Happy New Year!
Making algebra accessible to all students has been an increasingly significant goal over the last two decades. The recently released Common Core State Standards for Mathematics continue to emphasize the importance of algebra K-12. One goal of the Investigations curriculum is to make the foundations of algebra explicit to teachers and students. This newsletter focuses on how the curriculum engages K-5 students with early algebraic ideas, what it looks like the classroom, and how the curriculum supports teachers in taking on this work.
Karen Economopoulos and Keith Cochran
Co-Directors of Investigations in Number, Data, and Space
Myriam Steinback
Director of Investigations Workshops for Transforming Mathematics
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An Introduction to Early Algebra in Investigations
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"For many adults, algebra means symbolic notation and equations. We may have done well or poorly in high school algebra, but either way, we often have no sense that those equations we manipulated had any meaning beyond what was required to pass the course. Yet ... students in kindergarten through fifth grade are thinking about ideas
that are at the heart of algebra. In the course of working
on arithmetic, students notice regularities that might, in later years, be expressed with symbols and equations.
These ideas offer opportunities for rich mathematical investigation and discussion." (Russell et. al., 2006.)
In Investigations early algebra ideas are addressed in two major ways:
(1) work within the counting, number, and operations units focusing on generalizations that arise in the course of students' study of numbers and operations, and (2) one unit at each grade level about patterns, functions and change.
One reason for such explicit and in-depth attention is that "the work of generalizing and justifying in the elementary classroom has the potential of enhancing the learning of all students. ... Teacher collaborators report to us that students who tend to have difficulty in mathematics become stronger mathematical thinkers through this work. ... At the same time, students who generally outperform their peers in mathematics find this content challenging and stimulating." (Algebra in the Revision) Learn more about how these strands were developed, the research that informed our thinking, and the ways our work informed and furthered the work of the research community.
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The Patterns, Functions, and Change Strand
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The Investigations curriculum includes a unit focused on patterns, functions, and change at each grade. The result is "a coherent K-5 strand that starts with repeating patterns and number sequences in Grades K and 1, connects to functional relationships beginning in Grade 2, and focuses on linear and nonlinear change in Grades 3-5. Students study relationships that follow rules (such as the relationship between the number of windows in a building and the number of floors if the building has a fixed number of windows per floor) and relationships that do not follow rules (such as the relationship between temperature and time in Grade 3 and between plant growth and time in Grade 4). They work extensively with ways of representing [and describing] these relationships: in words, with numbers, [and] with tables and graphs... These units reinforce and connect with work in other units on multiplication, ratio, area, volume, and graphing." (Algebra in the Revision)
In Grades K-3, students use colored cubes to explore repeating patterns.
In grade 4, students think about a penny jar that starts with some number of pennies and then increases by a certain number of pennies each day.
For a detailed description of this strand K-5, read Patterns, Functions, and Change. |
The Generalized Arithmetic "Strand"
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Embedded in the number and operations units at each grade, this strand focuses on generalizations that arise in the course of students' study of number and operations. Specific lessons focus on general claims, for instance whether order matters when you count. The big ideas include properties of numbers and operations, and using models, representations, and contexts to investigate and justify general claims. "Teachers learn to help students articulate these generalizations and challenge them to consider the questions: Does this generalization apply to all numbers (in the domain under consideration)? Why does it work? How do you know?". (Algebra in the Revision)
Learn more about what it looks like when students use/explore such generalizations through video and student work.
For a detailed description of this strand K-5, read Early Algebra.
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Algebra-Related Curriculum Features
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How does Investigations help teachers identify and explore generalizations that arise in the course of students' study of number and operations? In every grade, students investigate general claims. "In each of the number and operations units, an essay, Algebra Connections in This Unit, highlights several generalizations and includes examples of how students think about and represent them. Investigation and discussion of some of these generalizations are built into unit sessions; at other times, 'Algebra Notes' alert the teacher to sessions in which these ideas are likely to arise." (Algebra in the Revision)
To learn more about these and other algebra-related features of the curriculum, and to view examples, see Early Algebra Components.
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Connecting Arithmetic to Algebra, by Investigations authors Susan Jo Russell, Deborah Schifter, and Virginia Bastable.
This book answers the questions "How can we better prepare elementary students for algebra? More importantly, how can we help all children, not just those who excel in math, become ready for later instruction? The answer lies not in additional content, but in developing a way of thinking about the mathematics that underlies both arithmetic and algebra." (Download a flyer and/or a sample chapter.)
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Investigations Workshops for Transforming Mathematics offers professional development that provides teachers and math leaders the opportunity to explore early algebraic ideas for themselves and with their colleagues.
Developing Mathematical Ideas is a case-based professional development curriculum designed to help teachers think through the major ideas of K-7 mathematics and examine how children develop those ideas. Two modules address algebra in the elementary classroom:
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- We are currently booking Investigations Workshops for Summer 2012. For information about hosting or attending a workshop contact Peter Swanson at 617-873-9600 or peter_swanson@terc.edu.
- This fall we offered Using Investigations to Implement the Common Core Institutes in three locations: Massachusetts, North Carolina, and Arizona. This new Institute focuses on how the newly developed companion materials align with the Common Core State Standards for Mathematics and support teachers as they implement the CCSS in their classrooms and schools. The feedback has been positive:
"I want to thank you for this workshop. It seems like so much of our day is 'on to the next thing', new mandates, how tasks are just things to be completed. It was a pleasure to spend the day with people who are thinking carefully and deeply about what will make sense. It was a breath of fresh air. Thank you for all your hard work. It is greatly appreciated."
"Such a well done day. I appreciate how thoughtful TERC has been in thinking about how to implement the CCSS. ... I also appreciated the time to talk about what we are doing on a school & district level."
If you are interested in hosting or attending such an Institute, which also includes 3 follow-up webinars, contact Peter Swanson at 617-873-9600 or peter_swanson@terc.edu.
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