Operations and Algebraic Thinking

November 15, 2012

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 most of 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?

• What would each of these 3 look like in a Grade 1 classroom?

Methods

• Direct Modeling

• Separate (Result Unknown)• There were 8 seals playing. 3 seals swam away.

How many seals were still playing?• A student would….. • A set of 8 objects is constructed. 3 objects are

removed. The answer is the number of remaining objects.

Methods

• Counting Strategies

• Separate (Result Unknown)• There were 8 seals playing. 3 seals swam away.

How many seals were still playing?• A student would….. • Start at 8 and count backwards 3 numbers. The

number they end on would be their answer.

Methods

• Invented algorithms /derived strategies

• Separate (Result Unknown)• There were 8 seals playing. 3 seals swam away.

How many seals were still playing?• What would students do? • “4 plus 4 is 8, so 8 minus 4 is 4. But I am only

taking away 3 so there should be 5 seals playing.”

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.

How students solve problems

• Does it matter what students strategy is? Why?

• What does it look like for students to be proficient with a problem type?

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?

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?