Intervention: reteaching, review, and problem solving

In my role as interventionist I’m writing the middle school math curriculum for our Title 1 after school program. Circumstances outside of my control have prevented us from rolling out the program sooner, but invitations will be going out next week and we anticipate a start date within the next two weeks.

I know students will need review and reteaching of concepts, but I also want them to develop a repertoire of problem solving strategies so I will be including James Tanton’s Curriculum Inspirations. which is supported by the Mathematical Association of America. It’s a fabulous resource that includes a series of 10 strategies with examples.

Since the first lesson will focus on multiples and factors students will be introduced to Tanton’s think aloud on the Divisible by 13 task using the problem solving strategy Engage in Successful Flailing. They’ll solve the problem using the attached.


Of course students will have opportunities to practice LCM. Then they will have a chance to explore multiples and 3 consecutive numbers problems from Don Steward’s site.

consecutive numbers

Several teachers will be implementing the after school program, so it makes sense for the curriculum to be delivered using Canvas. Each lesson begins with a content page that identifies the learning objectives and mathematical practices. The page also includes a resource link so teachers have access to answer keys stored in google drive.


As the title of this post suggests I want to balance review, reteaching, and problem solving.  Students need to be exposed to challenging problems but they need to have access to them as well.

Math Intervention and The Remainders Game

I’m in a new building with a different role this year, math interventionist. A big part of my job is working with the 6th-8th graders during their homeroom periods, pushing in to the sixth grade classrooms for reteaching, plus creating math curriculum for the after school Title 1 program.

Officially the intervention period is not a replacement class, but I do see specific students twice a week supporting the day’s learning objectives while shoring up foundational skills and problem solving strategies.  A twice weekly intervention period is not an ideal situation, but it’s the best we can offer students right now. The period also gives me a chance to pilot some of the lessons/activities I’m considering for the after school program.

One great example which fits perfectly with factors is NRICH’s The Remainders Game. The object is to deduce what number between 1-100 the computer is thinking in the fewest moves possible. The player chooses the divisor between 2-10 and the computer gives a clue–its remainder. We’ve played the game several times as a class and I’m at a point where I can now share a Remainders Game worked solution which the after school tutors can follow, plus a Remainders Game worksheet so students can play independently.

Discuss the first move. What number should we divide by to narrow the list of possibilities?
remainders game
What possible numbers between 1 and 100 can the computer be thinking?

In this scenario we’re working with the numbers 18, 28, 38, 48, 58, 68, 78, 88, and 98. We don’t know what we’re going to do with those numbers yet but we need to keep track of our work. I led students to roughing out a table by hand. I later created the worksheet.

Introducing factors as a strategy comes into play when you’ve determined the possibilities. For example when the computer is thinking of an even number, is dividing by 2 a wasted move? Can finding the factors of each of the numbers help us rule out some of the possibilities?

For time’s sake each student was assigned a number to factor and we shared out. In the future students will be able to do this independently now that they know the process. Next was a great discussion on which number to divide by next.


We chose 4, and the computer told us there was a remainder of 0. What numbers did we eliminate? What number should we divide by next?

This task builds stamina while studying the composition of numbers. I think a scaffolded approach is giving my students access to a challenging problem.

Recap: Standards Based Grading Series Day 1

Today, I had the pleasure of hearing both Thomas Guskey and Lee Ann Jung share their expertise on Standards Based Grading in Arlington Heights, Illinois (#sblchat). The day-long conference is the first in a series of five face-to-face and online discussions on the components of SBG and implementing a standards based reporting process. If you aren’t familiar with Guskey and Jung, they are authorities on the topic from University of Kentucky, Lexington. Handouts from the session can be found here.

My district has not adopted SBG, however math educators in our middle schools have designed common assessments with SBG in mind. Some of us have implemented a quasi SBG approach given the constraints of our percentage-based report card system. Our district appears invested in SBG by sending several representatives from the elementary schools and a few representatives from the middle schools to learn more about the topic.

Often teachers embrace a grading practice because that’s what they experienced when they were in school—myself included. It took me several years to evolve and I’m still evolving.

Guskey began with three guiding questions that continue to stretch my thinking.

Guiding questions

  • Why do we use report cards and assign grades to students’ work?
  • Ideally, what purposes should report cards or grades serve?
  • What elements should teachers use in determining students’ grades?

He also reminded us of the various purposes of grading.

Purposes of grading

  • Communicate achievement status to parents
  • Provide info to students for self-evaluation
  • Select, identify or group students
  • Provide incentives for students
  • Evaluate the effective of instructional programs
  • Document students’ lack of effort or inappropriate responsibility.

As educators we need to establish a common purpose on what grades are for, identify the purpose, then identify the method of communication or documentation. According to Guskey, ninety percent of parents agree the report card is for them.

If the purpose is to report on student achievement the grade can get muddied with more than a dozen grading elements. Educators need to have a common understanding on what counts for a grade.

grading elements

Jung focused on the importance of 1) sharing learning targets with students so they have clear expectations and 2) providing formative feedback so students have the opportunity to meet the standard.

A featured take away was her GPS analogy. When driving we care more about where we are now and the turns we need to make in order to reach our destination. A report card should reflect that as well. If the purpose of a report card is to reflect where the student is at the end of the marking period, it should not be based on averages.

Percentage grades and 1-4 scales

Guskey shared how grade reporting becomes more subjective when more categories or levels comprise the grading scale. For example is a student precisely 86% proficient on a concept? He may have earned 86% on the correct number of items on an assessment, but what best describes a student’s level of proficiency? Or, on a learning target if a student earns a series of four 1s and ends the marking period with two 4s on a 1-4 scale, what grade do we report? The heart of the matter is to use informed professional judgment instead of mathematical algorithms.

In the next session the two will share their thoughts on grading exceptional learners.

Perceptions on summer math packets vs. online refresher

Is it possible to transform our summer math packets into a more engaging, interactive experience?

At our last curriculum committee meeting I pitched the idea of retooling our summer math packets by offering an optional digital learning experience using Canvas. While there’s little debate that summer slide occurs, are math packets the answer? Until this year we had no alternative, but since our district has invested in Canvas and students in grades 6-12 are now 1:1 iPad, we now have the ability to leverage the technology and create an alternative that is hopefully better than worksheets.

To plant the seed I created a dummy Canvas course (feel free to poke around and enroll –even use a fake email and username if you want to remain anonymous). It contains one module on equations which consists of:


Here’s a section from the Overview page.homepage

Most of my colleagues haven’t dabbled in Canvas to the extent I have, so I merely wanted to present a framework which is a work in progress. While we tabled the idea for this summer, we still had a lively discussion. There was a lot a conversation centering on whether creating a self-paced course was worth the time and effort. Would students do it, would they wait till the last minute, would they take the time to watch the videos and do any extra work, would they complete an optional task? There were also questions on grading optional tasks, where would we find the time to create it, and would this minimize the summer slide.

With all the talk about students’ work habits, I realized I needed to get their input. We have developed some perceptions about our students so we need to either confirm or refute them.

I surveyed 103 7th grade standard math students, representing four classes. Not included in the 7th grade survey were two pre-algebra math classes and one algebra class. One colleague and I surveyed each of our two standard classes using Google Forms. I did my best to create an unbiased survey, but you’ll see where I missed the boat.


What pleasantly surprised me were the number of students who would voluntarily watch a tutorial if they needed a refresher. The next set of results surprised me a bit as well.


Seventy-five percent said they would not voluntarily complete extra worksheets if they needed a refresher. Not surprising for seventh graders. Is there a way to change this mindset? The second question in the above image is not reliable. While I’m impressed that 80% said they would go above and beyond if the task interested them, the response choices were too limiting. The manner in which the students completed the packet was of particular interest. Nearly 20% waited until the end of summer while nearly 60% completed the packet a bit at a time throughout the summer. This is quite different than what the teachers expected.

In the graph below about 68% think apps and internet resources are somewhat to very valuable to learning concepts. It is here where I would want to leverage the technology. GeoGebra, Desmos, and other virtual manipulatives would make an nice addition to summer review. So would other online activities such as SolveMe Puzzles, Refraction, Lure of the Labyrinth, 101 Questions, etc. I could even include puzzles and strategy games like Game About Squares, 2048, Set, and KenKen.


The above graph surprised me and it didn’t surprise me. I looked at the spreadsheet to see who answered a 1 or 2. The students I had who disagreed had strong math skills so perhaps they should experience something more. And if that’s the case, shouldn’t all students have the opportunity to experience something more?

survey 4

For the last graph, sixty percent said between 1/2 hour to one hour per week is a reasonable amount of time to spend on summer math review. Nearly every student who chose “Other” suggest no summer packet. One student said 15 minutes per week, and another said 1 page per day.

At the end of the survey I gave students an opportunity to share “anything else.” Overwhelmingly they said they didn’t want to do the packet. I get it they’re kids. Does that mean they also wouldn’t want to do a summer review on Canvas? Probably, although they did not specifically articulate that in the open ended response.

What to make of all of this.

Is it possible to make summer review more engaging and interactive? Absolutely.

Are we there yet? No, but maybe this time next year we will be.

I would love to learn how your school handles summer review. I do think an online format can offer more–if done properly.

Collaborative learning with Canvas discussions

This year I’ve been using the Canvas learning management system primarily as a work flow tool–because that’s what it is–primarily. It’s heavy on teacher productivity. Post, turn in, provide feedback, and electronically grade assignments. Create pages for resources or a week-at-a-glance. Build modules to deliver content. But what are the students doing?

For me the best feature Canvas has to offer is the discussion forum. It’s a place where student learning is visible. It’s a place where students learn from others and contribute to the learning of others.

To launch our 7th grade unit on percent I posted a discussion prompt using Robert Kaplinsky‘s Bed, Bath & Beyond coupon task from the 101Questions site.

BBB questions

I probably shouldn’t have led the students by saying, “Here are my questions, what are yours?” but I was impressed with their questions. Here are a few:


The students not only had to post two questions, they also had to respond to one other student. What impressed me more was how the online discussion minimized student status, gave all students a voice, and allowed them to contribute.


The following day students explored the answers to their questions with Kaplinsky’s Act 2. But my point is how the on-line discussion facilitated collaborative learning.

Here’s another example. Students shared their solutions to a small task, gave feedback, and later revised their work. Some posted as a narrative while others uploaded their work.

Percent rectangle

percent shade2

I plan to add more discussion forums in the future. Overuse can kill a good thing, but a well placed discussion is gold.

Partnering with parents, keeping them informed, Canvas

Since we returned from winter break I’ve been making a concerted effort to reach out to parents. Why the awakening? I had been so focused on implementing interventions (grade level lunch crew for students who don’t turn in assignments across multiple subjects, pulling students into math lab, etc.) that parents were left out of the loop. It came to a head when a parent never knew her child had been in lunch crew at the request of multiple teachers more than 10 times in one quarter.

I want to blame our RtI program for taking my eye off the ball. I wanted to hear, “Let’s bring in the parents when this isn’t working.” RtI has many shortcomings, but it’s my responsibility to communicate to parents. I didn’t. I should have called or emailed instead of relying on my online grade book.

Partnering with parents, keeping them informed
Two weeks ago I had a twenty minute phone call with a parent. She revealed intense frustration. Frustrated with the teachers; frustrated with her child; frustrated with herself. To repair the relationship I started contacting Mom daily, informing her if an assignment wasn’t turned in, if her child skipped lunch crew, and an occasional success when her child did turn in an assignment. The phone call and email communication have been replaced with notations in the assignment notebook. An assignment contract is now in place. Time will tell if this intervention will work, but at least the parent has been formally informed.

On Friday I contacted a few parents about their student’s eligibility for sports. One parent was quite frustrated because their child chooses to show them only what he wants to them to see. With most assignments on the iPad Mom is looking for transparency. The timing could not have been better.

Earlier in the week I sent an email notifying parents that our learning management system, Canvas, is now accessible to parents. Instructions on how to create an account were included in the email. So far four parents have created an account.

This post ends happily for me because this is the transparency the parent is looking for. My Canvas course is structured using pages. Each page provides an overview of the week, with links to in class practice, assignments, etc.

canvas week

When the parent creates the account he/she will also be able to see the feedback I’ve given to the student. For example, here’s a student who is not balancing equations with the one-step process.

student work
Textbook page in Word; pdf created for students to load into Notability.

I provide feedback.

student work feedback

Several days pass. The parent hasn’t created a Canvas account but must have talked to the student about correcting the work. The student resubmits.

student work2

I provide this feedback.

student work feedback2

Canvas is not the be all end all, but the transparency it offers does help keep parents informed.

5 Practices: applying lessons learned from chapter 2

5 Practices for Orchestrating Productive Mathematics Discussions

Chapter 2: Setting goals and selecting tasks

If I want my seventh graders to have a meaningful mathematical discussion I have to raise the level of thinking in the learning target. But it’s not over yet. I also need to select an appropriate task to support the learning goal.

As I wrote last week, I’m really enamored with this number puzzle.

Choose any number. Add the number that is 1 more than your original number. Add 11. Divide by 2. Subtract your original number. What is your answer?  Do the puzzle again for other numbers. Why do you get the answers that you get? Will this always work?

This puzzle fits nicely with our 7th grade expressions and equations unit and has the potential to be an anchor problem. Let’s look at how I can modify the learning target to raise the level of thinking and mathematical discussion.

Original target: Students will translate verbal phrases into an expression or equation.

I know my students can handle a more robust learning target. The number puzzle can certainly support it. This is still a work in progress, but here’s my revised target:

Revised target: After translating a verbal phrase into an equation, students will discover how grouping symbols and order of operations provide clarity.

When I gave the puzzle to the students, their first inclination was to simply follow the directions. Based on their work, their equation would look something like this:


This lesson now has a much better chance for meaningful mathematical discussion by focusing on grouping symbols, order of operations and how their use provides clarity. I could begin by saying, “Take out the equals sign and question mark. Make equivalent expressions using parentheses, brackets, fraction bars, etc.” We could then have a discussion as to which equations best represent the puzzle.


As I mentioned this is still a work in progress. If you have insights, please share.

Other bloggers discussing Chapter 2 can be found at the link below.