Transcript File

Framework for Depth of Procedural Understanding of Mathematical Solutions:
Assessing student explanations for conceptual understanding of procedural processes
I need your …
WELCOME * WELCOME
Dr. Vicki-Lynn Holmes
Ms. Emily Rowland
LiveScribe Pulse Smartpen
Assessing student
explanations for
conceptual understanding
of procedural processes
through technology –
Livescribe Smartpens.
http://www.livescribe.com/cgibin/WebObjects/LDApp.woa/wa/MLSO
verviewPage?sid=pwsn4T49njP1
Memorization -- know mathematical procedures on a basic,
memorization or rote level
Understanding – understanding concepts and applying
mathematical procedures
Problem solving / reasoning – Conceptual Knowledge use
knowledge to solve / reason through new (unique) problems
Depth of
Metacognition
All procedural knowledge is here!
Overview
Student and Teacher journals
Classroom observation
Student surveys
Student pencasts
Teacher Quotes:
“Wow! The class is full of chatter, and it is all
on task math chatter!”
“They surprised me by picking up the
instructions so quickly. I’ve noticed that the
kids are very excited about math.”
Student Quotes:
“It’s a tool that makes explaining easier.”
(#30)
“I think it does make me think because
when you write a problem, you have to
know why you’re doing what you’re doing.”
(#31)
“What I enjoyed least is the pen does not
erase.” (#24)
RESULTS
What to do with this?
On par with conceptual understanding of the procedural process?
PEMDAS
3x + 2 = 14
700 + pencasts later…
procedural knowledge
here only
Framework for Depth of Procedural Understanding
Basic
Understanding
•
•
•
•
•
•
major procedural
(“how to”)
simply speaking
what is being
written
steps are vague
descriptions lack
depth and
expansion
no or limited
vocabulary
cannot properly
teach to others
Extended
Understanding
•
•
•
•
•
•
- teleological semantic
understanding of
procedures
understanding of what
they are doing
major procedural
steps explained
Correct mathematical
form and
mathematical symbols
essential / moderate
vocabulary used
why explanations
limited
cannot properly teach
to others
Extensive Comprehension
•
•
•
•
•
Teleological and conceptual explanations
“Why” explanations of mathematics solutions are
clear
all vocabulary is present and correctly used
uses appropriate pictorial representations to aide
understanding
can properly teach to others
Framework for Depth of Procedural Understanding
Basic
Understanding
•
•
•
•
•
•
major procedural
(“how to”)
simply speaking
what is being
written
steps are vague
descriptions lack
depth and
expansion
no or limited
vocabulary
cannot properly
teach to others
Retta’s solving of 2x + 6 = 16
Basic understanding
Two x plus six equals sixteen. First I’m going
to subtract six from both sides, so this would
be ten. So now it’s two x equals ten. Now I’m
going to divide by two, so it’s x equals ten
divided by two, which is five. X equals five.
Framework for Depth of Procedural Understanding
Extended
Understanding
•
•
•
•
•
•
- teleological semantic
understanding of
procedures
understanding of what
they are doing
major procedural
steps explained
Correct mathematical
form and
mathematical symbols
essential / moderate
vocabulary used
why explanations
limited
cannot properly teach
to others
Retta’s solving of 2x + 6 = 16
Extended understanding
I have the equation two x plus six equals
sixteen. My goal is to isolate the variable, so
first I’m going to subtract six from both sides.
These cancel out and this equals ten. So two
x equals ten. Now, I’m going to do the inverse
of multiplication, division. So divide by two,
these cancel out and divide ten by two and
the answer is x equals five
Framework for Depth of Procedural Understanding
Extensive
Comprehension
•
•
•
•
•
Teleological and conceptual
explanations
“Why” explanations of
mathematics solutions are
clear
all vocabulary is present and
correctly used
uses appropriate pictorial
representations to aide
understanding
can properly teach to others
Retta’s solving of 2x + 6 = 16
Extensive comprehension
In order to solve the equation two x plus six equals
sixteen, we need to isolate the variable x. The first
thing I have to do is get rid of the six by itself
because if you do the inverse of order of
operations, you would do addition first instead of
multiplication first. So, the inverse of addition is
subtraction, so I’m going to subtract six from
positive six and they have a sum of zero. So they
would cancel out and what is done to one side
must be done to the other side of the equals sign
in order to keep both sides equivalent. So I
subtract six from here as well. Sixteen minus six
equals ten. So now I have two x equals ten. Now I
have to do the inverse of multiplication which is
division. So I’m going to divide two x by two and
they will cancel out and then I’m going to divide the
other side by two in order to keep them equivalent.
So I now have x equals ten divided by two, which
simplifies to be, x equals five.
Linear Regression
ANOVA
• Confirmed that our three
categories were significant
based on a p-value< .05
• Omnibus p<.001
• Tukey Post Hoc <.001
• Significant Positive
Correlations
Procedural
Extensions
• Beta Effect Sizes:
• Vocabulary usage (effect size β = .18),
• stating procedural processes (effect size β
=41), and
• making extensions / connecting ideas to
previous mathematical concepts (effect
size β =53).
• These three factors explain 83% of
depth of knowledge (r =.91).
Vocabulary
Thus having students include
mathematics vocabulary, stating
procedural processes and
increasing the amount of
extensions deepen understanding
of procedural processes.
TEACHERS
-Have a solid baseline of guidelines to gage their
students procedural knowledge
-Increase teacher effectiveness through
pedagogy
-Have a deeper understanding of the stages that
students should work through for success
-Identify and correct misconceptions
STUDENT KNOWS
-To continually use vocabulary
-Speak aloud steps
-Make connections to other aspects of
mathematics
-Understand the “why” of each step conducted in
their solution
700 + pencasts later…
Framework for Depth of Procedural Understanding of Mathematical Solutions:
Assessing student explanations for conceptual understanding of procedural processes
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