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Math for America San Diego
Developing Algebraic Reasoning from Quantitative Reasoning
Dr. Osvaldo SotoGenevieve Esmende
Who we areMath for America San Diego – Noyce Program
Genevieve EsmendeNoyce Master Teaching Fellow
Dr. Osvaldo SotoSenior Program Associate
Workshop Goals• What is quantitative reasoning?
• Why should we encourage it?
• How do we get students to reason quantitatively?
• How do we develop algebraic reasoning?
Do the Math
• Take time to solve the problems.– What are you noticing about your work as you
progress through the problems.– As you work on the problems, think about
strategies or difficulties students might encounter.
– What Common Core Standards for Mathematical Practice are you using as you solve the problems?
Motion Problems 1
1) From two towns, A and B, two cars left at the same time toward each other – one from Town A and the other from Town B. The speed of the first car is 70 mph, and the speed of the second car is 80 mph.
a) If the distance between the two towns is 300 miles, how long does it take for the two cars to meet?
b) If the distance between the two towns is 750 miles, how long does it take for the two cars to meet?
c) If the distance between the two towns is 525 miles, how long does it take for the two cars to meet?
d) If the distance between the two towns is 650 miles, how long does it take for the two cars to meet?
Motion Problems 2
2) A biker and a motor cycler left from the same location at the same time, and in the same direction; the biker at the speed of 12 mph, and the motor cycler at 40 mph.
a) In how many hours will the distance between them be 140 miles?
b) In how many hours will the distance between them be 42 miles?
c) In how many hours will the distance between them be 105 miles?
Sharing our work: Modeling Teaching Actions
• Please attend to how we structure the presentation of solutions.
• What points come out of our conversations?
• Which ways of thinking emerge?
Standards for Mathematical Practice
1) Make sense of problems and persevere in solving them.2) Reason abstractly and quantitatively.3) Construct viable arguments and critique the reasoning of
others.4) Model with mathematics.5) Use appropriate tools strategically.6) Attend to precision.7) Look for and make use of structure.8) Look for and express regularity in repeated reasoning.
Standards for Mathematical Practice
1) Make sense of problems and persevere in solving them.2) Reason abstractly and quantitatively.3) Construct viable arguments and critique the reasoning of
others.4) Model with mathematics.5) Use appropriate tools strategically.6) Attend to precision.7) Look for and make use of structure.8) Look for and express regularity in repeated reasoning.
Reasoning quantitatively
• Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
From the Common Core Standards for Mathematical Practice #2
Reasoning quantitatively
“Because the central goal is to focus on quantities and how they relate in situations, and because this represents a major mathematical change of focus for many students, it is important to open discussions with questions that lead to discussions of quantities, not numbers. A useful opening question can be a general one, like “what is going on here?” The goal is to get students to describe situations as they see them.”
Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York: Erlbaum.
Quantities vs. Numbers
Snowflakes
“What is going on here?”
Quantities vs. Numbers
8 ÷ 4 = 2
What is the meaning of the 2?
Student StrategiesFrom two towns, A and B, two cars left at the same time toward each other – one from Town A and the other from Town B. The speed of the first car is 70 mph, and the speed of the second car is 80 mph. If the distance between the two towns is 300 miles, how long does it take for the two cars to meet?
Student StrategiesFrom two towns, A and B, two cars left at the same time toward each other – one from Town A and the other from Town B. The speed of the first car is 70 mph, and the speed of the second car is 80 mph. If the distance between the two towns is 300 miles, how long does it take for the two cars to meet?
Making Sense of the Problem
• The students started with problems where the result was a “friendly” whole number solution. Students used their prior knowledge about motion to help them solve the problem. Students were able to show far they have traveled at each hour by creating their own mental image to guide their understanding of the problem.
Student StrategiesA biker and a motorcyclist left from the same location at the same time, and in the same direction; the biker at the speed of 12 mph, and the motorcyclist at 40 mph. In how many hours will the distance between them be more than 105 miles?
Making Sense of the Problem
When the solution resulted in an “unfriendly” number, this gave students the need to reason quantitatively and proportionally in order to find the solution.
Student StrategiesA biker and a motorcyclist left from the same location at the same time, and in the same direction; the biker at the speed of 15 mph, and the motorcyclist at 55 mph. In how many hours will the distance between them be more than 170 miles?
Leaving the computation open
Repeated Reasoning
Students reason repeatedly and quantitatively until they notice a pattern which lead them to write an equation based on the quantity that is varying.
Work Problems
• How are the strategies for solving work problems similar or different to the motion problems?
Work Problems
1) Josh can paint a room in 3 hours. It takes Gabe 6 hours to paint the same room. In how many hours will they complete the job if they work together?
2) Carmela can mow a lawn in 3 hours. Mia can mow the same lawn in 2 hours. In how many hours will they complete the job if they work together?
Student Strategies of Work Problems
Josh can paint a room in 3 hours. It takes Gabe 6 hours to paint the same room. In how many hours will they complete the job if they work together?
Student Strategies of Work Problems
Josh can paint a room in 3 hours. It takes Gabe 6 hours to paint the same room. In how many hours will they complete the job if they work together?
Student Strategies of Work Problems
Josh can paint a room in 3 hours. It takes Gabe 6 hours to paint the same room. In how many hours will they complete the job if they work together?
Student Strategies of Work Problems
Josh can paint a room in 3 hours. It takes Gabe 6 hours to paint the same room. In how many hours will they complete the job if they work together?
Student Strategies of Work Problems
Carmela can mow a lawn in 3 hours. Mia can mow the same lawn in 2 hours. In how many hours will they complete the job if they work together?
Student Strategies of Work Problems
Carmela can mow a lawn in 3 hours. Mia can mow the same lawn in 2 hours. In how many hours will they complete the job if they work together?
Applying to the classroom• Give students time to make sense of the problems.• Don’t rush to the equation. Give students time to
see the necessity for an equation.• Teacher facilitates the discussion to connect
students strategies to the development of an equation.
• Differentiating the level difficulty of the problem: by changing the numbers in the context or changing the context.
In Our Experience…• Students access images for motion more easily than
work situations (painting a house, filling a pool, etc.)
• To encourage algebraic reasoning, introduce working situations gradually.
• Repeated quantitative reasoning promotes the formation and internalization of algebraic reasoning.
• Naming algebraic properties should come last.
References
• Problems taken from: Harel, G. (2013). The Kaputian program and its relation to DNR-based instruction: A common commitment to the development of mathematics with meaning, In The SimCalc Vision and Contribution, (Fried, M., & Dreyfus, T., Eds.), Springer, 438-448.
Take two and call us in the morning…
Thank you!
Dr. Osvaldo SotoSenior Program Associate
Genevieve EsmendeNoyce Master Teaching Fellow
Evaluation
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