Lecture Powerpoint 13

Division

• Division is the focus of this session, exploring its relationship with multiplication and various problem-solving strategies.

Agenda

• The agenda encompasses division, including different types of division problems, the connection between multiplication and division, and strategies for solving division problems. Also included are math identity and agency, and growth mindset. These topics delve into how students perceive themselves as math learners and how their beliefs affect their learning approach and outcomes.

Learning Intentions

• Develop an understanding of the relationship between multiplication and division, recognizing division as the inverse operation of multiplication.

• Understand how a student’s mathematical identity and agency affect their learning. This involves exploring how students' beliefs about themselves as math learners and their ability to take action influence their engagement and success in mathematics.

• Understand how mindset can affect a child’s learning, differentiating between fixed and growth mindsets and their impact on learning and problem-solving.

Developing Division Skills

• Students will perform better with division if they possess:

  • A conceptual understanding of multiplication (i.e., understanding the “groups of” concept), recognizing that multiplication involves combining equal groups.

  • Fluency in multiplication basic facts, allowing for quick recall of multiplication facts to aid in division.

  • An understanding of the connection between multiplication and division, viewing division as the process of finding a missing factor in a multiplication problem.

Connecting Multiplication and Division Visually

• Draw a representation for the multiplication expression 5 \times 6. This could be an array, equal groups, or another visual model to represent the product.

• Discuss with a partner how a division situation can be seen within the picture. For example, how can the visual representation be used to solve 30 ÷ 5 or 30 ÷ 6?

Counting and Constructing Equal Groups

• This section focuses on problem situations for multiplication and division, emphasizing the importance of understanding different problem types to develop flexible problem-solving skills.

• Three different problem types:

  • Unknown Product: Same amount of something for each object (e.g., ears on bears). Make and count equal-sized groups (multiplication), understanding that multiplication involves finding the total number when you know the number of groups and the size of each group.

  • Group Size Unknown: Dealing out to find how many in each group; this is “Fair share” division (partitive division), recognizing that partitive division involves dividing a total into a known number of groups to find the size of each group.

  • Number of Groups Unknown: “Scooping out” a certain-sized group to find how many groups there are (measurement or quotative division), understanding that quotative division involves dividing a total into groups of a known size to find the number of groups.

Make and Count Equal Sized Groups

• Example: Children are playing with blocks and cars, making a garage. They park 3 cars in each space. There are 4 parking spaces. How many cars are in the garage?

Construct Groups and Count the Size of the Equal Group (Fair Share or Partitive Division)

• Example: The children have 12 cars and 4 parking spots. If there are the same number of cars in each parking spot, how many cars are in each parking spot?

  • We know the total (12 cars).

  • We know the number of groups (4 parking spots).

  • We don’t know the size of each group (fair share the 12 cars to find out).

Construct the Size of Group and Count Number of Groups (Measurement or Quotative Division)

• Example: There are 12 cars. We park three cars in each parking spot. How many parking spots will we need?

  • We know the total (12 cars).

  • We know the group size (3 cars).

  • We don’t know how many groups (scoop out three cars for each parking spot).

Multiplication/Division Story Situations

• Table 3 showcases multiplication and division situations including:

  • Equal Groups of Objects

  • Arrays of Objects

  • Compare

• Each category includes scenarios for Unknown Product, Group Size Unknown/Unknown Factor, and Number of Groups Unknown/Multiplier Unknown.

• Notes:

  • Equal groups problems can be stated in terms of columns, exchanging the order of A and B, to describe the same array.

  • In row and column situations, number of groups and group size are not distinguished.

  • Multiplicative Compare problems appear first in Grade 4 with whole-number values.

  • In Grade 5, unit fractions language may be used.

Structure of Contexts

• Values (Known and Unknown)

• Contextual Meaning

• Structural Meaning

• Number of Groups

• Size of Each Group

• Total Amount

• Equation: (Number \space of \space Groups) \times (Size \space of \space Each \space Group) = Total \space Amount

Example Problem: Cookies

• Aneeq baked 3 pans of cookies. Each pan has 12 cookies. How many cookies did Aneeq bake?

  • Values: 3, 12, ?

  • Contextual Meaning: Number of pans of cookies, Cookies per pan, Cookies baked total

  • Structural Meaning: Number of Groups, Size of Each Group, Total Amount

Example Problem: Markers

• Jet had 4 boxes of markers. Each box had 6 markers in it. How many markers did Jet have in all?

  • Values: 4, 6, ?

  • Contextual Meaning: Number of boxes of markers, Markers per box, Markers in total

  • Structural Meaning: Number of Groups, Size of Each Group, Total Amount

Example Problem: Halloween Candy

• Gloria brought Halloween candy to share with her friends at school. She has 40 pieces which she wants to share equally between 5 friends. How many pieces of candy will each friend get?

  • Values: 5, ?, 40

  • Contextual Meaning: Number of friends, Pieces of candy per friend, Candy in total

  • Structural Meaning: Number of Groups, Size of Each Group, Total Amount

Example Problem: Apples

• Caroline has 32 apples. She wants to put 8 apples in each basket. How many baskets does she need?

  • Values: ?, 8, 32

  • Contextual Meaning: Number of baskets, Apples per basket, Apples in total

  • Structural Meaning: Number of Groups, Size of Each Group, Total Amount

Word Problem Posing

• Pose a word problem for 48 ÷ 4 = ?

• Determine which representation best matches the problem situation.

• Did you pose an “equal sharing” or “fair shares” dealing out problem situation (Partitive division)?

• Did your problem situation involve repeatedly measuring out or packaging a group of a specific size? (Measurement or quotative division)

Partitive vs. Measurement Division

• Partitive Division: “know the number of partitions”

• Measurement Division: “know the size to measure out”

• 48 ÷ 4 = 12 (total amount ÷ number of shares = size of each share/group)

• 48 ÷ 4 = 12 (total amount ÷ size of each share/group = number of shares/groups)

WI State Standards for Multiplication and Division (Grade 3)

• Focus on Operations and Algebraic Thinking (3.OA).

• Standards include:

  • 3.OA.A.1: Interpret products of whole numbers, focusing on understanding the meaning of multiplication as combining equal groups and using visuals to support this understanding.

  • 3.OA.A.2: Interpret whole-number quotients of whole numbers, focusing on understanding the meaning of division in terms of fair sharing and measurement.

  • 3.OA.A.3: Use multiplication and division within 100 to solve word problems, ensuring problems are authentic and relevant to students' lives.

  • 3.OA.B.4: Apply properties of operations as strategies to multiply and divide, exploring different strategies to solve complex problems.

  • 3.OA.B.5: Understand division as an unknown-factor problem, linking division to multiplication and reinforcing the inverse relationship between these operations.

  • 3.OA.C.6: Use multiplicative thinking to multiply and divide within 100, encouraging flexible thinking and diverse problem-solving approaches.

  • 3.OA.D.7: Solve two-step word problems using the four operations, building on single-step problems to develop more advanced problem-solving skills.

  • 3.OA.D.8: Identify arithmetic patterns and explain them using properties of operations, promoting reasoning and justification in mathematics.

Developing Division Skills (Reiterated)

• Conceptual understanding of multiplication.

• Fluency in multiplication basic facts.

• Understanding of the connection between multiplication and division.

Math Identity, Agency, and Identity Affirming

• Topics include:

  • Mathematics Identity, focusing on students' beliefs and perceptions about themselves as math learners.

  • Agency, exploring students' ability to take action and be in control of their learning in mathematics.

  • Identity Affirming, examining how teachers can support and validate students' math identities in the classroom.

Productive vs. Unproductive Beliefs

• Examples are given; productive beliefs empower student participation, while unproductive beliefs limit potential.

Beliefs About Access and Equity in Mathematics

• Highlights Unproductive and Productive beliefs about access and equity in mathematics, relating to student abilities, equity vs. equality, language proficiency, cultural considerations, poverty, tracking, and expectations for all students.

Mathematics Identity

• Beliefs about oneself as a mathematics learner.

• Perceptions of how others perceive them.

• Beliefs about the nature of mathematics.

• Engagement in mathematics.

• Perception of self as a potential participant in mathematics.

Identity & Motivation

• Understanding the strengths and motivations that serve to develop students’ identities should be embedded in the daily work of teachers.

• Mathematics teaching involves not only helping students develop mathematical skills but also empowering students to seeing themselves as being doers of mathematics.

Agency

• Agency is our identity in action and the presentation of our identity to the world.

• Social and behavioral expectations are associated with agency.

• If you believe you are a good mathematician (identity) you will behave as one (agency).

Identity Affirming

• Identity-affirming behaviors influence the ways in which students participate in mathematics and how they see themselves as doers of mathematics.

Supporting Teaching

• Mathematics teaching should leverage students’ culture, contexts, and identities to support and enhance mathematics learning.

Guiding Students to a Healthy Math Identity

Growth Mindset

• The slide shows the differences:

  • Those with growth mindsets believe their abilities can be developed through dedication and hard work. Embracing a growth mindset can lead to increased motivation, resilience, and achievement in mathematics.

  • A growth mindset can be cultivated through specific strategies and interventions, like praising effort and learning from mistakes.

• Fixed vs. Growth Mindset characteristics regarding skills, challenges, effort, feedback, and setbacks, illustrating how different mindsets affect a student's response to learning opportunities.

Research on Growth Mindset

• Students who believe their intelligence can increase focus on learning, effort, and persistence, leading to improved academic outcomes and a more positive attitude towards learning.

• Teaching a growth mindset raises achievement scores and student investment and enjoyment of school by fostering a belief that intelligence is not fixed but can be developed through effort and learning.

• Teachers with growth mindsets impact student achievement by creating a classroom environment that encourages risk-taking, effort, and perseverance.

• Mindsets predict math and science achievement, highlighting the crucial role of mindset in academic success.

Mindset & Math

• Students tend to have more of a fixed view of math skills than any other intellectual skills, indicating a need to promote a growth mindset in mathematics education specifically.

• Many Americans consider themselves bad at math, suggesting a widespread fixed mindset about math abilities in the culture.

• If kids do not believe they can improve, they won’t bother trying, emphasizing the importance of instilling a growth mindset to encourage effort and engagement in math.

The Mathematics of Hope

Effective Mathematics Teaching Practices

• Establish mathematics goals to focus learning, ensuring that learning activities are aligned with specific objectives.

• Implement tasks that promote reasoning and problem-solving, encouraging students to think critically and apply their knowledge.

• Use and connect mathematical representations (physical, contextual, visual, verbal, symbolic), providing students with different ways to access and understand mathematical concepts.

• Facilitate meaningful mathematical discourse, creating opportunities for students to share their thinking and learn from one another.

• Pose purposeful questions, guiding students to deepen their understanding and make connections between concepts.

• Build procedural fluency from conceptual understanding, ensuring that students understand the underlying concepts before developing computational skills.

• Support productive struggle in learning mathematics, encouraging students to persevere through challenges and learn from their mistakes.

• Elicit and use evidence of student thinking, using formative assessment to inform instruction and address student needs.

How Many Feet?

• A task where children figure out how many feet are under the table without peeking.

Support Productive Struggle in Learning Mathematics

Promoting Growth Mindset