Class Materials 1-25
The class materials will give a sense
of the issues we have been discussing with our students in
grades 3 and 4. Perhaps they will inspire some teachers to try
out new ideas in their own classes.
But the materials are not recipes. The
quality of classroom discussions depends greatly on the
teachers’ ability to listen, to engage students, to draw
out the implications of their statements, and to see how
particular activities fit into the grande scheme of mathematics
and the students’ evolving understanding.
Any rote enactment of lessons
(”First, do step 1, then step 2, then collect the
papers...”, etc.) is doomed to fail.
Preparation and organization certainly
help, but teaching, like playing jazz, relies on creative
improvisation and careful listening.
Click on the lesson title to download a
copy.
Lesson 1 Symbols
In this class we will engage the children in an informal discussion about different kinds of symbols. Insights from this class will constitute a basis for the introduction of mathematical symbols as part of general symbol systems and as means of communication.
In this class we will engage the children in an informal discussion about different kinds of symbols. Insights from this class will constitute a basis for the introduction of mathematical symbols as part of general symbol systems and as means of communication.
Lesson 2 Comparisons
Symbols: Comparing Numbers
In this class we will explore: mathematical symbols for greater than (>), less than (<), equals to (=) and is not equal to (≠); comparative terms (more than, less than, longer than, shorter than, bigger than, smaller than). We will also focus on ambiguous situations where the choice of the attribute to be compared determines the comparison symbol to be used.
In this class we will explore: mathematical symbols for greater than (>), less than (<), equals to (=) and is not equal to (≠); comparative terms (more than, less than, longer than, shorter than, bigger than, smaller than). We will also focus on ambiguous situations where the choice of the attribute to be compared determines the comparison symbol to be used.
Lesson 3 Comparisons and Attributes
In this class we will work with comparison statements and will continue to explore: mathematical symbols for greater than (>) and less than (<) and comparative terms (more than, less than, longer than, shorter than, bigger than, smaller than, taller than, shorter than, wider than, narrower than). We will focus on ambiguous situations where the choice of the attribute to be compared determines the comparison symbol to be used.
In this class we will work with comparison statements and will continue to explore: mathematical symbols for greater than (>) and less than (<) and comparative terms (more than, less than, longer than, shorter than, bigger than, smaller than, taller than, shorter than, wider than, narrower than). We will focus on ambiguous situations where the choice of the attribute to be compared determines the comparison symbol to be used.
Lesson 4 Comparing Length: Heights – I
In this lesson we will help children to conceive of the difference between two heights as an object with a dimension of its own.
In this lesson we will help children to conceive of the difference between two heights as an object with a dimension of its own.
Lesson 5 Comparing Discrete Quantities
In this lesson we will help children to adopt line segments previously used to represent heights as a legitimate representation for amounts of discrete quantities. We will also start work on the composition of measures.
In this lesson we will help children to adopt line segments previously used to represent heights as a legitimate representation for amounts of discrete quantities. We will also start work on the composition of measures.
Lesson 6 Comparing Heights – III
In this class children will work on the functional representation of two unknown heights and on the composition of the shorter height plus the difference between the heights as equal to the second height.
In this class children will work on the functional representation of two unknown heights and on the composition of the shorter height plus the difference between the heights as equal to the second height.
Lesson 7 Comparing Functions: Candy Boxes
This class centers on the possible amounts of candies two children, John and Maria, have. They each have the same, unspecified number of candies inside their own candy box. John has, in addition, one extra candy and Maria has three extra candies. What are the possible total candies they might have?
This class centers on the possible amounts of candies two children, John and Maria, have. They each have the same, unspecified number of candies inside their own candy box. John has, in addition, one extra candy and Maria has three extra candies. What are the possible total candies they might have?
Lesson 8 Comparing Functions: Candy Boxes Repeat
This class centers on the possible amounts of candies two children, John and Maria, have. They each have the same, unspecified number of candies inside their own candy box. John has, in addition, one extra candy and Maria has three extra candies. What are the possible total candies they might have?
This class centers on the possible amounts of candies two children, John and Maria, have. They each have the same, unspecified number of candies inside their own candy box. John has, in addition, one extra candy and Maria has three extra candies. What are the possible total candies they might have?
Lesson 9 Total Candies [N X 2 + 4]
This class follows the discussion from the Candy Boxes classes. The main challenges will be (a) is to work from the total number of candies in order to determine how many each child had and (b) to develop a general notation for finding the total amount of candies from the amount of candies in each box. These are demanding problems for many third grade students, but they will feel proud when they understand the problem and develop new notations.
This class follows the discussion from the Candy Boxes classes. The main challenges will be (a) is to work from the total number of candies in order to determine how many each child had and (b) to develop a general notation for finding the total amount of candies from the amount of candies in each box. These are demanding problems for many third grade students, but they will feel proud when they understand the problem and develop new notations.
Lesson 10 Number Lines: Locations
Students place themselves at points on the number line. Main contexts: Mason stairs, age, money, temperature, and pure number.
Students place themselves at points on the number line. Main contexts: Mason stairs, age, money, temperature, and pure number.
Lesson 11 Changes and Shortcuts On The Number Line
A number line is stretched against the wall or across the classroom going from about —10 to +20. The children themselves become numbers on the line and perform different "line dances", and by doing so further explore additive operations and begin to explore negative numbers.
A number line is stretched against the wall or across the classroom going from about —10 to +20. The children themselves become numbers on the line and perform different "line dances", and by doing so further explore additive operations and begin to explore negative numbers.
Lesson 12 Partial and Total Changes
Students learn that two partial changes are equivalent to a single total change. On the number line, this corresponds to the idea of a shortcut. Three notations are emphasized: words, number lines with hopping arrows, and numerical expressions.
Students learn that two partial changes are equivalent to a single total change. On the number line, this corresponds to the idea of a shortcut. Three notations are emphasized: words, number lines with hopping arrows, and numerical expressions.
Lesson 13 Multiple Number Lines
Students learn that two partial changes that start at different points on the number line are equivalent. At the end, they will work with notation for variables (N + 5 – 3 or N + 2).
Students learn that two partial changes that start at different points on the number line are equivalent. At the end, they will work with notation for variables (N + 5 – 3 or N + 2).
Lesson 14 The N-Number Line
Students work with the table they built in the previous class for multiple number lines, focusing on the notation for variables (N + 5 – 3 or N + 2).
Students work with the table they built in the previous class for multiple number lines, focusing on the notation for variables (N + 5 – 3 or N + 2).
Lesson 16 The
Piggy Banks
The whole lesson revolves around a multipart story problem that involves changes in two quantities over several days. The initial quantities are equal yet unknown. Then transformations are applied to the quantities. Students are asked to compare the quantities throughout the week even though only their relative amounts can be determined.
The whole lesson revolves around a multipart story problem that involves changes in two quantities over several days. The initial quantities are equal yet unknown. Then transformations are applied to the quantities. Students are asked to compare the quantities throughout the week even though only their relative amounts can be determined.
Lesson 17 The
Piggy Banks II
This lesson continues exploring children’s work on the Piggy Banks Problem. In the previous lesson the students represented the problem using their own ideas. Now they systematically work with the instructor.
This lesson continues exploring children’s work on the Piggy Banks Problem. In the previous lesson the students represented the problem using their own ideas. Now they systematically work with the instructor.
Lesson 18 Guess
My Rule I
Two children create secrete rules for transforming input numbers. The teacher uses a doubling rule.
Two children create secrete rules for transforming input numbers. The teacher uses a doubling rule.
Lesson 19 Guess My Rule II
Two children create secrete rules for transforming input numbers. The teacher uses a doubling or tripling rule.
Two children create secrete rules for transforming input numbers. The teacher uses a doubling or tripling rule.
Lesson 20 Three
Heights Problem
In this class we will explore: (a) How the children deal with comparisons, (b) How they draw inferences from comparisons, and (c) How they represent comparisons between three unknown amounts.
In this class we will explore: (a) How the children deal with comparisons, (b) How they draw inferences from comparisons, and (c) How they represent comparisons between three unknown amounts.
Lesson 21 Comparison Problems and Tables
In this last class of the term we will review previous concepts and representations as applied to the solution of verbal comparison problems and to work on function tables.
In this last class of the term we will review previous concepts and representations as applied to the solution of verbal comparison problems and to work on function tables.
Lesson 22 All
Things Being Equal
The equals sign signifies that amounts on each side are the same. We will use line segments on grid paper to demonstrate all assertions.
The equals sign signifies that amounts on each side are the same. We will use line segments on grid paper to demonstrate all assertions.
Lesson 23 All
Things Being Equal II: Different ways to make five
The equals sign signifies that amounts on each side are the same. The students will use Unifix blocks and the corresponding equations to represent equalities between additive amounts.
The equals sign signifies that amounts on each side are the same. The students will use Unifix blocks and the corresponding equations to represent equalities between additive amounts.
Lesson 24 All
Things Being Equal III
The students will write equations to represent verbal statements and successive transformations that maintain or do not maintain the equality.
The students will write equations to represent verbal statements and successive transformations that maintain or do not maintain the equality.
Lesson 25 The Dots
Problem
We present to the students a problem dealing with a growing pattern over time. To begin, there is one dot, and with each passing minute four dots form around the first dot.
We present to the students a problem dealing with a growing pattern over time. To begin, there is one dot, and with each passing minute four dots form around the first dot.
