K.3.2 Measurement Ideas

Grade: 
K
Subject:
Math
Strand:
Geometry & Measurement
Standard K.3.2

Compare and order objects according to location and measurable attributes.

Benchmark: K.3.2.1 Compare Using Attributes & Location

Use words to compare objects according to length, size, weight and position.

For example: Use same, lighter, longer, above, between and next to.

Another example: Identify objects that are near your desk and objects that are in front of it. Explain why there may be some objects in both groups.

Benchmark: K.3.2.2 Order Using Measurable Attributes

Order 2 or 3 objects using measurable attributes, such as length and weight.

Overview

Big Ideas and Essential Understandings 

K.3.2 Essential Understandings

Measurement is one of the most widely used applications of mathematics. Kindergarten students  develop measurement concepts as they directly compare objects using the attributes of length, weight, size, and position. They order two or more objects using the attributes of length, weight and size. Kindergartners use the words above, below, next to, and between to describe the relative position of an object.

All Standard Benchmarks

K.3.2.1       Use words to compare objects according to length, size, weight and position.

K.3.2.2       Order 2 or 3 objects using measurable attributes, such as length and weight.

Benchmark Cluster 

Benchmark Group A

K.3.2.1            Use words to compare objects according to length, size, weight and position.

K.3.2.2            Order 2 or 3 objects using measurable attributes, such as length and weight.

What students should know and be able to do [at a mastery level] related to these benchmarks:

  • Use words such as shorter, longer/taller, and about the same to compare the length of objects. Use this knowledge to order two or more objects according to length.
  • Use words such as larger, smaller, and about the same to compare the size of objects. Use this knowledge to order two or more objects according to size.
  • Use words such as heavier, lighter, and about the same to compare weight of objects. Use this knowledge to order two or more objects according to weight.
  • Describe the relative position of an object using above, below, between and next to.

 

Work from previous grades that supports this new learning includes:

  • Young children have a natural curiosity about people and things in their world. This curiosity leads to finding out how things are related or how they fit together.
Correlations 

NCTM Standards

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

 

PreK-2 Expectations

  • Recognize, name, build, draw, compare, and sort two- and three-dimensional shapes.
  • Describe attributes and parts of two- and three-dimensional shapes.
  • Investigate and predict the results of putting together and taking apart two- and three-dimensional shapes.

 

Common Core State Standards:

Describe and compare measurable attributes.

  • K.MD.1. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
  • K.MD.2. Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

Classify objects and count the number of objects in each category.

  • K.MD.3. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Misconceptions

Student Misconceptions 

Students may think...

  • large objects weigh more.
  • the length of an object changes when its position is changed.
  • words used to describe relative position are interchangeable.

Vignette

In the Classroom

Vignette

The students in Mr. R.'s class are describing differences as they directly compare people and objects.

Mr. R:  Today we are going to figure out how things are different.

Would someone like to help with this activity?

Reid, will you come to the front of the room?

Reid walked up and stood beside Mr. R. 

Mr. R:  How are Reid and I different?

Connor:  You are a lot bigger, Mr. R!

Mr. R:  What do you think Connor means when he says I am a lot bigger than Reid?

Annie:  I think it means you're older than Reid.

Connor:  No...it's not about old.  Mr. Reid is bigger!

Mr. R.:  Let's think about this class. Could bigger mean older? taller? longer? Talk to your Math Talk Partner about this question.

Mr. R. listened as the students shared with their partners. Most students focused on the "taller" but a pair started to compare the size of their feet. A few students stood and compared their heights.  

Mr. R.:  When does bigger mean older?

Josiah:  My big sister is older than me.

Mr. R.:  When does bigger mean longer?

Xiong:  Mr. R., your feet are bigger than my feet.  That means they are longer. A lot longer!!

Mr. R.:  When does bigger mean taller?

Connor:  Oh....I mean you are taller.  I said you were bigger than Reid but you are taller than Reid.

Mr. R.:  When you compare how tall Reid and I are you are talking about our heights.

Mr. R.: Think again about our heights-how tall we are. Can someone use another word to compare our heights?

Amanda:  Reid is not taller, Mr. R. You are.

Mr. R.:  Is there another way we can say that Reid is not taller than Mr. R.?

Carson:  Reid is shorter than you, Mr. R.

Mr. R.:  Yes, Reid is shorter than Mr. R. and Mr. R. is taller than Reid.  We are comparing our heights and finding out who is taller and who is shorter.

Mr. R. writes the following on the board:

                        Mr. R. is taller than Reid.

                        Reid is shorter than Mr. R.

Mr. R. reads the sentences one at a time and asks the students to repeat them.

Mr. R. now pairs students and asks them to find out who is taller and who is shorter.  He moves from pair to pair asking, Who is taller? Who is shorter? and How do you know?  He works with students to use the language of comparison.   __________ is taller than________.  _________ is shorter than __________. I am taller than _______.  I am shorter than _______.   __________ is taller than me.   ___________ is shorter than me.

The students in Mr. R.'s class directly compared people according to the attribute of height.  Students' natural language did not convey the relationship between the height of Reid and Mr. R.  Mr. R. asked questions and elicited responses that would appropriately describe the height of one person as it related to the height of another person.  This scenario will be repeated many times as these kindergartners directly compare length, size and weight of objects.  Once students understand the process and language of comparisons of height, size, length and weight they will directly compare more than two objects.

For example, a child might compare these pencils.

Ross:  First I found the longest and the shortest. Then I put this one in the middle. It is not the longest or the shortest. It goes in between.

Resources

Instructional Notes 

Teacher Notes

  • Students may need support in further development of previously studied concepts and skills.
  • Kindergartners should sort objects into groups based on a comparison to a known object. For example, using a pencil as a referent, students would sort objects into three groups--longer than the pencil, shorter than the pencil, and about the same length as the pencil.
  • Kindergarten students need to compare/order real objects rather than pictures on paper.
  • Students should be asked to compare lengths of non-rigid objects including pieces of yarn or jump ropes.
  • Students need to compare the weights of large, light objects and small heavy objects.
  • Asking students to compare the weight of objects they cannot see keeps the focus on the attribute of weight rather than on the size of the object. Placing objects of differing weights in covered boxes, closed paper bags or in socks, enables young children to compare the weight without identifying the object.
  • Good questions, and good listening, will help children make sense of the mathematics, build self-confidence 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)

 

Students may need support in further development of previously studied concepts and skills

Instructional Resources 

NCTM Illuminations

Additional Instructional Activities

Longer, Shorter, About the Same

Students sort objects as longer, shorter, or about the same as a reference object.  The reference object can be changed to produce different sorts. Students could also put objects in order from shortest to longest. This activity can be modified to a different measurement such as weight or size. Consider using objects such as straws, pieces of string, strips of paper, etc.

Measurement Hunt

Students use a reference object (strip of paper, straw, piece of string) to find objects in the room.  One day, students may hunt for 6 objects that are longer than their reference object. The next day, they might hunt for objects that are shorter or about the same length as their reference object. Students can record their measurements by drawing pictures or writing the name of the objects they found on the measurement hunt. This activity could be modified to address the measurement areas of weight and general size.

Additional Instructional Resources    

Dacey, L., Cavanagh, M., Findell, Carol., Greenes, C., Scheffield, L., & Small, M. (2003). Navigating through measurement in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L.  (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

New Vocabulary 

"Vocabulary literally is the key tool for thinking."

Ruby Payne

 

New Vocabulary

attributes

bigger

larger

smaller

little

longer

shorter

taller

heavier

lighter

above

below

between

 

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 Use:    Multiple and varied opportunities to use the words in context.  These 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 non-examples.

Example/Non-example 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

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.

Professional Learning Communities 

Reflection - Critical Questions regarding the teaching and learning of these benchmarks.

What are the key ideas related to comparing objects according to length, size, weight and position at the kindergarten level? How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to compare objects according to length, size, weight, and position successfully?

When checking for student understanding when comparing objects 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 comparing objects according to length, size, weight, and position.  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 ordering objects using measurable attributes at the kindergarten level? How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to ordering objects using measurable attributes successfully?

When checking for student understanding of ordering objects using measurable attributes 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 ordering 2 or 3 objects using measurable attributes. 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 first grade level?

Materials - suggest articles and books for book study with PLC.

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K.. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8. (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 K-6). 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, K-6. Portsmouth, NH: Heinemann.

Sammons, L., (2011).  Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

References 

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, Marilyn. (2007). About teaching mathematics:  A k-8 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 K-8. (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 K-6). Sausalito, CA: Math Solutions.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. 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 pre-k-grade 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, K-6. 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 k-8. 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 K-5. 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., Bay-Williams, 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 student-centered mathematics grades K-3. Boston, MA: Pearson Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

Assessment

  • Using boxes or socks filled with objects of varying weights, students select two, then compare and describe the weight of each. Students may be prompted with the following questions: Are they the same weight? Is one heavier? Is one lighter?

Solution:          Correctly identifies the heaviest or the lightest.

Benchmark:     K.3.2.1

  • Comparing Length 

Materials: a box of familiar objects such as pencils, toothbrushes, forks, spoons, crayons, paint brushes, etc.

Instructions: After drawing draw two items from the box, students compare and then describe the length of the objects. Students may be prompted with the following questions: Are they the same length? Is one shorter? Is one longer?

The activity can be repeated using weight as the measurable attribute.

Solution:

Benchmark:     K.3.2.1

  • Ordering

Students are given sets of objects to order according to length (shortest to longest or longest to shortest) or weight (lightest to heaviest or heaviest to lightest).

Solution:          Correctly orders objects according to length or weight.

Benchmark:     K.3.2.2

  • Using Cuisenaire Rods, students pick 3 rods and order them from shortest to longest or longest to shortest.

Solution:          Correctly orders rods from shortest to longest or longest to shortest.

Benchmark:     K.3.2.2

Differentiation

Struggling Learners 

Struggling learners should compare two objects to determine which is longer or larger before comparing and ordering three or more objects. 

When comparing the weight of two objects the difference in weight should be obvious when the objects are held. 

Concrete - Representational - Abstract Instructional Approach

Adapted from The Access Center: Improving Access for All K-8 Students

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based 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 on-task.   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 instruction-Strategies that work for k-8 classrooms!. Thousand Oaks, CA.: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. 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., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L.  (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

English Language Learners 

English Language Learners need the vocabulary embedded as they compare objects. Identifying the attribute by which objects are being compared is critical as skills in language and measurement are being developed.

Serial ordering should be done with three objects in order to develop the language needed to describe ordering objects according to an attribute from least to greatest and greatest to least.

Words used to describe position need to be illustrated using physical objects that students can move.

Word banks need to be part of the student learning environment in every mathematics unit of study.  Refer to these throughout instruction.

Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as: bigger, larger, smaller, little, longer, shorter, taller heavier, lighter, next to, between, above, below, next to, 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:

_______________ is longer than _______________.

__________________ is shorter than __________________.

_________________ is heavier than _________________.

I know this is the shortest 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 Resource:

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning 

Given a stack of five connecting cubes and a stack of ten connecting cubes, students find objects that are longer/taller than five cubes and shorter than ten cubes.

Students can order more than three objects according to an attribute--length, size, weight, or height.

 

Additional Resources

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms!. Thousand Oaks, CA.: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Parents/Admin

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 real-world 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...         

directly comparing two or more objects according to the attributes of length, weight and size.

connecting the language of longer than/shorter than, heavier than/lighter than, or bigger than/smaller than as objects are being compared.

ordering objects according to the attributes of length, weight and size.

asking students to describe the ordering of objects according to an attribute. Why is this object here?  If I gave you ______ where would it be placed?

describing the relative position of one object as it relates to another.

asking students to place an object relative to the position of another object.

 

Additional Resources:

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8, 2nd edition.  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.

Parents 

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 self-confidence 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