Sheltering Open Response Questions

My math department analyzed some MCAS data a few weeks ago and found some interesting information. There are 6 open response questions on the 10th grade math MCAS test: each worth 4 points for a total of 24 points. Every student who scored Advanced or Proficient in our district earned at least 15 points on the open response questions. On the other hand, students who scored needs improvement earned an average of 11 points, and students who failed earned an average of only 3 points. Out of 24. We have some work to do, especially with our English Language Learners.

My student smart goal for the year is to work with my ELLs on open response questions. This is my first attempt at sheltering an MCAS open response question:

Word Doc Version. PDF Version.

The original MCAS question can be found here. You will notice that I didn’t change any of the words, word order, or questions. This is important to me when sheltering an assignment. This question is text-heavy and quite a challenge for my students, but I want to keep the integrity of the question as much as possible. while providing scaffolding. The goal is to slowly remove the scaffolding as the year goes on.

I bold, underline, or italicize key math words and try to do this consistently on assignments. I also gave students highlighters when we worked on this so they could practice identifying they parts they thought were most important. And they don’t just highlight the words that I have made stand out. They notice a lot. Also, many of the directions say to show OR explain how you found your answer, but I want students to practice both.

Before starting, my co-teacher and I pre-taught the word “represents” because it was a word we didn’t think they would know. She made a couple slides showing these two images and we talked about them:

As a class, we read the intro and first question. We paused to talk about: “expression”; “minimum”; and “enough.” Students then worked on simplifying Leo’s expression on their own, and we came back together for the explaining. The sentence frames have been a big hit so far. Many of my students don’t know where to start, or are just plain scared to start because they don’t want to say/write something wrong. The frames have been providing them with an invaluable structure, and most are willing to attempt to complete the sentence when it’s already started.

We continued this pattern throughout the question: we read together, students were given independent work time, and finally we came back together to wrap up each piece. In all, this question took about one hour to complete, divided into two 30-minute chunks. At the end, our students were genuinely surprised and happy to see how much work and writing they had completed. This is definitely something my co-teacher and I will continue to work on with our students, and when possible, start to remove the guidelines. Please let me know if you have any other tricks for us to try!

Explaining Your Steps

This year I am co-teaching an SEI (Sheltered English Immersion) College Prep Algebra 1 course. It is comprised of 16 students, from grades 9-12, whose first language is either Spanish or Portuguese. Most are currently at Wida Level 1 or 2. A few of these students attended 8th grade in our district last year, but the majority moved to America after January 2016. Due to the wide range of math and language abilities, my co-teacher and I are trying to find a balance between teaching new skills, remediating basic math skills, and teaching appropriate vocab/language.

We started the year with simplifying expressions using the order of operations and then moved onto solving equations. One structure we have developed is to have students solve a problem, and then explain the steps they followed. We provide transition words, sentence frames, and an example, so that students have a starting point. After students solve and write about their process, we pair them up and have them practice reading aloud their explanations. We also tried have half the room solve #1, and the other half solving #2, then pairing them and explaining a problem the other student had not solved yet, with the goal being that they could still understand. Both ways seemed to work equally well and I think we will continue to experiment as the year goes on.

I really like this structure as I think it is making students feel more confident in their solving ability as well as their understanding of these math phrases. We say them aloud all the time in class, so it’s important that students recognize them when they are spoken, and can implement them as well.

Here are the handouts:

PDF Version. DOCX Version.

PDF Version. DOCX Version.

#TMC16 – My Favorite

Get out your calendars and mark them now! Twitter Math Camp 2017 is being held from July 27th-30th at Holy Innocents’ Episcopal School in Atlanta, Georgia.

Having just returned from my 4th Twitter Math Camp (TMC) experience in Minneapolis, I feel the need even more to book my entire summer plans around this camp. I like to spend time in the summer with my family and friends; go to the beach; read a book for fun; nap; go to the drive-in, etc. But I also NEED to spend four days in person with my MTBoS family. The people and sessions at TMC invigorate my passions and spirit and convince me that I can conquer anything in the upcoming school year.

I can’t possibly recap everything I took away from TMC16, but here are some of my favorite takeaways. I hope they become some of your favorites too! Grab a drink, there are a lot 🙂

Favorite Pre-TMC Outing: After arrival, I adventured to the Minnehaha Falls with a small group. The weather wasn’t the best, but we had a fabulous time exploring the falls and walking to the confluence of the Mississippi River and Minnehaha Creek.

Favorite New Desmos Calculator Feature: Desmos now has audio capabilities for visually impaired and blind students. Use Command F5 for the voice option, and Option + T for the audio trace. Desmos will read the expression being typed, and then play a graph used a pitched audio representation. Kudos to Desmos for striving to be fully accessible to all users!

Favorite New Desmos Activity Builder Feature: Card sorts! Ask Desmos and you shall receive! By turning on the “Labs” option once you’re logged into Desmos, you now have the option to build card sorts within the activity builder platform. I made a cart sort for Quadratic Equations, and I can’t wait to make more and then also steal from the crowdsourced list. My group loved that we can input card sorts already created; ask students to sort in more than one way; narrow responses by asking for a specific number of cards in a pile; and ask students to analyze why someone else sorted the cards a different way. The possibilities are truly endless, and you can check out the card sort bank here.

card sort3

Favorite Passionate Educator Title: Chief Evangelist. In her keynote speech, Sara VanDerWerf challenged us to become “Chief Evangelists” for our passions in math education. Sara said, “Sharing your best with others who can benefit is a responsibility and opportunity that falls to everyone” and “To be good at evangelizing, you’re gonna have to practice.” You also need to prepare mini-elevator speeches for each of your passions so you’re ready to share, and tweak them depending on your audience (students, parents, teachers, leaders). I’m going to spend some time this summer narrowing down my list of math education passions to figure out what I want to advocate for.

Favorite Dorm Life: While not all aspects of Dorm Life at Augsberg College were glamorous (looking at you, shower stalls), I had a complete ball living with some TMC-ers for four days. Waking up and having morning bathroom chats about math ed, doing the wobble in common areas late at night, and watching the bachelorette with a huge crew were all completely memorable TMC experiences.

Favorite ELL Strategy: The snowball activity is a great way to get students writing, reading, and speaking in math class. Have students answer a prompt on paper, crumple up the paper, and throw it somewhere in the room. Each student then finds a new paper,IMG_0528
reads the response, and either 1) Adds a new idea 2) Contributes 3) Corrects something
written. After going through the cycle three times, ask students to share ideas out loud
from whatever paper is in front of them. It’s anonymous, non-threatening, and fun for students. And again, it provides them with an outlet for individual think time, following by writing, reading, and speaking. Priceless.

Favorite PD Strategy: To assess participant’s understanding during professional development, I highly recommend using the “Filling in Circles” strategy modeled by Michelle. Start by identifying the key concepts of the session, or in our case, the barriers to implementing responsive stations. Then, have participants continually reflect on the topics and fill in the circles as their learning and understanding grows. Facilitator is able to see what topics need to be discussed more, and participants are able to ask better questions about what they want/need to know. Very easy and very powerful.

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Favorite Mapping Tool: Popplet. Use it as a mind-mapping tool or to help students think/learn visually. We used it to map skills and identify gaps.

Favorite Restaurant: Pizza Luce! I had Baked Potato pizza both times we went there because it was just SO GOOD. They were also very accommodating of our large groups.

Favorite Shared Experience: Jonathan talked about how he created a shared
experience in his school by creating “Varsity Math” for his calculus and statistics

I'm on the team!

I’m on the team!

students. He branded them with shirts, stickers, and inspirational speeches; and the kids just LOVED it. They bought in. They felt like they were part of a special community… because they were. Jonathan even graciously invited all of TMC16 to join the team. How can we create shared experiences for the students in our own communities?

Favorite JLV Reminder: In Jose Vilson‘s keynote speech on “TMC16, Race, and What We’re Not Talking About,” he challenged us to lead hard conversations and be okay with feeling uncomfortable. He also reminded us that we have students who are much more capable of talking about this stuff than we are; often because they have less filters. He told us to “get out of their way” but provide an outlet to let it happen. This was a much needed reminder for me, because I often feel like I should/need to have all the answers, so when I don’t, I avoid the conversation. I know I need to work on this, and my students can probably help me. Watch Jose’s keynote here.

Favorite “Getting Triggy With It” Activity: Kristen led an excellent session on how to make trig and the unit circle not a mystery. Grab all her resources here! My favorite activity was using one triangle drawn on patty paper to construct the unit circle. Simple approach but nicely shows how all the key points are determined.

See Rachel’s tweet for pictures of activity in action:

Favorite Verb: Edmund Harris and Chris Shore reminded us that modeling is a verb. It’s something kids should be doing, not something given to them. Modeling is a: Creative. Active. Process.

Favorite Physical Activity: Sara graciously brought her Backwards Bike to camp, and let me ride it as much as I wanted. If you’re not familiar with backwards bikes, watch this video. Even though I came home with several bike battle wounds, I had an absolute blast trying to ride this thing. Even after just a few times, I felt like I was making progress and coming up with new strategies to try out. Now I’m off to find someone to make one for me.

Favorite Project: Sam shared a project he does with students called “Explore Math” so they can explore math outside of school and see its beauty. He wrote about the project on his blog and shared the website he asked students to explore. It’s a “low stakes, high reward” activity. Some kids will do the bare minimum, but others will take it to levels that Sam wasn’t even expecting. His recommendation is to keep it open, keep the mini explorations mini, and don’t compare projects.

Favorite Pre-Assessment: Don’t have one yet, need to make them! Michelle led us through an eye-opening morning session about identifying the gaps in students’ understanding and then usingtrifold responsive stations to address those gaps using differentiation. I’m excited to follow Michelle’s instructions to create appropriate pre-assessments. There should only be one skill per question and as short as possible. Focus on what pre-skills students need to know in order to be successful with new content, don’t worry about the would-be-nice-to-know. The goal of the pre-assessment is so you can figure out where students are at, and provide them with learning opportunities if they don’t know, and learning opportunities if they do know (enrichment). Elissa wrote a great recap of the entire three days.

Favorite Call to Action: Tracy Zager‘s keynote speech titled “What do we have to learn from each other?” was inspiring and community-driven. She stressed that we need to stop pitting content and pedagogy against each other; we need to stop pitting elementary and high school teachers against each other. Neither of these things is productive for our community. We all have an important role in building our students’ conceptual understanding, and we need to work together to get it done. Tracy’s call to action is to analyze whom you are following on Twitter, and make sure you have a variety of contacts you can reach out to for support and to ask questions. Watch Tracy’s keynote here.

Favorite Fraction Problems: After Tracy’s talk, I pushed myself to attend Brian Bushart‘s session on fractions: a place I knew I would feel out of my comfort zone as a high school teacher. One of the reasons Brian said fractions are so hard for students, is due to practices that simplify or mask the meaning of fractions.

By finding a common denominator, you aren’t comparing fractions anymore. You’re now only comparing the whole number numerators. Cross multiplying is an example of masking; you’re getting rid of the fractions and comparing whole numbers. This masks the fact that you’re still comparing two fractions. Neither of these strategies takes into account the size of the fractions and therefore rob students of sense making. Brian then shared a bunch of strategies for how to deal with this, and I will lead you directly to his documents to learn more.

Favorite Fraction Big Idea: Another huge idea that Brian threw at us is the difference in how whole numbers and fractions are treated as adjectives and nouns. Look at the slides below for comparison.

Whole Numbers:


I’ve never really thought about it this way before, so this was a *mind blown* moment for me, and others at my table. Many students don’t actually gain enough understanding about fractions to realize that fractions are actually numbers and can be represented on a number line. They get stuck at adjectives (1/2 a cake) instead of moving onto nouns. This is where we need to get!

Favorite “Make It Stick” Strategy: In her session, Anna talked about the various ways she uses strategies from Make It Stick in her classroom. My favorite strategy she discussed was Calibration. The goal is to “replace a subjective experience or feeling with an objective gauge outside ourselves.” It stems from the “Illusion of Knowing” in that we think we know something, but really we only have a familiarity with it. The book recommends providing more opportunities for students to test themselves, review again, and test again. Quizzes need to be low stakes. I chose this as my favorite, because it ties in nicely with my morning session theme of helping students to fill in gaps.

Favorite Dylan Kane Confession: Dylan Kane‘s keynote speech titled “More than Resources” was one of the most honest and open talks I’ve ever heard. Dylan’s confession that he thought he was doing a good job when he started, but then realized he could be doing much better, really stuck with me. His big lesson learned was: “My intuition isn’t very good, because we see what we want to see.” Dylan was stealing all the great resources from the MTBoS, but realized that great resources do not equal great teaching. He challenged us to think about what will specifically work with our own students; and deliberately practice what we want to get better at. I haven’t come across a video of Dylan’s keynote yet, but you can access his resources here.

Favorite Upcoming Books to Read:
Tracy Zager: Becoming The Math Teacher You Wish You’d Had [Expected: December 2016]
Denis Sheeran: Instant Relevance, Using Today’s Experiences in Tomorrow’s Lesson [Expected: August 2016]

Favorite Song: Greg answered a call from the twitterverse to write a song about the cubic formula. He answered with the most epic sister act version ever… enjoy:

Favorite Student Quote: I know what you’re thinking, there were no students at TMC16 mathematicianso how can I have a favorite student quote? Well, in Annie‘s flex session on “Mathematicians: More than just white dudes” she shared this student quote: “Are there any mathematicians like me?” This question led to her creation of the Mathematician’s Project, where she showcases one mathematician every Friday (as long as they aren’t an old, white, rich, dead man) in order to show her students that anyone can be a mathematician. She includes their name, date of birth, ethnicity, background biography, major accomplishments, and math specialty. She even polled her students to see the types of people they wanted to learn about, and had students write their own mathematician bios. The shift in her classroom culture was unmeasurable.

Favorite Icebreaker: Amy taught us an amazing new icebreaker that I can’t way to play with my students when school starts called “Go Ahead – Break the Ice.” Break students into small groups, and give them three minutes to collectively come up with a favorite book, movie and game. Then, have students list all the ways they came to the decisions they did. This leads into a great discussion on group norms and how to work with other people. Some of the decision-making strategies were: “strong arming, time pressure, majority rules, brainstorm, survey, throw out ideas until they stick, pickiest gets the choice, narrow the choices, help those who aren’t speaking up, make sure everyone has a voice, etc…” It was a really fun activity to get to know your group, and have time to talk about group dynamics.

Favorite Day of the Year: Hannah loves celebrating birthdays and she shared some great ideas for celebrating in the classroom. She does birthday shoutouts on the board and buys cheap birthday seat covers. She sees increased positivity in her classroom culture and her students love it. She also uses birthdays to talk about what is and is not a function:

Favorite My Favorite: I can’t really put into words out much Glenn means to me in this community. Watch his talk here, and be as thankful as I am that he didn’t turn around.

Thank you to everyone who helped make my experience at TMC16 an amazing one! Much MTBoS love ❤

And of course, the end of camp song and dance:

Ambiguous Sports Graph

“Oh shit, I think I just changed my mind.” -Student

Sports Graph

Today in class we talked about the above graph. For 45 minutes. One graph. And almost every single one of my students was engaged the entire time. Here’s what happened…

During the summer I attended an institute at Boston University on facilitating productive math conversations. We learned how to set up norms for math talk and how to use talk moves (student and teacher ones) to help us facilitate the conversation. Luckily, I attended this institute with a fantastic colleague, and we have been able to share strategies as we start to implement more math talk in our classrooms. My colleague has used the original graph for years, but now it’s more clear what we want the students to talk about and how we want them to do it.

Student goals for this lesson:
• Describe the behavior of the function using precise mathematical vocabulary
• Make a choice about the sport that can best be modeled by the given graph
• Defend their own choice and respectfully critique the choice of others if they do not agree
• Reconsider their choice if they have not taken into account the time that elapses in each section of the graph

Teacher goals for this lesson:
• Use the talk moves
• Stay neutral – don’t pick a sport to keep the kids wondering
• Say very little

I started the lesson by projecting the ambiguous graph on the front board:

Sports Graph

I asked students to “Turn and talk” with their partner about the behavior of the graph. Essentially, what is happening here? I asked volunteers to share with the class. Some pairs made up their own stories. We heard about objects, cars, runners, and animals. They were accelerating, then something happened to cause the speed to change quickly. Then “object” returned to a constant speed and stopped abruptly. There was a great debate over the vertical line at the end of the graph, with many students sharing that an object cannot have more than one speed at one moment in time. Misconceptions became clear as some students referred to the distance of the object from the starting point. Some thought the object stopped and turned around. Students jumped in to clarify. We didn’t move on until everyone understood the difference between a distance/time graph vs. speed/time graph.

Then, I threw this slide at them:

Sports Graph with Sports

Without talking, students were told to choose the sport they believe is best modeled by the graph. Convince me, I said. Students wrote their responses on this interview grid, then swapped ideas by interviewing two other students (See this post for more on the interview grid strategy).

By this point, students could barely keep their ideas inside anymore. Sharing with only two classmates was not good enough; they wanted, needed, to discuss their ideas with the whole class. I asked for one volunteer to start off. It was really hard to choose. And then they talked. I didn’t say much, just kept the conversation going by calling on students and using some of the talk moves. My above mentioned colleague observed one my lessons and wrote down some of the phrases I used to keep it going:

-Say more about ________
-Can someone rephrase what ________ said?
-Can someone add-on to what ________ said?
-Does anyone agree or disagree? Why?
-Are you changing your mind? Why?

I barely talked about the actual characteristics/behavior of the graph, because I didn’t have to. If a student said something incorrect, another student shared his thoughts to help clarify. Some students came up to the board to draw additional graphs to support their thinking. The students led the conversation and I just went with it. The goals above were always on my mind, and eventually we met them in each class.

So what did the students say? I didn’t write anything down because I wanted to give my full attention to the conversation. Here is a snapshot of their interview grids:

The student work is a great representation of the different ideas discussed during our conversation. Many students stuck with their original sport, but many were influenced by the thoughts of their peers and changed their minds. Students wanted to know what is the right sport, but, there isn’t only one right answer I told them. Many students came to this conclusion on their own as well. If we had more time, I would’ve asked students if they could think of another sport that could be modeled by this graph.

We will follow up this activity with the Desmos Function Carnival. I think students are really going to like this unit.

Update: The “Which Sport?” graph was originally published in the The Language of Functions and Graphs by Malcolm Swan.

The Interview Grid

At a Classroom Discussions institute I attended last week at Boston University (more to come on that later), one of the ideas that stuck with me most was “The Interview Grid.” The presenters learned about it in the book, Common Core Standards in diverse classrooms: Essential practices for developing academic language and disciplinary literacy, and it’s a great way to initiate classroom conversations.

Screen shot 2014-08-18 at 7.48.14 PM

The teacher poses a question to the class. The question should elicit varying responses… Compare/contrast, Explain why, Choose the best model, Convince me type questions, etc. Give students about 2-3 minutes to write an answer and then have them interview classmates. The students should listen to their classmates’ thoughts (each interview should be about 3 minutes total), and paraphrase the response onto their own paper. After speaking with a couple interviewees, students are given a chance to expand, adapt, edit their original response (2-3 more minutes). Total time = 10-12 minutes

Screen shot 2014-08-18 at 7.49.05 PM

The interview grid gives students the opportunity to externalize their own thinking, listen to others, deepen their own reasoning, and helps students work with the ideas of others. It also provides students with the chance to practice three of the domains for learning language: speaking, listening, and writing (a plus for all our students but especially our English Language Learners!). Another idea I had is to turn the document into a Google Form so that students can type their responses and submit to teacher electronically. Can’t wait to try this activity out!

Math Strategies for English Language Learners

In Massachusetts, all core subject area teachers and administrators of English Language Learners, must receive a Sheltered English Immersion (SEI) Endorsement from the state. One way for educators to receive this endorsement is by taking a RETELL (Rethinking Equity and Teaching for English Language Learners) course. One of the main goals of the course is to provide teachers with a repertoire of methods and strategies they can use to help students practice the four domains for learning language: reading, writing, listening, and speaking.

These are a few of my favorite strategies I learned from the course:

Reading Strategy: Partner Reading for Comprehension
Partner Reading

This strategy made the problems more manageable because students were able to have valuable discussions regarding the content before actually solving the problem. Since Partner #2 had to comment and respond to Partner #1’s questions, they had to pay close attention to what was being read. Most of the partners were able to choose the most important pieces from each word problem based on what Partner #1 had commented on during part two of the reading strategy. Giving my ELLs the opportunity to read aloud with a mainstream student allowed them to practice their expressions and ask for explanations. The strategy helps support both partners as they learn from each other’s observations and questions regarding the text. The students were able to determine what phrases were necessary for successful completion of the problem, and those that were not needed.

Writing Strategy #1: Cut and Grow
The Cut-n-Grow strategy provided students with an opportunity to see exemplar/non-exemplar student work samples and focus on improving their own open response questions. First, students looked at a student work sample that received a score of 2 on a standardized open response question. They cut the question apart and wrote additional explanations to turn the score into a 4. Then, students repeated the process for an open response question they had previously solved, to improve upon their own writing.

Cut and Grow

Many of my ELLs tend to leave open response questions blank on exams, so it’s important that we explicitly practice answering these questions. The strategy gave students a second chance at improving their work, and they responded very well to this strategy. The physical act of cutting and pasting pieces of the response, and then adding their revised sentences, really helped show students how to edit and model good writing.

This strategy can work with any writing sample the students produce in class, such as journal entries, AP open response answers, exit tickets, etc.

Writing Strategy #2: Write Around
Students should be divided into groups of 3 or 4. Each student starts with a blank sheet of paper and writes one sentence of a word problem. Then, the student passes the paper to the right. After reading what is written, students continue to add sentences until a word problem is created (approx 4 sentences). Each team will choose one problem to write on a large whiteboard or paper to show the rest of the class during a gallery walk. The gallery walk gives students the chance to make observations regarding other students’ work. Students can then choose one or more of the problems to solve. Teachers can scaffold this activity by providing students a list of must-haves for each word problem. For example, students might have to include the following in a quadratics word problem: a setting, the type of object being thrown/launched/dropped, height at which object starts, and the speed or distance the object travels. Each student would take turns providing one of these details.

My biggest takeaway from this course is that we all need to explicitly teach strategies for reading, writing, speaking, and listening in our disciplines. We cannot sit back and “let the English teacher handle it.” English Language Learners are trying to learn a new language at the same time we are expecting them to learn our content. It’s our responsibility to provide opportunities in our lessons to support both goals.