Understanding Fourth Graders Mathematical Thinking: Issues and Insights for Teaching Fractions - PowerPoint PPT Presentation

1 / 66
About This Presentation
Title:

Understanding Fourth Graders Mathematical Thinking: Issues and Insights for Teaching Fractions

Description:

Video Clip: Fourth grader Ebony. What does she understand about fractions? ... about this problem compare to Ebony's? Ebony's and Crystal's understanding ... – PowerPoint PPT presentation

Number of Views:610
Avg rating:3.0/5.0
Slides: 67
Provided by: susane6
Category:

less

Transcript and Presenter's Notes

Title: Understanding Fourth Graders Mathematical Thinking: Issues and Insights for Teaching Fractions


1
Understanding Fourth Graders Mathematical
Thinking Issues and Insights for Teaching
Fractions
  • Susan Empson
  • The University of Texas at Austin
  • Smart Start Conference - July 13, 2006

2
Guiding Questions
  • What does it mean to understand fractions?
  • What kinds of problems help children develop
    their understanding of fractions?
  • How do you use the details of childrens thinking
    in your teaching?

3
Reflection
  • List two things that struck you as important from
    Dr. Frankes talk yesterday
  • Share with your table
  • Choose one thing from the table to report one
    sentence only

4
Some main points
  • Children use what they already understand to
    solve new problems. This leads to new
    understanding.
  • Understanding is generative
  • Teaching involves
  • Listening to childrens problem solving,
  • Figuring out what they understand, and
  • Building on that understanding

5
I. What does it mean to understand fractions?
6
Video Clip Fifth grader Ally
  • What does Ally understand about fractions? What
    does Ally not understand?
  • Ally is an average fifth grader. What do you
    think accounts for how she thinks about
    fractions?
  • What does a teacher need to know to help Ally
    develop a deeper understanding of fractions?
  • Discuss and record your answers on Allys
    Mathematical Thinking in handout packet

7
Video Clip Fifth grader Ally
8
Circle the bigger fraction
9
Write as improper fraction or mixed number
10
1. What does Ally understand about fractions?
11
What she said
  • 1 is bigger than 4/3 because its a whole
    number
  • 1/7 is bigger than 2/7, because usually (with
    fractions) you go down to the smallest number to
    get to the biggest number
  • 1/2 is bigger than 3/10, because you just change
    the bottom number 1 more digit and it would be 1
  • 1/2 is bigger than 4/6, same reason

12
Her understanding
  • Acts uncertain
  • Uses nonsensical rules
  • Believes all fractions are smaller than 1
  • Relies on surface features of the symbol rather
    than understanding meaning of fractions to create
    equivalent mixed numbers and improper fractions
  • Not generative
  • Generative -- leads to new concepts, strategies,
    procedures, and so on

13
2. What do you think accounts for how Ally thinks
about fractions?
  • A problem with Ally?
  • But so many students seem to have problems with
    fractions! Shes average.
  • A problem with the curriculum?
  • What is a typical approach to teaching fractions?
  • Does it support development of generative
    understanding?

14
A common curriculum approach to fractions
15
If children learn fractions by doing lots of
exercises like this one, what are they likely to
think about fractions?
  • How much is shaded?

16
They think
  • Fractions are pieces
  • Fractions are always smaller than a whole
  • Fractions values are determined by counting parts

A fourth is a little pie shape.
4/3? Thats impossible!
Its 1/3 because 1 part out of 3 parts is
shaded.
17
Lets watch two more students solving fractions
problems
  • Here is the problem they solve
  • Neither student has had direct instruction on
    adding fractions with unlike denominators

Two sisters, Iris and Kathryne, are eating
cookies. Iris has 3/4 of a cookie. Kathryne has
1/2 of the same sized cookie. If they put their
pieces together to give to their mom, will it
make more or less than 1 whole cookie? How much
will it be?
18
Video Clip Fourth grader Ebony
  • What does she understand about fractions?
  • Record your observations on Video Notes
    handout

19
Video Clip Fifth grader Crystal
  • What does she understand about fractions?
  • How does Crystals way of thinking about this
    problem compare to Ebonys?

20
Ebonys and Crystals understanding
  • Relationship between halves and fourths
  • 2 fourths can be put together to make 1 half
  • Halves can be cut into fourths
  • To add fractions, need to combine like units
    (fourths, halves)
  • Fractions can add to more than 1 whole
  • Understanding of concepts is somewhat separate
    from understanding of symbols (Crystal)
  • Used what they understood about fractions to
    generate new strategies for adding fractions

21
II. What kinds of problems help children develop
their understanding of fractions?
22
Lets solve some problems
  • Purpose
  • To practice listening to and understanding each
    others thinking

23
Think-aloud problem-solving activity
  • Pair up
  • One of you solves problem, thinking aloud as you
    go
  • Read problem carefully
  • Then just start talking about the problem
  • Hmm. Ive never solved a problem like this one
    before. I think Ill try Nope, that didnt
    work
  • Job is to keep going till its solved or youre
    stuck
  • OK if unsure, make mistakes.
  • Other person listens
  • Say strategy back to first person, using your own
    words
  • Job is to understand what first person is
    thinking
  • Dont help! (Listen, and do your best to
    understand.)
  • OK to ask clarifying questions as other person
    works
  • If time, switch roles and solve a second way

24
Problem 1
  • 3 children want to share 2 candy bars equally.
    How much can each child have?

25
Sample childrens strategies
I cut the candy bars in half, to see if it would
work and it did. Everybody gets a half. Then I
cut the last half in three parts. Everyone gets
another piece.
26
Each child gets 1 third from the first candy
bar.
27
Each child also gets 1 third from the second
candy bar. Thats 2 thirds for each person.
28
Mental strategy
I know that everyone can share each candy bar
and get 1/3 of a candy bar. Theres 2 candy bars,
so that 1/3, 2 times. Its 2/3.
29
Mental strategy
2 3 2/3
30
Problem 2
  • Eric and his mom are making cupcakes. Each
    cupcake gets 1/4 of a cup of frosting. They are
    making 20 cupcakes. How much frosting do they
    need?

31
Sample childrens strategies
5 cups
4 cups
1 cup
2 cups
3 cups
so 5 cups altogether.
32
So, 5, 6, 7, 8 -- thats 2 cups.
9, 10, 11, 12 -- thats 3 cups.
13, 14, 15, 16 -- thats 4 cups.
17, 18, 19, 20 -- thats 5 cups.
4 of these is 1 cup
so 5 cups altogether.
33
1/4 1/4 1/4 1/4 1 1/4 1/4 1/4 1/4
1 1/4 1/4 1/4 1/4 1 1/4 1/4 1/4 1/4
1 1/4 1/4 1/4 1/4 1
5 cups
Q Whats a number sentence for this problem?
A 20 x 1/4 5 (there are others)
34
Problem 3
  • Ohkee has a snowcone machine. It takes 2/3 of a
    cup of ice to make a snowcone. How many snowcones
    can Ohkee make with 4 cups of ice?

35
Sample childrens strategies
4

3
1
2
5
6
Ohkee can make 6 snow cones.
36
2/3 plus 2/3 is 1 and 1/3. If I add 1 and 1/3
three times, I get 4. I remember this from
another problem. So there are six 2/3s in 4. The
answer is she can make 6 snow cones.
37
Q Whats a number sentence for this problem?
A 4 2/3 6 (there are others)
38
Problem 4
  • 4 children are sharing 10 pancakes, so that each
    child gets the same amount. How much pancake can
    each child have, if they eat all the pancakes?

39
Sample childs strategy
1
1
1
1
1
1
1
1
1
1
Each child gets 1 fourth from each pancake.
There are 10 pancakes. So each child gets 10
fourths altogether.
40
Problem 5
  • 12 children are sharing 9 pineapple cakes, so
    that each child gets the same amount. How much
    cake can each child have, if they eat all the
    cakes?

41
What do teachers need to know to develop
fractions?
  • What types of problems are these?
  • What kinds of strategies do children use to solve
    these problems?
  • What is the mathematics that can be learned by
    solving and discussing these problems?
  • What are the fundamental concepts of fractions?
  • How do you help children coordinate concepts and
    fraction symbols?

42
What do teachers need to know to develop
fractions?
43
Problem types for fractions
  • Equal Sharing (with remainder, answer 1)
  • 2 children want to share 5 cookies equally. How
    much can each child have?
  • 4 children want to share 10 candy bars so that
    each one gets the same amount. How much can each
    child have?
  • Equal Sharing (answer
  • There is 1 brownie for 4 children to share
    equally. How much brownie can each child have?
  • 3 children want to share 2 candy bars equally.
    How much can each child have?
  • (Division is total divided by number of groups)

44
Problem types for fractions, contd
  • Addition (combining like units)
  • Janie has 3/4 of a gallon of blue paint left over
    from painting her room. John has 2/4 of a gallon
    of the same blue paint left over from painting a
    table. How much blue paint do they have?
  • Equal Groups
  • Eric and his mom are making cupcakes. Each
    cupcake gets 1/4 of a cup of frosting. They are
    making 20 cupcakes. How much frosting do they
    need?
  • (Backwards sharing context) 6 friends shared some
    cookies. Each person got 2 2/3 cookies. How many
    cookies did they have altogether?
  • Division (total divided by the size of a group)
  • Okhee has a snow cone machine. It takes 2/3 of a
    cup of ice to make a snowcone. How many snowcones
    can Ohkee make with four cups of ice?

45
Whats the mathematics in childrens solutions to
these problems?
  • Write down 1 thing that children can learn about
    fractions by solving problems like these
  • Hint Think about the strategies you used
  • Link to Arkansas framework?

46
Whats the mathematics in childrens solutions to
these problems?
  • Meaning of fractions -- what does 1/3 mean?
  • 1 thing shared equally by 3 people, each person
    gets 1/3
  • 1 candy bar for every 3 people
  • 1 3 1/3
  • 1 part, with 3 equal parts to make a whole
  • These meanings generalize to improper fractions
    too
  • 4/3 is
  • Fractional units can be combined
  • 1 third from one candy bar plus 1 third from
    another candy bar is 2 thirds
  • 1/3 1/3 2/3

47
Whats the mathematics in childrens solutions,
contd
  • Fractional units can be combined no matter how
    many there are
  • 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4
    1/4 1/4 10/4
  • Fractional numbers fill in the whole-number
    line
  • 2 1/4 cookies is more than 2 cookies but less
    than 3 cookies
  • A fractional amount can be expressed in many ways
  • See Fundamental Concepts of Fractions in
    handout packet

48
Video clips Equal sharing strategies
There are 6 cakes at Anthonys party. 8 children
have to share the cakes equally. How much cake
can each child have?
If each child at the party brings a friend, how
much cake can each child have?
49
III. How do you use this information in
instruction?
50
Three approaches to teaching fractions
  • Introduce procedures and explain concepts
  • Emphasis on student discovery, with no conceptual
    analysis of discoveries
  • Discuss and extend concepts and procedures that
    come up in childrens problem solving
  • (from Saxe et al., 1999)

51
Developing childrens understanding
  • Use problem contexts to elicit childrens
    thinking about fractions
  • Equal Sharing good place to start
  • Then problems that involve combining like
    fractional units
  • Equal Groups
  • Division (Total divided by size of group)
  • Ask children to solve problems in ways that make
    sense them. This helps children develop
    fundamental concepts with understanding

52
Developing childrens understanding, contd
  • Ask children how they solved problems. Probe
    their understanding.
  • Introduce symbols, number sentences, and
    mathematical language to go with strategies
  • Dont rush using and manipulating symbols

53
For example, instead of starting here
54
start with an Equal Sharing problem
  • Have materials for children to create fractions
    (nothing fancy)
  • By drawing
  • By folding or cutting
  • Set expectation that children solve in way that
    makes sense to them (i.e., that builds on their
    understanding)
  • Share and discuss strategies
  • Use your own judgment about what to do and when
    to do it, by listening to children

There are 3 candy bars for 4 children to share
equally. How much candy bar can each child have?
55
Solving problems and recording thinking
  • Problem solving
  • notebook
  • - messy part
  • - neat part

56
Classroom video clip Listening to the details of
childrens thinking
12 children want to share 9 pineapple cakes so
that everyone gets the same amount. How much cake
can each child have?
  • First, solve and discuss at your table.

57
Classroom video clip Listening to the details of
childrens thinking, contd
  • Then, watch teacher interact with two 5th graders
    who have solved this problem
  • What do these boys understand about fractions?
  • What does the teacher do to find out what the
    boys understand?
  • What would you do next with these boys?
  • There is no one right answer!

58
Helping children symbolize fractions
  • Let children use physical materials to create
    fractional amounts (draw, fold, cut, shade)
  • Use fraction words 2 thirds of a candy bar
  • a third a third
  • See handout in packet

59
Symbolizing fractions, contd
  • Relate unknown fractions to well known fractions,
  • such as 1/2 or 1/4
  • its more than a fourth, but less than a
    half
  • its smaller than a quarter
  • Use language that emphasizes relationship of
  • fractional quantity to unit instead of number of
    pieces.
  • how many of this piece would fit into the
    whole candy bar?
  • instead of how many pieces is the candy bar
    cut into?

60
Writing problems
  • To elicit childrens understanding of fractions
  • To steer development of fractions

Write an equal sharing problem that a child could
solve entirely by repeated halving.
Write an equal sharing problem that could involve
the fractions 2/5 and 4/10 in the possible
solutions.
61
Possible problems
  • Problems where of sharers is 2, 4, 8, (power
    of 2)
  • Example 8 children are sharing 6 quesadillas so
    that everyone gets the same amount. How much can
    one child have?
  • 10 children are sharing 4 packages of modeling
    clay equally. How much clay can each child have?
    20 children, 8 packages

Write an equal sharing problem that a child could
solve entirely by repeated halving.
Write an equal sharing problem that could involve
the fractions 2/5 and 4/10 in the possible
solutions.
62
Continuing your learning
  • Now lets plan for using problems like these back
    at your school

63
Problems
6 children are having breakfast at a pancake
restaurant. The waitress brings them 20 banana
pancakes to share. If everyone gets the same
amount, and they eat all of the pancakes, how
much pancake can each child have?
Tom has ___ dog biscuits. His dog, Harmony, eats
___ biscuits a day. How many days will it take
for Harmony to eat all of the dog biscuits? (7,
1/4) (12, 1 1/3)
64
To consider
  • How do you think your students will solve these
    problems?
  • How will you pose these problems to your
    students?
  • Pull out a few students?
  • Give to whole class? Etc.
  • What could you learn from each student as you
    listen to their strategies?

65
Website
  • http//www.edb.utexas.edu/empson

66
Mathematical proficiency
  • Thinking mathematically involves
  • Conceptual understanding
  • Procedural fluency
  • Strategic competence
  • Adaptive reasoning
  • Productive disposition
Write a Comment
User Comments (0)
About PowerShow.com