A Presentation by Arthur Howard · A Presentation by Arthur Howard Providence Classical School...
Transcript of A Presentation by Arthur Howard · A Presentation by Arthur Howard Providence Classical School...
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Addition of Fractions The Unrecognized Problem
A Presentation by
Arthur Howard Providence Classical School
Spring, TX
at the
Conference for the Advancement of
Mathematics Teaching
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A Little Pretest
Question 1:
1
2+
1
3= ?
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A Little Pretest
Question 2: A family travels from Houston to San Antonio.
Because of traffic, they only average 40 mph
going there.
On the return trip the traffic is much better, so
they average 50 mph.
What is their average speed for the entire trip?
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A Little Pretest
Question 3:
1
2?
1
3
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Answer:
Question 1:
1
2+
1
3=
2
5
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1
2
1
3
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2
5
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6
12+
9
18=
15
30
1
2
1
2
1
2
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1
2+
1
3=
3
6+
2
6=
5
6
a
b+
c
d=
ad + bc
bd
We add the parts, but the whole is the same.
1
4
1
4
2
4
1
2
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a
b+
c
d=
a + c
b+ d
Parts and wholes are both added.
Å
a
bÅ
c
d=
a + c
b + d
1
4+
1
4=
2
8
Parts are the same, but the whole is larger.
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“Crazy” Addition
Makes sense when sets of things are combined.
Operation must be done with original data.
Key concept is accumulation.
4
6Å
1
2=
5
8
2
3Å
1
2=
3
5
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Farey Fractions
John Farey, Sr. (1766 – 1826)
British geologist
Sequence of reduced fractions
Called this “sum” the mediant
Rational Mean
1
4
1
3
1
2
2
7
2
5
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96 hits
317 at bats
Jose Altuve
0.303 hits/at bat
Å
0 hits
4 at bats =
96 hits
321 at bats
0.299 hits/at bat
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Mixtures
5 liters of a 40% acid solution.
20 liters of a 50% acid solution.
What is the concentration of the new solution?
2 L of acid
5 L of solutionÅ
10 L of acid
20 L of solution =
12 L of acid
25 L of solution
48% solution
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Answer:
Question 2: A family travels from Houston to San Antonio.
They average 40 mph going there.
They average 50 mph coming back.
What is their average speed for the entire trip?
Assume the distance each way to be 200 miles.
200mi @ 40 mph is
200 mi
5 hr 200mi @ 50 mph is
200 mi
4 hr
200 mi
5 hrÅ
200 mi
4 hr=
400 mi
9 hr= 44.4 mph
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A More Likely Scenario
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A More Likely Scenario
• Leave San Antonio.
• Stop in Schulenburg for pie and coffee.
• Drive to Beltway 8.
• Take Beltway 8 to I-45.
• Take I-45 to home.
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A More Likely Scenario
100 miles from San Antonio to Schulenburg. Average speed 60 mph.
70 miles to Beltway 8 (West). Average speed 70 mph.
20 miles from Beltway 8 to I-45. Average speed 80 mph.
10 miles on I-45 to neighborhood. Average speed 60 mph.
100 miles
100 minutesÅ
70 miles
60 minutesÅ
20 miles
15 minutesÅ
10 miles
10 minutes
200 miles
3.75 hours » 53.3 mph
Å
0 miles
40 minutes
Stop for pie and coffee. Average speed 0 mph.
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Puzzler:
A car is to travel two miles on a straight track.
It averages 30 mph for the first mile.
How fast will it need to go in the second mile to
average 60 mph for the entire length?
30 mph =
30 mi
60 min=
1 mi
2 min
2 mi
2 min =
1 mi
2 minÅ
1 mi
0 min
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Does Anybody Else Know This Stuff?
• Mathematicians
• Science Teachers
Accumulated Average
Average “builds” throughout term
Weights grades based on size
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Accumulated Average
Each assignment has its own total point value.
28
30
46
50
45
45
56
60
10
15Å Å Å Å
=
185
200= 92.5%
89.1%
Average =total points accumulated
total points possible
Rational Mean of the scores
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The Answers:
Question 3:
1
2>
1
3
Are you sure?
1
2?
1
3
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1
2=
1
3
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1
2<
1
3
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Answer:
Question 3:
1
2?
1
3
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What We’ve Learned So Far
• Fractions always represent parts of a whole.
• Two entirely different ways of combining them. – One is the SUM.
– Other is the RATIONAL MEAN.
• Rational Means correctly calculate: – Sports averages
– Average of two rates
– Combinations of sets or solutions
– Accumulated Grading
• When comparing fractions, we have to have CONTEXT.
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Why Weren’t We Taught This?
• Curriculum largely fixed by 15th Century.
• Rational Mean is recent.
• Pythagorean Means:
– Arithmetic Mean (Statistics)
– Geometric Mean (Finance)
– Harmonic Mean (Statistics, Electronics, Music)
• Rational Mean (Finance)
a + b
2
ab
2ab
a + b
a + c
b + d
1
1
a+
1
b
æ
èçö
ø÷
2
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Are They Related?
Rational vs Harmonic
When distances are equal.
Rational vs Arithmetic
When times are equal.
When denominators are equal.
2(40)(50)
40 + 50 =
4000
90 » 44.4
3
1Å
8
1=
11
2
1
3Å
2
3=
3
6
50 mi
1 hrÅ
40 mi
1 hr=
90 mi
2 hr
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The Big Picture
Combining fractions is more complicated than
we imagined.
What – Now I have to teach TWO procedures for
adding fractions?
NO!
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Procedures
• Rules that get the answer
• Memorized without understanding
• Don’t know when to use them
• Misremember rule
• No idea of error
• Don’t know what to do about it
The Psychology of Learning Mathematics
Prof. Richard R. Skemp
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Teaching for Understanding
• Two types of understanding
– Habit Learning (Procedural Learning)
“Rules without reason”
– Intelligent Learning (Conceptual Learning)
“Knowing what to do, and why”
– Most math instruction based on procedures
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Why?
–Often much easier to teach
– Immediate rewards
–Less knowledge involved, so answer comes
more quickly
–Requires a new rule for each situation
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Conceptual Learning
– Takes longer to teach.
– Students sometimes resist it.
– Sometimes involves more content.
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Example
A
D=
1
2bh
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Extension
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Further Extension
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Conceptual Learning
– Takes longer to teach.
– Students sometimes resist it.
– Sometimes involves more content.
– Easier to remember.
– More adaptable to new tasks.
– Is effective as a goal in itself.
– Leads to willingness to study further.
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Failure to teach Conceptually
• Use procedures inappropriately
• Can’t transfer
• Baffled by word problems
• Math phobia
“I never could do math.”
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Sir Herman Bondi,
Austian/British Mathematician and Cosmologist
“The negative attitude to mathematics,
unhappily so common, even among
otherwise highly-educated people, is
surely the greatest measure for our
failure and a real danger to our
society.”
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A public unable to reason with figures is an
electorate unable to discriminate between
rational and reckless claims in public policy.
Lynn Arthur Steen: Numeracy
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Teacher Comments
• The more they listened and spoke, the more they contributed. … This was the best classroom atmosphere we have had so far this year.
• I’ve become much better at dealing with misconceptions. I now use them as great teaching opportunities.
• I am really pleased with how much more actively these pupils engage. They are much more prepared to volunteer for tasks, to develop their own methods, to listen and question.
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Pupil Comments
• I liked being able to argue and get my thoughts clear.
• I think I’m much better at sharing my ideas now. Everyone listens to each other’s ideas, even the teacher!
• I liked being able to explain my ideas. I didn’t expect everyone to want to know what I thought.
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What Should We Do?
• Start with real-world situations.
• Use manipulatives to teach concepts.
• Ask students to explain their thinking.
• Connect physical, pictorial and abstract.
• Different contexts, but the same concepts.
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Not a New Idea
First Lessons in Arithmetic Warren Colburn
1822
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For each of these problems, the
first number represents the
fraction of an inch that it rained
on Monday. The second number
is the fraction of an inch it
rained on Tuesday.
How many inches of rain fell in
the two days together?
Draw a circle around each +
sign on this sheet.
The first number represents the
fraction of a bag of marbles that is
red. The second number is the
fraction of another bag of marbles
that is red.
If the two bags are poured into a
bowl, what fraction of the marbles
in the bowl will be red?
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Asking Questions
What “action” do you see in this problem?
Does it sound like something we’ve done before?
Can you use objects or draw me a picture?
Is this sometimes/always/never true?
How do you know that?
Do you see a pattern?
Can you explain that in your own words?
What kind of answer should we expect?
Is this answer reasonable?
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Data Analysis Object Circumference Diameter Ratio
Jar Lid 32.6 10.2 3.20
Bicycle Wheel 210 66 3.18
Coffee Lid 68.6 20.2 3.40
Mixing Bowl 94.1 30 3.14
Saucer 57.5 17.9 3.21
Dinner Plate 82 26.5 3.09
What kind of “average” should we use here?
Median? 3.19 Arithmetic Mean? 3.09 Geometic Mean? 3.20 Harmonic Mean? 3.20 Rational Mean? 3.20
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Let students struggle with novel problems.
Fostering Analytic Thinking
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Fostering Analytic Thinking
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Tomorrow
9:45 - 11:15 am
Basic Student Understanding Through
Effective Questioning
Karen Hardin
Right here in this room.
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NCTM Principles and Standards
Reasonableness and Sense Making
• Explain their thinking.
• Believe that mathematics makes sense.
• Make and evaluate mathematical conjectures and
arguments.
• Appreciate the power of reasoning as a major part
of mathematics.
• Construct proofs of mathematical assertions.
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Common Core State Standards
Clear understanding of what students are
expected to learn.
Robust and relevant to the real world.
Fully prepare students for the future.
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CCSS Mathematical Tasks
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
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Our Task To create a classroom where
a robust curriculum is taught concepts are taught patterns are discovered sense making is expected thinking is honored learning is valued
Give an education that will last a lifetime!
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Selected Resources
• Rice University School Mathematics Project
– rusmp.rice.edu
• Relational and Instrumental Understanding
– Prof. Richard R. Skemp
– http://www.blog.republicofmath.com/archives/654
• The Psychology of Learning Mathematics
– Prof. Richard R. Skemp, Amazon.com
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Selected Resources
• Knowing and Teaching Elementary
Mathematics
– Liping Ma, Amazon.com
• First Lessons in Arithmetic
– Warren Colburn, Amazon.com
• Why Johnny Can’t Add: The Failure of the
New Math
– Morris Kline, Amazon.com
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Thank you for your time.