finding volume of cone

Report
Eric Bright
8th Grade Math
Charleston Middle School
[email protected]
ACCURATE ASSESSMENT IN
THE COMMON CORE ERA
Assessing student content mastery
Your Assessment Experience


Have you ever gotten a grade you didn’t deserve?
Why?
What final grade should the student with the
following grades get?
 F,
D, C, F, F, B, D, F, C, F, F, B, F, A, F, A
Your Assessment Experience

What about now?
Homework: F
 Homework: D


Homework: F
 Homework: F
Homework: F
 Homework: F

Test: B

Homework: D
 Homework: F
Quiz: C

Quiz: C



Test: B

Project: A



Homework: F
Homework: F
Final Exam: A
Purpose of Assessment


Bottom line: Does the student get it?
Purpose: To determine to what degree a student has
mastered content standards with a high degree of
validity and reliability.
 Validity
– the assessment measures what it is supposed
to measure
 Reliability – the assessment produces consistent results
across evaluators
Formative vs. Summative

Formative – Assessment FOR learning; assessment
that informs teaching and learning strategies for the
teacher and/or student
 May
be formal or informal (observations, effort,
participation, exit slips, etc.)
 May be more qualitative
 Includes meaningful feedback to students
 Purpose: To improve student learning.
Formative vs. Summative

Formative Checklist – The assessment should…
 Tie
directly to standards
 Focus on student learning needs
 Identify students’ current learning progress
 Give results that you can act on
 Be a regular part of instruction
 Quick and easy to give and grade

If you don’t use the data, stop gathering it!
Formative vs. Summative

Summative – An assessment that summarizes the
student’s mastery of a standard
 Usually
formal (test, quiz, multiple choice, short
response, word problems, projects, performance,
portfolios, etc.)
 May be more quantitative
 Purpose: To give a picture of how well a student has
mastered a standard at a specific time.
Formative vs. Summative

Summative Checklist – The assessment should…
 Tie
directly to standards
 Include multiple levels of learning (Bloom’s: remember,
understand, apply, analyze, evaluate, create)
 Summarize students’ overall learning progress
 Give results that all stakeholders can understand

Make sure you have offered students examples of
what meeting the standards looks like prior to a
summative assessment.
SPECIFIC ASSESSMENT TYPES










Extra Credit
Class Participation
Homework Completion
Homework Accuracy
Problem Solving
Pop Quiz
Quiz
Pre-Test
Post-Test
Performance Assessment with Rubric
Extra Credit

Does the activity address a standard?
Yes – Should be either formative or summative. This is
regular credit.
 No – Should not be summative since it is not the grade level
standard. This is formative at best.


Does the activity go deeper into or below the grade
level standard?
Deeper – Should not be summative since it is not the grade
level standard. It is enrichment and formative.
 Below – Should not be summative since it is not the grade
level standard. It is remediation and formative.

Class Participation


Definition: A grade based solely on the frequency of participation in
class.
What does a participation grade measure?



Content mastery? No, those with high content mastery may not
participate and those without content mastery may participate
frequently. This is not summative.
Willingness to participate? Yes, which is critical to success in the real
world, but does not reflect content mastery. Therefore, participation
grades should be formative.
Innovate: Keep track of participation with tally marks on a seating
chart. Log it weekly in the grade book, but count it as worth 0%.
Homework Completion


Definition: A grade based solely on the amount of work
completed and not the accuracy of that work.
What does a completion grade measure?
Content mastery? No, due to the lack of accuracy
assessment. Therefore it is not summative.
 Effort? Yes, which is critical to a student’s academic success,
but is still not summative. This is formative.


Innovate: Keep completion grades in the grade book,
but count them as worth 0%.
The “No Homework” Policy

Examine the research on homework.







HW has no little to no effect on elementary students and begins having
positive effects at the middle school level
Positive correlation between HW frequency and student achievement
Positive correlation between HW completion and student achievement
Positive correlation between HW that promotes self-regulation and
student achievement
Negative correlation between the relative amount of time spent on math
HW versus other subjects
Negative correlation between drill/practice HW and student
achievement
Conclusion: HW is important, but we need to rethink how we use it.
Homework Accuracy

What is the purpose of homework?
 To
practice skills? Then it is formative, not summative.
 Consequence
of not doing: Nothing. Natural consequences
show up on summative assessment.
 Innovate: Write the homework completion rate on the top of
each major summative assessment so students see the
relationship between their practice and achievement.
 Innovate: Give a homework quiz at the end of every class
period (or every other day) with two to four questions from
the homework. Use this quiz as a formative grade.
Homework Accuracy

What is the purpose of homework?
 To
get teacher feedback? Then it is formative, not
summative.
 Consequence
of not doing: Redo the homework so feedback
can be given.
 Innovate: Don’t write a grade on this or else the students just
toss it. Give feedback qualitatively instead of
quantitatively.
Homework Accuracy

What is the purpose of homework?
 To
learn content through discovery? Then it is formative,
not summative.
 Consequence
of not doing: Redo the homework.
 Innovate: Give less homework problems but have what is
assigned take more thought with higher levels of Bloom’s
taxonomy. Be less helpful which forces the students to think
for themselves.
Homework Accuracy

What is the purpose of homework?
 To
learn time management and organization? Then it is
formative, not summative.
 Consequence
of not doing: Create a homework completion
plan with parents to train students in self-regulation skills.
 Innovate: Focus more on the time spent on task rather than
the amount of homework completed. Ask parents to track or
report that data to show growth.
Homework Accuracy

Is it possible for homework to be summative?
 Yes,
but for it to be summative, students must have had
the chance to master the skills. They need time to
correct/revise homework before it is graded.
Unfortunately, the homework loses its value as a
formative assessment with this method.
Problem Solving Activity


Definition: An extended response situation requiring
multiple steps to solve, use of multiple skills, and
justification of reasoning and process.
What is the purpose of the problem solving activity?
 To
be exposed to new applications of content? This is
formative.
 To demonstrate mastery of content through application?
This is summative.
Pop Quiz

Summative assessment with no advanced notice.


Are there “pop” football games? We’re not out to trap the
students. Give students fair notice so that they can not only learn
the material, but also develop and apply good study skills. (This
assumes we teach self-regulation skills.)
Innovate: Instead of pop quizzes, recent research is showing
the benefits of practice quizzes. These formative
assessments function like a pop quiz, but are not for a
grade. They merely provide feedback on the learning
process prior to the summative assessment.

Give a quick four question quiz at the beginning of class as a
warm-up activity. “Grade” it and go over it together instead of
going over homework.
Quiz


Definition: A shorter assessment designed to assess
mastery of a small set of skills.
What is the purpose of the quiz?
Establish reliability of mastery through multiple data
entries? This is summative.
 Provide feedback to students about particular deficit skills
before a culminating summative assessment? This is
formative.


Could it be a mixture of both?
Pre-Tests


Definition: A test given before a unit of study to
ascertain content already mastered.
Why give a pre-test? All reasons are formative.
Differentiation – Students who have already mastered
content can move deeper into that content.
 Identify student leaders – Students with content mastery can
be used to promote mathematical discourse.
 Show growth – Establishes a base line of where students are
to compare with the post-test at the end of the unit.

Post-Test


Definition: A test designed to show mastery over a
whole unit of study.
How does each type of test show mastery?
 Multiple
choice
 Short response (written or symbolic)
 Extended response
 PBA or Project
Performance Assessment with Rubric

Projects or Extended Response Items


The rubric must address mastery of standards to be summative.
Sample bad rubric:
3D Shape Children’s Story Book
5 pts
4 pts
3 pts
2 pts
PRESENTATION
1 X Posture, eye contact, grammar, pacing, clearness
of speech
Excellent
Good
Fair
Poor
REQUIREMENTS
2 X Has all shapes with theme and story that flows.
Excellent
Good
Fair
Poor
NEATNESS AND CONSTRUCTION
3 X Book well built and illustrations neat and colored.
Excellent
Good
Fair
Poor
CREATIVITY
3 X Well thought out and original theme
Excellent
Good
Fair
Poor
PROJECT
1 X Overall looks great with well thought out theme.
Excellent
Good
Fair
Poor
Performance Assessment with Rubric

A better rubric part 1:
3D Shape Children’s Story Book
DEFINITION OF CYLINDER
Student accurately defines in his own words
DEFINITION OF CONE
Student accurately defines in her own words
DEFINITION OF SPHERE
Student accurately defines in his own words
VOLUME OF CYLINDER
2 pts
1 pts
0 pts
Mastery level
understanding
Good understanding
but copied some of the
definition
Does not show
understanding
Mastery level
understanding
Good understanding
but copied some of the
definition
Does not show
understanding
Mastery level
understanding
Good understanding
but copied some of the
definition
Does not show
understanding
Yes
No
Yes
No
Yes
No
Student accurately gives the formula
VOLUME OF CONE
Student accurately gives the formula
VOLUME OF SPHERE
Student accurately gives the formula
Performance Assessment with Rubric

A better rubric part 2:
3D Shape Children’s Story Book
2 pts
1 pts
0 pts
Mastery level
understanding
Good understanding
but computation errors
Does not show
understanding
Yes
Partially
No
Mastery level
understanding
Good understanding
but computation errors
Does not show
understanding
Yes
Partially
No
Mastery level
understanding
Good understanding
but computation errors
Does not show
understanding
Finding the volume makes sense in the context of story/problem
Yes
Partially
No
MATHEMATICAL PRECISION
Student maintains precision by using π≈ 3.14 and rounding final
6 or 7 problems
solved with
precision
4 or 5 problems solved
with precision
< 4 problems solved
with precision
FINDING VOLUME OF CYLINDER
Student accurately finds volume and justifies solution with work X 2
Finding the volume makes sense in the context of story/problem
FINDING VOLUME OF CONE
Student accurately finds volume and justifies solution with work X 2
Finding the volume makes sense in the context of story/problem
FINDING VOLUME OF SPHERE
Student accurately finds volume and justifies solution with work X 2
solutions to two decimal place x 2
Performance Assessment with Rubric

A better rubric part 3:
3D Shape Children’s Story Book
2 pts
1 pts
0 pts
Mastery level
understanding
Good understanding
but computation errors
Does not show
understanding
Yes
Partially
No
Mastery level
understanding
Good understanding
but computation errors
Does not show
understanding
Yes
Partially
No
Mastery level
understanding
Good understanding
but computation errors
Does not show
understanding
Finding the radius makes sense in the context of the story
Yes
Partially
No
FINDING VOLUME OF COMBINATION
Mastery level
understanding
Good understanding
but computation errors
Does not show
understanding
Finding the volume makes sense in the context of the story
Yes
Partially
No
FINAL GRADE
/50 pts
X2=
__________%
FINDING RADIUS OF CYLINDER OR CONE
Student accurately finds radius and justifies solution with work
Finding the radius makes sense in the context of the story
FINDING HEIGHT OF CYLINDER OR CONE
Student accurately finds height and justifies solution with work
Finding the height makes sense in the context of the story
FINDING RADIUS OF SPHERE
Student accurately finds radius and justifies solution with work
Student accurately finds volume and justifies solution with work
ASSESSMENT POLICIES

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

Late Work Policy
Cheating
Giving Zeroes
Group Grades
Partial Credit
Test Retakes
Common Assessments
Late Work

Innovate: Avoid late work in the first place:
Give students advanced notice of any out-of-class
summative assessments.
 Have benchmark due dates for those assessments.
 Reduce the amount of out-of-class summative assessments.


If an assessment is turned in late:

Be flexible depending on the circumstances, but have a
written policy in place such as:
Minus 10% to grade, but only accepted up to a week late.
 Note late work as a formative assessment and track it with
individual students and parents. Grade assessment normally, but
only accept work up to a week late.

Cheating

What constitutes cheating?
Formative: Explaining what to do or just giving an answer
without explaining why we do it.
 Summative: Explaining what to do or just giving an answer.


If students cheat, what should be the consequences?
Formative: Perhaps nothing except notifying parents.
Natural consequences will occur on summative assessments.
 Summative: Take a different version of the assessment.

The “No Zero” Policy

Zero is an outlier and therefore we should only give
50% instead.
 False.
If the student really knows 0% of the content,
the zero is the best reflection of their content mastery.
It is an outlier because the outcomes (grades) are not
equally likely. 60% passing is a low standard.
 Giving a lower bound of 50% skews grades much
higher. Think of a student with scores of 60, 10, 80,
10.
The “No Zero” Policy

Any zero should be redone until the student passes.
 False
and impossible. There is a time component to
mastery. We expect mastery by a certain time. If
students had the whole year to master content, they
could save all assessments for the last day.
 Giving one chance for a retake assessment is
reasonable since students learn at different rates, but
beyond that either means bad teaching in the first
place or that a student truly has not mastered the
content.
The “No Zero” Policy

The “No Zero” policy came about because teachers
gave zeros for formative assessments and then
counted it toward a student’s grade (summative).
 This
is not an inappropriate use of the zero. It is an
inappropriate use of formative assessment.
Zeros

Innovate: Use the zero, but use it correctly!

Why do we give a zero for summative grades?
 Incomplete?
Then we don’t know how well a student has
mastered that standard, so force the student to
complete the assignment. If they refuse, our best guess
is that they do not understand the topic, and the zero
stands.
 Total lack of mastery? Then the zero is the most
accurate representation of student mastery.
Zeros

Innovate: Use the zero, but use it correctly!

Why do we give a zero for summative grades?
 Cheating?
This does not accurately assess what a
student has learned. A better consequence is to retake
a similar assessment.
 Late? (When does late become incomplete? One
day?) This does not accurately assess what a student
has learned.
 Why
is it late? If cheating, see that consequence. If effort,
note that as a formative assessment.
Group Grades


Definition: Giving the same grade (or slightly modified
grades) to each student in a group.
What does a group grade measure?


Content mastery? It can, but one student may have achieved
mastery while getting a poor grade due to someone else’s lack
of mastery. This is not summative for each student.
Innovate: Have students discuss ideas in a group, but…



Don’t let them write anything down until they are on their own.
Have them throw away their group work before filling out the
summative assessment.
Use group work only as a formative assessment or discovery task.
Partial Credit



Consider the following work on an algebra assessment:
Was the mistake an algebraic mistake? This was a
computation error, not an algebra error.
Innovate: If we are assessing the algebra standard, the
student appears to understand inverse operations.
Perhaps 3/5 points.
Test Retakes

Should students be able to retake tests? What is the
purpose of the retake?


If a student retakes a test, should the new grade be
averaged with the previous grade or should the new grade
replace the previous one?


Purpose: Assess mastery. Yes, a retake might show new mastery
of content.
Averaging acknowledges the struggle, but does not necessarily
show the student’s current level of mastery.
How many times can a student retake a test?

It is impractical to allow multiple retakes. Since a summative
assessment is tied to a time frame, giving one chance to retake an
assessment reinforces that timeliness and also student
responsibility.
Common Assessments


Definition: Identical assessments that are given by different teachers who
teach the same course.
Purpose of Common Assessments:


Establish inter-grader reliability for assessments.
Count as a Type II assessment for teacher evaluations.





Type I – MAP, PARCC, Universal Screener
Type II – District, grade level, or course-wide assessment adopted and approved by
the school district
Type III – Teacher created
Give a springboard for discussing student mastery for the purposes of lesson
revision.
Possible common assessments include: weekly or mid-chapter quizzes, unit or
chapter tests, quarter or semester exams
GRADING SYSTEMS



Grading on the Curve
Total Points vs. Weighted Categories
“Standards-Based” Grading
Grading on the Curve

Changing student grades methodically for a better
grade distribution.
 This
does not accurately assess student mastery of
content if we have a clear picture of what mastery is.
If we don’t know what mastery looks like, that must be
satisfied before we can assess.

Rather than making your grade distribution match
the normal curve, ask yourself why the grades are
distributed they way they are. This is a formative
exercise.
Grading on the Curve

Are there too many low grades?





Was it too soon to expect mastery? Eliminate the summative
grade and give the assessment later. Use this as a formative
assessment.
Was the material poorly taught? Eliminate the summative grade
and re-teach.
Was the assessment too difficult? Eliminate the summative grade
and give a better written assessment.
Was the assessment accurate? Give students options for
remediation, but move on in the curriculum.
Are there too many high grades?


Was it the assessment too easy? Eliminate the summative grade
and give a better written assessment.
Was the assessment accurate? Celebrate your students!
Points vs. Percents


When talking about how to record grades of the same
weight, this is irrelevant except for rounding differences.
For example, these are the same grade:
60%, 80%, 100%, 80% yields average of 80%
 3/5, 4/5, 5/5, 4/5 yields 16/20 = 80%


These have a slight rounding error due to “not nice”
denominators:
65%, 71%, 59%, 71% yields 67%
 11/17, 12/17, 10/17, 12/17 yields 66%


Moral: Choose the denominator wisely.
Points vs. Percents


When in reference to weighted categories versus
straight points, the differences are aesthetic
because every grading system is weighted.
Consider a typical “points” system:
 Homework
worth 5 points each
 Quizzes worth 50 points each
 Tests worth 100 points each

Is a test worth 20 times as much as homework?
Points vs. Percents

If there are 2 tests and 2 quizzes per quarter, but
homework every day that gives us:
 225
points of homework (43%)
 100 points from quizzes (19%)
 200 points from tests (38%)
Points vs. Percents

Think long-term. During the whole quarter say you
typically have:
8
Homework summative assessments (This is purely for
demonstration purposes! HW should be a formative
assessment!)
 4 Problem solving summative assessments
 4 Quizzes
 2 Unit tests
 1 Project
 1 Quarter Exam
Points vs. Percents

Now consider this weighted system:
 Homework
worth 10% (This is purely for demonstration
purposes! HW should be worth 0%!)
 Problem Solving worth 10%
 Quizzes worth 20%
 Unit Tests worth 30%
 Project worth 10%
 Quarter Exam worth 20%
Points vs. Percents

It is the same as this points system:
Homework worth 100 points each
 Problem Solving worth 200 points each
 Quizzes worth 400 points each
 Unit Tests worth 1200 points each
 Project worth 800 points
 Quarter Exam worth 1600 points


Hint: Making everything out of 100 makes it easier for
the students. So instead of saying a quiz is out of 400
points, tell students their grade counts four times.
Points vs. Percents

The difference is how you get to the end grade:
HW1 PS1 HW2 PS2 QZ1 HW3 QZ2 HW4 TST1 HW5 PS3 HW6 PS4 QZ3 HW7 QZ4 HW8 TST2 Proj Q Ex
Points for
assignment
100 200 100 200 400 100 400 100 1200 100 200 100 200 400 100 400 100 1200 800 1600
Joe Bob (%) 80% 75% 70% 85% 60% 100% 85% 90% 93% 60% 70% 50% 70% 60% 70% 65% 80% 88% 85% 96%
Joe Bob (pts)
80 150
70 170 240 100 340
90 1116
60 140
50 140 240
70 260
80 1056 680 1536
Weighted Grade
80% 78% 75% 78% 69% 71% 77% 78% 84% 83% 83% 82% 82% 81% 81% 80% 81% 80% 80% 83%
Points Grade
80% 77% 75% 78% 71% 74% 77% 78% 84% 83% 82% 81% 81% 79% 78% 77% 77% 80% 80% 83%

In this case, both grades end with a final grade of
83% because we made sure the weight of the
points matched the weight of the categories.
Points vs. Percents
The difference is how you get to the end grade:

86%
84%
82%
80%
78%
Weighted Grade
76%
Points Grade
74%
72%
70%
68%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
“Standards-Based” Grading

Standards-based grading usually doesn’t actually mean standards-based
grading. It usually means grading with a 4-3-2-1 rubric or something
similar with:





4 – Exceeds standard
3 – Meets standard
2 – Meets standard with assistance
1 – Does not meet standard
Example Report Card:

8th Grade Pre-Algebra: 3





Number System: 4
Expression and Equations: 2
Functions: 3
Geometry: 2
Statistics and Probability: 4
Note: You can have this same break down of
grades with a regular percent grading system by
simply making your category of grades follow
the Domain name and weighting summative
assessments appropriately via points.
“Standards-Based” Grading

Fact: 4-3-2-1 are not equally likely.




4 may represent 90% – 100% accuracy
3 may represent 80% – 90% accuracy
2 may represent 60% – 80% accuracy
1 may represent 0% – 60% accuracy


Even using objective benchmarks, they are still not equally likely.
Problem: You can’t average the scores, but we need to.




Geometry scores of 0%, 90%, 90%, 90% average to 67.5%
SB scores of 1, 4, 4, 4 average to 3.25 (meets)
Which one signals to the parent there is a problem?
To get an overall Geometry score we need to average to account for
subcategories within Geometry (area, volume, Pythagorean Theorem,
etc.)
“Standards-Based” Grading

Potential Solution: Power Law Average


The Power Law is basically a predictor of how the student
would score on the next assessment based on previous
performance. So scores of 1,2,3,4 might yield a 4 while
scores of 4,3,2,1 might yield a 1.
Problem: Power Law only works for individual skills

Most assessments cover a multitude of skills.

Getting a Geometry score of 4 on the first assessment does not
mean anything about the Geometry score on the next assessment
if the first assessment covered 2D geometry while the second
assessment covered 3D.
“Standards-Based” Grading


Problem: How do you deal with assessments that
incorporate multiple standards or skills? You would
need multiple grades for the same assessment(s).
Problem: There is more information reported
(usually) with SB grading, but it is still not useful.
Does a 2 in Geometry mean I need help with
transformations, volume, or the Pythagorean
theorem?
“Standards-Based” Grading


Bottom Line: 4-3-2-1 is just as flawed as a
traditional grading system.
Solution: Use good assessment practices in whatever
grading system you use and many of the problems
that the SB Grading movement is trying to tackle
will be resolved.
My Grading System
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Weighted Categories that are assessed based on standards mastery
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Homework Completion (0%) – Daily
Homework Accuracy (0%) – Each specific skill or set of skills
Mastery Task (0%) – Each specific skill or set of skills
Problem Solving (10%) – 4 to 8 per quarter
Weekly Quiz (30%) – 4 to 6 per quarter
Unit Pre-Test (0%) – 3 per quarter
Unit Post-Test (40%) – 3 per quarter
Quarter Project (10%) – 1 per quarter
Quarter Exam (10%) – 1 per quarter
Enrichment (0%) – As needed based on Pre-Tests
Remediation (0%) – As needed for progress monitoring
But this is not perfect! We’re working to change it!
Getting Accurate Grades
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Keep in mind the purpose of assessments (formative
and summative).
We can’t grade the way we want or the way we’ve
always done it.
Eric Bright
8th Grade Math
Charleston Middle School
[email protected]
ACCURATE ASSESSMENT IN
THE COMMON CORE ERA
Assessing student content mastery

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