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2021 Mathematics Standards:

High Level Changes, K-8

Visit www.dpi.wi.gov/math to access standards.

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Land Acknowledgement

https://dpi.wi.gov/amind/tribalnationswi

At the beginning of each of our webinars, we do a land acknowledgment to recognize and respect the Indigenous people and lands we are on. After millennia of Native history, and centuries of displacement and dispossession, acknowledging original Indigenous inhabitants is complex. Many places in Wisconsin have been home to different Native Nations over time, and many Indigenous people no longer live on lands to which they have ancestral ties. We directly recognize the American Indian nations and tribal communities’ past, present, and future as being the stewards of the land and their continued relationship to their traditional territories.” I acknowledge in (city) that I am on traditional XXX homeland.

Individuals should know the nearby native nations and work toward reconciliation where land has been taken and culture erased. I would invite you to share in the chat the land you are currently on and ask yourself What do I know about the of the land? Who were the original caretakers of the land? What are the impacts of colonialism? On the people? On the land? On natural resources?

The Wisconsin Department of Public Instruction (DPI) - American Indian Studies Program in partnership with CESA 12 is offering a unique opportunity to participate in a series of webinars to continue your journey of personal and professional development around First Nations Studies.

The monthly 2-hour webinar lecture series workshops will begin in November 2021 and continue through June 2022. The lecture series will feature various Native American scholars in the fields of history, literature, education, among other academic content areas. At each session, you will have the opportunity to hear from and learn from Indigenous authors and speakers.

https://drive.google.com/file/d/1UwK5WPzQrAgvkCmZbHq4jLdJuspa8wWJ/view?usp=sharing

https://login.myquickreg.com/register/event/event.cfm?eventid=30367

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Learning Facilitators:

Continuing to Share Our Stories

"As you venture into exploring how your own strengths can empower you to identify your own capabilities and, in turn, find strengths in your students, your proficiency as a change agent emerges" (Kobett and Karp 2020, 35).

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2021 Mathematics Standards Series

Webinar 1: Introduction to Wisconsin 2021 Standards for Mathematics

Webinar 2: High-Level Changes in the Standards - Part 1 (Grades K-8)

Webinar 3 : High-Level Changes in the Standards - Part 2 (High School)

Webinar 4: Aligning Curriculum to the 2021 Standards

Webinar 5: Supporting the 2021 Standards in the WI Mathematics Community

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Webinar Series Objectives

Gain a general understanding of the 2021 Wisconsin Standards for Mathematics
Be inspired to deepen your understanding of the new standards
Consider plans for your school/district/organization to support educators in understanding these revised standards within curriculum and instructional materials
Connect with the organizations in the WI mathematics community to create an awareness of opportunities for collaboration and further exploration
Center Wisconsin students throughout the work

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How to Use this Learning Resource

Learn from slides and slide notes. Consider learning collaboratively
Facilitate learning based on these slides for your colleagues
Make a copy and modify to meet local needs
Contact DPI with questions and feedback (see last slide)

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Foundations of the work:

Collaboration is key

Form a team that includes grade-level teachers, district mathematics specialist, educators with specific expertise about particular student groups, math coach, and C&I director

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Webinar 2 Outcomes

Make further connections to the WI Revised Mathematics Vision and its four components
Reflect on the 5 shifts as they relate to the K-8 revisions and educational equity
Understand the K-8 revisions at a high level
Introduce appendices as they relate to K-8 standards and a 2010-2021 Standards Comparison document
Plan for sharing/continuing learning with colleagues

Supplies:

Wisconsin Standards for Mathematics (2021)

Note T aking Tool

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Norms for Collaboration

Equity is central to the work

Stay engaged
Experience discomfort
Speak your truth and own the impact
Listen respectfully to all ideas
Expect and accept non-closure

I would invite you to take a moment to look at each one with a short pause for reflection. The revised standards are one component of the systems related to learning and teaching of mathematics across the state. I would like to make a call out to “stay engaged” in this second session on our journey together in understanding the Wisconsin Standards for Mathematics. While the standards may not appear to have significant changes stay engaged in hearing about how the standards can support our work in elevating students engagement and achievement in mathematics. We can look at numbers and we can listen to students, families, and other stakeholders to know there is room for improvement. Stay engaged in thinking about how standards support the work. If you are working through these modules through the recording it might be helpful to pause for a group discussion.

Discuss these norms. Consider:

What do the norms mean?
Which norms do you struggle with as an individual? How can the group support you?
Which norms are you strong with as an individual? How could this support the group?
How will the group check-in on their use of the norms?

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Boundaries for this Learning

This learning does:	This learning does not:
Overview revisions to the standards in terms of big ideas	Examine changes in the standards word-by-word
Emphasize learning and planning for action	Ask you to change what happens in your classroom tomorrow

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Meaningful Revisions

There’s been a slight change in plans.

OR

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Revisiting: WI Vision for Mathematics

vision

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Reflection: WI Vision

Question:

What does it mean for students to wonder, reason, and understand mathematics as they experience, interact and relate to the world?

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5 Shifts

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Broader Definition:

What does it mean to know and do math?

Subitizing
Recording Kindergarten Thinking
Estimation Strategies
Standard Algorithms

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Explicit Inclusion of Subitizing

Knowing how many without counting.

Perceptual Subitizing

Conceptual Subitizing

Our first example is about the explicit inclusion of subitizing in the 2021 WI Standards for Mathematics. Subitizing is all about knowing how many without counting. You see here that there are two types of subitizing. Perceptual subitizing is the first type, during which a quantity is seen and quickly named. Research shows that individuals are actually born with the ability to perceptually subitize very small quantities, but we must support and grow the ability or we lose it. Perceptual subitizing can be supported by first recognizing what are called regular patterns, like a finger pattern or a dot pattern that we often see on dice when playing games. Then later, irregular patterns can extend the ability to subitize and see the quantity in different arrangements. The second type of subitizing is conceptual subitizing. Conceptual subitizing is recognizing that a collection of objects is composed of smaller collections and then quickly combining those smaller collections. Again this is done without counting. It’s basically applying perceptual subitizing to see groups within groups. Experiences can be found with regular patterns - like on a bead rack or number frame where the groups might be highlighted by space or color change. Experiences can also be found in irregular patterns, like the snack of goldfish crackers you see on this slide. I might see a group of 3, another group of 3 and 1 more, so I know there are 7 crackers. Someone else may see a group of 4 and a group of 3 to help them know there are 7. Each individual subitizes in a way that is meaningful to them.

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Subitizing: Moving Beyond Counting by Ones

Helps extend counting and develop number meaning which supports number sense
Abilities to subitize develop through instructional experiences, not maturation
Starts with perceptual subitizing
Moves toward sophisticated conceptual subitizing
Goal is unitizing quantities supporting place value and

multiplicative thinking

Catalyzing Change, an important resource for the writing team, shares about the research and reasons why subitizing is important, especially for our youngest learners. Mathematical proficiency requires a strong understanding of numbers as quantities and the ability to reason with those quantities. At the same time many educators find that children, even in the upper grades count by ones as their “go to” strategy when solving problems. Including and emphasizing subitizing in learning and teaching helps students know from an early age that counting can be used to find “how many”, but knowing about groups can also be used to find “how many”. If the use of groups within groups begins early rather than an overreliance on counting by ones, students will build the foundation for unitizing quantities, that will support place value and multiplicative thinking as well as support overall number sense.

Catalyzing Change - EC and Elementary, pages 82-84

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Subitizing in the Standards

M.K.CC.B.5	Perceptual subitizing up to 5
M.K.OA.A.3	When composing and decomposing within 10, subitize objects anchored to 5
M.1.OA.C.5	When adding and subtracting within 20, use conceptual subitizing

Here you see how subitizing was added to the standards. The kindergarten Counting and Cardinality standard is a brand new standard. When you look at the actual standards document, you’ll notice that the cluster statement that standard 5 falls under was also revised. In the 2010 standards, Cluster statement B used to read “Count to tell the number of objects.” In the 2021 standards, the cluster statement reads simply “Tell the number of objects”. This opens up students ways of knowing mathematics to be more than counting.

The use of subitizing was added to an existing standard in Kindergarten, Operations and Algebraic Thinking. In 2010, it was a standard about the decomposition of numbers less than or equal to ten. The revised standard still includes this goal, but now also includes conceptual subitizing as students compose and decompose numbers of the same size.

Finally, a grade 1 OA standard was also expanded with coherence in mind. The 2010 standard was about “relating counting to addition and subtraction”. The revised standard is now about “using counting and subitizing strategies to explain addition and subtraction”. Again, the writing team carried the theme of not always having to count to know how many into grade 1.

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Recording Thinking in Kindergarten

2010

Record with a drawing

or equation

2021

Record with a drawing or numbers

Our second example of a broader definition of mathematics involves how kindergarteners are asked to record their thinking. There were a few kindergarten standards from 2010, either from the Operations and Algebraic Thinking domain or the Number and Operations in Base Ten domain, that included the language “record with a drawing or equation”. The 2021 standards broaden how kindergarteners can be successful in showing their understandings about number relationships by the standard using the language “record with a drawing or numbers”.There are many numeric ways to record thinking that come prior to the abstractness of equation writing and are very appropriate for kindergarten. Drawings - labeled with numbers like the pencil drawing you see here, or number bonds that use drawings, numbers or a combination of the two, are some examples of ways kindergarteners may successfully record their numeric thinking. Finally, while some kindergarteners MAY use equations to record their thinking, the new wording of these 2021 standards gives more space and time for our youngest students to deeply begin to understand the operations of addition and subtraction without using the equal sign in a potentially procedural way.

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Estimation Strategies

2010 Rounding Standards

3.OA.7
3.NBT.1
4.OA.3
4.NBT.3
5.NBT.4

2021 Additional Estimating Standards

M.3.OA.D.7
M.3.NBT.A.1
M.4.OA.A.3
M.4.NBT.A.3
M.5.NBT.A.4

Strategies:

Mental Math
Benchmark Numbers
Compatible Numbers
Rounding

As you see here on this slide, the 2010 standards in grades 3-5 included a specific mention of rounding in five different content standards. While rounding is certainly a helpful estimation strategy in some contexts, the 2021 standards now broaden these standards. Rather than “use place value understanding to round whole numbers”, students are now being asked to “generate estimates for problems in real-world situations, using strategies such as mental math, benchmark numbers, compatible numbers, and rounding”. These revisions go deep with the mathematics and use estimation strategies that make sense for the situation rather than a rote practice of rounding, especially to “any place” as one 2010 standard said, since rounding a number in the hundred thousands to the nearest ten, for example is rarely, if ever, helpful. I will also mention that these are not the only content standards that include estimation. The 2010 standards had many other standards that included estimation in different domains and these have also been maintained in the 2021 standards.

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Standard Algorithm

2010

Using the standard algorithm:

Grade 4 NBT: Add and subtract multi-digit whole numbers

Grade 5 NBT: Multiply multi-digit whole numbers

Grade 6 NS: Divide multi-digit whole numbers

Grade 6 NS: Add, subtract, multiply, and divide multi-digit decimals

Grade 4 NBT: Add and subtract multi-digit whole numbers using strategies or algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Grade 5 NBT: Multiply multi-digit whole numbers using strategies or algorithms based on place value, area models, and the properties of operations.

Grade 6 NS: Divide multi-digit whole numbers using strategies or algorithms based on place value, area models, and the properties of operations.

Grade 6 NS: Divide multi-digit decimals using strategies or algorithms based on place value, area models, and the properties of operations.

2021

Our next example to consider is standards that involved the use of the standard algorithm in the 2010 standards. When a grade 5 standard, for example, asked students to multiply multi-digit whole numbers using the standard algorithm, it was never the intention that the standard algorithm would be the only way in which students would solve these problems. Students weren’t being asked to leave all of their previous understandings behind. So in reality, these revised standards that you see on the right, are not really different, but they do now clarify the intent of these standards and explicitly broaden the definition of what it means to know and do mathematics. The goal of these standards from 4th to 6th grade is to have students use methods for solving problems that make sense for the context and numbers involved.

Some of you may have noticed that the fluency part of these standard was not included on this slide. The fluency discussion is considered a separate example and will be addressed a little later in this webinar.

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Grade 4: Like and Related Denominators

Like Denominators	3/8 + 2/8	2010 Grade 4 2021 Grade 4
Related Denominators	1/2 +1/4	2010 Grade 5 2021 Grade 4
Unlike Denominators	3/4 +5/6	2010 Grade 5 2021 Grade 5

Our final example of broadening what it means to know and do mathematic involves operating with fractions. In the 2010 standards, there was a clear distinction between when students would add and subtract with like denominators and when they would add and subtract with unlike denominators. As you can see on the slide, the change happened from grade 4 to grade 5. The 2021 revision of a fourth grade standard now extends the learning to include “like and related” denominators. If students spend an entire year adding and subtracting with fractions that have common denominators, avoid the thinking that is necessary when fractions don’t have the same denominator, misconceptions can easily be formed. Students think of just “keeping the same number on the bottom” and the operations can become very procedural. If carefully chosen problems with related denominators are also used, 4th graders will really see the depth of what is happening when adding and subtracting fractions. Time therefore, will not be needed in Gr. 5 to undo that unintended learning.

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Reflection: Broader Definition

Question:

How might the broader definition of what it means to know and do mathematics support student success?

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Clarity

Multiple Ideas Within One Standard
Fraction Language
Meaning of Fluency

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Multiple Ideas Within One Standard

M.1.OA.C.6 Use multiple strategies to add and subtract within 20.

Flexibly and efficiently add and subtract within 10 using strategies that may include mental images and composing/decomposing up to 10.
Add and subtract within 20 using objects, drawings or equations. Use multiple strategies that may include counting on; making a ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14) ; decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14) ; decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

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Unit Fractions

M.3.NF.A.1 Understand a unit fraction as the quantity formed when a whole is partitioned into equal parts and explain that a unit fraction is one of those parts (e.g., 1/4).

Understand fractions are composed of unit fractions, for example, 7/4 is the quantity formed by 7 parts of the size 1/4.

3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

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2/5 x 3 OR 3 x 2/5 ?

M.4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a whole number times a fraction.

Represent a whole number times a non-unit fraction (e.g., 3 x 2/5) using visual fraction models and understand this as combining equal groups of the non-unit fraction (3 groups of 2/5) and as a collection of unit fractions (6 groups of 1/5), recognizing this product as 6/5.

4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a)/b.)

The second fraction example that I would like to share with you is a fourth grade standard about the multiplication of fractions, specifically about multiplying a whole number times a fraction. Again this is an example of a revision that was done with clarity in mind. In 2010 the phrase “multiply a fraction by a whole number” was used to describe the mathematics. This was easily misinterpreted. If some educators looked at the phrase “a fraction by a whole number” and assumed it meant the fraction would come first in the expression, like ⅖ x 3, they would be mistaken. “A fraction by a whole number” actually means that it would be 3 x ⅖. Since fourth grade is the first time that students are multiplying with fractions, we want to support students’ conceptual thinking about 3 groups of ⅖, possibly relating this mathematics to repeated addition (⅖ + ⅖ + ⅖) and connect to other important work of the grade as they are focused in grade 4 on adding fractions with like or related denominators. But if an educator thinks that the phrase meant ⅖ x 3, then conceptually thinking of ⅖ of the quantity 3 is not quite as accessible for that initial work with multiplication of fractions. Again, this standard is maintaining the same important learning, but now is more clear.

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Fluency Means...

“Flexibly and efficiently add and subtract within 20.”

Fluency was another idea that benefited from clarity in the 2021 standards. The term fluently is used throughout grades K-7 in the 2010 standards and is often misunderstood to mean that the mathematics is done quickly. Fluency actually means that students are flexible, having a variety of strategies to approach problems, are accurate because they use the strategy in a way that makes sense mathematically, and are efficient as they implement their strategy without struggling. In order to be clear, the 2021 standards have changed the word “fluently” to the phrase “flexibly and efficiently”. Since accuracy is implied in all of the standards, accuracy was not included in the phrase that replaces fluently. And actually, if a student is truly flexible and efficient, then they are using an approach that makes sense to them and will be able to use reasoning to determine if their answer is accurate. This new phrase is consistent across the standards and now describes more accurately what fluency actually means.

This visual of the three post it notes is from Illustrative Mathematics which is considered to be high quality instructional materials as aligned to the 2010 standards so we can see there are already materials using the flexible and efficient definition of fluency.

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2021 Addition and Subtraction

Same meaning, new language.

M.K.OA.A.5	Flexibly and efficiently add and subtract within 5...
M.1.OA.C.6	Flexibly and efficiently add and subtract within 10...
M.2.OA.B.2	Flexibly and efficiently add and subtract within 20...
M.2.NBT.B.5	Flexibly and efficiently add and subtract within 100...
M.3.NBT.A.2	Flexibly and efficiently add and subtract within 1000...
M.4.NBT.B.4	Flexibly and efficiently add and subtract multi-digit whole numbers

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Multiplication and Division

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

M.3.OA.C.6 Use multiplicative thinking to multiply and divide within 100.

Use the meanings of multiplication and division, the relationship between the operations (e.g., knowing that 8 x 5 = 40, one could reason that 40 ÷ 5 = 8), and properties of operations (e.g., the distributive property) to develop and understand strategies to multiply and divide within 100.
Flexibly and efficiently use strategies, the relationship between the operations, and properties of operations to find products and quotients with multiples of 0, 1, 2, 5, & 10 within 100.

M.4.OA.D.6 Flexibly and efficiently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations [e.g., knowing that 7 x 6 can be thought of as 7 groups of 6 so one could think 5 groups of 6 is 30 and 2 more groups of 6 is 12 and 30 + 12 = 42 (informal use of the distributive property)].

Now let’s take a look at the multiplication and division standards. We’ll start with the third grade fluency standard, 3.OA.7. The 2010 standard shows students becoming fluent with all multiplication and division problems within 100, or what’s typically considered the basic facts, by the end of third grade. Other than two introductory standards in second grade about equal groups, this is the first grade level in which the conceptual understandings of multiplication and division are found. That means that students only have one academic year to understand these operations deeply and become fluent. If we compare that to addition and subtraction, students develop their understanding of those operations across kindergarten, grade 1 and grade 2 before fluency is expected.

With that in mind, our 2021 revised standards first of all clarified the third grade standard by creating a part a and part b. Part a has language about “using the meanings of multiplication and division to develop and understand strategies to multiply and divide” which were important to explicitly articulate and then part b speaks to becoming flexible and efficient. You’ll notice that the revised third grade standard looks for flexibility and efficiency with the products and quotients of multiples of specific factors. These factors were chosen carefully so that students will leave third grade with the building blocks they will need so that the rest of their fluency can be built on the relationships to these earlier factors.

Please notice that although the factors are listed in numerical order, that does not imply an instructional sequence. The standards do not recommend that students begin with fluency of 0 as a factor.

So the 2021 standards are providing more space and time for students to dig deeply into the relationship between multiplication and division and the properties associated with them. This is especially true of the distributive property that is so powerful in developing flexibility and efficiency. The fluency is then continued in grade 4 with a brand new standard in its own cluster. That standard brings in the rest of the products and quotients within 100 so that students develop flexibility and efficiency with all basic facts by the end of grade 4.

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Final Fluency Progression

M.5.NBT.B.5	Flexibly and efficiently multiply multi-digit whole numbers...
M.6.NS.B.2	Flexibly and efficiently divide multi-digit whole numbers...
M.6.NS.B.3	Flexibly and efficiently add, subtract, multiply, and divide multi-digit decimals...
M.7.EE.B.4	Flexibly and efficiently apply the properties of operations and equality to solve equations of the form px + q = r and p(x + q) = r...

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Omissions

1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations

8 + ? = 11, 5 = _ - 3, 6 + 6 = _.

3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

Already in standards M.1.OA.A.1

and M.3.OA.A.3

The last example I would like to share with you today around clarity are omissions. Here you see two parallel standards. One is about determining the unknown whole number in an addition or subtraction equation and the other is about that same idea but with multiplication and division. These standards were omitted since the relationship between the 3 whole numbers in an addition or subtraction equation can actually be fully considered while solving the various problem situations associated with the first grade OA standard you see listed here in the blue box. The same is true for third grade. The standard you see here was omitted since the 3 whole numbers in a multiplication or division equation can be fully considered while solving the various problem situations in the third grade OA standard listed.

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Reflection: Clarity

Task/Question(s):

How might the clarity that has been brought to the K-8 standards support student success?
Watch this linked interview between Jo Boaler and mathematician, Francis Su. What do you find interesting about Francis’ view on flexibility?

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From Maria Hernandez Modeling Webinar 5.5.21�North Carolina School of Science and Mathematics

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Mathematical Modeling:

An Iterative Process

Adapted from Ontario Curriculum and Resources, 2020

K-8 Process

This brings us to mathematical modeling. I would like you to think about that garden pathway you just saw on the right. The one that meandered, felt real, and was leading you to an unknown destination. That is the essence of mathematical modeling.

True mathematical modeling provides authentic connections to real-life situations. It is an iterative process that starts with messy real-life problems that may have several different solutions that are all correct. Mathematical modeling requires the modeler to be critical and creative as well as make choices, assumptions, and decisions. Through this process, students create a mathematical model that describes a situation using mathematical concepts and language. The model can be used to solve a problem or make decisions and also can be used to deepen understanding of mathematical concepts.

Mathematical modeling, as I said, is an iterative process, meaning that students can move between and across, as well as return to each of the components within the process. So this is not a linear process. The K-8 visual you see here shows paths that students may take, but the arrows are certainly not meant to limit where the modeling process takes someone.

The Standards for Mathematical Practice come into play throughout mathematical modeling, supporting students’ social-emotional development. Along with this idea, modeling also helps move students toward both a positive mathematical identity and a strong sense of agency as thinkers and doers of mathematics while they grapple with problems that matter to them and find places within the process to showcase their strengths.

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Mathematical Modeling vs. Mathematical Model

Mathematical Modeling: An iterative and interconnected process of using mathematics to represent, analyze, make predictions (future), and provide insight (past or present) into real-life situations.

Mathematical Model: A representation of a problem, situation, or system using mathematical concepts. For example, a number line is a model to show the order and magnitude of numbers.

Mathematical Modeling can fall into both of our revision categories from this webinar. This slide can help us think about the clarity the 2021 standards bring to mathematical modeling. What is mathematical modeling vs. what is a mathematical model can be often misinterpreted. Throughout a student’s mathematics education, the word ‘model’ is used in many ways. Manipulatives as models, demonstrations as modeling, role modeling, and conceptual models of mathematics are a few of the ways and they are valuable tools for teaching and learning. However, they are different from the practice of mathematical modeling. Mathematical modeling, whether in the community, the workplace, or in school, uses mathematics to answer big, messy, reality-based questions.

Now I’ll turn things over to Mary to make some connections to the actual revisions that were made around mathematical modeling and how they can support a broader definition of what it means to know and do mathematics.

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Mathematical Modeling:

An Iterative Process

Adapted from Ontario Curriculum and Resources, 2020

K-8 Process High School Process

This diagram, with its four components, offers several supportive features for teachers of students in kindergarten through grade eight:

The diagram includes components that use familiar language to describe the process.

The K-8 diagram of the modeling process supports our youngest learners and their teachers as they engage in the modeling process as a natural extension of noticing and wondering about situations they come across daily, both in school and outside the classroom. They Understand the Problem, Analyze the Situation, Create a Model, and Analyze and Assess the Model. Then within each of the components, educators and their students dig into what is included in that part of the modeling process. For example, Analyze the Situation includes making assumptions (or thinking about “what matters”) and defining variables (or thinking about what changes and remains the same).

The visual clearly differentiates the action of mathematical modeling from the product of a mathematical model.

This diagram shows the majority of the interactive process of modeling happening within the real-life situation and then the creation of a mathematical model lying outside of that situation, but clearly connected to the process. It is also important to note that mathematical models can often be used outside of mathematical modeling, when they are not part of this iterative process, but instead are used to simply represent or solve a problem.

The components are not numbered.

Not numbering the modeling components reinforces that the process is not linear or experienced step by step. Modeling starts with understanding the problem, but after that students may move back and forth between components as their thinking and findings lead them. This back and forth will continue until the modelers are satisfied that their solution will help their audience.

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Wasn’t Modeling Already in the Standards?

2010 Standards

K-12: Standard for Mathematical Practice 4 (SMP4)
High School Conceptual Category
HS Modeling Standards (*)

2021 Standards

K-12: Standard for Mathematical Practice 4 (SMP4)
Grade Band SMP4
K-8 Introductions
K-12 Progression
High School Conceptual Category
MS and HS Modeling Clusters (M)

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Mathematical Modeling

“...one of the most important uses of modeling in the classroom is for the experience of creating a model all one’s own” (Godbold, Malkevitch, Teague, and van der Kooij 2016, 54).

The quote you see here is from the GAIMME Report. When I think of this quote, it reminds me of the strong connections between mathematical modeling and equitable teaching. Mathematical modeling can strengthen connections between students’ lived experiences and mathematics, providing opportunities to develop students’ appreciation for the role mathematics plays in real life. Students have the opportunity to solve problems that are important to them and use mathematics authentically when working toward a solution.

GAIMME Report:

Godbold, Landy, Joe Malkevitch, Dan Teague, and Henk van der Kooij. 2016. “High School.” In Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME). 45-70. United States of America: Consortium for Mathematics and Its Applications (COMAP, Inc.) and Society for Industrial and Applied Mathematics (SIAM).

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A K-12 Progression

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Reflection: Mathematical Modeling

Question(s):

What does it mean for students to experience mathematical modeling?
How might a K-12 progression support student success with mathematical modeling? With mathematics overall?

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Appendix 1: Tables

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Appendix 2: Glossary

“Children build their own “working definitions” based on their initial experiences … the concepts will become more precise, and the vocabulary with which we name the concepts will, accordingly, carry more precise meanings. Formal definitions generally come last” (Education Development Center 2020).

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2010 - 2021 Comparison

K-8 Version available today
Shows word-level changes in grade-level standards
To be used as a scaffold beside existing curriculum
Of short term value

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2010 - 2021 Comparison

K-8 Version available today
Shows word-level changes in grade-level standards
To be used as a scaffold beside existing curriculum
Of short term value

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Learning with Colleagues

Copy these slides and notes. Modify to meet local needs.

Who needs to be part of the planning team?
Who needs to learn the information?
When will the learning happen? What existing systems and structures for professional learning can you use?
How will you know if the learning was successful?
What steps will you take if more learning needs to happen?
How will you plan for changing staff (new teachers)?

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Ongoing Learning

Webinar 1: Introduction

Webinar 2: High-Level Changes - K-8

Webinar 3: High-Level Changes - HS

Webinar 4: Aligning Curriculum

Webinar 5: Support - Our Community

Virtual Office Hours

4:00 - 5:00 PM

October 27
November 23
December 8
February 2

Click here for Archived Webinars and PL Modules

Click here for Virtual Office Hours Flyer

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Contact Information

Julie Bormett, Mathematics Consultant

julie.bormett@dpi.wi.gov | 608-266-7921

Mary Mooney, Mathematics Consultant

mary.mooney@dpi.wi.gov | 608-266-9368