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Learning to Think Mathematically

With the Rekenrek

A Resource for Teachers

A Tool for Young Children

Jeff Frykholm

Overview

- In this seminar we will examine the Rekenrek, a simple, but powerful, manipulative to help young children develop mathematical understanding.
- Specifically, we will explore…
- A rationale for the Rekekrek
- The mathematics of the Rekenrek
- Many activities that show how the Rekenrek can improve students understanding and proficiency with addition and subtraction, number sense, and our base-10 system

Introduction

- Solve the following addition problems mentally.
- As you come up with the answer, be aware of the strategy that you used to determine the answer.
- Most likely, your brain will make adjustments on these numbers very quickly, and you will use an informal strategy to find the result. Of course, some of you will know these by rote memorization as well.

8 + 7 = ?

- What mental adjustments did you make as you solved this problem?
- Double 8, subtract 1? (8 + 8 = 16; 16 - 1 = 15)
- Double 7, add 1? (7 + 7 = 14; 14 + 1 = 15)
- Make 10, add 5? (8 + 2 = 10; 10 + 5 = 15)
- Make 10 another way? (7 + 3 = 10; 10 + 5 = 15)
- Other strategies?
- Next problem…

5 + 8 = ?

- What mental adjustments did you make as you solved this problem?
- Make 10, add 5? (5 + 5 = 10; 10 + 3 = 13)
- Make 10 another way? (8 + 2 = 10; 10 + 3 = 13)
- Use another fact? (If 8 + 4 = 12, then 8 + 5 = 13)
- Other strategies?
- Next problem…

9 + 7 = ?

- What mental adjustments did you make as you solved this problem?
- Make 10, add 6? (9 + 1 = 10; 10 + 6 = 16)
- Other strategies?

Questions…

- If we use these strategies as adults, do we teach them explicitly to young children?
- Should we?
- If so… how?

With the Rekenrek

- The Rekenrek is a powerful tool that helps children see “inside” numbers (“subitize”), develop cardinality (one-to-one correspondence), and work flexibly with numbers by using decomposition strategies.

What is the Rekenrek?

- The rekenrek combines key features of other manipulative models like counters, the number line, and base-10 models.
- It is comprised of two strings of 10 beads each, strategically broken into groups of five.
- The rekenrek therefore entices students to think in groups of 5 and 10.
- The structure of the rekenrek offers visual pictures for young learners, encouraging them to “see” numbers within other numbers… to see groups of 5 and 10.
- For example…

A group of 10

3 more

- With the rekenrek, young learners learn quickly to “see” the number 7 in two distinct parts: One group of 5, and 2 more.
- Similarly, 13 is seen as one group of 10 (5 red and 5 white), and three more.

Constructing a Rekenrek

- What do we need?
- A small cardboard rectangle (foam board)
- String/Pipe cleaners
- 20 beads (10 red, 10 white)
- For younger children, one string with 10 beads may be sufficient.

How to Make a Rekenrek

Step one: Cut 4 small slits in the cardboard

Step 2: String beads

- 20 beads 10 Red, 10 White
- Two strings of 10 beads
- Tie a knot in the end of each string, or use pipe cleaners with snugly fitting beads.

Step 3: Strings on Cardboard

- Slip the ends of the string through the slits on the cardboard so that the beads are on the front of the cardboard, and the knot of the string is on the back side.

Activities with the Rekenrek

- Show me… 1-5; Show me 5-10
- Make 10, Two Rows
- Flash Attack!
- Combinations, 0 - 10
- Combinations, 10 - 20
- Doubles
- Almost a Double
- Part-Part-Whole
- It’s Automatic… Math Facts
- The Rekenrek and the Number Line
- Subtraction
- Contextual Word Problems and the Rekenrek

An Introduction

- Students should be introduced to the Rekenrek with some simple activities that help them understand how to use the model.
- These activities also give teachers a chance to assess the thinking and understanding of their students.
- For example:
- Show me 1 - 10
- Make 5; Make 10 (with two rows)
- Emphasize completing these steps with “one push”
- After students can do these exercises with ease, they are ready for other more challenging activities. The following lesson ideas are explained fully in the book.

Flash Attack

- Objectives
- To help students begin to “subitize” -- to see a collections of objects as one quantity rather than individual beads
- To help students develop visual anchors around 5 and 10
- To help students make associations between various quantities. For example, consider the way a child might make the connection between 8 and 10.
- “I know there are ten beads in each row. There were two beads left in the start position. So… there must be 8 in the row because 10 - 2 = 8.”

Lesson Progression

- Start with the top row only; Push over 2 beads: “How many do you see?”
- Repeat: push over 4 beads; 5 beads
- Now… instruct students that they have only two seconds to tell how many beads are visible.
- Suggested sequence:
- 6, 10, 9, 7, 8, 5, 3, 4
- Ready?

Basic Combinations 0-10 (p21)

- Lesson Objectives
- Relational View of Equality
- One of the strengths of the rekenrek is its connections to other forms of mathematical reasoning. For example, equality.
- Our children are only used to seeing problems like…
- 4 + 5 = ? 6 + 3 = ?
- Part-Part-Whole relationships
- Missing Addend problems
- Continue to build informal strategies and means for combining numbers

Basic Combinations 0-10

- Modeling the activity:
- “We are going to work together to build numbers. Let’s build the number 3. I will start. I will push two beads over on the top row. Now you do the same. Now… how many beads do you need to push over on the bottom row?”
- Use both rows to keep the addends clearly visible
- Suggested sequence (begin with 5 on the top row)
- “Let’s make 8. I start with 5. How many more?”
- “Let’s make 9. I start with 6. How many more?”
- “Let’s make 6. I start with 5. How many more?”
- “Let’s make 6 again. I start with 3. How many more?”
- E.g., Build on previous relationship. Use doubles.
- Listen to the thinking/explanations of students.

Combinations, 10-20 (p. 23)

- This activity works the same as the previous, but we use numbers between 10 and 20
- Vary the presentation of the numbers, using context occasionally
- Take advantage of this opportunity to help students develop sound understanding of the addition facts.
- Examples…
- “Let’s make 15. I start with 8. How many more?”
- Think through the reasoning used here by the children. What strategies might they use?

Combinations, 10-20 (p. 23)

- Teaching strategies:
- Use the same start number 3-4 times in a row, changing the initial push. This will establish connections between fact families.
- Build informally on the doubles.
- If students are counting individual beads, stop, and model groups of 5 and 10, perhaps returning to previous activities in the book.

1+1=2 2+2=4 3+3=6 4+4=8Doubles (p. 25)

- Objectives
- Help students visualize doubles (e.g., 4+4; 6+6)
- Help students use doubles in computation
- The visualization is key:

Developing Understanding of the Doubles

- Ask students what they notice about these visualizations. They might see them as vertical groups of two… as two horizontal lines of the same number of beads… even numbers… etc.
- As students are ready, teachers should include the doubles between 6 and 10, following a similar teaching strategy. In this case, students should know to use their knowledge of a double of 5 (two groups of 5 red beads = 10) to compute related doubles.
- For example, consider 7 + 7 = 14. Students are likely to see two sets of doubles. First, they will recognize that two groups of 5 red beads is 10. Next, a pair of 2’s is 4. Hence, 7 + 7 = (5 + 5) + (2 + 2) = 10 + 4 = 14

Almost a Double

Objectives of the lesson

- Students should use their understanding of doubles to successfully work with “near doubles”, e.g., 7 + 8
- Students can begin to recognize the difference between even numbers (even numbers can be represented as a pair of equal numbers) and odd numbers (paired numbers plus one).

Near Doubles: Developing Ideas

- Develop this idea by doing several additional examples with the Rekenrek. Ask students to use the Rekenrek to “prove” whether or not the following are true.
- Have students visually identify each component of these statements:
- Does 6 + 7 = 12 + 1?
- Does 3 + 2 = 4 + 1?
- Does 4 + 5 = 8 + 1?
- Does 8 + 9 = 16 + 1?

Part-Part-Whole (p. 29)

Objectives of the Lesson

- To develop an understanding of part-part-whole relationships in number problems involving addition and subtraction.
- To develop intuitive understandings about number fact relationships.
- To develop a relational understanding of the equal sign.
- To develop confidence and comfort with “missing addend” problems.

Cover Remaining BeadsDeveloping Ideas

- Push some beads to the left. Cover the remaining beads.
- Ask students: “How many beads do you see on the top row?”
- Ask: “How many beads are covered (top row)?”
- Listen for answers like the following:
- “5 and 1 more is 6. I counted up to 10 … 7, 8, 9, 10. 4 are covered.”
- “I know that 6 + 4 = 10. I see 6, so 4 more.”
- “I know that there are 5 red and 5 white on each row. I only see one white, so there must be 4 more.”
- Next… move to both rows of beads.

It’s Automatic:Math Facts

Objectives in Learning the Math Facts

Quick recall, yes. But… with understanding, and with a strategy!

- To develop fluency with the addition number facts through 20.
- To reinforce anchoring on 10 and using doubles as helpful strategies to complete the math facts through 20.
- Students can model the number facts on one row of the rekenrek (like 5 + 4), or model facts using both rows (which they have to do when the number get larger, e.g., 8 + 7)

An Activity: Subtraction

- With two or three of your colleagues, create a plan for teaching subtraction with the Rekenrek
- Be prepared to share your method

Story Problems and the Rekenrek

Examples

- “Claudia had 4 apples. Robert gave her 3 more. Now how many apples does Claudia have?”
- “Claudia had some apples. Robert gave her 3 more. Now she has 7 apples. How many did she have at the beginning?” (Here the Start is unknown.)
- “Claudia had 4 apples. Robert gave her some more. Now she has 7 apples. How many did Robert give her?” (Here the Change is unknown.)
- “Together, Claudia and Robert have 7 apples. Claudia has one more apple than Robert. How many apples do Claudia and Roberthave?” (This is a Comparison problem.)

An Activity

- Select one of these problem types from CGI
- Create three problems
- Find the solutions to the three problems using the Rekenrek
- Exchange your problems with another group. Try to solve the other group’s problems by modeling each step with the Rekenrek

In Summary…

- Ideas, Insights, Questions about the Rekenrek?
- How it can be used to foster number sense?
- Strengths?
- Limitations?

The Rekenrek and a Relational View of Equality

- The process of generalization can (should) begin in the primary grades
- Take, for example, the = sign

What belongs in the box?

8 + 4 = + 5

How do children often answer this problem? Discuss…

8 + 4 = + 5

- 3 research studies used this exact problem
- No more than 10% of US students in grades 1-6 in these 3 studies put the correct number (7) in the box. In one of the studies, not one 6th grader out of 145 put a 7 in the box.
- The most common responses?
- 12 and 17
- Why?
- Students are led to believe through basic fact exercises that the “problem” is on the left side, and the “answer” comes after the = sign.
- Rekenrek use in K-3 mitigates this misconception

Rather than viewing the = sign as the button on a calculator that gives you the answer, children must view the = sign as a symbol that highlights a relationship in our number system.

- For example, how do you “do” the left side of this equation? What work can you possibly do to move toward an answer?

3x + 5 = 20

- We can take advantage of this “relational” idea in teaching basic facts, manipulating operations, and expressing generalizations in arithmetic at the earliest grades
- For example…
- 9 = 8 + 1
- 5 x 6 = (5 x 5) + 5
- (2 + 3) x 5 = (2 x 5) + (3 x 5)

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