Transcript mbamp.ucsc.edu

Learning to Think Mathematically
With the Rekenrek
A Resource for Teachers
A Tool for Young Children
University Of California Math Project @ UCSC
Adapted from J. Frykholm
Overview
• First 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:
With the Rekenrek, young learners learn quickly to “see”
the number 7 in two distinct parts: One group of 5, and 2
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.
A group of 10
3 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?
Example 1: One Second
How many Red beads?
Example 2
Example 3
Example 4
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.
Doubles (p. 25)
• Objectives
• Help students visualize doubles (e.g., 4+4; 6+6)
• Help students use doubles in computation
• The visualization is key:
1+1=2
2+2=4
3+3=6
4+4=8
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
7 + 7 = 14
Seen as, perhaps…
2 groups of 5, plus 4
Almost a Double
Objectives of the lesson
• Students should use their understanding of
doubles to successfully
doubles”, e.g., 7 + 8
work
with
“near
• 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…
2+1
1
2+1 = 3
3+2
4+3
5+ 4
4+1 = 5
6+1 = 7
8+1 = 9
Use a pencil
to separate
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.
Developing Ideas
• Push some beads to the left. Cover the remaining beads.
• Ask students: “How many beads do you see on the top
row?”
Cover
Remaining
Beads
• 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 Robert have?” (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)