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Supporting the TEKS Mathematical Processes with Exemplars Performance Tasks and Rubric at the Second Grade Level

Friday, May 13th, 2016

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 3 of a six-part series that features a problem-solving task linked to a  Math TEKS Unit of Study and a student’s solution in grades K-5. Evidence of all seven Mathematical Process Standards will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress toward being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 3: Observations at the Second Grade Level

The third anchor paper and set of assessment rationales we’ll review in this series is taken from a second grade student’s solution for the task, “A New Hamster Toy.” This is one of a number of Exemplars tasks aligned to the Problem Solving with Money Unit.

“A New Hamster Toy” would be used toward the end of the learning time allocated to this standard. This task provides second grade students with an opportunity to apply different strategies to determine if there is enough money to buy a hamster toy for $2.25. The task does not provide the symbolic notation for $2.25, $0.05, or 5¢. Students need to bring this understanding to their solutions, which provide the teacher with an opportunity to assess if they can correctly notate money. This task also provides students with the opportunity to use comparison and to solve a problem that includes two steps. Students need to determine the popcorn bag sales for one day, determine the total sales for five days and compare that total to $2.25.

When forming their solutions, students have a variety of strategies to consider. Some examples include using actual money to model the bag sales and total bag sales, diagramming the bags and/or money earned, creating a table to indicate popcorn sales for one or five days, using a printed number line, creating a number line or a tally chart.

Second Grade Task: A New Hamster Toy

Some students want to earn two dollars and twenty-five cents to buy a toy for their class hamster. The students decide to sell small bags of popcorn at snack time for five cents each. The students sell ten bags every day for five days. Do the students earn enough money to buy a toy for their class hamster? Show all your mathematical thinking.

Math TEKS Alignments

Exemplars Problem Solving with Money Unit

The Problem Solving with Money Unit involves knowing the relationship between U.S. coins, how to use skip counting and other methods to find the value of a collection of coins, and answer questions such as:

  • If you have saved a given amount of money and you know a toy costs a certain amount, how can you figure out how much more you need to save to buy the toy?
  • If you have a collection of coins worth 30 cents and put a quarter in your bank every week, how much money will you have in three months?

The standards covered in this Unit include:

2.5 Number & Operations:

  • 2.5A determine the value of a collection of coins up to one dollar
  • 2.5B use the cent symbol, dollar sign, and the decimal point to name the value of a collection of coins.
Mathematical Process Standards
  • 2.1A apply mathematics to problems arising in everyday life, society, and the workplace;
  • 2.1B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
  • 2.1C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
  • 2.1D communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;
  • 2.1E create and use representations to organize, record, and communicate mathematical ideas;
  • 2.1F analyze mathematical relationships to connect and communicate mathematical ideas; and
  • 2.1G display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Supporting the TEKS Mathematical Processes with Exemplars Performance Tasks and Rubric at the First Grade Level

Thursday, April 7th, 2016

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 2 of a six-part series that features a problem-solving task linked to a Unit of Study for the Math TEKS and a student’s solution in grades K–5. Evidence of all seven Mathematical Process Standards will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress towards being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Processes is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the Math TEKS can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Processes are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in-hand while reviewing a piece of student work.

Blog 2: Observations at the First Grade Level

The second anchor paper and set of assessment rationales we’ll review in this series is taken from a first grade student’s solution for the task “A Birdbath.” In this piece, you’ll notice that the teacher has “scribed” the student’s oral explanation. This practice was also used with the Kindergarten task that was published in the first blog. Scribing allows teachers to fully capture the mathematical reasoning of early writers.

“A Birdbath” is one of a number of tasks aligned to the Strategies for Addition and Subtraction Unit designed by Exemplars for the new Math TEKS. This task could be used toward the end of the learning time allocated to this Unit. “A Birdbath” provides first grade students with an opportunity to apply different strategies to find the sum of addends six and 14 by decomposing six into five and one and decomposing 14 into 10 and four, or by finding the sum of six and four and adding that sum to 10. The student can use counters, ten frames, a Rekenrek, number lines or a tally chart to support her/his numerical thinking.

First Grade Task: A Birdbath

Leah counts the birds that came to her birdbath. In the morning, Leah counts six birds that came to her birdbath. In the afternoon, Leah counts fourteen birds that came to her birdbath. Leah says nineteen birds came to her birdbath. Is Leah correct? Show all of your mathematical thinking.

Math TEKS Alignment:

Exemplars Strategies for Addition & Subtraction Unit

The Strategies for Addition and Subtraction Unit involves understanding the processes of addition and subtraction in order to solve problems and answer questions such as:

  • If we know all of the parts, how can we find the whole?
  • If we know the whole and one of the parts, how can we find the missing part?
  • Given an equation, can you create an addition or subtraction situation to match it? How can you prove it matches the equation?

The standards covered in this Unit include:

1.3 Number & Operations:

  • 1.3A use concrete and pictorial models to determine the sum of a multiple of 10 and a one-digit number in problems up to 99.
  • 1.3D apply basic fact strategies to add and subtract within 20, including making 10 and decomposing a number leading to a 10.
  • 1.3E explain strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences.
  • 1.3F generate and solve problem situations when given a number sentence involving addition or subtraction of numbers within 20.

1.5 Algebraic Reasoning:

  • 1.5D represent word problems involving addition and subtraction of whole numbers up to 20 using concrete and pictorial models and number sentences.
  • 1.5G apply properties of operations to add and subtract two or three numbers such as if 2 + 3 =5 is known, then 3 + 2 = 5.
Mathematical Process Standards:
  • 1.1A Apply mathematics to problems arising in everyday life, society, and the workplace;
  • 1.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
  • 1.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
  • 1.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
  • 1.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

 

Supporting the TEKS Mathematical Processes with Exemplars Performance Tasks and Rubric at the Kindergarten Level

Tuesday, February 9th, 2016

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 1 of a six-part series that features a problem-solving task linked to a Unit of Study for the Math TEKS and a student’s solution in grades K–5. Evidence of all seven Mathematical Process Standards will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress towards being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Processes is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with scoring rationales that demonstrate the alignment between the Exemplars assessment rubric and the Math TEKS can be insightful for educators. Anchor papers and scoring rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Processes are embedded in the scoring rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 1: Observations at the Kindergarten Level

The first anchor paper and set of assessment rationales we’ll review in this series is taken from a Kindergarten student’s solution for the task “Boots.” You will notice that the teacher has “scribed” the student’s oral explanation. This method allows teachers to fully capture the mathematical reasoning of early writers.

“Boots” is one of a number of Exemplars tasks aligned to the Counting and Cardinality Unit designed by Exemplars for the new Math TEKS. This task could be used toward the end of the learning time allocated to this Unit. Prior to “Boots” being given, students have already completed a number of tasks with questions that state, “How many ears?”, “How many shoes?”, “How many balloons?”, etc. “Boots” gives students an opportunity to bring a stronger understanding of the concept how many to their solutions.

Kindergarten Task: Boots

Five students wear boots to go outside for recess. When the students come in from recess they must put all boots on a rubber mat to dry. The teacher counts seven boots on the mat. The teacher thinks some boots are missing. Is the teacher correct? Show and tell how you know.

Task Alignments

The Counting and Cardinality Unit involves understanding numbers and how they are used to name quantities and to answer questions, such as:

  • How many balls is the clown juggling?
  • Do you have enough cups for each member of your group to have one?

The TEKS standards covered in this Unit include K.2 Numbers and Operations:

  • K.2A count forward and backward to at least 20 with and without objects.
  • K.2B read, write, and represent whole numbers from 0 to at least 20 with and without objects or pictures.
  • K.2C count a set of objects up to at least 20 and demonstrate that the last number said tells the number of objects in the set regardless of their arrangement or order.
  • K.2D recognize instantly the quantity of a small group of objects in organized and random arrangements.

Mathematical Process Standards: K.1A, K.1B, K.1C, K.1D, K.1E and K.1G

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.

 

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

Tuesday, October 27th, 2015

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.

Task: A New Aquarium

(More Accessible Version)

Joseph has a new rectangular aquarium. The aquarium has a length of four feet, a width of two feet, and a height of two feet. What is the volume of Joseph’s new aquarium? An aquarium holds one inch in length of fish for each twelve square inches of the area of the base of the aquarium. Joseph can buy fish in two different sizes—about three inches in length or about five inches in length. About how many three-inch fish can Joseph put in the new aquarium? About how many five-inch fish can Joseph put in the new aquarium? Show all your mathematical thinking. 

Danielle’s Biography

Danielle Descarfino is a fifth grade teacher at P.S. 94 in Sunset Park, Brooklyn. She graduated from Fordham University with a Masters of Science in Teaching English to Speakers of Other Languages. Danielle grew up in Tappan, New York, and currently lives in Brooklyn. She was inspired to become a teacher after spending time as a volunteer teaching English at an orphanage and community center in Salvador, Brazil.

Preparing for the New Math TEKS: Using Rubrics to Guide Teachers and Students

Tuesday, October 6th, 2015

By: Ross Brewer, Ph.D., Exemplars President

As you begin preparing your staff to focus on the new math TEKS this year, rubrics should play a key role in terms of helping your teachers and students achieve success with the new standards.

 What are rubrics?

A rubric is a guide used for assessing student work. It consists of criteria that describe what is being assessed as well as different levels of performance.

Rubrics do three things:

  1. The criteria in a rubric tell us what is considered important enough to assess.
  2. The levels of performance in a rubric allow us to determine work that meets the standard and that which does not.
  3. The levels of performance in a rubric also allow us to distinguish between different levels of student achievement among the set criteria.

Why should teachers use them?

The assessment shifts in the new math TEKS pose challenges for many students. The use of rubrics allow teachers to more easily identify these areas and address them.

For Consistency. Rubrics help teachers consistently assess students from problem to problem and with other teachers through a common lens. As a result, both teachers and students have a much better sense of where students stand with regard to meeting the standards.

 To Guide Instruction. Because rubrics focus on different dimensions of performance, teachers gain important, more precise information about how they need to adjust their teaching and learning activities to close the gap between a student’s performance and meeting the standard.

To Guide Feedback. Similarly, the criteria of the rubric guides teachers in the kind of feedback they offer students in order to help them improve performance. Here are four guiding questions that teachers can use as part of this process:

  • What do we know the student knows?
  • What are they ready to learn?
  • What do they need to practice?
  • What do they need to be retaught?

How do students benefit?

Rubrics provide students with important information about what is expected and what kind of work meets the standard. Rubrics allow students to self-assess as they work on and complete problems. Meeting the standard becomes a process in which students can understand where they have been, where they are now and where they need to go. A rubric is a guide for this journey rather than a blind walk through an assessment maze.

Important research shows that teaching students to be strong self-assessors and peer-assessors are among the most effective educational interventions that teachers can take. If students know what is expected and how to assess their effort as they complete their work, they will perform at much higher levels than students who do not have this knowledge. Similarly, if students assess one another’s work they learn from each other as they describe and discuss their solutions. Research indicates that lower performing students benefit the most from these strategies.

Rubrics to Support the New Math TEKS.

Exemplars assessment rubric criteria reflect the TEKS Mathematical Process Standards and parallel the NCTM Process Standards. Exemplars rubric consists of four performance levels (Novice, Apprentice, Practitioner (meets standard) and Expert) and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation).

Our rubrics are a free resource. To help teachers see the connection between our assessment rubric and the TEKS Mathematical Process Standards, we have developed the following document: Math Exemplars: A Perfect Complement for the TEKS Mathematical Process Standards aligns each of the Process Standards to the corresponding sections of the Exemplars assessment rubric.

It’s never too young to start.

Students can begin to self-assess in Kindergarten. At Exemplars, we encourage younger students to start by using a simple thumbs up, thumbs sideways, thumbs down assessment as seen in the video at the bottom of the page.

Our most popular student rubric is the Exemplars Jigsaw Rubric. This rubric has visual and  verbal descriptions of each criterion in the Exemplars Standard Rubric along with the four levels of performance. Using this rubric, students are able to:

  • Self-monitor.
  • Self-correct.
  • Use feedback to guide their learning process.

How to introduce rubrics into the classroom.

In order for students to fully understand the rubric that is being used to assess their performance, they need to be introduced to the general concept first. Teachers often begin this process by developing rubrics with students that do not address a specific content area. Instead, they create rubrics around classroom management, playground behavior, homework, lunchroom behavior, following criteria with a substitute teacher, etc. For specific tips and examples, click here.

After building a number of rubrics with students, a teacher can introduce the Exemplars assessment rubric. To do this effectively, we suggest that teachers discuss the various criteria and levels of performance with their class. Once this has been done,  a piece of student work can be put on an overhead. Then, using our assessment rubric, ask students to assess it. Let them discuss any difference in opinion so they may better understand each criterion and the four performance levels. Going through this process helps students develop a solid understanding of what an assessment guide is and allows them to focus on the set criteria and performance levels.

Deidre Greer, professor at Columbus State University, works with students at a Title I elementary school in Georgia. Greer states that in her experience,

The Exemplars tasks have proven to be engaging for our Title I students. Use of the student-scoring rubric helps students understand exactly what is expected of them as they solve problems. This knowledge then carries over to other mathematics tasks.

At Exemplars, we believe that rubrics are an effective tool for teachers and students alike. In order to be successful with the learning expectations set forth by the new math TEKS, it is important for students to understand what is required of them and for teachers to be on the same “assessment” page. Rubrics can help.

To learn more about Exemplars rubrics and to view additional samples, click here.

Understanding Mathematical Connections at the Fifth Grade Level

Tuesday, August 18th, 2015

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

This blog is the final post 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.

In the first blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of today’s mathematics curriculum. We also reviewed the Exemplars rubric and offered strategies for teachers to try in their classroom to help their students become more proficient in making mathematical connections:

As part of the other blogs in this series, we reviewed solutions from a first grade student and third grade student to observe how they successfully included mathematical connections as well as the other problem-solving criteria of the Exemplars rubric in their work.

Blog 4: Mathematical Connections at the Fifth Grade Level

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 at the Third Grade Level

Wednesday, August 5th, 2015

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

This blog is Part 3 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.

In the first blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of mathematics curriculum. We also reviewed the Exemplars rubric and offered strategies for teachers to try in their classrooms to help their students become more proficient in making mathematical connections.

As part of the second blog, we reviewed a first grade solution and how this student successfully included mathematical connections as well as the other problem-solving criteria of the Exemplars rubric in his or her work.

Blog 3: Mathematical Connections at the Third Grade Level

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 First Grade Level

Monday, July 20th, 2015

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

Summer Blog Series Overview:

This blog represents Part 2 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.

In the previous blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of mathematics curriculum.

We also reviewed the Exemplars rubric and offered the following strategies for teachers to try in their classroom to help their students become more proficient in making mathematical connections:

  1. Develop students’ abilities to use multiple strategies or representations to show their mathematical thinking and support that their answers are correct.
  2. Encourage students to continue their representations.
  3. Explore the rich formal language of mathematics.
  4. Incorporate inquiry into the problem-solving process.
  5. Encourage self- and peer-assessment opportunities in your classroom.

Blog 2: Mathematical Connections at the First Grade Level

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

Wednesday, June 24th, 2015

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. As part of this series, student work will be examined at Grades 1, 3 and 5.

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 Grades 3 and 5 will follow in subsequent posts of this series.)

 

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