2.2.1 Patterns

Grade: 
2
Subject:
Math
Strand:
Algebra
Standard 2.2.1

Recognize, create, describe, and use patterns and rules to solve real-world and mathematical problems.

Benchmark: 2.2.1.1 Patterns

Identify, create and describe simple number patterns involving repeated addition or subtraction, skip counting and arrays of objects such as counters or tiles. Use patterns to solve problems in various contexts.

For example: Skip count by 5s beginning at 3 to create the pattern

3, 8, 13, 18, ... .

Another example: Collecting 7 empty milk cartons each day for 5 days will generate the pattern 7, 14, 21, 28, 35, resulting in a total of 35 milk cartons.

Overview

Big Ideas and Essential Understandings 

students making patterns

Standard 2.2.1 Essential Understandings

Second graders expand their work with patterns to include number patterns. 

They connect number to patterns made with objects and examine the relationship between the numbers to predict what come next. For example,

•          ••          ••••     •••••••

1          2           4            8                  ?                      ?                                 

Second graders identify, create, complete, and extend number patterns involving addition, subtraction, skip counting, and arrays of objects. In addition, they describe the rule for a number pattern.

All Standard Benchmarks

2.2.1.1            
Identify, create and describe simple number patterns involving repeated addition or subtraction, skip counting and arrays of objects such as counters or tiles. Use patterns to solve problems in various contexts.

Benchmark Cluster 
Benchmark  Group A

2.2.1.1
Identify,create and describe simple number patterns involving repeated addition or subtraction, skip counting and arrays of objects such as counters or tiles. Use patterns to solve problems in various contexts.

What students should know and be able to do [at a mastery level] related to these benchmarks:
  • Recognize patterns in numbers and apply the pattern to predict what comes next in a number pattern.
  • Describe the rule for a given number pattern.
  • Use a given rule to extend or complete a number pattern.
  • Use patterns to solve problems.
Work from previous grades that supports this new learning includes: 
  • Sort and categorize objects according to attributes.
  • Creating, extending and completing repeating patterns.
  • Creating, extending and completing growing and shrinking patterns.
Correlations 

NCTM Standards

Understand patterns, relations, and functions
Pre-K-2 Expectations:

  • sort, classify, and order objects by size, number, and other  properties;
  • recognize, describe, and extend patterns such as sequences of  sounds and shapes or simple numeric patterns and translate from one representation to another;
  • analyze how both repeating and growing patterns are generated.

Common Core State Standards:  None

Misconceptions

Student Misconceptions 
Student Misconceptions and Common Errors

Students may think . . .

  • all patterns are repeating patterns.
  • patterns involve only pictures, objects, or movements.
  • there is no relationship between numbers in a number pattern.
  • growth patterns only get bigger, not realizing patterns can shrink.

Vignette

In the Classroom

Ms. N.:            Today we are going to explore some patterns with toothpicks. Please take a few minutes to explore with the toothpicks and build some patterns.

Free exploration of 5 minutes or so will help students focus when using the materials during the lesson. Ms. N. has the students share what patterns they made during their exploration time.

Ms. N.:            Now I would like you to make a triangle with your toothpicks. What do we know about triangles?

Student:          They have three sides.

Student:          They are closed shapes.

Ms. N. :           Anything else?

Student:          They have three angles.

Ms. N.:            Nice job. Those are the main attributes of a triangle. Now please make a triangle using three toothpicks.

Ms. N.:            What do you notice about the triangles you made?

toothpick triangle

Student:          All the sides are the same.

Ms. N.:            Tell me more.

Student:          They are all the same length 'cause the toothpicks are all the same.

Ms. N.:            What do we call triangles that have all sides of equal length?

Student:          Equilateral triangles.

Ms. N.:            What helps you remember the word "equilateral?"

Student:          Well, it sounds kind of like "equal" which means "the same" and we learned    that "lateral" means "sides."

Ms. N.:            I like the way you used chunking of those words to help you remember what they mean! Powerful thinking! We will be building on to our triangle, and keeping track of the number of triangles we make and the number of toothpicks we use. Any suggestions about how we can organize that information?

Students:         We could use a t-chart.

Ms. N.:            That's one of our favorite tools, isn't it? Would you please come up and draw a t-chart on the board? Everyone else can draw one in their math notebooks.

Ms. N.:            How are we going to label the two columns?

Student:          Well, we want to keep track of triangles and the toothpicks we use to build them, so we could have "triangles" and "toothpicks" for our labels.

Ms. N.:            That will work.

student drawing t-chart

Ms. N.:            Let's fill in the information for the triangle we have built. Dominique, would you please come up and write it on the board? Everyone else should be recording in your math notebooks.

Ms. N.:            Now, I would like you to build a second triangle like this:

toothpick parallelogram

Student:          It looks like a parallelogram.

Ms. N.:            Tell me more.

Student:          The outside shape has four sides and they are parallel.

Ms. N.:            What do you mean the sides are parallel?

Student:          Well, this side and this side will never meet (pointing to opposite sides). And this side and this side will never meet. That means they're parallel. 

Ms. N.:            Nice explanation!

Ms N.:             What information should we put in our t-chart for our two triangles?

Alice:               We have to put in two triangles and five toothpicks.

Alice comes up and fills in the chart.

Alice fills in the chart.

Ms. N.:            Do you see any patterns in the chart?

Student:          The triangles went up by one.

Student:          The toothpicks went up by two.

Ms. N.:            Why did the number of toothpicks only increase by 2, not 3? I thought every triangle has three sides.

Student:          Because they're sharing a side.

Ms. N.:            Tell me more.

Student:          This side was already there, so we just added 2 more.

We just added two more.

Ms. N.:            I get it. Please explain to your neighbor why you only added two toothpicks to make the second triangle. Okay, let's add another triangle. Make sure you build so your triangles form a straight row instead of curving around into a hexagon.

Hey, it's a trapezoid!

Student:          Hey, it's a trapezoid!

Ms. N.:            Tell me more.

Student:          Because these sides are parallel and these sides aren't!

Ms. N.:            Oh, I see it now.

Ms. N.:            Who can come and fill in the information for our three triangles on the board?

Travis fills in the information on the class chart.

Ms. N.:                        What patterns do you see now?

Student:          The triangles went up by one again and the toothpicks went up by two. 

Student:          It's the same as before!

Ms. N.:            Why did the toothpicks only go up by two again?

Student:          Because they share a toothpick.

Ms. N.:            Please build the next two triangles and record the information on the chart in your math notebook.

chart in math notebook

Ms. N.:            Now, let's think about the patterns in the numbers on the t-chart, and see if we can figure out how many toothpicks it would take to build 10 triangles?  Thoughts?

Student:          We could build them with the toothpicks.

Ms. N.:            We could build them. Can anyone think of a more efficient way to figure it out?

Student:          We could just fill in the t-chart. We know the patterns the numbers make so we can just add one in the triangle column and add two every time in the toothpick column.

Student:          We could just double the number of toothpicks for 5 triangles, because double 5 is 10 so it would be double 11 makes 22.

Ms. N.:            Interesting ideas. Let's build the ten triangles and fill in the chart as you go. 

Ms. N. knows these second graders can continue to build triangles and update their charts as they go. She knows third graders would be using the patterns on a T-chart (input-output table) to find the number of toothpicks needed for ten triangles. Second graders are finding the patterns in the number representing triangles and in the numbers representing toothpicks.

Ms. N.:            What do you notice about the numbers?

Student:          The numbers for triangles kept getting bigger by one.

Student:          The numbers got bigger by two for the toothpicks.

Ms. N.:            Someone suggested it would take twenty-two toothpicks for ten triangles because it took eleven toothpicks for five triangles and we could just double that amount.  How many toothpicks did it take for ten triangles?

Student:          It took twenty-one toothpicks.          

Ms. N.:            Is that what everyone got?

Students nod.

Ms. N.:            Why didn't the number of toothpicks double when the number of triangles doubled? Did we all make a mistake?

No responses.

Ms. N.:            Look at your charts. Is there any evidence that doubling the number of triangles means the number of toothpicks also double?  What happened to the number of toothpicks when you went from one to two triangles or from two to four triangles? Is there a pattern in the numbers?  Talk to the people at your table. Be ready to share your thinking.

Ms. N. asks the students to look for a doubling pattern in the numbers on the chart as a follow up to a student's conjecture about doubling the number of toothpicks needed to make five triangles as a way to determine the number of toothpicks needed to make ten triangles. The conjecture was not based on a pattern within the chart but on the relationship five to ten. At this point, all Ms. N. wants is the second graders to see there is not a doubling pattern in the chart. Knowing that the pattern is "add two" in the toothpick column is sufficient at this time.

Resources

Instructional Notes 

Teacher Notes

  • Students may need support in further development of previously studied concepts and skills.
  • Patterns are a way for young students to recognize order and to organize their world. Teachers should help students develop the ability to form generalizations by asking questions such as, "How could you describe this number pattern?" or "How can it be repeated or extended?" or "How are these patterns alike?"
  • It's important to guide students in discovering the pattern in a sequence of numbers. In the beginning, using number lines or hundreds charts to mark the numbers in a sequence will help students find the pattern and state the rule. They can circle the numbers on the number line and draw arrows from one circled number to the next, with the "rule" written above. Students could color the number or place colored counters on them when using a hundreds chart. Once the numbers are marked, students need to look for patterns. The pattern will be the same number being added or subtracted over and over. Once students find the rule, invite them to share how they know.

This could be a discussion for the sequence 16, 19, 22, __, __, __ .

What is the rule?  (add 3)
How do you know?  (16 + 3 = 19 and 19 + 3 = 22)
What are the unknown numbers? 25, 28, and 31
How do you know? (22 + 3 = 25, 25 + 3 = 28, 28 + 3 = 31)

  • Keep the focus on the change from one number to the next. What is the change from _____ to ______? Is the change the same from one number to the next in this sequence? What does that tell us about the pattern rule?

Questioning

  • 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)

Instructional Resources 
NCTM Illuminations

Multiple Patterns In this lesson, students explore patterns which involve doubling.  They use objects and numbers in their exploration and record them using a table.

Exploring Other Number Patterns  Students make and extend numerical patterns using hundreds charts in this lesson.  They also explore functions at an intuitive level.  This lesson integrates technology.

Growing Patterns  In this lesson, students explore growing patterns.  They analyze, describe, and justify their rules for naming patterns.  Since students are likely to see growing patterns differently, this is an opportunity to engage them in communicating about mathematics.

Looking Back and Moving Forward The final lesson in this unit reviews the work of the previous lessons.  Students explore patterns in additional contexts and record their investigations.  Students rotate through center activities.

Calculating Pattern Students use an Internet-based calculator that is linked with an interactive hundred chart to create, extend, and record numerical patterns in different ways. By connecting the two representations, students observe the numerical patterns as that are created.

Additional Instructional Resources    

Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra 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. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston: Pearson Education.

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

Von Rotz, L., & Burns, M. (2002). Grades k-2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.

An additional helpful website
Many lesson ideas and literature connections are available at:
http://mathwire.com/algebra/growingpatterns.html

New Vocabulary 

growing patterns - show an arithmetic change (constant change of the same amount);  e.g. 1, 3, 5, 7, 9, . . .  (add two);  50, 45, 40, 35, 30, . . .  (subtract 5).  Growing patterns increase or decrease in a predictable and describable way.

number pattern - A list of numbers that follow a certain sequence or pattern.
For example: 1, 4, 7, 10, 13, 16, 19 ... starts at 1 and jumps 3 every time. (mathisfun.com)

number sequence - A list of numbers, often generated by a rule.  For example:  1, 3, 5, 7, 9 . . . starts at 1 and adds 2 each time.

repeated addition - Adding equal groups. For example:  5 + 5 + 5 + 5 is 5, 10, 15, 20.

repeated subtraction - Subtracting equal groups.  For example:  12 - 2 - 2 - 2 is 12, 10, 8, 6.

"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 non-examples.

Frayer Model

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.

Example /non-example chart

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 understanding number patterns at the second grade level?  How do student misconceptions interfere with mastery of these ideas?

What kind of patterns should second graders see in an instructional setting?

Give some examples of number patterns that second graders should encounter.

What experiences do students need in order to develop an understanding of number patterns?

When checking for student understanding, 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 number patterns. 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 second grade level?

Professional Learning Community Resources

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

Blanton, M. (2008).  Algebra and the elementary classroom, transforming thinking, transforming practice.  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.

Hyde, A. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. 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.

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

  • This pattern continues.  Draw the next two figures.  Use numbers to describe the pattern.  Write a rule for the pattern.

continuen this pattern

Solution:          a. Correctly draws the next two figures.
b. 1, 3, 5, 7, 9
c. Add 2         
Benchmark:     2.2.1.1

  • Continue this number pattern. Write the rule.

57, 54, 51, ___, ___, ___


Solution:          a. 48, 45, 42   
b. Rule:  Subtract three (-3)
Benchmark:     2.2.1.1

  • Are the rules of these two patterns the same or different?  Explain your thinking.

2,  5,  8,  11                 6,  9,  12,  15

Solution:          The rule is the same.  It is add three in both patterns. The starting
number is different in the two patterns.
Benchmark:     2.2.1.1

  • Which numbers come next in the pattern?

8          12        16        20        _____ 

A.  30, 36
B.  24, 28
C.  22, 24
D.  28, 32

Solution:          B. 24, 28
Benchmark:     2.2.1.1

  • Devin is skip counting.  He underlines the numbers that he says on the hundreds chart.  What is Devin skip counting by?

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Solution:          Devin is counting by six.
Benchmark:     2.2.1.1

  • What is the pattern rule? 

636      626      616      606

A.  Add 100
B.  Subtract 10
C.  Subtract 100
D.  Add 10

Solution:          B. Subtract 10
Benchmark:     2.2.1.1

Differentiation

Struggling Learners 

Students might use connecting cubes to show the numbers in a sequence. This visual can help them see the pattern and determine the type of pattern (growing or shrinking). 

Use a hundreds chart to mark the numbers in a sequence to help students see the pattern and make predictions as to what comes next.        

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. 

Concrete Triangle

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 

ELL students may need support in understanding the word rule.  Share that the word rule has an everyday usage, like the classroom rules as well as a mathematical usage. In math it tells us to "add or subtract" and "how many."  Add 2 and Subtract 5 are also rules.

Use concrete examples to establish the idea of number patterns, such as a hundreds chart or a calendar.

Linking informal language with formal mathematical vocabulary is another strategy for helping children learn new vocabulary. Modeling the verbal description of patterns and having students copy written descriptions will help them learn the vocabulary associated with patterns.

  • 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.

Frayer Model

  • 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:

The rule for this pattern is ____________________________.

The next numbers in this pattern are ________________  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 k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning 

Investigate Pascal's Triangle, a number pattern created by a French mathematician. They could discuss patterns in the numbers of the triangle and write the next two rows to extend it.  

Investigate square numbers (1, 4, 9, 16 . . . ) and triangular numbers (1, 3, 6, 10, 15, . . .).

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 
Administrative/Peer Classroom Observation

Students are . . .         

Teachers are . . .         

recognizing patterns in sequences of numbers.

questioning students, keeping the focus on their understanding of number patterns.

describing number patterns they see, both orally and in writing.

modeling descriptions of patterns and monitoring student descriptions for accuracy and mathematical validity.

using rules to predict what comes next in a given number pattern

facilitating the organization of information about patterns using tables and charts.

identifying pattern rules.

facilitating the generalization of patterns, such as "What is the pattern?  Is it growing, shrinking, or repeating?  What will come next?"

using number patterns to solve real-world problems, such as skip-counting by 2s to find how many eyes 19 people have.

asking:

Why?

How do you know?

Will that always be the case?

 

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.

 

Additional Resources

For Mathematics Coaches

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

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