Kim Hughey

A couple other compilations of review games:

Tina Cardone, Sarah Hagan, Global Math Department

Celebrating the Victories

At the beginning of January, I did a re-set with my class because I felt like students weren’t learning and neither the students nor I were enjoying being in class. 5 weeks later, it’s the end of a tough week and I am feeling discouraged again.

However, I wanted to take some time today to focus on the positive changes that both the students and I have seen since the re-set. A week ago, we had a class meeting to celebrate the positives that have come out of the work we have done as a class to make a more productive and learning environment. Here's the student feedback that I got during that meeting.

Survey

Directions: Here is a list of new things that we have tried. Rate each one with a symbol:

+ Has really helped the class improve

✓ Has kind of helped the class improve

— Has not helped the class improve

Results:

What we have tried	+	✓	—	Weighted Total +: 3 points ✓: 1 point -: 0 points
New math games (Climb the ladder,  solve-crumple-toss, auction)	27	10	1	91
Seats in tables instead of rows	27	8	2	89
Not using the warning/lunch  reflection system	26	11	1	89
New bulletin boards with space for  you to decorate how you want	27	7	3	88
More group work	24	14	0	86
Do Now questions to help us get to  know each other	23	11	4	80
Team captains handing out smiley/sad faces during the Do Now	19	12	3	69
Team captains helping us stay  focused during class	13	21	4	60

 

Student Responses:

In your opinion, do you think class is better than it was before winter break?

 

8C:

• Maybe because I’m starting to get use to this and it is more comfortable and nice
• YES. Because I feel like it really helps me focus
• It was ok because things got better
• Yes a lot because we are now learning more before we were learning a lot even though we aren’t perfect right now we will continue to get better
• Yes because everyone is being more and more focused and they are working harder and better than before
• Yes
• I think yes because we settled down after we got the seats but I still think its not working well that much so we should switch seats every month
• Yes because before we did we couldn’t do no project and we never paying attention and we always got a minus on clipboard. But now we get more work done and learn more in math class and it more fun now.
• Yes
• Yes I do because before I was bored with the lesson but now there more fun and I think people pay more attention
• Yes because most people are working more and I now understand more of the math and am able to help more people understand it too. And I getting to participate more in class!! I love it! But can we step it up like the math to improve 8^th grade math a little harder
• Yes because class is going much better

8B:

• Yes, because now everyone listens to her but before the break everything was a mess
•  I think it’s ok. I feel like it was better when it was no group.
• Yes, I think class has been better before winter break because we have improved as a whole class
• Yes, I think it has been better now than before winter break because now it’s more focus, and engaged people and it’s less chatty than before
• Yes because more people are concentrating than before and also because a lot people are getting their grades up
• Yes because we got more work done and less people getting in trouble and helped me understand more classwork and helped me stay on task more
• It was better because before we didn’t let you talk and we made all this noise
• I think class wasn’t better before winter break because people keep getting into trouble. I think that the class is way better than it was before.
• Yes
• Sure, because we are better than before
• No
• Yes because everyone is participating more
• Yes because some rules that we made are working out
• I think it’s better than ever!

Shout-out Certificates:

The kids also did a great job recognizing each other. This is just one example of the certificates that they filled out.

Productive Struggle

Here is part of the quiz I gave on Friday:

The first time you go to the gym, there is a fee you have to pay to sign up.  You also have to pay every time you use the gym.  If you use the gym 5 times, you will have paid $70.  If you use the gym 10 times, you will have paid $110.

a) What is the cost per time you use the gym?

b) What is the sign up fee? 

To give a little bit of context, we have been working with linear functions for the last five weeks or so.  I have completely overhauled the unit from last year in order to (hopefully) have students develop significant conceptual understanding and then leverage that as we work towards fluency. In order to reach this goal, part of what I am doing is giving students more tasks where there is not a set procedure for them to follow and supporting them through productive struggle.

With this quiz question, about 1/3 of the students aced it, and 2/3 had no idea. There was very little middle ground. The majority of students who were able to solve this problem are also the ones who have been persevering through problems all week-talking through them with their partners, asking questions, and sharing ideas with the class.

I was frustrated that despite spending significant time on this type of problem in class, 2/3 of the students were not able to answer this question. My solution was to do more of these problems with more scaffolding.

But then I went to talk to one of my co-workers and asked for his advice. He had two pieces of advice for when the class is so starkly split in understanding of a concept.
1) Raise the level of demand/difficulty, at least for part of the task. This evens the playing field and doesn't leave 1/3 of the students frustrated because they already get it.
2) Take a break and come back. Use the break time to come at the concept from a different angle or do something totally different. It should be different enough that it's not only the same kids feeling successful.

This was exactly the advice I needed to help me keep on this path of students as sense-makers, productive struggle, and building conceptual understanding.

Introducing Student-Generated Questions

On Thursday, I am going to be introducing the idea of student-generated questions to my classes.

Inspired by Mary Bourassa and Alex Overwijk, we will start class by projecting the picture below. Students will write three questions that they have based on this picture. After having a couple students share out their questions, I will pose the question of why I am asking them to generate questions. Some of my reasons are that asking questions is part of what mathematicians do, in generating questions students connect natural curiosity to math class, students may be more invested in answering their own or their peers’ questions, making space for student-generated questions gives students power and voice in the classroom, and being able to ask good questions is an increasingly important skills. However, I am interested in hearing what reasons students will give.

 

Once we are done with that discussion, I will introduce the kids to number bracelets. They will do a couple examples in order to understand how they work. Then, we will go through a modified version of the Question Formulation Technique, which was developed by the people at the Right Question Institute.

Here are the steps in the Question Formulation Technique:

1. Question Focus (jumping off statement most often set by the teacher)
2. Produce Questions (four rules: ask as many questions as you can; do not stop to discuss, judge or answer the questions; write down every question exactly as it is stated; change any statement into a question)
3. Improve Your Questions (categorize questions into closed and open, and then re-write at least one in each category so it would be in the other category)
4. Selection of Priority Questions (groups choose their top 3)
5. Next Steps (teacher and/or students decide what to do next with the questions)
6. Reflection Activity (debrief the process)

For the first day, we will only be doing steps 1, 2, 4, and 5. In pairs, students will generate questions about number bracelets following the rules. They will then choose their three favorite questions and investigate (at least) one of them. My hope is that this will serve as enough of an introduction to the process and then the next time we do a student-generated questions task we will follow the whole protocol.

Thoughts or suggestions for supporting kids in generating interesting questions?

(Cross-posted at BetterQs)

Goals for Year Three

School starts on Tuesday and I have been making a lot of decisions about the year. Here are my goals:

1. My students and I will build a culture of student ownership in my classroom.

At TMC15, Christopher Danielson told us to “Find what you love. Do more of that.” What I love is student ownership. I believe that it is the main avenue to joy in my classroom. Most of the not-strictly-math pedagogy that I have focused on for the past two years has been in this domain. Here are my goals, based on three areas of focus:

Standards Based Grading:

Students will understand their comfort level with the big ideas and skills of eighth grade and use that knowledge to prioritize how they use their learning time.

Group Work:

a) Students will clarify, develop, question, integrate, and add to each other’s ideas in ways that no one person could alone.

b) Students will be able to identify what they need from their classmates and what their classmates need them from them.

c) Students will not see me, the teacher, as the only source of knowledge and power in the room, but instead believe in their classmates and themselves.

Student Generated Questions:

Students will generate and answer their own mathematical questions.

With my focus on group work this past year, I found it effective to develop the skill over time. Therefore, I decided to map out how we would progress through each of these foci over the course of the year.

This is a little bit hard to read, so I have converted it here:

	September	October	November	December
Group Work	- Private Think Time - Question/Stuck Protocol - Table Work Norms - Turn & Talk	- On Task Check Groupworthy Tasks: - Partner/Group Work Norms - Partner Roles - Partner Work Protocol - Tasks Cards - Partner Metacognition: What did you need your partner for? What did they need you for?
SBG	- Intro to Grading Philosophy - Scoring a Write-Up - Revising a Write-Up	- Understanding the Progress Report - Calculating Your Grade		- Own tracking of math standards in the unit
Revision Day	- Quiz Corrections/Revising a Project			Own Problems &
Student-Generated Questions	- Using Question Formulation Technique for problem-solving with prompts that are visuals/problem stems			- Identify what standards were used in solving the question

	January	February	March	April	May
Group Work	- Groups Chosen by Me - Group Roles - Participation Quizzes	- Weekly Random Groups		- Students Choose Groups to Stay in for 1 Month - Choose How to Split Up Responsibilities (Group Roles or Not)	- Multi-Day Project Where Groups Budget Time
SBG			- Planning Revision Day Choices
Revision Day	Interview/Revising MTR/Other Type of Evidence
Student-Generated Questions			- Class: Identify corresponding standards to questions and choose based on that	- Individual: Identify corresponding standards to questions and choose based on that

2. I will give more distributed practice.

Last year, I was frustrated because students would seemingly demonstrate understanding of a skill or concept at the end of a unit and then when I came back to it weeks/months later, they did not demonstrate that same understanding. One of my big takeaways from PCMI was the necessity of distribute practice for real learning to happen. I am not ready to do something as dramatic as completely spiralling my curriculum in order to get distributed practice. Instead, I am going to approach this from two avenues. The first is sequencing such that the most important concepts in eighth grade are seen early on and then reviewed/developed using connections in subsequent units. The second is to totally revamp how I do homework.

Here is my plan for homework:

Content: Each homework will have three parts—lagging practice problems, skills-based problems that get at the big ideas for the year, and a higher cognitive demand problem that requires some sort of writing. The homework will be relatively short. I am currently thinking a total of five problems.

Format: I will give homework in weekly cycles. Each week, I will create one homework and three variations that have the same problem types.  After one or two nights in the cycle, we will spend part of a double block somehow going over the homework. I will only collect the last homework. The expectation is that students may not be confident with all of the problems at the beginning of the cycle, but over the course of the cycle they will make progress with what they initially could not do.

Grading: Homeworks 1-3 will be graded for completion and I will teach students a protocol for what to do when they get stuck on a problem in order to still show their thinking and get credit for completion. Homework 4 in the cycle will be graded for accuracy.

3. I will do a better job at building procedural fluency from conceptual understanding.

Out of all of my goals, I have the least concrete plan for what this will look like right now. This will come as I get more deeply into planning each of my units. However, I know that part of this process will be starting units with meaty, high cognitive demand tasks rather than only ending units with these tasks. My goal is that these tasks will do the following: draw upon student intuition and prior understanding that will be essential to the new content they are learning and/or create a need for that new content. I will also try to have longer arcs of building understanding before pushing to fluency.

4. I will plan my lessons farther in advance in order to maintain some work-life balance.

My first year, I had my lessons done by the time I needed to teach them. This past year, my goal was to have copies made before I left the building at least once a week. This year, I think I can do better. I will also have a co-teacher in my classroom a couple periods a week, so earlier planning will be essential for us to be on the same page.

There are three action steps for this goal:

a) By Sunday noon, have an outline for the week with objective for the day and short description of the lesson

b) Have a rough draft of the lesson two nights before it is taught

c) Have copies made before leaving school at least three days a week