K.3.1 Shapes
Recognize basic two and threedimensional shapes such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, cones, cylinders and spheres.
Sort objects using characteristics such as shape, size, color and thickness.
Use basic shapes and spatial reasoning to model objects in the realworld.
For example: A cylinder can be used to model a can of soup.
Another example: Find as many rectangles as you can in your classroom. Record the rectangles you found by making drawings.
Overview
Standard K.3.1 Essential Understandings
Kindergarten students sort and classify objects using attributes including shape, size, color and thickness. They recognize, name, compare and sort two and threedimensional shapes including squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, cones, cylinders and spheres.
Kindergarten students make connections between two and threedimensional shapes and their physical world. They use basic shapes and spatial reasoning to model realworld objects.
All Standard Benchmarks
K.3.1.1
Recognize basic two and threedimensional shapes such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, cones, cylinders and spheres.
K.3.1.2
Sort objects using characteristics such as shape, size, color and thickness.
K.3.1.3
Use basic shapes and spatial reasoning to model objects in the realworld.
K.3.1.1
Recognize basic two and threedimensional shapes such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, cones, cylinders and spheres.
K.3.1.2
Sort objects using characteristics such as shape, size, color and thickness.
K.3.1.3
Use basic shapes and spatial reasoning to model objects in the realworld.
What students should know and be able to do [at a mastery level] related to these benchmarks:
 sort objects using characteristics such as shape, color, size, etc.
 recognize basic two and threedimensional shapes.
 recognize and use words that describe spatial relationships such as above, below, inside, outside, touching, next to, far apart.
 use basic shapes and spatial reasoning to model realworld objects.
Work from previous grades that supports this new learning includes:
 experience shape through puzzles, books, mobiles and toys
 intuitive sense of position
NCTM Standards
Analyze characteristics and properties of two and threedimensional geometric shapes and develop mathematical arguments about geometric relationships.
PreK2 Expectations
 Recognize, name, build, draw, compare, and sort two and threedimensional shapes.
 Describe attributes and parts of two and threedimensional shapes.
 Investigate and predict the results of putting together and taking apart two and threedimensional shapes.
Common Core State Standards
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
K.G.1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
K.G.2. Correctly name shapes regardless of their orientations or overall size.
K.G.3. Identify shapes as twodimensional (lying in a plane, "flat") or threedimensional ("solid").
Analyze, compare, create, and compose shapes.
K.G.4. Analyze and compare two and threedimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length).
K.G.5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
K.G.6. Compose simple shapes to form larger shapes. For example, "Can you join these two triangles with full sides touching to make a rectangle?"
Misconceptions
Student Misconceptions and Common Errors
Students may think . . .
 a change in orientation means a change in shape.
 a threedimensional objects can be named by the shape of one of the faces. For example, a box can be called a rectangle.
 all triangles are equilateral.
 squares are not rectangles.
Vignette
In the Classroom
Vignette 1
In this vignette series, Ms. Sharpe leads students in sorting activities based on observable attributes.
Ms. Sharpe asked a group of children to join her at the table.
There was nothing on the table except a colorful shoe box. Ms. Sharpe picked it up and shook it gently creating a rattling sound.
"I wonder if you can guess what is in here," Ms. Sharpe said. "Shall we open it and see? Sarah, will you open the box and gently tip everything out on the table for us all to see?"
Sarah was thrilled. She carefully opened the box and did what she was asked. Out fell quite a mixture of colorful things. There were some toy cars, some balloons (which needed blowing up), and some colorful buttons (all with two holes in them). It was quite natural for the children to want to touch the objects. They began to do what Ms. Sharpe had thought the lesson would be about; they began to group things together.
"Jose, can you tell us why you are doing that?" Jose was putting some red cars together. He was reluctant to speak but kept on with his task. Kim mouthed 'cars' to Ms. Sharpe, who shrugged her shoulders. Then Jose added a red balloon to his group.
"They are all red," said Jose.
"I am finding blue toys," said Sarah.
The children were happy sorting the colors, even Jeni who was also collecting red things. Eventually she and Jose put their groups together.
Ms. Sharpe was delighted. She now took the opportunity to stress some mathematical language. With a hint of praise in her voice, she said, "Jeni and Jose have collected a SET of RED TOYS."
The other children were prompted to describe their collections in a similar way.
Later...
Ms. Sharpe called over a group of children to the demonstration table. This time her sorting box contained geometrical shapes. There were four shapes, in four colors and two sizes. These had all been part of the incidental play the children had experienced since the day they came to school. They knew them well as building blocks, but not as mathematical shapes. She tipped the contents of the sorting box out onto the table. The children were eager to touch, so she asked each person to pick up a shape.
"Tyrone, put the shape you have over here. Good. Now has anyone else got a shape like the one Tyrone has put here?"
They all had a good look. Some even touched Tyrone's shape and compared it to their own. "I've got one. It is the same color, but it is smaller." "I've got the same size, but a different color." "So have I."
And it went on. Eventually all the shapes were sorted into four sets. The children had sorted by shape, but as yet Ms. Sharpe had not introduced any names. Instead she asked the children to look around the room and see if they could find other things which belonged in the sets they had made. Of course she knew there would be plenty because she had been collecting containers and packages from her home.
"What have you found?"
"An eraser."
"Where will you put it then? Good. What have you found?"
"A Corn Flakes box."
"Where does that belong? Good. Has anyone found anything for this set?"
"I have. It's a new pencil." It rolled away from the others in the set as soon as it was placed on the table.
Everybody found something to go into the sets. Ms. Sharpe then said, "We had better make some labels for our sets today. These are cubes..." She invited the children to say each word after her. "These are rectangular prisms...These are cylinders...and these are triangular prisms."
The children all repeated the names and the labels were made for the display. The children then went to another part of the room where Mrs. Sharpe had made a puzzle for them. She had placed some solid shapes in some "feely" bags and the children had to feel each bag in turn and name the shape in the bag.
Vignette 2
Sorting People: Who Fits My Rule (1997 MN Framework)
"Guess My Rule" is a classification guessing game in which players try to figure out the common characteristic, or attribute, of a set of objects.
The game can be played in many different contexts, but the students first play it in a context very familiar to them: they sort themselves. To play the game, the rule maker (who may be you, a student, or a group of students) decides on a secret "mystery rule" for classifying a particular group of people. For example, classification rules for people might be everyone who is wearing orange, or everyone who is missing a front tooth.
The rulemaker starts the game by giving some examples of people who fit the rule (for example, by having two students who are wearing orange stand up). The guessers then try to find other individuals who might fit the rule: "Can Sandy fit your rule?"
With each guess, the individual named is added either to the group that does fit or to the group that does not fit the rule. Both groups must be clearly visible to the guessers so they can make use of all the evidence, both what does and does not fit, as they try to figure out what the rule might be.
Two guidelines are particularly important to stress during play:
"Wrong" guesses are clues that are just as important as "right" guesses. Errors are not just mistakes but can be important sources of information.
When students think they know what the rule is, they ''test" their theory by giving another example, not by revealing the rule. Requiring students to add new evidence, rather than making a guess, serves two purposes. It allows students to test their theories without revealing their guess to other students. It also provides more information and more time to think for students who do not yet have a theory of their own.
When students begin choosing rules themselves, they sometimes think of rules that are either too vague (students wearing different colors) or too difficult for other students to guess (students with a piece of thread hanging from their shirts). You can guide and support students in choosing rules that are not so obvious that everyone will see them immediately, but not so hard that no one will be able to figure them out. The students should be clear about who would fit their rule and who would not fit. This eliminates rules like wearing different colors which everyone will probably fit. It's also important to pick a rule about something people can see. That is, one rule for classifying might be "likes baseball," but no one will be able to figure out this rule just by looking.
Since classification is a process used in many disciplines, you can easily adapt the game to other subject areas. Animals, states, historical figures, geometric shapes, types of food, and countless other items can all be classified in different ways.
Being able to see how things go together improves with experience. As you and your students play this game repeatedly in different contexts, you will find yourselves becoming more observant and more flexible in your thinking about similarities and differences.
SciMathMN Minnesota K12 Mathematics Framework
About this Activity
 This unit focuses on collecting, organizing and interpreting categorical data, and emphasizes student work in classification and data analysis.
 This particular activity encourages children to redefine their thinking in response to new data.
 Children feel free to do their best thinking in a safe environment where their ideas are valued. The "rules" in the game go a long way to maintaining a thinking curriculum, rather than one based on the "right" answer.
Where do we go from here?
"Guess My Rule" naturally extends to the sorting of concrete objects, and later to the graphical representation of data on charts, graphs, and Venn diagrams. Students first sort on one characteristic, then proceed to sort on two or more characteristics.
Resources
Teacher Notes
 Students may need support in further development of previously studied concepts and skills.
 Students should be provided with both examples and nonexamples of shapes. Nonexamples help students eliminate unimportant features and focus on relevant features.
 Students may confuse two and three dimensional shapes. For example, while on a classroom shape hunt for rectangles, a student records the filing cabinet, the door, and wooden blocks as rectangles when they are actually rectangular prisms. The faces are rectangles. Note the difference for students and ask, "What part of the door is a rectangle?" Students should find many rectangular faces when examining a door and responding to this question.
 Geometric experiences (not developmental stages) are the greatest factor in advancing students geometric knowledge. Exploration allows children to talk about, explore and engage with content at the next level while increasing experiences at their current level. These explorations provide students with the best chance of advancing geometric thought.

The van Hiele Levels of Geometric Thought
Adapted from: Van de Walle J. (2006), and Van de Walle, J., & Lovin, L. (2006).
The van Hiele Levels of Geometric Thought hierarchy describes the way that students learn to reason about shapes and other geometric ideas. There are five levels in the hierarchy that are deemed to be sequential. Students need to move through each prior level before moving on to the next. Student thinking about particular concepts will likely be at different levels at any given time. The levels describe how we think and what types of geometric ideas we think about, rather than how much knowledge we have. Movement through the levels depends on the types and amount of experiences students have with geometry. Instruction that takes place at a level higher than the students' functional level will be ineffective. The goal in kindergarten is to provide experiences for students at level 0.
Level 0  Visualization
 The objects of thought at level 0 are shapes and what they "look like."
 Students may be able to talk about the properties of the shapes, but the properties are not thought about explicitly. They characterize individual shapes based on appearance. "It is a square because it looks like a square." At this level, students think shapes "change" or have different properties when rotated or rearranged.
 The products of thought at level 0 are classes or groupings of shapes that
 seem to be alike.
Level 1  Analysis
 The objects of thought at level 1 are classes of shapes rather than individual
 shapes.
 Students are able to think of properties (number of sides, angles, parallel sides etc) of a shape rather than focusing on the appearance of a shape. A student operating at this level might list all the properties the student knows about a shape, but not discern which properties are necessary and which are sufficient to identify the shape. "It is a square because it has square corners and the sides are the same." Though students see properties of shapes, they cannot make generalizations about how different shapes relate to one another. Students at this level will not see the relationship of a square to a rectangle.
 The products of thought at level 1 are the properties of shapes.
Level 2  Informal Deduction or Abstraction
 The objects of thought at level 2 are the properties of shapes.
 Students develop relationships between and among properties. Shapes can be classified using minimal characteristics. "Rectangles are parallelograms with a right angle." Students at level 2 will be able to follow an informal deductive argument about shapes and their properties. While many adults remain at level 0 or level 1, with appropriate experiences, most students could reach level 2 by the end the elementary grades.
 The products of thought at level 2 are relationships among properties of geometric objects.
Level 3  Deduction
 The objects of thought at level 3 are relationships among properties of geometric
 objects.
 Students work with abstract statements about geometric properties and make conclusions based on logic. This is the level of a traditional high school geometry course.
 The products of thought at level 3 are deductive axiomatic systems for geometry.
Level 4  Rigor
 The objects of thought at level 4 are deductive axiomatic systems for geometry.
 Students operating at this level focus on axiomatic systems, not just deductions within the system. This is usually at a level of a college geometry course.
 The products of thought at level 4 are comparisons and contrasts among different axiomatic systems of geometry.
Note: In some literature the van Hiele Levels of Geometric Thought are
labeled 15 rather than 04.
For further information on the van Hiele Levels of Geometric Thought
 Dynamic Paper Need a set of pattern blocks where all shapes have oneinch sides? You can create all these things and more with the Dynamic Paper tool. Place the images you want, then export it as a PDF or as a JPEG image for use in other applications.
 Good questions, and good listening, will help children make sense of the mathematics, build selfconfidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...
(Adapted from They're Counting on Us, California Mathematics Council, 1995)
NCTM Illuminations
Patch Tool
Quilters and other designers sometimes start by producing square patches with a pattern on them. These square patches are then repeated and connected to produce a larger pattern. Create your own patch using the shapes and the tool provided.
Buttons! Buttons!
In the following lesson, students participate in activities in which they focus on connections between mathematics and children's literature. Specifically, students use clues to sort and classify buttons.
Additional Instructional Resources
Findell, C., Small, M., Cavanagh, M., Dacey, L., Greenes, C., & Scheffield, L. (2001). Navigating through geometry in prekindergartengrade 2. Reston, VA: National Council of Teachers of Mathematics.
Richardson, K. (1999). Understanding geometry. Bellingham, WA: Lummi Bay Publishing.
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
square, circle, triangle, rectangle, trapezoid, hexagon, cube, cone, cylinder, sphere,
"Vocabulary literally is the
key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Multiple and varied opportunities to use the words in context. These
Use: opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and nonexamples.
Example/Nonexample Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word 
definition 
illustration 
word square

definition a 4sided figure with 4 equal sides and 4 equal angles

illustration 
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004) Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Reflection  Critical Questions regarding the teaching and learning of these benchmarks.
What are the key ideas related to basic two and threedimensional shapes at the kindergarten level? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to recognize two and threedimensional shapes?
When checking for student understanding of twoand threedimensional shapes first grade level, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
Examine student work related to a task involving two and threedimensional shapes. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
What are the key ideas related to sorting objects/shapes at the kindergarten level? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to sort and classify objects/shapes successfully?
When checking for student understanding of sorting and classifying objects and shapes at the kindergarten level, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
Examine student work related to a task involving sorting and classifying. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
How can teachers assess student learning related to these benchmarks?
How are these benchmarks related to other benchmarks at the kindergarten level?
Professional Learning Community Resources
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prekgrade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach, grades K8. (2nd Ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K6). Sausalito, CA: Math Solutions
Fuson, K., Clements, D., & Beckmann, S. (2010). Focus in kindergarten: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K6. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prekgrade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k2. Sausalito, CA: Math Solutions Publications.
Burns, Marilyn. (2007). About teaching mathematics: A k8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K8. (2nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k2.Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration prekgrade 2. Reston, VA: National Council of Teachers of Mathematics.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K5. Sausalito. CA: Math Solutions.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching studentcentered mathematics grades K3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons.Portsmouth, NH: Heinemann.
Assessment
 Point to each shape and ask the student, "What is the name of this shape?" Note the shapes students are able to identify. If students are unable to identify shapes, say "Point to a circle..." Continue with other shapes.
Solution: Correctly identifies shapes.
Benchmark: K.3.1.1
 Sorting Collections (K.3.1.2)
Provide students with a collection of items (buttons, toys, attribute blocks, etc.). Ask students to think of a way that they could sort the items. Then, direct students to move their items into groups. Students should be able to explain their sorting rule. If the student needs prompts, ask, "How are these alike?" while pointing to each group. Next, ask the student to sort the same collection in a different way. Note the characteristic (color, shape, size, thickness, texture) that is used to sort.
Solution: Correctly sorts and resorts according to attributes.
Benchmark: K.3.1.2
Differentiation
Learners who are emergent in their thinking need to sort collections that have very obvious differencescolor, size, shape, etc. They may need to be directed to sort a collection by a specific attributesort these buttons according to color. These students may not always be able to explain why they sorted and may need appropriate words to describe collections that were correctly sorted.
Students need handson experiences with shapes. Finding a shape that is the same/different as a given shape helps them focus on characteristics of a shape.
Concrete  Representational  Abstract Instructional Approach
The ConcreteRepresentationalAbstract Instructional Approach (CRA) is a researchbased instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.
The CRA approach is based on three stages during the learning process:
Concrete  Representational  Abstract
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and ontask. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources
Bender, W. (2009). Differentiating math instructionStrategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k2. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroomA guide for teachers k8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
English Language Learners may confuse the names for two and threedimensional shapes. It is important to have models of shapes available as well as descriptions of shapes that students have helped develop.
These students may be able to sort objects but lack the words to describe the sort. It might help to direct initial sorting activitiessort this collection by colors.
 Word banks need to be part of the student learning environment in every mathematics unit of study.
 Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.
 Sentence Frames
Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions. Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.
Sample sentence frames related to these benchmarks:
This is a _______________. It has ______________. 
I sorted by _____________________. I have a ____________ group, a _________ group and a ____________ group. 
This is not a ______________. It has ______________. 
I see a ____________ in the classroom. I know it's a _______ because ____________. 
 When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.
Additional ELL Resources
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k2. Sausalito, CA: Math Solutions Publications.
Show students many squares that are different sizes and have different orientations. Encourage students to describe the characteristics of a square. Ask, "What do all squares have?" As students analyze the square, record the properties they find. Ask questions such as, "Will all squares have these properties?" Students may agree on a conjecture such as a square must have 4 sides that are all the same length. Continue the process with other shapes. As students examine triangles, they may notice different types of triangles and classify them differently.
Students need to sort objects that are alike in some way such as a set of blue objects or a set of toy cars. This will force them to look for other less obvious attributes.
Sorting and then resorting many times is important as students develop critical thinking skills.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k8 classrooms!. Thousand Oaks, CA.: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k2. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Parents/Admin
Administrative/Peer Classroom Observation
What should I look for in the mathematics classroom? (adapted from SciMathMN 1997)
What are students doing?
 Working in groups to make conjectures and solve problems.
 Solving realworld problems, not just practicing a collection of isolated skills.
 Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
 Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
 Recognizing and connecting mathematical ideas.
 Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
 Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
 Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
 Connecting new mathematical concepts to previously learned ideas.
 Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
 Selecting appropriate activities and materials to support the learning of every student.
 Working with other teachers to make connections between disciplines to show how math is related to other subjects.
 Using assessments to uncover student thinking in order to guide instruction and assess understanding.
Students are . . . 
Teachers are . . . 
sorting objects and shapes according to attributes and explaining the sort. 
asking students to justify the placement of objects or shapes in specific groups. 
using shapes to model real world objects. 
encouraging grossmotor activities where students make shapes with their bodies or walk to make shapes. 
finding and naming shapes in their world. 
helping students translate between two and threedimensional shapes 
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k8. (2nd ed.). Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build selfconfidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...
Adapted from They're counting on us, California Mathematics Council, 1995
ReadALoud Books:
A Pair of Socks by Stuart J. Murphy, Illustrator Lois Ehlert
Circus Shapes by Stuart J. Murphy
Cubes, Cones, Cylinders, & Spheres by Tana Hoban
Sorting by Henry Pluckrose