Title: Understanding Fourth Graders Mathematical Thinking: Issues and Insights for Teaching Fractions
1Understanding Fourth Graders Mathematical
Thinking Issues and Insights for Teaching
Fractions
- Susan Empson
- The University of Texas at Austin
- Smart Start Conference - July 13, 2006
2Guiding 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?
3Reflection
- 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
4Some 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
5I. What does it mean to understand fractions?
6Video 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
7Video Clip Fifth grader Ally
8Circle the bigger fraction
9Write as improper fraction or mixed number
101. What does Ally understand about fractions?
11What 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
12Her 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
132. 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?
14A common curriculum approach to fractions
15If children learn fractions by doing lots of
exercises like this one, what are they likely to
think about fractions?
16They 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.
17Lets 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?
18Video Clip Fourth grader Ebony
- What does she understand about fractions?
- Record your observations on Video Notes
handout
19Video Clip Fifth grader Crystal
- What does she understand about fractions?
- How does Crystals way of thinking about this
problem compare to Ebonys?
20Ebonys 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
21II. What kinds of problems help children develop
their understanding of fractions?
22Lets solve some problems
- Purpose
- To practice listening to and understanding each
others thinking
23Think-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
24Problem 1
- 3 children want to share 2 candy bars equally.
How much can each child have?
25Sample 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.
26Each child gets 1 third from the first candy
bar.
27Each child also gets 1 third from the second
candy bar. Thats 2 thirds for each person.
28Mental 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.
29Mental strategy
2 3 2/3
30Problem 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?
31Sample childrens strategies
5 cups
4 cups
1 cup
2 cups
3 cups
so 5 cups altogether.
32So, 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.
331/4 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)
34Problem 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?
35Sample childrens strategies
4
3
1
2
5
6
Ohkee can make 6 snow cones.
362/3 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.
37Q Whats a number sentence for this problem?
A 4 2/3 6 (there are others)
38Problem 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?
39Sample 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.
40Problem 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?
41What 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?
42What do teachers need to know to develop
fractions?
43Problem 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)
44Problem 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? -
45Whats 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?
46Whats 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
47Whats 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
48Video 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?
49III. How do you use this information in
instruction?
50Three 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)
51Developing 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
52Developing 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
53For example, instead of starting here
54start 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?
55Solving problems and recording thinking
- Problem solving
- notebook
- - messy part
- - neat part
56Classroom 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.
57Classroom 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!
58Helping 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
59Symbolizing 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?
60Writing 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.
61Possible 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.
62Continuing your learning
- Now lets plan for using problems like these back
at your school
63Problems
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)
64To 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?
65Website
- http//www.edb.utexas.edu/empson
66Mathematical proficiency
- Thinking mathematically involves
- Conceptual understanding
- Procedural fluency
- Strategic competence
- Adaptive reasoning
- Productive disposition