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Chapter 11 (Invented Strategies (The environment (Things to keep in mind…

- - - - sharing ideas is safe
      - they will have to persevere
      - Be ready for a productive struggle
    - - How kids learn
        
        Word problems are the primary vehicle for developing invented strategies
        
        When teaching standard/invented algorithms, they should have
        
        Certainty (precise procedures)
        
        Transparency (process is understood)
        
        Efficient
        
        Generalizable
        
        Students gain a superior sense of numbers and why math works when avoiding the standard algorithm
        
        Productive struggle in early stages means far fewer issues later on
    - - Encourage kids to clarify previous knowledge and and make attempts to construct new ideas.
      - Promote curiosity and openness to new ideas
      - Expect and encourage students to discuss ideas with each other
      - Discuss right and wrong ideas with respect, take wrong ideas seriously.
      - Don’t immediately give the right answer
      - Use anonymous samples for kids to critique
      - Give context. Context is superior to just having the numbers
  - - - Useful representation. For the problem 6x34 a kid could double 34 and get 68 and add three sets of 68. This doesn’t count as repeated addition apparently.
      - Partitioning. Students can take 4x27 and make a tree of calculations. 4x20 and 4x7 amalgamated yield the solution.
      - Compensation strategies. Often split a number in half before doubling the other number. There are a variety of things to do with this kind of math
      - Cluster problems. Using a variety of guess and check equations based off the original equation
      - Standard algorithm of multiplication
        
        Probably the hardest algorithm to understand.
        
        Area models-begin with these. They’re highly useful.
        
        Partial Products- It’s a big area model that focuses on the ones, tens, and hundreds blocks.
        Lattice Multiplication- the big box with all the diagonal lines.
        
        Open arrays- It’s that square/rectangle that becomes divided in pieces to demonstrate the largest numbers in multiplication
        
        The partial products written record- a large source of errors.
    - - An empty number line can be quite useful. Using tick marks helps.
        
        It can be used to model a students thinking.
      - Think addition- when you take two numbers and figure out the differences between them. Work well with subtraction.
    - - Developed in 3-5th grade
        Fair share problems are easier to solve with 10 sticks.
      - Missing Factors Strategies
        You have a goal of, say, 318, and you have to figure out how many sevens are in it. You guess with multiplication how many sevens, sometimes going under the amount and sometimes going over. Often an invented strategy.
      - The standard algorithm of division: Start with models. Partition/Fair Share Model- Kids are confused when I say “One four goes into the 5” in 571. They wonder why I ceased looking into the 71. Use context to help. Can say “5 cartons with 7 boxes extra, and 1 more bar.”
      - More models
        
        Explicit trade method. The Long division where the first step is normal. The second step, where you’re subtracting two digit numbers in the second column (say, 37-31) and two digits in each subsequent column.
        
        Repeated Subtraction- similar to multiplication where you put in the zeroes. Don’t put numbers at the top. Rather, subtract normally and put the number of times the first digit went in at the same time. However, on the side, write the answer with however many zeroes for the place value. You’ll get your hundreds, your tens, and your ones. Circle the remainder at the bottom and add up the side so that you’ll get the answer.
        
        Computational Estimation
        
        Used when you don’t need a precise calculation. To teach, use
        
        Real examples
        
        Estimation language. Not the word “guess!”
        
        Don’t reward or emphasize the student who is the closest. Focus on how reasonable the answer and thinking is.
        
        Accept a range of estimates (or offer a range as an option)
        
        Context
        
        Focus on Flexible Methods, not answers. You can ask them if it over/under a certain amount, about how much, or if it is between certain numbers.
      - Be explicit in teaching each part of Long Division. There are four steps
        
        Share and record the number of pieces in each group.
        
        Record the number of pieces shared in all. Multiply to find the number
        
        Record the number of remaining pieces by subtracting
        
        Trade (if necessary) for smaller pieces, and combine any same sized pieces that are there already. Record the new number in the next column
      - A Good Idea to keep in mind
        
        When working with long division. Divide and always assume fewer than what you expect so that you undershoot and arrive close to the answer.