Mathematical Rigor in Pre-K-5 Standards Analysis
Curriculum Frameworks Key
Shifts:
Mathematical Rigor in Pre-K – 5
Standard Analysis Protocol
Session Objective
For participants to be able to analyze content standards using the three aspects of
Mathematical Rigor (Conceptual Understanding, Procedural Fluency, and
Application (Capacity)).
2
Agenda
Session Agenda
01
Mathematical Rigor
Item 1 Mathematical Rigor
Mathematical Rigor
5
02
Grade 3 Example
Item 2 Grade 3 Example
Standard Analysis
3.MD.C.5 -
Recognize area as an attribute of plane figures and
understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one
square unit” of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares
is said to have an area of n square units.
What
Conceptual Understanding
does this standard describe?
What
Procedural Fluency
does this standard describe?
What authentic opportunities exist for the
Application
of this standard to real-
world problems?
7
Standard Analysis
3.MD.C.5 -
Recognize area as an attribute of plane figures and
understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one
square unit” of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares
is said to have an area of n square units.
What
Conceptual Understanding
does this standard describe?
8
Standard Analysis
3.MD.C.5 -
Recognize area as an attribute of plane figures and
understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one
square unit” of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares
is said to have an area of n square units.
What
Procedural Fluency
does this standard describe?
9
Standard Analysis
3.MD.C.5 -
Recognize area as an attribute of plane figures and
understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one
square unit” of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares
is said to have an area of n square units.
What authentic opportunities exist for the
Application
of this standard to real-
world problems?
10
Conceptual Understanding – Students will be able to….
•
Explain why area is measured with squares, like distance is measured
with lengths. We use a square because a square is unique – its sides
are the same length and if the lengths are 1 unit then the area is 1
square unit.
•
Explain why area can be determined by covering a shape with
squares, like a distance can be measured by laying down equal
lengths next to each other and counting the total.
11
Procedural Fluency– Students will be able to….
•
Arrange squares so that the whole shape is covered without gaps or
overlaps to measure area.
•
Use/write the term “square units” when describing area of a shape.
•
Count the squares that cover a shape using groups or other methods
to keep track of counting – Count using groups and/or multiplication
to determine the total.
12
Application– Students will be able to….
Measure the area of a classroom using square floor tiles.
Explain the possible errors a person may make while measuring an
area with square tiles.
Compare the results of measuring an area with square inches versus
square feet.
Explain why it is necessary to use the term “square units” when
describing area versus not including units.
13
03
Grade 2 Example
Item 3 Grade 2 example
Standard Analysis – Example 1
2.OA.C.3
Determine whether a group of objects (up to 20) has
an odd or even number of members, e.g., by pairing objects
or counting them by 2s; write an equation to express an even
number as a sum of two equal addends.
What
Conceptual Understanding
does this standard describe?
What
Procedural Fluency
does this standard describe?
What authentic opportunities exist for the
Application
of this
standard to real-world problems?
15
Standard Analysis – Example 1
2.OA.C.3
Determine whether a group of objects (up to 20) has
an odd or even number of members, e.g., by pairing objects
or counting them by 2s; write an equation to express an even
number as a sum of two equal addends.
What
Conceptual Understanding
does this standard describe?
16
Standard Analysis – Example 1
2.OA.C.3
Determine whether a group of objects (up to 20) has
an odd or even number of members, e.g., by pairing objects
or counting them by 2s; write an equation to express an even
number as a sum of two equal addends.
What
Procedural Fluency
does this standard describe?
17
Standard Analysis – Example 1
2.OA.C.3
Determine whether a group of objects (up to 20) has
an odd or even number of members, e.g., by pairing objects
or counting them by 2s; write an equation to express an even
number as a sum of two equal addends.
What authentic opportunities exist for the
Application
of this
standard to real-world problems?
18
Standard Analysis – Example 1
2.OA.C.3
Determine whether a group of objects (up to 20) has
an odd or even number of members, e.g., by pairing objects
or counting them by 2s; write an equation to express an even
number as a sum of two equal addends.
Create student learning objectives for this standard that describe
student learning for each aspect of Rigor:
Conceptual Understanding
Procedural Fluency
Application
19
04
Grade 1 Example
Grade 1 Example
Standard Analysis
1.NBT.C.6
- Subtract multiples of 10 in the range 10–90 from
multiples of 10 in the range 10–90 (positive or zero
differences), using concrete models or drawings and strategies
based on place value, properties of operations, and/or the
relationship between addition and subtraction; relate the
strategy to a written method and explain the reasoning used.
21
Wrap-up
•
Do we regularly teach to objectives that align to all three
aspects of rigor?
•
Are we prioritizing one aspect of rigor over the others?
•
Do we regularly assess students’ ability to meet objectives
that align to all three aspects of rigor?
•
Do our curricular materials sufficiently include tasks that
align to all three aspects of rigor?
22
Session Approximate Time: 2 1/4 hrs
Explore the three aspects of Mathematical Rigor - Conceptual Understanding, Procedural Fluency, and Application Capacity in analyzing content standards for grades Pre-K to 5. Dive into examples, standard analysis, and real-world application opportunities provided by the Massachusetts Department of Elementary and Secondary Education.
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Presentation Transcript
Curriculum Frameworks Key Shifts: Mathematical Rigor in Pre-K 5 Standard Analysis Protocol
Session Objective For participants to be able to analyze content standards using the three aspects of Mathematical Rigor (Conceptual Understanding, Procedural Fluency, and Application (Capacity)). 2 Massachusetts Department of Elementary and Secondary Education
01 Mathematical Rigor 02 Grade 3 Example Agenda 03 Grade 2 Example 04 Grade 1 Example
01 Mathematical Rigor
Mathematical Rigor Massachusetts Department of Elementary and Secondary Education 5
Item 2 Grade 3 Example 02 Grade 3 Example
Standard Analysis 3.MD.C.5 - Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. What Conceptual Understanding does this standard describe? What Procedural Fluency does this standard describe? What authentic opportunities exist for the Application of this standard to real- world problems? 7 Massachusetts Department of Elementary and Secondary Education
Standard Analysis 3.MD.C.5 - Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. What Conceptual Understanding does this standard describe? 8 Massachusetts Department of Elementary and Secondary Education
Standard Analysis 3.MD.C.5 - Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. What Procedural Fluency does this standard describe? 9 Massachusetts Department of Elementary and Secondary Education
Standard Analysis 3.MD.C.5 - Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. What authentic opportunities exist for the Application of this standard to real- world problems? 10 Massachusetts Department of Elementary and Secondary Education
Conceptual Understanding Students will be able to. Explain why area is measured with squares, like distance is measured with lengths. We use a square because a square is unique its sides are the same length and if the lengths are 1 unit then the area is 1 square unit. Explain why area can be determined by covering a shape with squares, like a distance can be measured by laying down equal lengths next to each other and counting the total. 11 Massachusetts Department of Elementary and Secondary Education
Procedural Fluency Students will be able to. Arrange squares so that the whole shape is covered without gaps or overlaps to measure area. Use/write the term square units when describing area of a shape. Count the squares that cover a shape using groups or other methods to keep track of counting Count using groups and/or multiplication to determine the total. 12 Massachusetts Department of Elementary and Secondary Education
Application Students will be able to. Measure the area of a classroom using square floor tiles. Explain the possible errors a person may make while measuring an area with square tiles. Compare the results of measuring an area with square inches versus square feet. Explain why it is necessary to use the term square units when describing area versus not including units. 13 Massachusetts Department of Elementary and Secondary Education
Item 3 Grade 2 example 03 Grade 2 Example
Standard Analysis Example 1 2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. What Conceptual Understanding does this standard describe? What Procedural Fluency does this standard describe? What authentic opportunities exist for the Application of this standard to real-world problems? 15 Massachusetts Department of Elementary and Secondary Education
Standard Analysis Example 1 2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. What Conceptual Understanding does this standard describe? 16 Massachusetts Department of Elementary and Secondary Education
Standard Analysis Example 1 2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. What Procedural Fluency does this standard describe? 17 Massachusetts Department of Elementary and Secondary Education
Standard Analysis Example 1 2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. What authentic opportunities exist for the Application of this standard to real-world problems? 18 Massachusetts Department of Elementary and Secondary Education
Standard Analysis Example 1 2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. Create student learning objectives for this standard that describe student learning for each aspect of Rigor: Conceptual Understanding Procedural Fluency Application 19 Massachusetts Department of Elementary and Secondary Education
Grade 1 Example 04 Grade 1 Example
Standard Analysis 1.NBT.C.6 - Subtract multiples of 10 in the range 10 90 from multiples of 10 in the range 10 90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 21 Massachusetts Department of Elementary and Secondary Education
Wrap-up Do we regularly teach to objectives that align to all three aspects of rigor? Are we prioritizing one aspect of rigor over the others? Do we regularly assess students ability to meet objectives that align to all three aspects of rigor? Do our curricular materials sufficiently include tasks that align to all three aspects of rigor? 22 Massachusetts Department of Elementary and Secondary Education
