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Operations and Algebraic Reasoning. Algebra…. Where have you seen students use or apply algebraic reasoning? Where have you seen students struggle with algebraic ideas? . Refreshing our memory…. Glossary, Table 1 – take it out if you have it . Problem Types: Agree or Disagree.
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Algebra… • Where have you seen students use or apply algebraic reasoning? • Where have you seen students struggle with algebraic ideas?
Refreshing our memory… • Glossary, Table 1 – take it out if you have it
Problem Types: Agree or Disagree • The problem types are research-based and come from research with young children doing these tasks.
Problem Types: Agree or Disagree • This idea of problem types are all over Investigations curriculum in various grades.
Problem Types: Agree or Disagree • When we think about problem types with addition and subtraction it does not matter at all about how students “solve” tasks (e.g., manipulatives, drawing, counting, number lines).
Problem Types: Agree or Disagree • Writing tasks to fit a specific problem type is a tasks that my teachers can do.
Problem Types and their history • Cognitively Guided Instruction • Problem Types (Types of tasks) • Is that all there is to CGI ?????? • Does it matter how students solve these problems? Why or why not?
Problem Types and their history • Cognitively Guided Instruction • Problem Types (Types of tasks) • Methods in which students solve tasks • Decisions that teachers go through to formatively assess students AND then pose follow-up tasks
Methods • Direct Modeling • Counting Strategies • Algorithms or Derived Facts • There were 8 seals playing. 3 seals swam away. How many seals were still playing? • Susan is 5 years old. Mark is 15 years old. How many times older is Mark than Susan?
Methods • Direct Modeling
Methods • Counting Strategies
Methods • Derived Facts or Algorithms
Direct modeling, counted or invented strategy? • There were 8 seals playing. 3 seals swam away. How many seals were still playing? • The student starts at 8 on a number line and count backwards 3 numbers. The number they land on is their answer. • The student puts 3 counters out and adds counters until they get to 8. The number of counters added is their answer.
Direct modeling, counted or invented strategy? • There were 8 seals playing. 3 seals swam away. How many seals were still playing? • The student draws 8 tallies and crosses out 3. The number left is their answer. • The student starts at 3 and counts up until they get to 8. As the student counts they put a finger up (1 finger up as they say 4, 5, 6, 7, 8). The number of fingers up is their answer.
Direct modeling, counted or invented strategy? • Susan is 5 years old. Mark is 15 years old. How many times older is Mark than Susan? • A student draws 5 tallies and circles them. They then draw another 5 tallies and circle them and then count their 10 tallies. They do this one more time and count 15 tallies.
Direct modeling, counted or invented strategy? • Susan is 5 years old. Mark is 15 years old. How many times older is Mark than Susan? • A student writes the equation 5x3 = 15 and also the equation 15 divided by 5 = 3.
How students solve problems • Does it matter what strategy students use? Why? • What does it look like for students to be proficient with a problem type? Does the strategy that they use indicate they are proficient?
Common Core Connection • “Fluently add and subtract” • What do we mean when students are fluent? • Fluently (Susan Jo Russell, Investigations author) • Accurate, Efficient, Flexible • What do these mean? • Where do basic facts tests fit in?
Factors and Multiples • Three cruise ships are in port today. They arrive back to port and leave the same day. The Allure of the Seas arrives every 3 days. The Oasis of the Seas arrives every 4 days. The Quantum of the Seas arrives every 6 days. • Over the next 200 days, on what days will 2 of the ships be in port at the same time? When will 3 of the ships be in port at the same time?
Factors and Multiples • Where is the algebra with this type of work? • In the following case- • Where is there “algebraic reasoning”? • How does the teacher promote “algebraic reasoning?”
Task Modification • Investigations Unit– examine a number sense unit • Look for “opportunities” to modify tasks to match “more difficult” task types • Modify/write tasks • What is an appropriate size of numbers? • What are the task types? • How would you assess?
Teaching experiment… • Select students who are struggling • Pose a few problems for a problem type • Observe and question • Pose a follow-up task that “meets them where they are”
Working with Large Numbers • On your own solve 4,354 – 3,456 + 455 in three different ways • Write a story problem to match this problem. • Pick one of your strategies… how did algebraic reasoning help you complete the task?
4,354 – 3,456 + 455 • Gallery Walk • Explore various strategies and explanations • Any commonalities or frequently occurring ideas?
4,354 – 3,456 + 455 • Sharing out strategies • How can estimation help us BEFORE we start? • Rounding…. Rounding to which place helps us get the best estimate? • What is the point of rounding?
