Standards-based assessment and Instruction

# Archive for March, 2018

## 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?

The four-part blog series below 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. As part of this series, student work will be examined at Grades 1, 3 and 5.

## 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.

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

## Dynamic Math Learning at The Phoenix School

Thursday, March 29th, 2018

Written By: Barbara McFall, Head of School

Extending Exemplars problem, "Park Play Area," to an irregular quadrilateral that will become a plan for one of our community’s potential park spaces … real-life math building on an Exemplars problem!

Fall is the time of year when my students begin pestering me for “real” word problems. They love solving problems that include the names of their classmates and using a collaborative experience in which they can bounce ideas off one another as they work through more complex problems. My go-to resource for problems that create mathematical thinkers of my students is Exemplars.

At The Phoenix School, for many years (first in paper form, now digital) Exemplars has been a vital component of our math program. No more memorization and regurgitation that flies in – and out! — of kids’ brains. Exemplars work solidifies concepts, which are the foundation of mathematical learning, and allows for multiple solutions as well as encouraging creative thinking. It has the flexibility that enables us to extend concepts on which we have been working … and it is so well organized that it is easy to determine which sets of problems we need to best showcase particular math concepts.

Working Exemplars problems requires students to do much more thinking than in traditional mathematical learning formats. It teaches them to analyze information and focus on what is most important in getting to the solution. Exemplars problems also encourage students to use previous knowledge to work through solutions. When students discover how mathematical systems work, they can solve almost anything.

I choose a variety of math problems at all levels to begin. Varying math topics and levels help students learn to be flexible in their thinking and to transfer learning from one topic to another with ease. To create excitement, I change the names of the people in the problems to the names of students in my class. This leads to wonderful interactions which, in turn, leads to sharing ideas and helping one another. Each student gets a folder of problems. I put them on label paper so they can stick them in their math journals, one problem at a time. I don’t specify in which order students should work the problems – leaving it open-ended makes the math more engaging, encouraging students to ask questions of each other.

Some students like to work alone, while others prefer to connect with partners to work on problems they have in common. Using Exemplars legitimizes working with partners. It teaches students to share strategies, talk to others, and collaborate. Traditionally, teachers have considered the sharing of math work to be “cheating,” but I have found the learning to be so much better for all when my students share solutions. Math at The Phoenix School often becomes a social event as we collaborate on a particularly complex challenge, especially when we turn to using materials to represent a solution. Everyone’s ideas are considered and sometimes debated with passion!

I discuss different approaches to solving problems with students and expect them to document their work using charts/graphs/tables, words, numbers, equations, pictures/diagrams, and personal reflection. This helps students see math in multiple ways, which increases the learning. Requiring visual representations helps students organize their thinking so that, as I say to them, “Your paper talks to me. I can tell exactly what you were thinking as you worked through the problem.” For my older students, I always use the more challenging form of each problem. It is easy to extend these problems into algebra since so many lend themselves to finding the pattern and taking it to the algebraic equation.

From our youngest students to our oldest, we find that Exemplars problems engage, challenge, and create mathematical thinkers at every level. Our math program is truly strengthened by the Exemplars component.

## 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?

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?

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.

(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.)

## Plainville Community Schools and Exemplars Math

Wednesday, March 28th, 2018

Written By: Phil Sanders, Math/Science Instructional Leader

Plainville Schools has had a long-standing relationship with Exemplars. We have been using their tasks for more than ten years. It wasn’t until two years ago, however, that we really decided to put the Exemplars problem-solving methods to full use. With the Smarter Balanced Assessment, we noticed a lack of growth in our students on the Performance Task. This year, we decided to address this issue. Exemplars was a perfect fit.

Our district started by sending our Math Leadership team to an Exemplars Institute with Deb Armitage. This session gave the team a comprehensive understanding of how to incorporate Exemplars problem-solving methods successfully into the classroom. We learned that we could use the problems in several ways: Teachers could use them as instructional material for students in small groups as well as for whole-class instruction. These problems could also be used to provide formative information to guide the instruction and understanding of how to solve this type of performance task. Most of our teachers, for example, use the Exemplars tasks in the middle of a unit to provide information and then at the end as a summative assessment. The information we gather helps to differentiate our instruction and provides us with intervention grouping for both higher and lower levels of understanding. Our teachers appreciate being able to pick problems that are aligned to the Common Core Standards; students like the problems because they are challenging and allow them to showcase their mathematical thinking and understanding. After a problem has been completed, we usually have a gallery walk, where the students get the opportunity to explain their thinking and use of models.

Exemplars materials were first introduced to teachers during our August professional development. We had teachers at each grade level solve a task and create anchor sets to be used for instruction. To introduce the problems to students, we made PowerPoint slideshows for many of the tasks. These slideshows had interesting pictures, which helped build students’ level of understanding. During the school year, students regularly asked if they were going to see a slideshow for math that day! As the school year progressed, we slowly reduced the reliance on the slideshows as students gained confidence in reading and in developing their understanding of the problems.

Our goal for the past year was to increase student understanding of performance tasks. We saw excellent improvement in students’ understanding of how to break down a problem and model their thinking. I can remember many blank papers in years past when we administered problem-solving tasks. This year we had not one single blank paper! All students were able to find an entry point to begin the task.

We attribute the success to several factors. We feel that grouping students in three different ways has removed the stigma of this type of math problem. Using the small group, paired and individual methods for students to solve tasks provides multiple ways to approach students’ learning needs. We also used the Exemplars anchor papers to show students how problems can be solved in different ways. The anchor papers helped teachers gain a better understanding of the material as well. This year, we dedicated our Math PLC to Exemplars and problem solving. Teachers from each grade level across the district gathered once a month to look at student work and discuss their successes and ways to improve instruction.

We have found Exemplars to be very helpful in meeting our goals.

## Exemplars in the Classroom: “They Want to Become Experts.”

Tuesday, March 27th, 2018

Written By: Danielle Descarfino, Fifth Grade Teacher at P.S. 94 in Brooklyn

#### Getting Started

From the beginning of the school year, I used Exemplars problem-solving tasks regularly to create routines that have helped my fifth grade students grow and succeed. Following the Problem-Solving Procedure is a central part of this.

Although each task is different, the procedure helps kids internalize a framework for approaching a problem. I provided each student with his or her own color copy (in a sheet protector for safe keeping.) Each time we begin an Exemplars task, the students take out their Problem-Solving Procedures and refer to it. I also have a poster-sized version prominently displayed in the classroom, which I hold up and point to while guiding and facilitating tasks.

#### Building Background Knowledge

My class is made up of English Language Learners and former English Language Learners, so I anticipate that reading and understanding the problem may be especially challenging for them. We read the problem together, I ask questions to activate their background knowledge, and I often provide pictures that help them visualize the problem.

For example, we recently completed “A New Aquarium,” a 5.MD.C.5a task involving volume. We had been working on this math concept for only a few days and this was our first volume Exemplars task. Before reading the problem, I displayed a photo of an aquarium on the Smart Board and discussed the following questions with the class:

• What is an aquarium?
• What type of solid is this aquarium?
• How could you figure out how much space this aquarium takes up? What steps would you take?

Although many students initially were not familiar with the word “aquarium,” after this discussion, they understood that an aquarium is a fish tank and a rectangular prism, which meant that we would be calculating its volume to find out how much space it takes up. Using visual aids and background questions to ensure that students understand the situation in the problem has been very helpful when completing Exemplars with English Language Learners.

#### Differentiation

We always utilize the differentiated Exemplars tasks. Students are aware of which problem-solving group they are in and know where to sit when it is time for an Exemplars task. One group gets the More Accessible Version; they are guided through the problem as they work with the Special Education teacher at a kidney-shaped table. The other two groups receive the Grade Level and More Challenging versions and sit with their groups in desk clusters, like a team of problem solvers.

For the Grade Level and More Challenging groups, we discuss background information, read the problem out loud, annotate it, and write our “I have to find …” statements. Then the students go on to work with their groups to complete the task while the teacher takes on the role of a facilitator, conferring with groups. Students share ideas, address misconceptions, and explain their mathematical reasoning to one another as they solve.

#### Motivating Students

I love hanging Exemplars tasks on bulletin boards. I think it’s useful for students to look at the page and see all of the different ways their classmates organize and express their mathematical thinking through equations, representations, and writing.

From day one, I have made it clear that it is expected that their finished work clearly communicate their problem-solving steps to the reader. Not only should the students make an effort to write neatly, but they should also organize their problem-solving steps on the page in a way that makes sense. Sometimes if a student is not showing all of their steps or it is unclear, I’ll say, “I am confused. When I look at your paper, I don’t understand the steps you took to solve the problem.” When the students have the understanding that a goal is to communicate their math thinking to a reader, it helps them create a higher-quality finished product.

Another great way to motivate students is through mathematical connections. I have given a strong emphasis to connections, as I initially noticed that once students solve the problem, they feel like they are done! This is not the case, because noticing mathematical connections, patterns, and alternate strategies really helps students understand mathematics on a deeper level and practice critical thinking skills.

To help them stretch their thinking, I discourage students from writing “boring” connections, like “This number is greater than that number” or “John ate the least amount of pizza.” Instead, I encourage them to use mathematical language, create a second representation, show steps to solving with alternate strategies, convert fractions/decimals/percents, or extend the problem by adding to the story in the original problem. Once they get the hang of it, they start being more creative, going above and beyond to make more complex math connections. During the volume unit, I taught students how to use grid paper to make scaled models of rectangular prisms. When completing these tasks, many students decided to build models to represent the rectangular prisms in the task and attach them to make 3-D Exemplars. They looked great, and the students loved making them!

#### Peer Assessing

At the beginning of the year, I explained each portion of the Exemplars rubric to the students. The rubrics are very student-friendly and I find that they inspire students to want to become Experts.

Each time I assess Exemplars, I use the rubric along with a sticky note full of feedback. The sticky note always contains one “Glow,” something the student did well, and one “Grow,” something the student could improve upon. At the beginning of the year, I let the students know that when they become more comfortable with Exemplars, they would learn how to peer assess. After a few months, I told the students that they were ready to peer assess one another’s work. They were so excited! This made them feel proud that they had reached a new level of expertise in problem solving and feel empowered that they were now trusted to assess a classmate’s work.

To peer assess, they do exactly as the teacher has done all year: complete the student rubric and use a sticky note to write “Glow and Grow” feedback. An example of this can be seen below. From the start, I was so impressed at how well the students were able to assess one another’s work with Exemplars. I found that regularly providing students with written feedback and referring to the rubric when expressing expectations is a great way to model peer-assessment. Furthermore, the experience of assessing Exemplars helps students get new ideas from their classmates and become more aware of how their own work will be graded.