I need some focus!
Helping Calculus students navigate mathematical writing
David Clark
Grand Valley State University
Mathfest 2015
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Outline
Portfolio Problems
A good, solid set of
interesting problems
Focuses
Assessment
Specific goals to
help students focus
Math vs. Writing,
Minor vs. Significant,
Revision
Goal: Help students navigate the conventions and expectations in
professional communication.
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Portfolio problems
Let

1
 2
x sin
f (x) =
x
0
if x 6= 0
if x = 0.
Determine whether f 0 (0) exists.
You are hired to work for 30 days (without any breaks). You
can choose how to be paid:
$1000 per day
$0.01 on day 1 and doubling every day
$1,000,000 on day 1 and halving every day
What is your best choice? How much do you earn each
day, and how much have you earned in total on each day?
Include relevant graphs.
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Focuses
Goal: Help students focus on specific aspects of professional writing
and mathematical content
Organize major writing ideas into broad categories.
Select key categories for each problem.
Organize to build throughout the course.
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Focuses
Goal: Help students focus on specific aspects of professional writing
and mathematical content
Organize major writing ideas into broad categories.
Select key categories for each problem.
Organize to build throughout the course.
Examples:
Notation Proper use of mathematical notation and terminology.
Justification Convince me that you know why your work is correct.
Organization Arrange your work in a sensible way.
Methods Use specific mathematical techniques correctly.
Figures Include well-formatted figures when helpful.
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Focus examples: Calc 1
Let

1
 2
x sin
f (x) =
x
0
if x 6= 0
if x = 0.
Determine whether f 0 (0) exists.
Methods: Carefully use the definition of f 0 (0).
Notation: Use limit notation and the definition of the derivative
properly. Don’t drop a limit until you’ve evaluated it!
Organization: First clearly state your answer and then prove that
it is correct. Center important equations on their own line.
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Focus examples: Calc 1
Let

1
 2
x sin
f (x) =
x
0
if x 6= 0
if x = 0.
Determine whether f 0 (0) exists.


Methods: Carefully use the definition of f 0 (0).




 Notation: Use limit notation and the definition of the derivative
properly. Don’t drop a limit until you’ve evaluated it!



Organization: First clearly state your answer and then prove that



it is correct. Center important equations on their own line.
Implementation:
State a few broad focuses, each
central to the solution or writing.
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Focus examples: Calc 1
Let

1
 2
x sin
f (x) =
x
0
if x 6= 0
if x = 0.
Determine whether f 0 (0) exists.
Methods: Carefully use the definition of f 0 (0).
Notation: Use limit notation and the definition of the derivative
properly. Don’t drop a limit until you’ve evaluated it!
Organization: First clearly state your answer and then prove that
it is correct. Center important equations on their own line.
Implementation:
Give additional explanation specific to
the problem, focusing on a particular
writing goal.
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Focus examples: Calc 1
Let

1
 2
x sin
f (x) =
x
0
if x 6= 0
if x = 0.
Determine whether f 0 (0) exists.
Methods: Carefully use the definition of f 0 (0).
Notation: Use limit notation and the definition of the derivative
properly. Don’t drop a limit until you’ve evaluated it!
Organization: First clearly state your answer and then prove that
it is correct. Center important equations on their own line.
Implementation:
Focus your grading on mastery of
these focuses.
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Focus examples: Calc 2
You are hired to work for 30 days (without any breaks). You
can choose how to be paid:
$1000 per day
$0.01 on day 1 and doubling every day
$1,000,000 on day 1 and halving every day
What is your best choice? How much do you earn each
day, and how much have you earned in total on each day?
Include relevant graphs.
Justification: Fully justify steps related to sequences and series.
Notation: Properly use the terminology and symbols for
sequences, series, and sequences of partial sums.
Organization: Arrange your answers in a clear, relevant order of
your choosing.
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Assessment: m/S grading
Each part of a solution is:
Math
or
Writing
– and –
Minor
m
or
Significant
S
Focuses specify minor vs. significant.
Helps students understand feedback.
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Advice
Revision: Focuses and m/S grading help students. . . focus.
Use focuses and m/S grades to guide
conversations and comments.
Allow students to revise Significant errors with
reflection.
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Advice
Structure: Organize focuses to build throughout semester.
Organization:
Early: Center important equations on their own line.
Summarize your conclusion in one clear
sentence.
Mid: First clearly state your answer and then
prove that it is correct.
Late: Arrange your answers in a clear, relevant
order of your choosing.
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Advice
Using m/S grading: Be careful and intentional.
Grade each focus separately with m/S.
Give S’s only on focuses.
Avoid “stacking” penalties. Use a non-linear scale.
10/10: Perfect
9/10: A few m’s
7/10: One S
5/10: Multiple S’s
Create a style guide organized around focuses.
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Questions?
clarkdav@gvsu.edu
Happy Abel day!
Niels Henrik Abel: Born August 5, 1829
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