# Presentation: 665454

## Early Steps Count: Teaching Arithmetic to Prepare Students for Algebra

- Linda Plattner
- Archived Information

**Notes:**
Used to be math teacher and am still happiest when I am making a living in a way that has something to do with math

On the personal side, about 7 years ago I married a man I am still crazy about and between us we have 5 kids.  2 of them are now tax payers and 3 of them are sophomores in 3 different universities scattered along the East Coast. 

As you might guess my husband is happiest when I am making a living doing anything. 

For the next 90 minutes we are going to have some fun with what a lot of people call basic math.

## Teaching Subtraction

**How are all of these problems similar?**
**How would you explain these problems if you were teaching second grade? **

** ****62**
_**-49 **_

** ****53**
_**-26 **_

** ****72**
_**-16**__ _

** ****91**
_**-79**__ _

## Ways of Teaching Subtraction

**Ways of Teaching Subtraction**

## Ways to Decompose 

- 40          13

-          53
- 40     10     3

-         26
- 20     3     3

**Conceptual and procedural understanding are intertwined**

**Ways to Decompose **

**Notes:**
Chinese teachers found other ways to decompose . They broke 53 into 40 +13. How is this helpful?

How is it helpful to think of 53 as 40 +10 + 3? 
What about 26 as 20 + 3 + 3?

## HOW would you teach subtraction?

** ****62**
**-**_**49**_

**How can you decompose these numbers to complete the problem?**

**Practice in Decomposing **

**Notes:**
Work with your neighbor to decompose 62-49. Work on chart paper.

How does decomposing help one become successful in Algebra?
(2x +3) (x +4)  (2x +11x + 12)

## How would you teach? 

**Most American teachers said they would use manipulatives in ways that paralleled their understanding**
- Develop a concrete idea of subtraction
- Build understanding of 1 ten = 10 ones
- One teacher wanted to build the idea of equivalent exchanges, which is equal to the Chinese idea of decomposing numbers.

*Knowing and Teaching*; Ma, Liping; cc 1999; Lawrence Erlbaum Ass. Inc.; pg. 26

**Notes:**
Concrete example-one American teacher would use beans and say they were dinosaur eggs because students were interested in dinosaur eggs

## Knowledge  Packet: Subtractionwith Regrouping

- as inverse operations

- Subtraction
- with regrouping of
- large numbers

**Knowledge  Packet: Subtraction****with Regrouping**

*Knowing and Teaching*; Ma, Liping; cc 1999; Lawrence Erlbaum Ass. Inc.; pg. 19

- Add and subtract
- within 10

- Composition of numbers
- within 100

- The rate of composing

- Composing & decomposing
- a higher value unit

- The composition of 10

- Addition
- without carrying

- Subtraction without
- regrouping

- Add and subtract
- within 20

- Subtraction w/regrouping
-  numbers > 20 and < 100

**Notes:**
Few Chinese teachers mentioned manipulatives, rather they talked about knowledge packets
Knowledge Packets--math topics group-by-group rather than topic-by-topic
Knowledge packets have some variability among teachers, although the core components are common
Contain central sequence-CONCEPT AND PROCEDURE from primary and simple to complex and advanced
Rectangle is the topic Dr. Ma raised
Supporting circles--directly connected, these are the concepts and procedures that support the central sequence

## Knowledge Packet for “Borrow and Trade” Approach

- Procedure for subtracting
- with borrowing

- Procedure for subtracting
- without  borrowing

- Subtraction facts

- Addition facts

- Addition and
-  subtraction are
- inverse operations

## Multidigit Multiplication: Dealing with Student Mistakes

**What do you do, when you get this:**

**  ****123**
*****_**456**_
**  ****615**
**  ****492**
**  **_**738**_
**1845**

**Instead of this?**

**    ****123**
***  **_**456**_
**    ****738 **
**  ****492**
_**615**_
**79335**

**Notes:**
What does Place Value have to do with Algebra:
Place value is the foundational to everything. It is the operating system for numbers. It is water to fish

## How did we explain it?

| Procedural | Conceptual |
| --- | --- |
|  |  |

**Notes:**
Procedural teachers--does not help Alg.
Describe the rule
Use lined paper
Use place holder, like asterisk
Conceptual teachers
1. explain rational
2. Subdivide into separate problems (distributive property) 

## Teaching Strategies

**Notes:**
Remember to talk about procedurally directed vs. conceptually directed, rather than American vs. Chinese

## Tr. Chen’s Approach

**    ****123**
***  **_**456**_
**  ****492**
**738**
_**    **__**615**_
**79335**

**Challenge: Find other ways to align the problem so that it is correct.**

**Notes:**
Could be tied to Fundamental Counting Principal

## Division of fractions

**People solve this problem in different ways. How do you solve it? Can you solve it in more than one way?**

**Imagine you are teaching fractions. What is a story you would make up to fit with this problem?**

**1  3/4 ÷ 1/2**

**Notes:**
Invert and multiply
Convert to decimals
Apply the distributive property
You don’t have to multiply (invert and multiply always works)
7   ÷ 1   =  7     or 3 1/2
    2       2
Confusion:  divide by 2, multiply by 1/2 

## Teacher’s Knowledge of Division by Fractions

**Calculated**

**answer**

**More than**

**one way**

**Provided**

**story**

**American**

**Chinese**

## Knowledge Packet for Dividing by Fractions

- Meaning of division
- with whole numbers

- Concept of
- inverse operations

- Concept of fraction

- Concept of unit

- Meaning of multiplication
- with whole numbers

-  Meaning of multiplication
- with fractions

- The meaning of
- division by fractions

*Knowing and Teaching*; Ma, Liping; cc 1999; Lawrence Erlbaum Ass. Inc.; pg. 77

**Knowledge Packet for Dividing by Fractions**

**Notes:**
Reference, do not review

## Profound Understanding of Fundamental Mathematics

- S

- M

- D

- G

- Place value

- Meaning of
- multiplication

- Meaning of
- addition

- Concept of inverse
- operations

## Review Procedure for Division

**In your handout packet**
- Is this approach more procedural or conceptual?
- Which algebraic concepts does it support?
- How could it be enhanced to better support algebraic thinking?
- Would you use this approach?  Why or why not?

## Final Points

**Americans talk about basic math; Chinese talk about fundamental mathematics.**
**How children are taught elementary mathematics sets them up for success or failure in Algebra. **
**The knowledge gap between American and Chinese teachers parallels the learning gap between American and Chinese children.**

**Notes:**
Fundamental because
Foundational (sets up stu. For further learning)
Primary: contains the rudiments of many important concepts in more advanced branches of the discipline.
Elementary - the beginning of stu. Learning; never erase 1+1 =2, just enhance and enrich

## Fall extension

**I will post lessons. As a group, we will discuss ways to strengthen the algebraic connections.**
**You will take one lesson from your text, identify the core math that is being taught and identify the algebraic concepts that could be included. You will post this lesson and your comments about how you could teach it.**** **