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Standards-based assessment and Instruction


Archive for the ‘Math’ Category

How PS 12, the Dr. Jacqueline Peek-Davis School, Took Student Math Scores “Ahead of the Pack”

Tuesday, December 11th, 2018

Exemplars changed my teaching,” says one teacher at Brooklyn’s PS 12, the Dr. Jacqueline Peek-Davis School. And it changed the school’s learning outcomes, too: So substantial were its improvements in math scale scores that it made New York City’s top 10 list of positive percentage changes for 2017.

Why do the school’s educators attribute that to Exemplars? Aldine Finnikin-Charles, a special education teacher at Brooklyn’s PS 12, credits the problem-solving procedure students learn as they work through our tasks. “That’s the best thing for them,” she says. “Exemplars changed the way my students look at problems and how they unpack them.” Challenging students with complex multi-step tasks, Problem Solving for the 21st Century: Built for the Common Core gives them a robust framework for developing solutions — so they’re well equipped to tackle such problems on standardized tests.

LaToya Garcia, assistant principal and former teacher at PS 12, cites the practice students get through regular classroom use. “Students in previous years had struggled with the test’s open response questions and the writing that was required with the math,” she says. That changed with Exemplars, since every task challenges students to communicate their thinking. “Exemplars really gives them the practice and support that they need,” says LaToya, “not only to be able to solve the problem, but also to solve it using multiple strategies, critically think about how they could expand the problem, and explain their thinking in a precise way.”

Nyree Dixon, the school’s principal during the 2017-2018 school year, says one of the keys to PS 12’s improvements has been Exemplars’ impact on student self-assessment. With Exemplars standards-based rubric to support them, she says, “Our children are not waiting for the teacher. They’re not waiting for ‘authority’ or a supervisor. They’re assessing their work themselves, and aligning it to the rubric.” As they do this, they develop critical skills of self-assessment and metacognition, which puts them, Nyree says simply, “ahead of the pack” — and enjoying a real improvement in learning outcomes, too.

We thank the Brooklyn PS 12 team for making us part of their school’s top 10 story, and for sharing their thoughts about how Exemplars helps their students succeed. 

A Mindset Shift in Problem Solving

Monday, October 1st, 2018

By: LaToya Garcia, Asst. Principal, & Kimberly Naidu, Third Grade Teacher, at P.S. 12, Brooklyn, NY

Exemplars Problem Solving Procedure has demonstrated that students can be very creative when they are allowed to choose their own methods for solving mathematical problems. Many educators, we believe, focus on enhancing mathematics skills to meet academic expectations, but fail to promote more critical thinking. Without the ability to inquire while thinking, students will not learn to make decisions.

At our school, Exemplars has given students the opportunity to learn from one another and provide opportunities to use multiple solution strategies for the same problem. The Problem Solving Procedure has supported students in understanding a problem, identifying a strategy to use, solve the problem, communicate findings in words, and trying to solve a problem in another way.

Exemplars has also helped to prepare our teachers for a more cognitively guided instruction. Our goal was to modify our current curriculum to help students become mathematically literate citizens. We realized that problem solving wasn’t about knowing the solution to the problem, it was about understanding that knowledge. In order to enhance learning, our teachers use a student’s prior knowledge, transfer that information to new situations and organize their students’ knowledge using model-based problems from the Exemplars Library. Teachers are able to plan activities to help students build their mathematical knowledge through problem solving and reflect on the process of mathematical problem solving. Our teachers find that students connect different meanings, interpretations and relationships to the four operations in math.

Our training with Exemplars helped to develop a more differentiated learning environment as well. We now integrate mathematics across our curriculum. For example, teachers use more math read-alouds in the classroom as a visual interpretation of mathematical problems. Teachers modify problems from the Exemplars Library by simply changing the character’s name to one students recognized from a literacy lesson. Students are also provided with manipulatives to show math situations. Manipulatives and other mathematical tools are made apart of our daily routines. Materials are made accessible in math centers and students are encouraged to self-select resources when needed. Our entire school community understands that our students need to improve in the area of communication. Students practice using math language to explain what they did while solving a problem and represent their math thinking through drawings and diagrams.

Exemplars has helped to guide and organize our students’ thought processes–a skill that can be transferred to all content areas. Throughout the school, students have begun to recognize the positive impact of a problem-solving strategy guided by specific steps. A third-grade student Michael C. commented, “Exemplars is fun. It helps you to do math and writing. While you’re doing Exemplars, you can also work on your writing.” Another third-grade student Desayee S. said, “What I like about Exemplars math is that it helps me to practice using different strategies for math, like diagrams, models and tables.”

Additionally, Exemplars has led to an increase in student-to-student conversation in students who had previously shied away from mathematics and students who are currently learning English as a second language.

As the year continues and students become even more experienced Exemplars math problem-solvers, we anticipate an improvement in students’ critical thinking and problem-solving skills. Students have already begun to develop a love for mathematics and are learning to persevere until they arrive at their answer.



10 Things Educators Love About Exemplars, Plus 1 Big Bonus

Monday, October 1st, 2018

We asked a powerhouse teaching team to share their Exemplars experience. In an inspirational conversation, a panel of educators at the Brooklyn’s PS 12 offered ten amazing ways that using Exemplars has transformed their school.

1. It encourages students to be flexible and creative in their approach to solving problems. “My students love the freedom Exemplars tasks give them,” says Kimberly Naidu, a third-grade teacher. “They’re never forced to use a specific strategy. They can create one or use one they’ve learned before.”

2. It builds students’ math vocabulary. Aldine Finnikin-Charles, an experienced special education teacher, says, “My students are always using math nicknames,” says Aldine, “so the tasks that require them to use at least two math vocabulary terms really help them a lot.”

3. It helps students become capable peer and self-assessors. PS 12 teachers ask students to consider their own work alongside Exemplars anchor papers. “It really helps them get a good understanding of how to meet the standard,” says Kimberly. “They see exactly what needs to be done.”

4. It spurs students to demonstrate their understanding. They learn to communicate their ideas clearly and effectively. Plus, it’s fun, says Cherron Knight, who teaches grades 4 and 5: “Our students love showing their strategies. They can always argue, ‘Why did you use that strategy?’ They love that part, explaining what they think.”

5. It motivates students to make new mathematical connections. “When kids bring their work up to me, I say, ‘Go back and see what else you can do. What’s a different way of showing it?’ They find that enjoyable,” says Kimberly, “because they can always do it another way.”

6. It lets students practice a process for solving problems — so they can internalize it and apply it to every task. “That’s the greatest thing,” says Aldine, “the best thing for them. They are able to become reflective thinkers.”

7. It gives students a clear understanding of what kind of work meets the standards. Kimberly says, “They really want to become Experts,” and because of the Exemplars rubric, “they know exactly what needs to be done to achieve that level.”

8. It empowers students as peer and self-assessors. With what Principal Nyree Dixon calls “these amazing rubrics” that are at the heart of Problem Solving for the 21st Century, “they’re not waiting for the teacher,” she says. “They are not waiting for quote-unquote ‘authority.’ They are assessing their work themselves,” and experiencing the benefits.

9. It inspires students to grow as problem solvers — and as leaders, too. When kids solve Exemplars tasks in a problem-solving classroom, they become the experts. And they become skilled collaborators and advocates, Nyree says. “They’re learning to work together and disagree or agree,” she explains. “They’re learning to take a stance and defend their responses and answers.”

10. It helps educators establish a classroom culture of problem solving. In a lot of schools, this constitutes a major shift in thinking. “We went through growing pains. We still do,” Nyree says. “We have to keep our minds open and be willing and ready to learn.” And we’re there to help them every step of the way. After all, every teacher in every school should be able to say, as Nyree does, “We do not ever, ever, ever lower our expectations.”

The PS 12 team reported another benefit, too: Their students made a dramatic jump in math scores on standardized tests, an improvement they attribute to Exemplars — but more importantly, students made enormous strides forward as problem solvers.

“I think Exemplars did a really, really good job in helping the students to think critically about the math that they were learning,” says LaToya Garcia, Assistant Principal. “Exemplars really gave them the practice and support that they needed to be able to not only solve the problem but also to solve it using multiple strategies, critically think about how they could expand the problem, and explain their thinking in a precise way.”

Our thanks to the team at PS 12 for sharing their insights. We’re thrilled to be a part of your students’ success.

Understanding Mathematical Connections at the First Grade Level

Thursday, March 29th, 2018

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

In today’s post, we’ll look at a first-grade student’s solution for the task, “Pictures on the Wall.” This anchor paper demonstrates the criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the Exemplars assessment rubric. It also shows a solution that goes beyond arithmetic calculation and provides the evidence that a student can reflect on and apply mathematical connections. The beauty of mathematical connections is that they often begin with the other four rubric criteria. In other words, the Exemplars rubric provides multiple opportunities for a student to connect mathematically!

In this piece of student work, you’ll also notice that the teacher has “scribed” the student’s oral explanation. Scribing allows teachers to fully capture the mathematical reasoning of early writers.

This blog will offer tips for the type of instructional support a teacher may provide during this learning time as well as the type of support students may give each other. Teacher support may range from offering direct instruction to determining if a student independently included mathematical connections in her or his solution. After reading this post, give the task a try in your own classroom along with the Exemplars rubric. You may view other Exemplars tasks here.

First Grade Task: Pictures on the Wall

There are sixteen pictures on a wall. The art teacher wants to take all the pictures off the wall to put up new pictures. The art teacher takes seven pictures off the wall. How many more pictures does the art teacher have to take off the wall? Show all your mathematical thinking.

Common Core Alignments

  • Content Standard 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
  • Mathematical Practices: MP1, MP3, MP4, MP5, MP6

Understanding Mathematical Connections at the Third Grade Level

Thursday, March 29th, 2018

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

In today’s post, we’ll look at a third-grade student’s solution for the task “Bracelets to Sell.” This task is one of a number of Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 3.OA.3. It would be given toward the end of the learning time dedicated to this standard.

In addition to demonstrating the Exemplars criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the assessment rubric, this anchor paper shows evidence that students can reflect on and apply mathematical connections successfully. For many students, mathematical connections begin with the other four criteria of the Exemplars rubric, regardless of their grade.

After reviewing our scoring rationales below, be sure to check out the tips for instructional support. Try these in your classroom along with the sample task and the Exemplars assessment rubric. How many mathematical connections can your students come up with?

3rd Grade Task: Bracelets to Sell

Kathy has thirty-six bracelets to sell in her store. Kathy wants to display the bracelets in rows on a shelf. Kathy wants to have the same number of bracelets in each row. What are four different ways Kathy can display the bracelets in rows on the shelf? Each bracelet costs three dollars. If Kathy sells all the bracelets, how much money will she make? Show all of your mathematical thinking.

 Common Core Alignments

  • Content Standard 3.OA.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
  • Mathematical Practices: MP1, MP3, MP4, MP5, MP6

Understanding Mathematical Connections at the Fifth Grade Level

Thursday, March 29th, 2018

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

In today’s post, we’ll look at a fifth grade student’s solution for the task “Seashells for Lydia.” This task is one of a number of Exemplars tasks aligned to the Number and Operations in Base Ten standard 5.NBT.2. It would be given toward the end of the learning time dedicated to this standard.

In addition to demonstrating the Exemplars criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the assessment rubric, this anchor paper shows evidence that students can reflect on and apply mathematical connections successfully. For many students, mathematical connections begin with the other four criteria of the Exemplars rubric, regardless of their grade.

After reviewing our scoring rationales below, be sure to check out the tips for instructional support. Try these along with the task and the Exemplars assessment rubric in your classroom. How many mathematical connections can your students come up with?

5th Grade Task: Seashells for Lydia

Lydia started collecting seashells when she was five years old. At age seven, Lydia had 12(10)2 seashells. At age nine, Lydia had 24(10)2 seashells. At age eleven, Lydia had 48(10)2 seashells. Lydia wants to collect 75(10)3 seashells. Lydia continues to collect seashells at the same rate. How old will Lydia be when she has 75(10)3 seashells? Show all of your mathematical thinking.

Common Core Alignments

  • Content Standard 5.NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
  • Mathematical Practices: MP1, MP3, MP4, MP5, MP6, MP7

Understanding Mathematical Connections

Thursday, March 29th, 2018

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

What is a mathematical connection? Why are mathematical connections important? Why are they considered part of the Exemplars rubric criteria? And how can I encourage my students to become more independent in making mathematical connections?

This blog represents Part 1 of a four-part series that explores mathematical connections and offers guidelines, strategies and suggestions for helping teachers elicit this type of thinking from their students. We find many students enjoy making connections once they learn how to reflect and question effectively.

A Brief Introduction to the Exemplars Rubric

The Exemplars assessment rubric allows teachers to examine student work against a set of analytic assessment criteria to determine where the student is performing in relationship to each of these criteria. Teachers use this tool to evaluate their students’ problem-solving abilities.

The Exemplars assessment rubric is designed to identify what is important, define what meets the standard and distinguish between different levels of student performance. The rubric consists of four performance levels — Novice, Apprentice, Practitioner (meets the standard) and Expert — and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation). Our rubric criteria reflect the Common Core Standards for Mathematical Practice and parallel the National Council of Teachers of Mathematics (NCTM) Process Standards.

The Importance of Mathematical Connections

Exemplars refers to connections as “mathematically relevant observations that students make about their problem-solving solutions.” Connections require students to look at their solutions and reflect. What a student notices in her or his solution links to current or prior learning, helps that student discover new learning and relates the solution mathematically to one’s own world. A student is considered proficient in meeting this rubric criterion when “mathematical connections or observations are recognized that link both the mathematics and the situation in the task.”

NCTM defines mathematical connections in Principals and Standards for School Mathematics as the ability to “recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics.” (64)

The Common Core State Standards for Mathematics (CCSSM) support the need for students to make mathematical connections in problem solving. Reference to this can be found in the following Standards for Mathematical Practice:

  • MP3: Construct viable arguments and critique the reasoning of others. “… They justify their conclusions, communicate them to others, and respond to the arguments of others.”
  • MP4: Model with mathematics. “… They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.”
  • MP6: Attend to precision. “Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose … They are careful about specifying the units of measure and labeling axes … They calculate accurately and efficiently express numerical answers with a degree of precision appropriate …”
  • MP7: Look for and make use of structure. “Mathematically proficient students look closely to discern a pattern or structure …”
  • MP8: Look for and express regularity in repeated reasoning. “… They continually evaluate the reasonableness of their intermediate results.”

The CCSSM also state, “The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word ‘understand’ are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations …” (Common Core Standards Initiative, 2015)

When students apply the criteria of the Exemplars rubric, they understand that their solution is more than just stating an answer. Part of that solution is taking time to reflect on their work and make a mathematical connection to share.

What Can Teachers Do to Help Students Make Mathematically Relevant Connections?

When students begin to explore mathematical connections, teachers should take the lead by providing formative assessment tasks that introduce new learning opportunities and provide practice, so they may become independent problem solvers. As part of this process, teachers will want to focus on five key areas to help students develop an understanding of mathematical connections.

(1) Develop students’ abilities to use multiple strategies or representations to show their mathematical thinking and support that their answers are correct. When students demonstrate an additional or new strategy or representation in solving a problem, a mathematical connection is made. The Common Core includes a variety of representations students can apply to solve a problem and justify their thinking. Examples include manipulatives, models, five and ten frames, diagrams, keys, number lines, tally charts, tables, charts, arrays, picture graphs, bar graphs, linear graphs, graphs with coordinates, area/visual models, set models, linear models and line plots. By practicing these different approaches, students will begin to create new strategies and representations that are accurate and appropriate to their grade level. This in turn opens the door for them to use a second or even third representation to show their thinking in a new way or to justify and support that their answer(s) is correct.

Using formative problem-solving tasks to introduce and practice new strategies and representations is part of the problem-solving process. Teachers should provide formal instruction so that students may grow to independently determine and construct strategies or representations that match the task they are given. An example of this can be seen in the primary grades when many teachers introduce representations in the following order: manipulative/model, to diagram (including a key when students are ready), to five/ten frames, to tally charts, to tables, to number lines. This order allows students to move from the most concrete to the more abstract representations.

(2) Encourage students to continue their representations. Mathematical connections may be made when students continue a representation beyond the correct answer. Examples of this can be seen when a table or linear graph is continued from seven days to 14 days or when two more cats are added to a diagram of 10 cats to discover how many total ears a dozen cats would have. Another example includes adding supplemental information to a chart such as a column for decimals in a table that already has a column indicating the fractional data. In this case, the student extends his or her thinking to incorporate other mathematics to solve the task. It is important to note that connections must be relevant to the task at hand. In order to meet the standard, a connection must link the math in the task to the situation in the task.

(3) Explore the rich formal language of mathematics. Mathematical connections may be made as students begin to use the formal language of mathematics and its connection to their representations, calculations and solutions. Mathematical connections can be seen in the following examples: two books is called a pair; 12 papers is a dozen, the pattern is a multiple of 10; 13 is a prime number so 13 balls can’t be equally placed in two buckets; and the triangle formed is isosceles. The input and output on a table can also help students generalize a rule with defined variables. Students will quickly learn that making connections promotes math communication (formal terms and symbols) and that using math communication promotes connections. Again, these connections must link the math in the task to the situation that has been presented.

(4) Incorporate inquiry into the problem-solving process. Asking students to clarify, explain, support a part of their solution to a math partner, the whole class, or a teacher, not only helps develop independent problem solvers but also leads to more math connections. In your discussions, use verbs from Depth of Knowledge 2 (identify, interpret, state important information/cues, compare, relate, make an observation, show) and from Depth of Knowledge 3 (construct, formulate, verify, explain math phenomena, hypothesize, differentiate, revise). By asking students questions that provide them the opportunity to show and share what they know, connections become a natural part of their solutions.

Instead of asking, “Do you see a pattern in your table?” say, “Did you notice anything about the numbers in each column in your table?” Try asking a primary student, “I know you have a cat. Would you like your cat to join the cats in your problem?” “What new numbers are you using?” “I heard you tell Maria that all the numbers in your second column are even. Can you help me understand why they are all even numbers?” Every time a student provides you with a correct answer to your or another student’s inquiry, stop and say, “Thank you for explaining/showing/sharing your thinking. You just made a mathematical connection about your problem.” If you hear a student make a mathematical connection outside of class, stop and comment, “You just made a math connection!” Some examples of these student connections may include, “Look, we are lined up as girl, boy, girl, boy, girl, boy for lunch. That is a pattern,” “In four more days it is my birthday,” “Art class is in 15 minutes because we always go to art at 10 o’clock,” “We can have an equal number of kids at each table because four times six equals 24,” “My dad says we have to drive 45 miles per hour because that is the speed limit, so I think I can write each student as ‘per student’” or “I think I can state all the decimals on my table as fractions.”

(5) Encourage self- and peer-assessment opportunities in your classroom. Encourage students to self-assess their problem-solving solutions either independently, with a math partner or with the support of their teacher. The more opportunity students have to use the criteria of the Exemplars assessment rubric to evaluate their work, the more independent they become in forming their solutions, which will include making mathematically relevant connections.

Exploring Authentic Examples of Mathematical Connections

In the next blog post of this series, we’ll look at a problem-solving task and student solution from Grade 1 to observe how mathematical connections have been effectively incorporated. We’ll also explore the type of support a teacher may provide during this learning time as well as the type of support students may give each other. (Solutions from Grade 3 and Grade 5 will follow in subsequent posts of this series.)


A Problem-Solving Lab to Support the Math Practices

Monday, October 31st, 2016

Written By: Donna Krachenfels & Debra Sander, Teachers from PS 54

The school administrators at PS 54 had a vision to create a math laboratory based on the eight Standards of Mathematical Practice. The idea was to create a setting in which students could focus on multi-step problem solving.

The Exemplars program has given our students many opportunities to build and strengthen their problem-solving skills. Students were also able to strengthen their close reading skills as they reread problems multiple times to identify and think about the relevant information necessary to find a solution. Collaboration allowed students to become confident in their problem-solving skills and increased their abilities to construct viable arguments as they defended their solutions and critiqued the solutions of their classmates. Students were not afraid to take risks as they tried different representations and strategies to solve problems. As a result of the Exemplars math program, our students became more confident and more independent problem solvers.

The math laboratory is in its second year at PS 54. Last year, our data saw increased math scores for the classes that participated in the problem-solving lab. This year, the trend continued and all general education students passed the state math exam.

Special thanks goes to Exemplars professional development consultant Deb Armitage for all of her help and support. She is a true math educator!

7 Things I’ve Learned on My Journey to Implementing Problem Solving in the Classroom

Monday, October 3rd, 2016

Written By: Suzanne Hood, Instructional Coach, Georgia

I’ve always believed in the power of students to use their own childlike curiosity to problem solve. These problem-solving experiences happen for our students naturally, through the math they use in cooking, playing games and playing with toys, among other things. Problem solving is a life-long skill all mathematicians use. The true power of a mathematician is the ability to see math in all situations and solve problems using a toolbox of proven strategies.

While I believe that students are innate problem solvers, I also believe that learned algorithmic thinking corrupts a child’s natural ability to problem solve and discourages perseverance. Although I have met many teachers who share my belief that problem solving should be the focus of the math, many struggle to create this culture in their classroom.

This is becoming more apparent—and the stakes of ignoring problem solving much higher—as we approach testing season. The classrooms that will likely fall behind in this new era are those who insist on teaching math through algorithmic thinking. Conversely, I am convinced that teachers who use problem solving to teach math, supported by materials like Exemplars, will have students who score proficiently on the state assessment and are more prepared for success beyond the classroom.

So how can teachers help their classrooms make this critical transition to problem solving? My personal story of transformation, which began after participating in one of Exemplars’ Summer Institutes, offers a path forward. This was when I realized two important things: first, I needed to work on my own personal proficiency in teaching problem solving. And second, I wasn’t alone; veteran teachers confessed their frustration in teaching problem solving, and many admitted their backgrounds did not include support in how to instruct students through the problem-solving process. Here are seven things I’ve learned on my journey to becoming an educator fully committed to teaching mathematics through a problem-solving approach.

1. Nurture a community of trust.

Based on my experience as a Mathematical Instructional Coach in Georgia, I believe it is essential to nurture relationships and establish a community of trust between teachers, so that discussions are authentic and all voices are included. Trust is a prerequisite for being able to assess the strengths, weaknesses and gaps of teacher readiness in the classroom. Only when teachers feel they are in an environment where they can share their knowledge, their doubts and their pedagogical weaknesses, will they be able to feel comfortable.

2. Establish a baseline of teacher readiness.

Evaluating teacher readiness and needs and getting them on the same page is an important first step. How can you get teacher teams to have collegial conversations when everyone has a totally different math background? Do all teachers even want a problem-solving classroom? Do they know what that means? Asking these questions can be illuminating, albeit tough. As such, using universally agreed-upon protocols such as those from the National School Reform Facility can establish a baseline to work from, encourage collaboration, and support an atmosphere of trust.

3. Assess student work so you can see where the gaps are.

One way to assess teacher acuity and readiness in teaching problem solving is by assessing student work using an Exemplars task. Here’s how it worked for me: At the first Professional Learning session, I asked teachers to bring classroom samples from their most recent classroom Exemplars task. As a community, we agreed to facilitate the discussion with the protocol Atlas – Learning From Student Work. As I observed teachers at the meeting, I noticed that while some teachers were proud to display their samples, others pretended to forget their samples or chose to stick their student work in their tote bag. As we used the Exemplars standards-based rubric to score our samples, it became clear that our understanding of the skills needed to meet the standards did not align. The journey began; teachers began to talk about problem solving.

4. As a team, align your mathematical beliefs towards problem solving.

When we began, we knew we shared some foundational mathematical beliefs. We also knew that we needed to solidify a shared understanding of how a mathematics culture transfers knowledge from the teacher to the student. We used the Math Framework (a document listing all the mathematical beliefs of the faculty) as a tool to target instructional strengths and weaknesses. As a team, we revised the document to build cohesion and a shared understanding of our beliefs. Next, I had the team read a book rooted in Vygotsky’s constructivist theory to increase our group’s understanding of the problem-solving trajectory. Because we had been working hard to build an atmosphere of trust, teachers felt safe sharing their struggles and personal hardships with teaching problem solving. We discovered that we shared similar experiences, and that we all believed our students would be successful at any problem if we just taught them the necessary skill set. The student samples, however, told a different story.

5. Create simple tools to help teachers and students internalize the standards and assess their progress.

At our next meeting, we reviewed Exemplars student work samples and discovered a misconception: we thought we knew how to teach problem solving, but we were actually teaching skills in isolation. Why? Quite simply, it turns out that many teachers lacked background knowledge about the Standards of Problem Solving. To facilitate the understanding of the standards, I created posters with clear icons for each standard. These anchor charts would support teachers and students. It worked. Now, teachers could explain each standard. Each classroom in our building displayed the posters. It was a great reference for both students and teachers. We made a replica of the posters into a small book that students put in folders for their own reference. Students used the folders as portfolios to track their problem-solving progress, and created data notebooks to reflect on their growth and set goals for their next Exemplars task. Using data notebooks empowered kids to self-reflect on their own progress.

6. Hold individual meetings with students to track progress and set goals.

Currently, I am encouraging teachers to hold one-on-one Exemplars conferences with their students. Individual conferences support differentiated instruction, meet students where they are, and set goals for the next problem-solving task. Although this approach makes some teachers uneasy at first, they become more confident over time. Allowing other teachers or coaches to observe and co-teach the process can lead to greater transparency and effect change in teacher practice.

7. You may not get the teacher of the year award, but you’ll still be changing students’ lives.

At the beginning of my career, I thought Oprah would call me to announce my Disney Teacher of the Year Award. While this hasn’t happened yet, I do have countless memories of the sparkle in a child’s eye when he or she announces, “I get it!” I believe I have the responsibility to show up every day prepared to change the lives of children and equip them with the skills to be life-long mathematicians. Exemplars provides the problem-solving tools necessary to guide teaching and build capacity for each child’s mathematical journey.

Great Minds Don’t Always Think Alike

Friday, January 15th, 2016

Written By: Jaclyn Mazzone, Fifth Grade Special Education Teacher at P.S. 94 in Brooklyn, NY

Working in special education, I help students with special needs as well as other students who struggle with math. One of the most beneficial features of Exemplars is the ability to differentiate easily for struggling learners. Some need just a little extra support through small group instruction. The more accessible version of Exemplars tasks is perfect for them!

The more accessible versions present the same problem-solving elements as the grade-level task and the more challenging task, but in a simplified form: they’re less wordy, involve simpler numbers, and require fewer steps to complete. At the same time, they’re still demanding enough that they challenge each student to think more deeply to determine what the question is asking, decide what information they need to know, choose a strategy to solve the problem, complete the mathematical computations, and show their work.

Repetition Is Key

Specifically in my class, I’ve noticed repetition is key. My students complete Exemplars at least once a week, so they’ve been able to build a routine. A colored version of the Problem Solving Procedure is provided for each student. It has become automatic for them to carry out the first two steps of the Procedure independently. They start out by reading the problem twice; the first time they just read it, and the second time they annotate, highlight/locate important information in the problem, and underline the question. Once they’ve determined what the problem is asking them to do, they then move on to writing their “I have …” and “I will …” statements independently. Next, as a group we read the entire problem together — their third time to read it — ensuring that the important information has been located, underlined, or highlighted, and that they understand what the problem is asking them to figure out. Many of my students are English Language Learners so language is also sometimes an issue for them. Therefore, we carefully break apart the math tasks sentence by sentence. We address any unknown vocabulary and I encourage them to use their prior knowledge to connect to the problem.

Small-Group Discussion and Scaffolding

Then, the students engage in a small-group discussion and discuss what the problem is about. They brainstorm different strategies that could be used to solve the problem and discuss why those strategies would work. Then I step in to scaffold and guide them to set up their chosen strategy correctly, whether it be a table, number line, diagram etc. Once I’ve supported them in setting up their chosen strategy and answered any questions they might have, they work with their small group to continue solving the problem. The students know to show all of their mathematical thinking on paper — this expectation has been instilled since day one of their work with Exemplars. As they work, I circulate around the room to provide one-on-one support where necessary.

After I allow approximately 20-25 minutes for them to complete the problem with their peers, we come back together as a group and discuss the different ways we solved it. The students then move on to the next step of the Problem Solving Procedure by making mathematical connections. I encourage them to try to extend the problem, write what they notice in their work, or try to find a pattern. Because connections are still rather challenging for them, I link the task to our current unit of study or math topics we have previously worked with to help them make those connections.

The Importance of Tools

Tools are important resource for my students. Working with learners in a small group, I am able to accommodate their different learning styles. Depending on the task, I use manipulatives such as connecting cubes, fraction bars, unit cubes, or sometimes even a basic multiplication table chart to further support their needs.

Peer Assessment

My students also peer-assess each other’s work. They swap papers with a partner and use the Exemplars student-friendly rubric to score their partner’s work. Since the student-friendly rubric addresses the same criteria as the teacher rubric in a simpler form, this practice has made them very familiar with the four levels of achievement and what’s required to become a Practitioner (meet the standard).

Peer assessment has also built up their self-confidence. As they assess each other’s work, I ask them to find a “Glow” statement — something their classmate did well — and a “Grow” statement — an area for improvement. In doing this, they notice that other students have challenges, too, and might even make the same mistakes as they do. Peer assessment has not always been an easy task for them, but through repetition and teacher feedback, the students have picked up on sophisticated math language to use when providing helpful Glow/Grow comments to peers.

An example of a student’s peer-assessment comments using the “Glow and Grow” strategy. 

Exemplars has been a very beneficial tool in my classroom to reinforce the mathematical concepts we learn throughout the year. Through careful scaffolding and support, my students have learned to persevere when problem solving, show all of their mathematical thinking, use different strategies to achieve the same answer, and provide peer feedback. My students consistently aim to achieve a level of Practitioner and some even aim for Expert. Exemplars has become a weekly activity in my classroom. The students truly find it enjoyable!

Jaclyn Mazzone’s Biography

Jaclyn Mazzone is a fifth-grade special education teacher at P.S. 94 in Sunset Park, Brooklyn. She has a master’s degree in special education from Touro College and a bachelor’s degree in childhood education from the College of Staten Island.


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