1. Chapter 05

2. Learning with Understanding:
Concepts and Procedures And
Multidigit Addition and Subtraction 2

3. Number Talk 50 – 27= Suggested Activity:
Today’s routine, another number talk, focuses on subtraction strategies. Careful choice of numbers will help elicit a wider range of strategies. Note the numbers in the above problem. These numbers were purposefully chosen for many reasons. After conducting the number talk, ask your students why they think these numbers were chosen. 50 and 25are easy numbers to think about. Also, twenty-seven can easily be converted to thirty. Some students may over-generalize a strategy they used in addition by adding the 3 onto the 27 and then take it away from the total. Take time to carefully explore (using manipulatives and pictures) why that strategy does not work.
Remember to ask for all the answers and write them on the board before anyone begins to share out their individual strategy, then invite students to defend their answer. This is also the perfect time to explore typical errors children would make. 3

4. Conversation in Mathematics
Discuss what the student meant by, “I would have remembered how to do it the right way.” 4

5. Concepts and Procedures
Pedagogy 5

6. Conceptual vs. Procedural Knowledge
Weak conceptual knowledge
Strong conceptual knowledge
Weak procedural knowledge
Strong procedural knowledge
Remind students that conceptual knowledge and Procedural fluency are the two branches that TOGETHER make up mathematical proficiency.
Discuss the difference between concepts and procedures.
Suggested Activity:
Invite students to generate an example of each:
Weak conceptual knowledge
Strong conceptual knowledge
Weak procedural knowledge
Strong procedural knowledge
An example of each is found in figure 5.1 on page 127 of the text. 6

7. Beliefs about Mathematics
Mathematics is a web of interrelated concepts and procedures.
One’s knowledge of how to apply mathematical procedures does not necessarily go with understanding of the underlying concepts.
Understanding mathematical concepts is more powerful than remembering procedures.
If students learn mathematical concepts before they learn procedures, they are more likely to understand the procedures when they learn them. If they learn the procedures first, they are less likely to ever learn the concepts.
Discuss each of these beliefs. 7

8. Constructing Mental Concepts
Naturalistic learning (Piaget)
Informal learning (Vygotsky)
Structured learning (Vygotsky)
Discuss each of these learning situations in which concepts are acquired
■ Naturalistic learning (Piaget). The adult role is to provide an interesting environment, and the child controls his choice of activity and actions.
■ Informal learning (Vygotsky). There is some adult intervention, but the child is still in control of the activity. This is where the teacher acts on those “teachable moments.”
■ Structured learning (Vygotsky). The adult is in control of the activity and gives direction to the child’s action. 8

9. Equilibration Schema Equilibrium Disequilibrium Assimilation
Accommodation
Zone of Proximal Development
Discuss each of these terms as it relates to the process of Equilibration 9

10. Developing Procedural Fluency
Efficiency
Accuracy
Flexibility
Discuss what is needed to become procedurally fluent. 10

11. Traditional vs. Alternative Algorithms
Compare and contrast traditional and alternative algorithms. Invite students to provide an example of each. The number talks you have had up until now should assist students in generating examples of alternative algorithms. 11

12. Multidigit Addition and Subtraction
Content
As you begin to study how students develop the earliest of number concepts, consider the role tools have in these phases. 12

13. Invented Algorithms related to Addition and Subtraction
Direct modeling with ones
Directly modeling with tens and ones
Incrementing Combining tens and ones
Making tens
Compensating
Discuss each of the ways children solve problems involving multi-digit addition and subtraction. Invite students to describe each of these using different tools. Discuss how to get students to pass from directly modeling with ones to directly modeling with tens and ones through questioning.
You may want to re-visit the number talk from earlier in this session and connect some of these algorithms to what your students shared. Any algorithm not shared, for example compensating using constant difference, should be explored as a class. Don’t forget to connect it to pictures and tools. 13

14. Standard Algorithms Suggested Activity 1
Have students Directly model with cubes what is actually happening in the standard algorithms of addition and subtraction. Also, have them make connections between the standard algorithm and some of the invented algorithms discussed.
Suggested Activity 2
Invite students to research some standard algorithms for addition and subtraction that are used in other countries. Compare them with the standard algorithms we use in the United States 14

15. Connecting the model with a pictorial and symbolic representation
This is probably one of the most important, yet neglected aspect of inventing an algorithm or developing conceptual understanding of the traditional algorithm.
Suggested Activity
Provide the following Problems and have students solve through direct modeling. Have them then draw a picture of what they built and somehow indicate what they did with their tools, such as crossing out when subtracting. They then should create a numeric representation of each step they built/drew. Finally, have them share their problem solving strategy with their classmates to practice articulating how they solved a problem that others could replicate.
1) Miguel had 27 toy cars. Tim gave him some more cars. Now he has 61 cars altogether. How many toy cars did Tim give Miguel?
2) Sam got 35 answers correct on his math test. Last week he got 42 answers correct. What is the difference between the two tests?
3) Mayra had a bag of valentines. Her classmates gave her 51 more. Now she has 75 Valentines. How many did she have to start with? 15