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Why A Focused Mathematics Curriculum Matters and How Exemplars Can Help Texas Educators

Monday, November 3rd, 2014

Written By: Dinah Chancellor, Exemplars Math Consultant

Prior to 2006, many states—including Texas—had a math curriculum that was perceived to be “an inch deep and a mile wide.” Teachers were required to teach a large number of math skills that spiraled from grade to grade and seemed both disconnected and fragmented. When Texas’ own Cathy Seeley became President of the National Council of Teachers of Mathematics (NCTM), she determined that a more focused mathematics curriculum that was built around fewer “big ideas” would give students and teachers the luxury of time—time to plumb the depths of major math concepts, and time to form a foundation of connected mathematical understandings.

Therefore, in 2006 NCTM published the Curriculum Focal Points—A Quest for Coherence. The Texas response to the Curriculum Focal Points was the new state assessment program—STAAR—the State of Texas Assessment of Academic Readiness taken by students in grades 3-8. STAAR focuses on fewer skills at each grade level and it is expected that these skills will be taught at greater depth. When the new math TEKS were written, released in April 2012 and implemented in the fall of 2014, the writing teams focused on fewer skills at each grade level. Teachers are expected to address these skills and understandings by teaching rich lessons in which students make critical connections between foundational big ideas in mathematics. Because of the need to teach a focused mathematics curriculum, it does not make sense to teach each of the new math TEKS in isolation.

To assist Texas educators in achieving this goal, Exemplars latest K–5 product, Problem Solving for the TEKS, groups the individual math TEKS student expectations into rich Units of Study. Four or more instructional tasks/formative assessments and one or more summative assessment is provided to address the big mathematical ideas within each Unit. Tasks are meant to supplement a school or district’s existing curriculum. Teachers may choose to use all or only a few of the instructional tasks/formative assessments in a Unit. The summative assessments include anchor papers that exemplify each of the performance levels in the Exemplars Rubric—Novice, Apprentice, Practitioner (meets the standard) and Expert.

A Look at a Sample Unit

The Place Value Unit represents one of eleven Units in the third grade. The math TEKS covered in this Unit include: 3.2A, 3.2B, 3.2C, 3.2D. These math TEKS were grouped together to provide a cohesive Unit that enables 3rd grade students to understand the Properties of Place Value and to apply this understanding to compare and order whole numbers. See the full list of Units of Study for K–5.

How Might a Teacher Use the Tasks in This Unit?

As the Place Value Unit progresses, a teacher may want to use one of the instructional tasks to teach students the expectations of the Exemplars Assessment Rubric. A lesson using the task “Tables for a Party” may include the following steps:

  •  Whole Group: Read the task together and ask students to underline the question, identify important information in the problem and summarize the task by restating what the question is asking them to do. Example: I need to find out how many tables need to be set up for 34 students with no more than 10 students sitting at each table. On the Exemplars Rubric, this step is scored in the category for Problem Solving—Does the student understand the problem?
  •  Small Group: Ask students to work together, think of a plan, and write it down. Example: I will draw a diagram of tables with students sitting at them. At this point, students will implement their plan to solve the problem.

Example:

(Refer to the task Planning Sheet for additional examples of solution strategies.)

 Students will check their plan to make sure it works and put a box around their answer.

Example:

On the Exemplars Rubric, this step is scored in the category for Problem Solving—Does the student have a plan? Does the student get the correct answer? It is also scored in the category for Reasoning and Proof—Does the student show a systematic implementation of the plan?

Small Group: Ask students to polish their papers—

  • Organize your solution.
  • Explain your plan and how it solved the problem.
  • Create a representation—such as a diagram with a key, use a model (such as manipulatives), use a table, use a number line.
  • Use mathematical vocabulary and/or symbolic representation.
  • Label your solution.
  • Show your answer. Put a box around it. Make sure it answers the question.

On the Exemplars Rubric this step is scored in the category for Communication—Does the student use at least two mathematical vocabulary words, at least two correct symbolic representations or one of each? It is also scored in the category for Representation—Is the representation correct and appropriate to the solution?

 Small Group: Finally, make a connection—

  • Make an observation.
  • Identify and describe a pattern.
  • Make a comparison between this task and other tasks. Explain how the math is similar.
  • Identify a rule.
  • Create a hypothesis or conjecture to test.
  • Solve the problem using a different strategy to prove the original solution is correct.
  • Recreate the problem and show a different solution.

On the Exemplars Rubric, this step is scored in category for Connections—Does the student include a mathematically relevant connection? Making connections requires students to look at their solutions and reflect.

Using Anchor Papers & Scoring Rationales

Anchor papers and assessment rationales are provided with every summative assessment task. These problem-solving tasks are given at the end of a Unit of Study to assess students’ understanding. A summative assessment must represent a student’s total independent solution. One Hundred Miles is the summative assessment for the grade 3 Place Value Unit.

Anchor papers and scoring rationales provide a great way to show both teachers (in professional development sessions) and students the expectations of the Exemplars Rubric; i.e. What a Practitioner (meets the standard) piece of student work looks like. Analyzing Exemplars anchor paper solutions and rationales at the Practitioner and Expert levels help students polish their own work and measure their own progress toward a specific goal. Analyzing the Novice and Apprentice samples can help identify for students where the work falls short of the goal and specifically how the papers could improve.

To view other sample tasks and anchor papers for grades K–5, you can sign up for a free 30-day Trial for Problem Solving for the TEKS.

 

A Problem-Solving Lab to Support the Math Practices

Friday, October 31st, 2014

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!

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Fifth Grade Level

Thursday, September 4th, 2014

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

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 6 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice 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 6: Observations at the Grade 5 Level

The final anchor paper and set of rationales we’ll review in this series is taken from a fifth grade student’s solution for the task, “Newspaper Layout.” This task is one of a number of Exemplars tasks aligned to the Number and Operations–Fraction Standard 5.NF.6.

“Newspaper Layout” would be used toward the end of the learning time allocated to this standard. This particular task provides provides fifth graders with an opportunity to apply different strategies to determine how much the mathematics department pays for each part of the layout and the total cost of the advertisement. The task requires students to bring prior conceptual understanding of area and multiplying with money to their solution. In assessing this task, teachers will be able to determine if their students can apply these concepts and multiply mixed numbers.

Students have a variety of strategies to consider in forming their solutions. Some examples include creating a diagram of the newspaper layout, using grid/graph paper to correctly scale the newspaper area layout, applying the formula for area and money calculations or using a table to record the necessary data to support two correct answers. Students may also demonstrate their conceptual understanding of decimals.

5th Grade Task: Newspaper Layout

The newspaper staff is designing a layout to advertise the mathematics department’s “I Love Math” celebration. The newspaper staff will charge the mathematics department for the advertising by finding the number of square inches for each part of the layout. Below is a diagram of the layout. The newspaper staff charges fifty cents per square inch. How much does the mathematics department pay for each part of the advertisement? What is the total cost of the advertisement?  Show all of your mathematical thinking.

Common Core Alignments:

  • Content Standard 5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Fourth Grade Level

Thursday, August 28th, 2014

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

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 5 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice 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 5: Observations at the Fourth Grade Level

The fifth anchor paper and set of rationales we’ll review in this series is taken from a fourth grade student’s solution for the task “Sharing Muffins.” This task is one of a number of Exemplars tasks aligned to the Numbers and Operations–Fractions Standard 4.NF.3c.

“Sharing Muffins” would be used toward the end of the learning time allocated to this standard. This task provides fourth graders with an opportunity to apply different strategies to determine the number of muffins needed for each of nine friends to have one and one-third muffins. In solving this task, there are a variety of strategies for students to consider. Some examples include using actual muffins to model one and one-third muffins per friend or diagramming the muffins using a table, tally chart or number line. In their solutions, students may replace each mixed number with an equivalent fraction. Addition, subtraction and multiplication of fractions may also be used.

Fourth Grade Task: Sharing Muffins

Nine friends are going to equally share some muffins. Each muffin is the same size. Each friend gets one and one-third muffins. How many muffins did the nine friends equally share? Show all your mathematical thinking.

Common Core Task Alignments

  •  Content Standard 4.NF.3c: Add and subtract mixed numbers with like denominators e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Third Grade Level

Wednesday, August 13th, 2014

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

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 4 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice 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 4: Observations at the Third Grade Level

The fourth anchor paper and set of assessment rationales we’ll review in this series is taken from a third grade student’s solution for the task, “Henry’s Lego Structure.” This task is one of a number of Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 3.OA.8.

“Henry’s Lego Structure” would be used toward the end of the learning time allocated to this standard. This particular task provides third graders with an opportunity to apply different strategies to determine how many Legos are needed to build a three-level structure and if “Henry” has enough Legos to build a fourth level. Students need to bring an understanding of the terms twice, three times and pattern to the task as well as the correct calculation. When assessing this task, teachers can observe which forms of calculation a student chooses to use and if s/he can solve a two-step problem.

There are a variety of strategies for students to consider in forming their solutions. Some examples include using actual Legos to model the structure, diagramming the structure, creating a table, tally chart or using a number line.

Third Grade Task: Henry’s Lego Structure

Henry wants to build a structure with his new Lego set. The Lego set contains five hundred Legos. The structure will be three levels high. The first level is made of twenty-seven Legos. Henry uses twice as many Legos for the second level as for the first level. Henry uses three times as many Legos for the third level as for the second level. How many Legos does Henry use to build his structure with three levels? If this pattern continues, does Henry have enough Legos in his new set to build a fourth level on his structure? Show all of your mathematical thinking.

 Common Core Alignments:

  • Content Standard 3.OA.8: Solve two-step problems using the four operations.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Second Grade Level

Friday, August 1st, 2014

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

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 3 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice 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 Measurement and Data Standard 2.MD.8.

“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 provides 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.

Common Core Alignments

  • Content Standard 2.MD.8: Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.
  •  Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the First Grade Level

Monday, July 21st, 2014

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

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 2 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice 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 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 Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 1.OA.6. This task would be used toward the end of the learning time allocated to this standard. “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 Bird Bath

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 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, MP2, MP3, MP4, MP5, MP6, MP7

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Kindergarten Level

Thursday, July 10th, 2014

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

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 1 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice 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 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 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 Standard K.CC.5. This task would be used toward the end of the learning time allocated to this standard. 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 solution.

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.

Common Core Task Alignments

  • Content Standard K.CC.5: Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle or, as many as 10 things in a scattered configurations: given a number from 1-20, count out that many objects.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6

Bridge the Gap Between Common Core, Your Classroom and the Real World

Tuesday, March 25th, 2014

Written By: Elaine Watson, Ed.D., Exemplars Math Consultant

To most nonscientists, mathematics is counting and calculating with numbers. That is not at all what a scientist means by the word. To a scientist, counting and calculating are part of arithmetic and arithmetic is just one very, very small part of mathematics. Mathematics, the scientist says, is about order, about patterns and structure, and about logical relationships.

By, Keith Devlin, Life by the Numbers

The word “scientist” above could be replaced by the word “doctor, lawyer, engineer, accountant, CEO, military officer, government worker, homeowner, citizen …” In other words, anyone who uses numbers to make decisions needs to look beyond the calculations and be able to discern what the numbers are telling them.

Math textbooks have developed “word problems” in response to the question so often asked by students as they learn to follow algorithms and solve equations in order to find the correct answer: “When am I ever going to use this in real life?” The question is often answered by the jaded teacher, who has heard it from each new generation of students in this way: “You’re going to use it on the test!” This answer seals the students’ belief that what they learn in math class is not applicable to the real world, but merely a set of exercises that need to be done in order to pass the course.

Past mathematics standards documents have focused on the hard content, the factual and procedural content students should learn, which is of course important. The focus on the soft content, the habits of mind and thought processes that are practiced by students when solving a problem, has traditionally either been relegated to the end of the standards document as an afterthought or omitted altogether.

The Common Core State Standards in Mathematics (CCSSM) recognize that the soft content, the practices students used to approach and solve a mathematical task, are as important as the hard standards. Soft does not mean unimportant. In the same way that a computer (hardware) cannot function effectively without appropriate software, CCSSM Content Standards cannot be accessed and used without students using the supporting Practice Standards.

The Practice Standards have to be learned, and practiced, alongside the Content Standards, but because of the “soft” nature of Practice Standards, they are harder to pin down. Phil Daro, one of the three authors of the CCSSM, describes the Practice Standards as “the content of a student’s mathematical character.”

It is important to remember that it is the students who practice the Practice Standards. Teachers should model the practices in their instruction, but more importantly, teachers should explicitly plan lessons that include teacher pedagogical moves, student activities and tasks that will elicit the Practice Standards in students.

The tasks created by Exemplars are excellent examples of rich problem-solving that naturally elicit the Practice Standards. Below we will look at the Grade 2 task “Barnyard Buddies” and discuss how it meets each of the eight Mathematical Practice Standards as well as content standard 2.OA.A.1.

Barnyard Buddies

A farmer has 8 cows and 10 chickens. The farmer counts all the cow and chicken legs. How many legs are there altogether? Show all your mathematical thinking.

CCSSMP.1 Make sense of problems and persevere in solving them.

There is no hint in this task as to how to go about solving the task. It is not a generic type of problem with which the student has had previous experience. The student must make sense of the task before being able to develop an approach for solving it. Some approaches may be more efficient than other approaches.

CCSSMP.2 Reason abstractly and quantitatively.

In order to solve the problem, students will need to use an approach in order to organize their thinking and keep track of the quantities involved.  One approach is to draw 4-legged animals and 2-legged animals and count.  Another approach is to create a table. Both of these approaches have created an abstraction (mathematical model) of the situation. The student work below shows how two students modeled the problem.

Student 1 created abstractions of the chickens (square with 2 legs) and cows (circles with 4 legs).

Student 2 simply drew the legs without the bodies, which was a step toward greater abstraction. She or he then went on to use an even more abstract approach by noticing that there was a pattern and deciding to use a table. This student work is also a good illustration of Practice Standard 8: Look for and express regularity in repeated reasoning.

CCSSMP.3 Construct viable arguments and critique the reasoning of others.

This task will elicit a lot of different ideas as to how to approach it. Students will need to persuade others as to why their approach will work the best. In order for students to exhibit this practice standard, a classroom culture needs to be developed where student discussion of their work is the norm. The teacher’s role is to encourage the discussion and question and guide as needed.

CCSSMP.4 Model with mathematics.

In order to solve this task, students will need to go through the steps of the Modeling Cycle. They formulate an approach, compute, and then check their answer to see if they have correctly counted all 8 cow’s legs and all 10 chicken’s legs. If their answer makes sense, they report it out. If it doesn’t make sense, they need to go back through the cycle, determining where they went wrong. Were their pictures correct? Did they have the right number of each type of animal and the correct number of legs on each type of animal? If they used a table, did they skip count correctly by 2 and by 4? Did they add correctly? The cycle continues until they are satisfied that their result is a viable answer for the problem.

CCSSMP.5 Use appropriate tools strategically.

Tools are not necessarily physical, like a ruler or a calculator. On this problem, the student’s drawing or table can be considered a tool, since it helps make sense of and solve the problem.

CCSSMP.6 Attend to precision.

Precision is needed in the drawings or table, in the counting, and in the addition. Students also need to be precise in labeling their answer. If a student answers with only a number without the label “legs,” they are not attending to precision.

CCSSMP.7 Look for and make use of structure.

The student needs to visualize the structure of the situation. In this case the structure involves a given number of animals with 4 legs and a given number of animals with 2 legs. That structure will inform how the student approaches and solves the problem. If the student notices that 4 consists of 2 copies of 2, this will help in counting, since he or she should be proficient at counting by 2s.

CCSSMP.8 Look for and express regularity in repeated reasoning.

The student is repeatedly adding 2 or adding 4 for a given number of times. The student can count by 2s while pointing to each chicken. For the cows, students can either count by 4s, or they can count by 2s when pointing to the cows and touching each of the two pairs of legs on every cow.

Support for Common Core Content Standards

In addition to eliciting the Common Core Practice Standards, Exemplars tasks are also aligned Common Core Standards for Mathematical Content.

To solve “Barnyard Buddies,” students need to model the situation by using some type of drawing to represent the 10 chickens and the 8 cows as well as the number of legs on each animal.  Creating such a representation is an early form of algebraic thinking. After developing the pictorial model, students then need to count the total number of legs. Most students will skip count by either 2 or 4. Some students may organize their counting by making groups of 10 (2 cows and 1 chicken or 5 chickens).  Whichever approach students use for counting, they are recognizing a numerical pattern, which is also an underpinning of algebraic thinking.  This type of thought process is best matched by the Common Core Domain Operations and Algebraic Thinking.  Within this Domain, “Barnyard Buddies” aligns with the cluster, Represent and solve problems involving addition and subtraction. The specific content standard addressed is 2.OA.1.

2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Download a copy of the “Barnyard Buddies” task complete with anchor papers and scoring rationales to try with your students!

 

A Common Core “Must Read” Paper

Friday, November 30th, 2012

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

Jay McTighe and Grant Wiggins have written a “must read” paper – “From Common Core Standards to Curriculum: Five Big Ideas,” in which they offer key ideas to guide the work of transforming the Common Core Standards to a functioning curriculum in a school or district. The paper highlights some of the important misconceptions that readers bring to the Common Core and focuses on important processes that will lead schools and districts to creating an effective curriculum that actually embraces the Common Core Standards.

You may download a copy of this important paper here.

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