kindergarten common core math

KINDERGARTEN MATH

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PUTTING THE “HOW” BEFORE THE “WHAT”

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There are 25 sheep and 5 dogs in a flock. How old is the Shepherd?

There are 25 sheep and 5 dogs in a flock. How old is the Shepherd?

Three out of four students will give a numerical answer to this problem.

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There are 25 sheep and 5 dogs in aflock. How old is the Shepherd?

25 because a shepherd has the word sheepin it so you have to take away the dogs

and you just get 25. !

I dont haveenough info. I cant

answer this question !

If he started out with 2they reproduced it would take a year(about) for each to be born. And thesame with the dogs. If he started

at the age of about 18

What else is there?

Unpacking the Standards For Practice “I am somewhat familiar with the Practice Standards.”

Productive Struggle, Problem Solving, an Understanding“I am starting to think about teaching with rigor.”

Lesson Resources (prompts, checklists, rubrics)“I have a few resources t0 use across many lessons.”

Everyday Math & Problem Solving Resources “I know where to find problem solving resources.”

Unpacking theStandards For Practice

Contentwhat

Contentwhat

Practicehow

Practicehow

1 2 3 4

5 6 7 8

Make sense of problems & persevere in solving them

Reason abstractly & quantitatively

Construct viable arguments & critique the reasoning of

others

Model with mathematics

Use appropriate tools

strategically

Attend to precision

Look for & make use of structure

Look for & express

regularity in repeated reasoning


Activity

For each Practice Standard, write one student friendly “I can” statement that

clearly and concisely “summarizes” the standard.

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R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide Visit go.solution-tree.com/commoncore to download this page.

Table 2.2: Mathematical Practices—Look-Fors as Classroom Indicators

Mathematical Practice Look-Fors: Classroom Indicators

Mathematical Practice 1: Make sense of problems, and persevere in solving them.

Students:tasks

Teacher: Provides adequate time with formative feedback for students to discuss problem pathways and solutions with peers

Mathematical Practice 2: Reason abstractly and quantitatively.

Students: Are able to contextualize or decontextualize problems

Teacher: Provides access to and uses appropriate representations (manipulative materials, drawings, or online renderings) of problems and asks questions focused on determining student reasoning

Mathematical Practice 3: Construct viable arguments, and critique the reasoning of others.

Students: Understand and use prior learning in constructing arguments

Teacher: Provides opportunities for students to listen to or read the conclusions and arguments of others—as students discuss approaches and solutions to problems, the teacher encourages them to provide arguments for why particular strategies work and to listen and respond to the reasoning of others and asks questions to prompt discussions.

Mathematical Practice 4: Model with mathematics.

Students: Analyze and model relationships mathematically (such as when using an expression or equation)

Teacher: Provides contexts for students to apply the mathematics learned

Mathematical Practice 5: Use appropriate tools strategically.

Students: Have access to and use instructional tools to deepen understanding (for example, manipulative materials, drawings, and technological tools)

Teacher: Provides and demonstrates appropriate tools (like manipulatives)

Mathematical Practice 6: Attend to precision.

Students: Recognize the need for precision in response to a problem and use appropriate mathematics vocabulary

Teacher: Emphasizes the importance of precise communication, including appropriate use of mathematical vocabulary, and emphasizes the importance of accuracy and efficiency in solutions to problems, including use of estimation and mental mathematics, when appropriate

Mathematical Practice 7: Look for and make use of structure.

Students: Are encouraged to look for patterns and structure (for example, when using properties and composing and decomposing numbers) within mathematics

Teacher: Provides time for students to discuss patterns and structures that emerge in a problem’s solution

Mathematical Practice 8: Look for and express regularity in repeated reasoning.

Students: Reason about varied strategies and methods for solving problems and check for the reasonableness of their results

Teacher: Encourages students to look for and discuss regularity in their reasoning

Source: Adapted from Kanold, Briars, & Fennell, 2012.

“A bad curriculum well taught is invariably a better experience for students than a good curriculum badly taught: pedagogy trumps curriculum. Or more precisely, pedagogy is curriculum, because what

matters is how things are taught, rather than what is taught.” - Wiliam

Khan Academy Does Angry Birds

How does the video represent the PA Core Standards for Mathematical Practice?

Productive Struggle, Problem Solving, an Understanding

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“It is hard to think of allowing - much less planning for - the children in your

classroom to struggle. Not showing them a solution when they are experiencing

difficulty seems almost counterintuitive. If our goal is relational understanding, however, the struggle is part of the

learning, and teaching becomes less about the teacher and more about what the children are doing and thinking.” - VDW

“Teaching mathematics through problem solving generally means that children solve problems to

learn new mathematics, not just to apply mathematics after is has been learned.”

- VDW

Doing Problems vs. Problem Solving

Tatyana’s mother is decorating a cake for Tatyana’s fifth birthday but she only has green and blue candles. If she wants to use exactly 5 candles on the cake, how

many green and blue candles could she use.

“Most, if not all, important mathematics concepts or procedures can best be

taught through problem solving.” - VDW

“I know that8 + 2 = 10 because…”

!

“I don’t know 8 + 2, but 8 + 1 = 9, and

then I can add on 1 more to get the

answer, which is 10.” !

“It would make sense to add 8 and 2

when…”

What is an understanding?

vs.

“An understanding can never be ‘covered’ if it is to be understood.”

- Wiggins & McTighe

!

• Give the answer and ask for the problem.

• Replace a number in a given problem with

a blank or a question mark.

• Offer two situations or examples and

ask for similarities and differences.

• Create a question in which children have

to make choices. - VDW

A variety of strategies you can use to

create open questions include the

following:

Strategy Standard Question Open Question

Replace a number in agiven problem with ablank or a questionmark.

Offer two situations or examples and ask for similarities or differences

Create a question so that children have to make choices.

23 + 68 = _____

Draw a triangle.

What number is10 more than 25?

?3 + 6? = _____

How are these triangles the same? Different?

A number is 10 greater than another number.

What could the number be?

• Measure a length

• Perform a specified or routine procedure

• Evaluate an expression

• Solve a one-step word problem

• Retrieve information from a table or graph

• Recall, identify, or make conversions between and among representations or numbers (fractions, decimals, and percents), or within and between customary and metric measures

• Locate numbers on a number line, or points on a coordinate grid

• Solves linear equations

• Represent math relationships in words, pictures, or symbols

• Compare and contrast figures

• Provide justifications for steps in a solution process

• Extend a pattern

• Retrieve information from a table, graph, or figure and use it solve a problem requiring multiple steps

• Translate between tables, graphs, words and symbolic notation

• Select a procedure according to criteria and perform it

multiple steps and multiple decision points

• Generalize a pattern

• Describe, compare, and contrast solution methods

• Formulate a mathematical model for a complex situation

• Provide mathematical justifications

• Solve a multiple- step problem, supported with a mathematical explanation that justifies the answer

• Formulate an original problem, given a situation

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• Design a mathematical model to inform and solve a practical or abstract situation

NOTE: Level 4 requires applying one approach among many to solve problems. Involves complex restructuring of data, establishing and evaluating criteria to solve problems.

Level 4 (Extended Thinking) requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections-relate ideas within the content area or among content areas-and have to select one approach among many alternatives on how the situation should be solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.

Table 1: Math Descriptors - Combined Webb Depth of Knowledge Levels for Mathematics (Webb, 2002), NAEP 2002 Mathematics Levels of Complexity, and Other Descriptors Related to NECAP GLEs.

(M. Petit, Center for Assessment 2003, K. Hess, Center for Assessment, updated 2005)

Level 1 Recall

Level 2 Skills/Concepts

Level 3 Strategic Thinking

Level 4 Extended Thinking

Examples represent, but do not constitute all Level 1 mathematics performances:

• Recall or recognize a fact, definitions, or term

• Apply a well known algorithm

• Apply a formula

• Determine the area or perimeter of rectangles or triangles given a drawing and labels

• Identify a plane or three dimensional figure


• Classify plane and three dimensional figures

• Interpret information from a simple graph

• Use models to represent mathematical concepts

• Solve a routine problem requiring multiple steps, or the application of multiple concepts

• Compare figures or��VWDWHPHQWV


• Interpret information from a complex graph

• Explain thinking when more than one response is possible

• Make and/or justify conjectures

• Develop logical arguments for a concept

• Use concepts to solve problems

• Perform procedure with


• Relate mathematical concepts to other content areas

• Relate mathematical concepts to real-world applications in new situations

• Apply a mathematical model to illuminate a problem, situation

• ConduFt a project that specifies a problem, identifies solution paths,

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Level 4 (Extended Thinking) requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections-relate ideas within the content area or among content areas-and have to select one approach among many alternatives on how the situation should be solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.

Table 1: Math Descriptors - Combined Webb Depth of Knowledge Levels for Mathematics (Webb, 2002), NAEP 2002 Mathematics Levels of Complexity, and Other Descriptors Related to NECAP GLEs.

(M. Petit, Center for Assessment 2003, K. Hess, Center for Assessment, updated 2005)

Level 1 Recall

Level 2 Skills/Concepts

Level 3 Strategic Thinking

Level 4 Extended Thinking


• Recall or recognize a fact, definitions, or term

• Apply a well known algorithm

• Apply a formula

• Determine the area or perimeter of rectangles or triangles given a drawing and labels

• Identify a plane or three dimensional figure


• Classify plane and three dimensional figures

• Interpret information from a simple graph

• Use models to represent mathematical concepts

• Solve a routine problem requiring multiple steps, or the application of multiple concepts

• Compare figures or��VWDWHPHQWV


• Interpret information from a complex graph

• Explain thinking when more than one response is possible

• Make and/or justify conjectures

• Develop logical arguments for a concept

• Use concepts to solve problems

• Perform procedure with


• Relate mathematical concepts to other content areas

• Relate mathematical concepts to real-world applications in new situations

• Apply a mathematical model to illuminate a problem, situation

• ConduFt a project that specifies a problem, identifies solution paths,

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Is there any place for drill and practice?!

• Drill is only appropriate when: ! • The desired concepts have been

meaningfully developed • Students have already developed (not

mastered) flexible and useful procedures • Speed and accuracy are [eventually]

needed - VDW

Lesson Resources

Questioning!

Observations do not have to be silent. Probing into student thinking through the use of questions can provide better data and more insights to inform instruction. As you circulate around the classroom to observe and evaluate students’ understanding, your use of questions is one of the most important ways to formatively assess in each lesson phase. Keep the following questions in mind (or on a clipboard, index cards, or a bookmark) as you move about the classroom to prompt and probe students’ thinking:

• What can you tell me about [today’s topic]? • How can you put the problem in your own words? • What did you do that helped you understand the problem? • Was there something in the problem that reminded you of another problem we’ve

done? • Did you find any numbers or information you didn’t need? How did you know that

the information was not important? • How did you decide what to do? • How did you decide whether your answer was right? • Did you try something that didn’t work? How did you figure out it was not going to

work? • Can something you did in this problem help you solve other problems?

NAME: Sharon V.

Estimates fraction quantities

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AB

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FRACTIONS

MATHEMATICAL PRACTICES

Understands numerator/denominator

Area models

Set models

Use fractions in real contexts

Make sense of problems and perseveres

Models with mathematics

Uses appropriate tools

Used pattern blocks to show

2/3 and 3/6

Showing greater reasonableness

Stated problem in own words

Reluctant to use abstract models

Names

Lalie

Pete

Sid

Lakeshia

George

Pam

Maria

Topic: !Mental Computation Adding 2-digit numbers

Not There Yet CommentsOn Target Above and Beyond

Difficulty with regrouping

Flexible approaches used

Counts by tens, then adds ones

Beginning to add the group of tens first

Using a posted hundreds chart

3-20

3-18

3-24

3-24

3-20

Can’t do mentally

Has at least one strategy

Uses different methods with

different numbers

3-21

Observation Rubric Making Whole Given Fraction Part

Above and Beyond Clear understanding. Communicates concept in multiple representations. Shows evidence of using idea without prompting.

On Target Understands or is developing well. Uses designated models.

Not There Yet Some confusion or misunderstanding. Only models idea with help.

Fraction whole made from parts in rods and in sets. Explains easily.

Can make whole in either rod or set format (note). Hesitant. Needs prompt to identify unit fraction.

Needs help to do activity. No confidence.

Sally !

Latania !

Greg

John S. Mary

Lavant (rod) !

Julie (rod) !

George (set) !

Maria (set)

Tanisha (rod) !

Lee (rod) !

J.B. (set) !

John H. (set)

“You used the red trapezoid as your whole?” “So, first you recorded your measurements in a table?” “What parts of your drawing relate to the numbers from the story problem?” “Who can share what Ricardo just said, but using your own words?”

Clarify Students’ Ideas

“Why does it make sense to start with that particular number?” “Explain how you know that your answer is correct.” “Can you give an example?” “Do you see a connection between Julio’s idea and Rhonda’s idea?” “What if...?” “Do you agree or disagree with Johanna? Why?”

Emphasize Reasoning

“Who has a question for Vivian?” “Turn to your partner and explain why you agree or disagree with Edwin.” “Talk with Yerin about how your strategy relates to hers.”

Encourage Student-Student Dialogue

Examples of teacher prompts for supporting classroom discussions.

Got it Evidence shows that the student essentially has the target concept or idea.

Not Yet Student shows evidence of major misunderstanding, incorrect concept or procedure, or failure to engage the task.

4 Excellent: Full Accomplishment Strategy and execution meet the content, processes, and qualitative demands of the task. Communication is judged by effectiveness, not length. May have minor errors

3 Proficient: Substantial Accomplishment Could work to full accomplishment with minimal feedback. Errors are minor, so teacher is confident that understanding is adequate to accomplish the objective.

2 Marginal: Partial Accomplishment Part of the task is accomplished but there is lack of evidence of understanding or evidence of not understanding. Direct input of further teaching is required.

1 Unsatisfactory: Little Accomplishment The task is attempted and some mathematical effort is made. There may be fragments of accomplishment but little or no success. N

EED

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Everyday Math & Problem Solving Resources

Activity

Explore the Everyday Math series and the other resources that are available to you.

Select one activity and analyze it by responding to our 3 questions…


What else is there?

Thank You!


kindergarten common core math

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