1. Presented by: Angela J. Williams
Algebraic Thinking
Presented by: Angela J. Williams
Welcome to Algebraic Thinking. This PD was developed to help us better understand our role as elementary teachers in higher mathematics. We will explore the BIG IDEAS we build in elementary school and where we can begin to build this understanding from a young age.

2. Why Teach Algebraic Thinking?
Large percentages of students struggle to PASS Algebra, much less UNDERSTAND!
Algebra is recognized as the GATEWAY to higher education and other career opportunities.
To become a successful member of our society in this technology-driven world requires problem-solving skills and abstract thinking.

3. Just What is Algebraic Thinking?
Mathematics teaching and learning that will prepare students with critical thinking skills.
The 8 Standards for Mathematical Practice in the CCRS are the “thinking tools” or “habits of mind” ALL students need to be successful in Algebra.
The CONTENT we teach is HOW we develop these thinking skills in our students and help them generalize the mathematical ideas that they will use in higher mathematics courses, including Algebra.
The connection between mathematical ideas and these thinking skills are KEY to developing what we call Algebraic Thinking!

4. Components of Algebraic Thinking
Mathematical Thinking
Problem-solving skills
Representation & Modeling
Quantitative Reasoning
Algebraic Ideas
Generalizations
Using mathematical language & vocabulary
Representing mathematical ideas visually and symbolically

5. Communication is KEY
Being able to “talk” about math allows students to build understanding at a deeper level.
Opportunities to make connections and see relationships between multiple representations of mathematical information (visually, symbolically, verbally, numerically) also deepens student’s mathematical understanding.
The ability to “talk” about their thinking and create, interpret, and translate among representations gives students a powerful tool for future mathematical learning.

6. What would this look like if we continued to build this pattern a third time?

7. How can we figure out what the 5th pattern would look like without actually building it with cubes?

8. The TASK is KEY!

9. Tasks Should… Be thought provoking and offer multiple solution paths
Encourage communication (talking, listening, visual representations, drawings, graphs, symbolic notation etc.)
Be interesting and engaging to students
Be focused on on mathematical understanding of numbers, properties, relationships, etc.

10. It ALL Begins with Counting & Cardinality
Early counting and number work with whole numbers, including representations, extends to all other mathematical concepts.
Operations (addition, subtraction, multiplication, division) grow from this understanding of whole numbers.
The understanding of the relationship between these operations is key to algebraic thinking and are developed over several grade levels.
Properties, such as the commutative and associative are built based on all of these experiences.
Progressions for the Common Core, 2011

11. What does this symbol mean?

12. Children’s Responses….
The answer comes next.
A command to carry out the calculation.
It means to use all of the numbers.
It means the total amount of two numbers.
It is the sign for the sum.
It means the answer is next.
It is the mark for showing the answer.
“Children’s Understanding of Equality and the Equal Symbol” by Cumali Oksuz

13. What are the different answers students might give for this question?
Put a number on the line to make the number sentence true = ___ + 5

14. The results of asking 288 1st through 6th Graders7 12 17
12 & 17 9%
49%
25%
10%
Carpenter, Franke, Levi, 2003)

15. Research Says…
The equality relationship denoted by the equal sign is a foundational concept that serves as a key link between arithmetic and algebra.
(Cumali Oksur)
In order to help students develop understanding of equality it is important to challenge their possible existing misunderstandings.
(Carpenter, Franke and Levi)

16. Benchmarks for Developing a Concept of the Equal Sign
Uncover a student’s current concept of equality.
The student accepts as TRUE a number sentence that IS NOT written in the form of a+b=c.
The student sees the equal sign as a relationship between the 2 sides using comparison of quantity.
The student can compare the mathematical expressions without actually carrying out the calculations.

17. A Context for the Equal Sign (=)
Appropriate tasks provide focus and challenge their current concept of equality while allowing the teacher to “see” their current thinking!
One way to do this is through TRUE/FALSE Number Sentences like these:

18. A 1st Grade Perspective

19. Activity: What’s in the Bag?
You NEED:
A balance
A bag of mystery items
Cubes and tiles
Directions:
Use your reasoning skills to determine what is in the bag without looking in the bag.
Be prepared to JUSTIFY your decision.

20. Relational Thinking

21. Relational Thinking – Your Turn!

22. Find the value of each shape. Explain your reasoning.+ + + = 40 + + = 29

23. How many ovals do you need to balance the scale?
Solve the problem. Justify your answer mathematically.

24. Revisiting Algebra Solve the problem conceptually, not procedurally!
Use red tiles to represent x and blue tiles to represent numbers.
You may use the scale to explain your reasoning.

25. Summing it UP!
Algebraic thinking begins in Kindergarten and continues throughout elementary school.
Children must have the cardinality and number sense developed in early grades and the ability to reason and justify their thinking in order to be prepared for Algebra.
The concept of equality and relational thinking are imperative for algebraic thinking.
Tasks should encourage thinking and reasoning and be designed to challenge students’ understanding of concepts such as properties of operations and the examination of strategies used to solve problems.
The language we use and the vocabulary we engage students in using can strengthen conceptual understanding or create misconceptions (i.e. equal means “is the same as” or equal means the answer comes next)