US Congress: is it representative? A math lesson

As I read Dan Meyer’s post, “A Response to Danny Brown,” and the stream of comments that followed, I began to think about middle school and the lost interdisciplinary units, or the singular, math related, social justice lesson that “matters”-a topic that appears to be dear to Danny Brown. He writes,

…I really want to model something *important*, in the sense that it *matters*, perhaps politically, or socially, perhaps having a *local* relevance for the students in the classroom

I hope there is still a place in middle school for cross-curricular thematic learning. I haven’t seen many IDUs play out in my 15 years of teaching. Some middle schools are more middle school than others. I only know what I’ve experienced.

The point of my post is to share a lesson I slightly modified from the October, 2015 issue of Mathematics Teaching in the Middle School. I do know one thing about middle schoolers–they always seek fairness.

The proportional reasoning lesson is titled US Congress is it Representative?  Students studying the US Constitution use proportions or percentages to examine equity, fairness, and how the racial and ethnic make-up of our country compares to the ethnic backgrounds of the current Congress.

The lesson begins by providing background knowledge on the US Census, which counts every resident of the US then, from that count, determines the number of seats each state has in the U.S. House of Representatives. The NCTM article did not include this background so this is where I deviated from the original lesson.

The most recent apportionment is below.



If your state lost or gained seats, that would make for good discussion. On a side note, I’m surprised Illinois did not lose more seats in the House!

One major part of the lesson is examining the population statistics and discussing why the sum of the various ethnicities exceeds the total. (The way the Census is counted residents can and do identify with more than one ethnicity.)


Students are then asked, “Based on this data, and the knowledge that there are 435 members of the House of Representatives and 100 members of the Senate, what should a “fair” racial/ethnic make-up of the House and Senate be?

Once they’ve determined what they think is “fair” the ethnic background of Congress is revealed for comparison.


Students use claim, evidence, and reasoning to argue whether the House and Senate are representative of the US population.

I’m with Danny in the sense that I want students to learn about “important” matters. But I also understand that what’s important to a middle schooler may not be the same as an adult.

In my role as math interventionist, I’ll use this lesson to review percentages in our after school math program, but it may get more traction if the lesson is presented by the social studies teachers. So I shared it the social studies teachers instead of the math teachers with the hope they’ll consider it when teaching the Constitution.

Time and timing are the biggest constraints when implementing an interdisciplinary lesson.

Justifying moving the decimal

I need your help. I’ve been thinking of ways to better connect decimal multiplication with fraction multiplication, especially in terms of justifying moving the decimal point. Sixth graders may recall counting the number of decimal places, but do they really know what they are counting and why they are counting it?

The following is work in progress and I would appreciate your input. How can my attempt at creating curiosity  be improved upon? Are the examples sufficient or should they be revised? What am I missing?


I’m anticipating students will say, “Count the decimal places,” so my follow up is this question.


I don’t think students will connect fraction multiplication with decimal place value so my next move will be to ask them to compare the two expressions.


Some students may recognize the two expressions are equivalent, but will they make the place value connection?


I prepared a second example that “culminates” with this slide.


Since the above problem has numbers in the ones place, some students may recognize 2 x 2 = 4, so it would be reasonable to place the decimal between the 5 and 6. I would certainly accept that, but my point is to discuss powers of ten. Yet I need to be careful. My sixth graders haven’t been exposed to negative exponents. Would it be appropriate to describe it as powers of tenths? Is this they best way to address the mathematics? I’ll likely begin with establishing a pattern before I present these slides.

I’m also thinking of creating a similar lesson which connects dividing decimals with dividing fractions using the common denominator method.  The students are familiar with the common denominator method, but I wonder if it will be too messy?

Mindset, abundance

Even though I’ve switched middle schools a colleague in my former building and I continue to commiserate on the implementation of Carol Dweck’s growth mindset research. Nurturing a growth mindset is a daunting task. It requires creating and sustaining the conditions for it with multiple learning opportunities, feedback that’s timely (and implemented!), and frequent formative assessments.

What my colleague and I have both feared, and Dweck  too, is that growth mindset has been reduced to the grit mentality of telling the students to work harder.

A growth mindset isn’t just about effort. Perhaps the most common misconception is simply equating the growth mindset with effort. Certainly, effort is key for students’ achievement, but it’s not the only thing. Students need to try new strategies and seek input from others when they’re stuck. They need this repertoire of approaches—not just sheer effort—to learn and improve.

Could this be a reason why the growth mindset effect size is only 0.19? Hattie thinks teachers treat their students with a fixed mindset. Is focusing our attention on effort a fixed mindset?

Now I wonder, should we be asking a different set of questions?


I am so conflicted right now I’m putting mindset in the back of my mind. Instead my focus is on abundance.

The biggest gift of abundance I can give students is patience. I have to remember they are children. They don’t share the same enthusiasm for learning as I do. Or worse, they’ve quit on themselves because of a self-concept they’ve developed. That patience will also serve me well when students learn implementing feedback is a requirement, not an option.

I can also give an abundance of learning opportunities that weave the concrete, representational, and abstract. For example, when re-teaching estimating percent of a number several students got the concept when I introduced a bar model. In fact one said, “Can I do this on the test?” Of course.

One final gift of abundance is continuous self-reflection. We put students on a treadmill and never give them time to step off and check their heart rate. How can they take ownership of their learning when we don’t give them time to reflect on the learning process?

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.