Early Algebra

James J. Kaput Center for Research and Innovation in Mathematics Education University of Massachusetts Dartmouth

Maria Blanton

the main focus of our early algebra work

  Understanding progressions in children’s functional thinking across grades

  Teacher learning - developing “algebra eyes and ears”

1st-graders act out the Handshake Problem

children’s functional thinking o  developing representational

tools o  keeping the independent

variable explicit o  transitioning from recursive

patterns to functional relationships

o  moving from words to symbols

THE STRING PROBLEM

Fold a piece of string to make one loop. While it is folded, make 1 cut. How many pieces of string do you have? Find the number of pieces of string for 2, 3, 4, and 5 cuts. Find a relationship between the number of cuts and the number of pieces of string.

1st-graders work on the String Problem

children’s functional thinking developing representational tools keeping the independent variable explicit

1st-grader’s function table: Handshake Problem.

“As children record numerals in a table, they also begin to work on ideas of correspondence between quantities because they are attending to where these numbers go in the table and the meanings embodied in the position of the numbers.… This process helps children begin to visually and cognitively look across columns and keep track of two quantities simultaneously.” (Blanton, 2008)

1st-grader’s representation for counting handshakes

children’s functional thinking developing representational tools making the independent variable explicit

3rd-graders’ representations: The Telephone Problem

By 3rd-grade, children can use representational tools in more abstract forms to actively reason about functional relationships.

children’s functional thinking transitioning from recursive patterns to functional relationships moving from words to symbols

o  1st-graders: recursive thinking; primitive covariational and

correspondence thinking “It’s 2 more each time; It’s like skip-counting by twos”

“Every time you make one more snip, it’s two more than the one before.”

o  by 3rd grade: correspondence relationships in natural language and

symbolic forms “You double the number and add one”

“n x 2 + 1 = number of pieces”

teacher learning o  transforming arithmetic tasks into

opportunities for algebraic thinking (e.g., varying a parameter, making known quantities unknown)

o  seeing algebra as a ‘way of thinking’, not a content strand or an ‘add-on’

o  integrating algebra across the elementary grades curriculum

o  developing a school culture of algebraic thinking 1st-grader demonstrates cutting string

connecting algebra across the curriculum: the telephone problem

o  Social studies - unit on immigration

How the Second Grade Got $8,205.50 To Visit the Statue of Liberty (Zimelman, 1992) - read aloud

o  Math - The Telephone Problem

o  Science - unit on sound

Students design telephones to act out and gather data on The Telephone Problem

The second graders at the Jefferson School have raised money to visit the Statue of Liberty. Thirteen friends are planning to go. They are very excited about the trip and worried that they might forget something! On the night before the trip, they call each other to double check on what they need to bring. Each friend talks to every other friend once. How many phone calls will be made? What if 100 friends were planning to go?

3rd-graders act out the Telephone Problem

building a school culture of algebraic thinking

o  School-wide algebraic thinking task

o  Math Night for parents

o  Monthly Math - focus on algebra

o  Math Buddy Project - teaming students in upper elementary grades with those in early elementary grades

o  Homework Club

o  Make children’s work visible across the school

o  Coordinate with other professional development initiatives

o  Coordinate with accountability testing

These can be a subtle way to bring teachers into early algebra

how research has guided curriculum decisions

With students

  Use of functional thinking tasks and the development of appropriate representational tools for this as early as first grade;

  Keeping the independent variable explicit from 1st-grade on to support young children’s understanding of co-variation, correspondence, and representational tools that enable functional thinking.

  Development of symbolic, algebraic notation as early as first grade; expectations that students be able to symbolize functional relationships as early as 3rd-grade (in contrast to CFP).

With teachers

  Use of tasks to support development of teachers’ ability to transform their own existing instructional resources; TPD should be generative and self-sustaining, not TPD-dependent

implications for the development of students’ mathematical reasoning and proof   Because of its emphasis on justifying conjectures in building generalizations, early algebra

is a natural context for developing students’ mathematical reasoning and proof.

  Children bring to formal schooling an innate tendency to generalize and justify (Mason, 2008; Maher, to appear); early algebra can foster this thinking

  Children’s arguments in early algebra settings typically begin with testing numerical cases, but can be developed into sophisticated forms that involve

(1) reasoning from the problem’s context,

(2) reasoning from representations, and

(3) reasoning with previously established generalizations, and

In expressing the number of people for t tables as 3t + 2, students noted that “3 came from the people that could sit ‘on the top and the bottom’ and the 2 came from the two sides.”

2nd-grader: a + b - b = a because b - b = 0 and a + 0 = a

Commutative Property of Addition

implications for pre-service and in-service mathematics teacher education   Elementary teachers are in the forefront of early algebra reform, but have not had

the experiences (as teachers and students) to prepare them for teaching early algebra;

  Yet they are some of the most creative professionals I’ve worked with -- and some of the ‘stars’ in our work with teachers are self-confessed math phobic!

  Pre-service teachers need undergraduate course work specifically in developing classrooms that support children’s algebraic thinking;

  In-service TPD needs to be systemic, long-term, and classroom-based. It needs to help teachers develop the tools for identifying and exploiting opportunities for algebraic thinking in their own classrooms.

WHAT ARE YOUR EXPERIENCES WITH HOW YOUR UNIVERSITY/TPD WORK PREPARES TEACHERS?

past and present treatment of early algebra in schools: shifts forward Past/Present   Shifts in early algebra focus from 1989 Curriculum and Evaluation

Standards to 2000 Principles and Standards supported increasing emphasis on algebra in elementary grades.

  Emphasis reflected in revisions of curricula (Investigations), state frameworks

But what does this mean for teachers’ daily classroom practices?

Locally   initiatives such as MA Curriculum Frameworks and MCAS have supported

early algebra.   But implementation has been slow, sporadic and while implementation

efforts have been systemic, have they been sustained?   TPD an issue: Elementary teachers reported that, in spite of the presence

of algebra in MCAS and Frameworks, they teach it less than 10% of time.

WHAT IS HAPPENING IN YOUR SCHOOLS (re curriculum, tpd, etc.) ?

future treatment of early algebra in schools: shifts back?

In grades PreK-8, focus on the Critical Foundations for Algebra -Proficiency with whole numbers -Proficiency with fractions -Particular aspects of geometry and measurement

‘Proficiency’ a euphemism for the (failed) traditional arithmetic-then-algebra approach?

A Grade 3 Focal Point:

Number and Operations and Algebra - Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts.

‘Algebra’ too implicit? How does this translate into teachers’ classrooms?

Is the goal of early algebra to serve the purpose of arithmetic?

Which way will the pendulum swing?

Early Algebra

Which way will the pendulum swing?

Early Algebra

Which way will the pendulum swing?

Early Algebra

Which way will the pendulum swing?

Early Algebra