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Eliciting Algebraic Thinking
Supporting Pk-3 Students’ development of algebraic thinking in the OA domain
Caitlin Duncan, May 2015
Agenda
Goals and Outcomes!
Algebra Review!
Symbolic Meaning!
Part-Whole and Unknown Addends!
Math Practices!
Final ReflectionCaitlin Duncan, May 2015
Why are we here?
Highlight some concepts that support algebraic thinking.!
Explore elements of algebraic thinking within the Operations and Algebraic Thinking Domain.!
Review instructional approaches that support algebraic thinking.!
Engage with activities that elicit algebraic thinking.
Caitlin Duncan, May 2015
When we leave . . .
We will understand algebraic concepts embedded in K-3 standards!
We can identify K-3 standards that relate to algebraic thinking.!
We have a toolkit of instructional strategies and activities that support algebraic thinking.
Caitlin Duncan, May 2015
What is algebra?Record of algebraic ideas date back to 2000 BC in ancient Babylon. !
Other records throughout history.!
al-Kwarizmi (820 AD) is credited as naming algebra. He wrote The compendious on Calculation by Completion and Balancing.!
The word “algebra” means restoration or completion.!
al-Kwarizmi was the first to think about algebra more abstractly.!
Focus on the idea of completing equations through balance of each side.!
Definition: The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.
Khan Academy - Origins of Algebra
Why does algebra matter?
Advanced math equals career opportunity.!
STEM jobs (17% growth) are growing at almost twice the rate of non-STEM jobs (10% growth).!
Simply taking advanced math has a direct impact on future earnings, apart from any other factors. Students who take advanced math have higher incomes ten years after graduating—regardless of family background, grades and college degrees.!
A little over half of all U.S. occupations require a significant level of “knowledge of arithmetic, algebra, geometry, calculus, statistics, and their applications.” Included in these nearly 500 occupations are about 45% of low skills jobs, about half of middle skills jobs, and over 80% of high skills jobs.9!
Many “blue collar” jobs also require advanced math: One study found that the math skills required by electricians, construction workers, upholsterers and plumbers match what’s necessary to do well in college courses.10 The International Brotherhood of Electrical Workers’ test for prospective apprentices includes algebra problems.
http://www.achieve.org/files/MathWorks-AllStudentsNeedAdvancedMath.pdf
“- regardless of family background grades and college degrees.”
“included . . . are about 45% of low skills jobs . . .”
“Many ‘blue collar’ jobs also require advanced math . . .”
What primary skills and concepts underlie algebra and
algebraic thinking?
Conceptual Understanding of and procedural fluency with . . . !
Operations!
Properties!
Symbols!
Quantity
Caitlin Duncan, May 2015
Understanding Symbols
Caitlin Duncan, May 2015
Understanding Symbols🙋❤️🎲
What does this string of symbols mean? Are you able to make sense of it?!
Teaching students to work with equations requires that students attach meaning to the symbols we use.
Symbols that indicate an action is required:!
➕➖✖️➗
Symbols that indicate a comparison should be made:!
< = >
Caitlin Duncan, May 2015
Supporting Understanding of Symbols
Language!
Think about the words you use associated with equation symbols.!
What kind of meaning do those words hold?!
What words might we use that evoke stronger understanding?
Caitlin Duncan, May 2015
Supporting Understanding of Symbols
Especially confusing is the equal sign.!
“Children as young as kindergarten may have appropriate understanding of objects but have difficulty relating this understanding to symbolic representations involving the equals sign. A concerted effort over an extended period of time is required to establish appropriate notations of equality. Teachers should also be concerned about children’s conceptions of equality as soon as symbols for representing number operations are introduced.”!
!
What words do you use to evoke understanding of =?!
! “. . . IS THE SAME VALUE AS . . . ”
Teaching Children Mathematics, Falkner, K., Dec. 1999, p. 233
Why is this important?
Without a firm grasp of what the symbols mean in equations, students lack the conceptual understanding needed to engage in the other four strands of proficiency.
5EXECUTIVE SUMMARY
Mathematical ProficiencyOur analyses of the mathematics to be learned, our reading of the research
in cognitive psychology and mathematics education, our experience as learnersand teachers of mathematics, and our judgment as to the mathematical knowl-edge, understanding, and skill people need today have led us to adopt acomposite, comprehensive view of successful mathematics learning. Recog-nizing that no term captures completely all aspects of expertise, competence,knowledge, and facility in mathematics, we have chosen mathematical profi-ciency to capture what we think it means for anyone to learn mathematicssuccessfully. Mathematical proficiency, as we see it, has five strands:
• conceptual understanding—comprehension of math-ematical concepts, operations, and relations
• procedural fluency—skill in carrying out proceduresflexibly, accurately, efficiently, and appropriately
• strategic competence—ability to formulate, repre-sent, and solve mathematical problems
• adaptive reasoning—capacity for logical thought,reflection, explanation, and justification
• productive disposition—habitual inclination to seemathematics as sensible, useful, and worthwhile, coupledwith a belief in diligence and one’s own efficacy.
The most important observation we make about thesefive strands is that they are interwoven and interdependent.This observation has implications for how students acquiremathematical proficiency, how teachers develop that profi-ciency in their students, and how teachers are educatedto achieve that goal.
The Mathematical KnowledgeChildren Bring to School
Children begin learning mathematics well before they enter elementaryschool. Starting from infancy and continuing throughout the preschool period,they develop a base of skills, concepts, and misconceptions. At all ages, stu-dents encounter quantitative situations outside of school from which theylearn a variety of things about number. Their experiences include, forexample, noticing that a sister received more candies, counting the stairs
Conceptual Understanding
StrategicCompetence
ProductiveDisposition
ProceduralFluency
AdaptiveReasoning
Intertwined Strands of Proficiency
Copyright © National Academy of Sciences. All rights reserved.
Adding it up : helping children learn mathematics; Kilpatric, J, 2001
How does this connect to algebraic thinking?
“Understanding equality as a relationship is important [because] . . . a lack of such understanding is one of the major stumbling blocks for students when they move from arithmetic to algebra.”!
Teaching Children Mathematics, Falkner, K., Dec. 1999, p. 234
Activities to Support Symbolic Understanding
Caitlin Duncan, May 2015
Naughty Bears !This activity emphasizes using either side of the equal sign to indicate the parts and the whole.
!Addresses the K.OA.2 standard (Both Addends Unknown problem type).
!Can be modified with quantity for older grades.
Let’s try it out! We’ll choose 10 as our number. Use the bears to model your story and really try to write an equation that matches the sequence of the story.
Caitlin Duncan, May 2015
Number Rack Detectives
Important Points Rekenreks (number racks)
!Emphasis: Equations can be written with the whole before the equal sign.
!First grade lesson supporting 1.OA.6 but can also support K.OA.3 or 2.OA.2
Caitlin Duncan, May 2015
Equal Sums, Different Arrays
Second grade activity that addresses 2.OA.4 and 2.OA.2
!Also supports 3.OA.7
!Emphasizes comparing equality of expressions through a visual model.
Caitlin Duncan, May 2015
Connections & Ideas
With table mates, spend a few minutes sharing connections and ideas you’ve made after playing some of these games.!
What ideas do you have for eliciting symbolic understanding in your instruction?!
What new ideas are you pondering?
Caitlin Duncan, May 2015
Implications for Pre-KGiven that research shows that very young children can understand equality but misunderstand the equality symbol, how can Pre-K teachers/programs set the stage for increased understanding?!
Comparison Activities!
Balance Concepts!
Fair Share Activities!
Language
Caitlin Duncan, May 2015
Part-Whole Relationships
Decomposition & Unknown Addends
Caitlin Duncan, May 2015
What does this look like?
K.OA.3 - Partners to 9.!
K.OA.4 - How many more to make 10?!
1.OA.4 - Understand subtraction as an unknown addend problem.
Caitlin Duncan, May 2015
Why is Part-Whole understanding important?
Supports Level 3 addition and subtraction strategies.!
Make a Ten [8+5=(8+2)+3=10+3=13]!
Doubles +/- 1 or 2 [7+8=(7+7)+1=14+1=15]!
Level 3 strategies support flexibility and procedural fluency.
Caitlin Duncan, May 2015
Why is the +/- relationship important?
Relating addition and subtraction equations to other known equations is another Level 3 method.!
Is addition easier than subtraction? YES!
Caitlin Duncan, May 2015
Are students more successful when counting forward or backward when using a counting method to add/subtract?!
9+4= ⃞!
13-4= ⃞!
“Learning to think of and solve subtractions as unknown addend problems makes subtraction as easy as addition (or even easier), and it emphasizes the relationship between addition and subtraction.”
Why is the +/- relationship important?
K-2 OA Progressions, p.15
How does part-whole and unknown addend understanding
work together?“Students can solve some unknown addend problems by trial and error or by knowing the relevant decomposition of the total.”!
Example:!
13-4= ⃞! ! ! ! 13!! ! ! 13!! ! 4+ ⃞=13!
! ! ! ! ! 4! ! ⃞! ! 4! + ⃞
K-2 OA Progressions, p.14
How does this understanding support algebraic thinking?Understanding subtraction as an unknown addend problem makes equations dynamic and flexible.!
Potential “rearrangements and manipulations are ever present.”!
“The standards emphasize purposeful transformation of expressions into equivalent forms that are suitable for the purpose at hand.”
Algebra Progressions, p. 4-5
Sort the Sum (Grade 1)
What’s Missing? Bingo (Grade 3)
Drop the Beans (Grade K/1)
Part-Whole & Unknown Addend Activities
Caitlin Duncan, May 2015
Activity Extensions - Sort the Sum
Use double 9 or double 12 dominoes for more possibilities.!
Careful record of equations!
9=4+5!
9=3+6!
Connect dominoes (visual model) to equations with expressions on each side) to support understanding of equations.
Supports understanding of
the equal sign.
==4+5 6+3
Caitlin Duncan, May 2015
Activity Extensions - Drop the Beans
Modify for kindergarten (work within 5) and second grade (work beyond 10)!
Connect to other representations & incorporate equations!
Number Frames app
Caitlin Duncan, May 2015
Activity Extensions - What’s Missing? Bingo
Geoboard app!
Provides a lasting record of thinking!
Can tie other concepts used on the geoboard (geometry, fractions, etc. to multiplication/division)
Caitlin Duncan, May 2015
Connections & Ideas
With table mates, spend a few minutes sharing connections and ideas you’ve made after playing some of these games.!
What ideas do you have for eliciting symbolic understanding in your instruction?!
What new ideas are you pondering?
Caitlin Duncan, May 2015
Math Practices“The algebra standards are fertile ground for the Standards for Mathematical Practice. Two in particular that stand out are MP.7, ‘Look for and make use of structure’ and MP.8 ‘Look for and express regularity in repeated reasoning.’”!
Example:!
! ! ! ! !
3(5x - 4) + 2 = 20
3 (5x - 4) = 18?
5x - 4 = 65x = 10?
?
Algebra Progressions, p. 3 and www.thinkmath.edc.ort
Why are we here?
Highlight some concepts that support algebraic thinking.!
Explore elements of algebraic thinking within the Operations and Algebraic Thinking Domain.!
Review instructional approaches that support algebraic thinking.!
Engage with activities that elicit algebraic thinking.
Caitlin Duncan, May 2015
When we leave . . .
We will understand what algebraic concepts relate to K-3.!
We can identify K-3 standards that relate to algebraic thinking.!
We have a toolkit of instructional strategies and activities that support algebraic thinking.
Caitlin Duncan, May 2015
Please take your number rack and a resource sheet.
Please take a minute to complete the evaluation.
Thank you for attending!
Questions?
Caitlin Duncan, May 2015
