moving forward

Moving Forward

Common Core in the

Next Generation

MaTHink Conference

Application

Today’s Agenda0 Become familiar with why math is so important in the

foundational years (K-2)

0 Understand cardinality

0 Introduce strategies that develop cardinality

0 Learn more about the Next Generation Science Standards (and how to read them)

0 Become familiar with how the Science and Math standards overlap

Norms

Silence all cell phones Step outside to make or take a call Keep sidebar conversations to

a minimum Have an open mind Participate fully!

Math Requires:0 Flexibility to other possibilities0 Logical thought0 Systematic thinking0 Abstraction0 Problem solving0 Complex thinking

Mathematical Practice 1

Math Key FluenciesGrade Required Fluency K Add/subtract within 5 1 Add/subtract within 10 2 Add/subtract within 201

Add/subtract within 100 (pencil and paper)

3 Multiply/divide within 1002 Add/subtract within 1000

4 Add/subtract within 1,000,000 5 Multi-digit multiplication 6 Multi-digit division

Multi-digit decimal operations 7 Solve px + q = r, p(x + q) = r 8 Solve simple 2x2 systems by

inspection

Progression: Where are we going with this?

Grades K – 5

Number & Operations: Base Ten

Grades 3 – 5

Number & Operations:

Fractions

Grades 6 – 8

The Number System

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Kinder• Counting sequence• Work w/ 11-19

focusing on place value

First• Extend counting

sequence• Extend place value

understanding

Second• Understand Place Value• Use Place Value when

adding and subtracting

Children’s Exposure to Number

0 Calendar0 Telephone numbers0 Clocks0 Measurement0 Money0 Concrete examples (length)

What are numbers anyway??0 Numbers are an infinite set

that are highly systematic which makes them useful as identifiers

Role Playing Vs. Cardinality

Role Playing0 Don’t trust quantities0 Unsure about the different “looks” 0 Unsure about quantity that is

hidden0 Focus on “counting right”0 Unable to determine if a quantity

is reasonable

Cardinality0 Understand the last number

counted tells how many there are in a set

0 Can use counting to match and find a set

0 Usually occurs by 4 to 4-1/2

Counting is more than reciting a rote sequence and recognizing numerals. Counting is finding out “how many.”

So what, if a student doesn’t have cardinality?

What skills did Cena have?

What skills did Cena Lack?

Cena is probably in what grade?

In what grade should we have intervened to remedy this type of misunderstanding?

What is Number Sense? “ …good intuition about numbers and their

relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Howden)

https://www.teachingchannel.org/videos/mingle-count-a-game-of-number-sense

Fluency Has 3 Components

Fluency- 1. Efficiency- ability to

work fairly quickly2. Accuracy-ability to

get the correct answer3. Flexibility- multiple

solution strategies determined by the problem

Fluency is the by-product of flexibility. Assessing fluency by occasionally using timed tests is acceptable. Using timed tests as an instructional tool to build fluency is ineffective.

Children’s progression to make sense of the formal symbolism we use in mathematics. (Bruner)

Enactive-using tangible items to model the problem (the Rekenrek, cubes, acting it out, etc)

Iconic-representing what they did in the enactive phase with an icon (tally marks, circles, etc. on paper)

Symbolic-writing the formal signs and symbols

Best PracticesA strategies review

A mathematician, like a painter or a poet, is a maker of patterns.

If his patterns are more permanent than theirs, it is because they are made with ideas.

Godfrey Harold HardyA Mathematician’s Apology

Brain Research Effecting Teaching and Learning (Sousa)

Creating and using conceptual Subitizing patterns help young students develop the abstract number and arithmetic strategies they will need to master counting.

Information is most likely to be stored if it makes sense and has meaning.

How We Learn Best

Memorize this eleven digit number:

25811141720

Now look for connection (relationship) within numbers

2 5 8 11 14 17 20

Brain Research cont’d

Too often, mathematics instruction focuses on skills, knowledge and performance but spends little time on reasoning and deep understanding.

Just as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisite for succeeding in mathematics.

Carol DweckThere is a growing body of evidence that students’

mindsets play a key role in their math and science achievement.

Fixed mindset vs. growth mindset

Essential Number Relationships

Spatial Relationship- recognizing how many without counting by seeing a visual pattern

One/Two More or Less- knowing which numbers are one/two more or less than any given number

Benchmarks of 5 & 10- how any given number related to 5 and 10

Part-Part-Whole- ability to conceptualize a number as being made up of two or more parts

Van De Walle, 2006

Using manipulatives effectively?

How many cubes are there? How did you know?How does this support development of number

relationships?

What does the Frame add in Building Essential Number Relationships

How did adding the frame support numeracy ?

Spatial Relationship

One/Two More or Less

Benchmarks of 5 & 10

Part-Part-Whole

What skills does this support?

How many now?“Tasks that encourage students to think in collections provide rich opportunities for them to construct abstract mathematical relationships and become powerful problem solvers.”

-Wheatly & Reynolds, 2010

How did adding the pattern support numeracy ?




Part-Part-Whole

The Rekenrekis a tool developed at the

Freudenthal Institute in the Netherlands by Adrian Treffers to support the natural mathematical development of children

in Dutch means “calculating frame” or “arithmetic rack.”

looks like a counting frame but is designed to move children away from counting each bead.

looks like an abacus but it is not based on place value.

Rekenreks

The Rekenrek has a built-in structure that encourages children to use their knowledge about numbers instead of counting one to one.

The built-in structure allows children the flexibility to develop more advanced strategies as well.

Tournaki Study

“The rekenrek, in addition to being a manipulative, also acts as a facilitator of

knowledge while students develop efficient thinking strategies. This process supports Gravemeijer’s (1991) argument that the materials themselves cannot

transmit knowledge: The learner must construct it.”

Building Early Numeracy

“What do you notice?” explore the tool learn the built in structure before you have them use

the tool. “Show Me”

Have them show a certain number.

Some may count one-by-one The structure of the tool

allows for more advanced strategies.

Flash forward Show the Rekenrek of a

certain number (1-10) for a few seconds

Have them determine which number was flashed.

When first starting allow enough time that children who need to can still count one-by-one.

Gradually shorten the time to encourage children to see groupings.

Growing Numeracy

Just One MoreShow a certain numberKids build theirs to

show“One More”“Two Less”

Peek-a- BooPush some beads over Cover the rest“How many are hiding?”

Number Talks ReviewRead this side

Turn and Talk: What are all the possible ways children will figure out how many?

How did the rekenrek support numeracy ?




Part-Part-Whole

Adding Context:

Attendance chartBunk bedsDouble-decker busBookshelvesA parking lotEgg cartons

Math Practice 4: Model with Mathematics… “mathematical meaning in their lived world”

Taking AttendanceHow many children are here today?

How did you figure it out?

The Double-Decker Bus

Math Practice 2: Reasoning Abstractly and Quantitatively… “Contextualizing and Decontextualizing”

Passenger Pairs matching game:

Moving from the bus story to a model of the context

Focus on Essential Number Relationships in the Classroom

0How did the teacher keep all the students engaged and on task?

0How did the teacher reinforce:0 Spatial Relationships0 One more/ two more0 Benchmarks of 5 & 100 Part-Part- Whole

Focus on RelationshipsWhen we focus on relationships, it helps

give children flexibility when dealing with their basic facts and extending their

knowledge to new task. When we build a child’s number sense it promotes thinking

instead of just computing.

The Vision

“Standards should be organized as progressions that support students’ learning over multiple grades. They should take into account how students’ command of the concepts, core ideas, and practices becomes more sophisticated over time with appropriate instructional experiences.” (NRC 2011, Rec 7)

Science and Engineering Practices

Disciplinary Core Ideas

Crosscutting Concepts

An Analogy between NGSS and a Cake

Baking Tools & Techniques(Practices)

Cake(Core Ideas)

Frosting(Crosscutting Concepts)

Baking a Cake(Performance Expectation)

Conceptual Shifts in the NGSS1. K–12 Science Education Should Reflect the Real World

Interconnections in Science

2. Using all Practices and Cross-Cutting Concepts to Teach all Core Ideas All Year.

3. Science Concepts Build Coherently across K-12

4. The NGSS Focus on Deeper Understanding and Application of Content

5. Science and Engineering are Integrated into Science Education K-12

6. Science Standards Coordinate with English Language Arts and Mathematics Common Core State Standards

Based on work by Tina Chuekell.stanford.ed

u

Math Science

ELA

M1: Make sense of problems and persevere in solving them

M2: Reason abstractly & quantitatively

M6: Attend to precisionM7: Look for & make

use of structureM8: Look for &

make use of regularity in repeated reasoning

S1: Ask questions and define problems

S3: Plan & carry out investigationsS4: Analyze & interpret data

S6: Construct explanations & design solutions

M4. Models with

mathematicsS2: Develop & use modelsS5: Use mathematics &

computational thinking

E1: Demonstrate independence in reading complex texts, and writing and speaking about them

E7: Come to understand other perspectives and cultures through reading, listening,

and collaborations

E6: Use technology & digital media strategically & capably

M5: Use appropriate tools strategically

E2: Build a strong base of knowledge through content rich texts

E5: Read, write, and speak grounded in evidence

M3 & E4: Construct viable arguments and critique reasoning of others

S7: Engage in argument from

evidence

S8: Obtain,

evaluate, &

communicate information

E3: Obtain, synthesize,

and report findings clearly and effectively in response to task and purpose

Commonalities Among the Practices in Science, Mathematics and English Language Arts

Practices in Mathematics, Science, and English Language Arts*Math Science English Language Arts

M1. Make sense of problems and persevere in solving them.

M2. Reason abstractly and quantitatively.

M3. Construct viable arguments and critique the reasoning of others.

M4. Model with mathematics.

M5. Use appropriate tools strategically.

M6. Attend to precision.

M7. Look for and make use of structure.

M8. Look for and express regularity in repeated reasoning.

S1. Asking questions (for science) and defining problems (for engineering).

S2. Developing and using models.

S3. Planning and carrying out investigations.

S4. Analyzing and interpreting data.

S5. Using mathematics, information and computer technology, and computational thinking.

S6. Constructing explanations (for science) and designing solutions (for engineering).

S7. Engaging in argument from evidence.

S8. Obtaining, evaluating, and communicating information.

E1. They demonstrate independence.

E2. They build strong content knowledge.

E3. They respond to the varying demands of audience, task, purpose, and discipline.

E4. They comprehend as well as critique.

E5. They value evidence.

E6. They use technology and digital media strategically and capably.

E7. They come to understanding other perspectives and cultures.

* The Common Core English Language Arts uses the term “student capacities” rather than the term “practices” used in Common Core Mathematics and the Next Generation Science Standards.

Practices in Math, Science, and ELA*

Scientific and Engineering Practices

1. Asking questions (for science) and defining problems (for engineering)

2. Developing and using models

3. Planning and carrying out investigations

4. Analyzing and interpreting data

5. Using mathematics and computational thinking

6. Constructing explanations (for science) and designing solutions (for engineering)

7. Engaging in argument from evidence

8. Obtaining, evaluating, and communicating information

Life Science Physical ScienceLS1: From Molecules to Organisms:

Structures and Processes

LS2: Ecosystems: Interactions, Energy, and Dynamics

LS3: Heredity: Inheritance and Variation of Traits

LS4: Biological Evolution: Unity and Diversity

PS1: Matter and Its Interactions

PS2: Motion and Stability: Forces and Interactions

PS3: Energy

PS4: Waves and Their Applications in Technologies for Information Transfer

Earth & Space Science Engineering & TechnologyESS1: Earth’s Place in the Universe

ESS2: Earth’s Systems

ESS3: Earth and Human Activity

ETS1: Engineering Design

ETS2: Links Among Engineering, Technology, Science, and Society

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Disciplinary Core Ideas

High School Grades — those not checked are offered in physical sciences, physics, or chemistry.

http://stem.browardschools.com/science/science_general/science-big-ideas-in-k-8/

http://stem.browardschools.com/science/science_general/science-big-ideas-in-k-8/

Closer Look at a NGSS (Grade 2)

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2.PS1 Matter and Its Interactions Students who demonstrate understanding can:

2-PS1-1. Plan and conduct an investigation to describe and classify different kinds of materials by their observable properties. [Clarification Statement: Observations could include color, texture, hardness, and flexibility. Patterns could include the similar properties that different materials share.]

The performance expectations above were developed using the following elements from the NRC document A Framework for K-12 Science Education:


Disciplinary Core Ideas Crosscutting Concepts

Planning and Carrying Out Investigations Planning and carrying out investigations to answer questions or test solutions to problems in K–2 builds on prior experiences and progresses to simple investigations, based on fair tests, which provide data to support explanations or design solutions. • Plan and conduct an investigation

collaboratively to produce data to serve as the basis for evidence to answer a question. (2-PS1-1)

PS1.A: Structure and Properties of Matter • Different kinds of matter exist and

many of them can be either solid or liquid, depending on temperature. Matter can be described and classified by its observable properties. (2-PS1-1)

Patterns • Patterns in the natural and

human designed world can be observed. (2-PS1-1)

Connections to other DCIs in this grade-level: will be available on or before April 26, 2013. Articulation of DCIs across grade-levels: will be available on or before April 26, 2013 Common Core State Standards Connections: will be available on or before April 26, 2013. ELA/Literacy – Mathematics –

Note: Performance expectations combine practices, core ideas, and crosscutting concepts into a single statement of what is to be assessed. They are not instructional strategies or objectives for a lesson.

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Kindergarten Performance Task Ideas:Weather Forecast

0 Make a chart showing what the weatherman said the weather is supposed to be for each day of the week.

0 You should tell the predicted temperature, rain or snow fall, and if it's supposed to be windy. (You will need to watch a 7 day forecast.)

0 Keep track: 0 What was the actual temperature? 0 Did it rain or snow? Was it windy? 0 Was the weatherman right? 0 Were the predictions (guesses) better in the

beginning of the 7 day forecast?0 How close were they? 0 This activity can be done as a whole class

keeping one chart. Local weather reports can be pulled up on the Internet and shown to the entire class.

0 You can also add variables by breaking the students in groups and showing different forecasts to see whose are more accurate.

1st Grade Performance Task0 Create different drums using different

materials for each drum. Please be sure that the drums make different sounds. When finished, all groups will play their drums for the class and explain what materials were used.

0 Keep track: 0 What was the diameter or circumference?0 How long did the vibration last?0 How far away can the sound be detected?0 What factors can you change about the

drum?0 Did diameter/ circumference affect the

volume of the drum? 0 Chart which drum students preferred?0 Create a bar graph of materials used.

2nd Grade Performance Task0 Choose a vegetable that your team would like to

grow in your school. Before you decide on a vegetable think about:

0 Keep Track0 How much light does my plant need to grow?0 How much water does my plant need to grow?0 What kind of soil will be best for my plant grow?0 How does the climate where you live affect the plants?0 What else should you think about to help your plant

grow?

0 Once you have decided on a vegetable and a good place for it, plant your vegetable.

0 Keep Track0 Make sure you keep track of how much water or plant

food you give it? 0 How many days it took to grow through the soil?0 How much it grows each week? 0 You may want to keep track of this data on a classroom

chart so everyone can see your plant's growth.

moving forward

Documents

number operations

progression of standards

number sense

number system domain

generation science standards

sequenceextend place

place valuefirstextend

digit decimal operations