operations and algebraic reasoning. algebra… where have you seen students use or apply algebraic...
TRANSCRIPT
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
• 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?
Approaches? Solutions?
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?