MA201
Example Presentation
Section 2.3: The Hindu-Arabic System
Presentation Outline
I.
Presentation Theme:
Theme: Understanding the role of the decimal point
(Where does the “oneths” place go?)
Why Challenging for Elementary Students:
The meaning of the position of decimal point can be confusing.
Mathematics for Elementary Teachers:
Understanding the “mirror symmetry” of our place value system.
II.
Mathematics and Sharing my understanding:
Explaining the mirror symmetry that exists in our place value
system and the misleading position of the decimal point.
Providing a solution that teachers can use to explain the regularity
of the place value system from hundreds to hundreths.
Presentation Outline
I.
Presentation Theme:
Theme: Understanding the role of the decimal point
(Where does the “oneths” place go?)
Why Challenging for Elementary Students:
The meaning of the position of decimal point can be confusing.
Mathematics for Elementary Teachers:
Understanding the “mirror symmetry” of our place value system.
II.
Mathematics and Sharing my understanding:
Explaining the mirror symmetry that exists in our place value
system and the misleading position of the decimal point.
Providing a solution that teachers can use to explain the regularity
of the place value system from hundreds to hundreths.
Presentation Outline
I.
Presentation Theme:
Theme: Understanding the role of the decimal point
(Where does the “oneths” place go?)
Why Challenging for Elementary Students:
The meaning of the position of decimal point can be confusing.
Mathematics for Elementary Teachers:
Understanding the “mirror symmetry” of our place value system.
II.
Mathematics and Sharing my understanding:
Explaining the mirror symmetry that exists in our place value
system and the misleading position of the decimal point.
Providing a solution that teachers can use to explain the regularity
of the place value system from hundreds to hundreths.
Presenting My Presentation Theme
(At least one slide clearly identifying
the presentation theme or topic.)
Presentation Theme: Understanding
the role of the decimal point.
Presentation Theme: Understanding
the role of the decimal point.
By exploring a common elementary school
misconception: “Where did the oneths
place go?”
Problem: Where did the
oneths place go?
Find the place value of 8.
a) 6.781
Trissy says, “tenths”.
Problem: Where did the
oneths place go?
Find the place value of 8.
a) 6.781
Trissy says, “tenths”.
Problem: Where did the
oneths place go?
Consider the following number:
3 2 1 .1 2 3
3: hundreds
2: tens
1:ones
1: tenths
2: hundredths
3: thousandths
Problem: Where did the
oneths place go?
Consider the following number:
3 2 1 .1 2 3
3: hundreds
2: tens
1:ones
1: tenths
2: hundredths
3: thousandths
Problem: Where did the
oneths place go?
Consider the following number:
3 2 1 .1 2 3
3: hundreds
2: tens
1:ones
1: tenths
2: hundredths
3: thousandths
Problem: Where did the
oneths place go?
Consider the following number:
3 2 1 .1 2 3
3: hundreds
2: tens
1:ones
1: tenths
2: hundredths
3: thousandths
Problem: Where did the
oneths place go?
Consider the following number:
3 2 1 .1 2 3
3: hundreds
2: tens
1:ones
1: tenths
2: hundredths
3: thousandths
Problem: Where did the
oneths place go?
Consider the following number:
3 2 1 .1 2 3
3: hundreds
2: tens
1:ones
1: tenths
2: hundredths
3: thousandths
Problem: Where did the
oneths place go?
Consider the following number:
3 2 1 .1 2 3
3: hundreds
2: tens
1:ones
1: tenths
2: hundredths
3: thousandths
Where did the ‘oneths’ go?
Topic Justification
For this specific problem:
I found this problem described in a journal
publication, “But what about the oneths?”
by Amy MacDonald in Australian
Mathematics Teacher, 2008.
Trissy was a real student who asked this
question.
Topic Justification
Quoted from the paper:
Many errors students make in mathematics are
rational, “logically consistent and rule based
rather than random.” The strategies constructed
in order to assist in solving an unfamiliar
problem are usually based on prior knowledge
and experience.
(It is natural to ask where the ‘oneths’ place is.)
Topic Justification
Quoted from the paper:
Many errors students make in mathematics are
rational, “logically consistent and rule based
rather than random.” The strategies constructed
in order to assist in solving an unfamiliar
problem are usually based on prior knowledge
and experience.
(It is natural to ask where the ‘oneths’ place is.)
Topic Justification
In general:
Elementary children have traditionally found place
value to be difficult to learn, and their teachers
have found it difficult to teach. Understanding
place value requires a child to make connections
among and sense of a highly complex system for
symbolizing quantities.
From Place Value: Problem solving and written assessment
by Sharon Ross in Research, Reflection, Practice
II. Sharing my
understanding
Explaining The Problem
• Mirror symmetry is a pattern in which there
is a vertical line of reflection.
The left-hand side and
the right-hand are mirror
images of each other.
Explaining The Problem
A student knows that there is a mirror
pattern, and expects that this patterns
occurs symmetrically around the decimal
point.
427.81
“If 7 is ones, then 8 is oneths.”
Explaining The Problem
427.81
“If 7 is ones, then 8 is oneths.”
However, ask Trissy:
What would 8 oneths mean?
Ones and ‘oneths’ would be the same!
Explaining The Problem
427.81
“If 7 is ones, then 8 is oneths.”
However, ask Trissy:
What would 8 oneths mean?
Ones and ‘oneths’ would be the same!
Explaining The Problem
427.81
The “mirror symmetry” occurs around the
ones place, not the decimal point.
7 = ones
2 = tens and 8 = tenths
4 = hundreds and 1 = hundreths
Ones and ‘oneths’ would be the same!
Solution for elementary students:
teach them the
Endless Base 10 Chain
Discussion For Teachers
• The decimal place is not exactly ‘in the
middle’. Is this different than what you
thought?
• Exercise: Suppose that you have
and
would like to multiply this by 1000. Explain
how you could solve this problem using
the endless base 10 chain.
Discussion For Teachers
• The decimal place is not exactly ‘in the
middle’. Is this different than what you
thought?
8
• Exercise: Suppose that you have 100 and
would like to multiply this by 1000. Explain
how you could solve this problem using
the endless base 10 chain.
Implications for Teachers
from “But what about the oneths?”
Amy MacDonald suggests:
• Use the endless Base 10 chain (Steinle &
Stacey) when explaining the structure of
decimal notation.
• Encourage students to recognize that the
symmetry of the decimal revolves around
the ones.
• The decimal point indicates the ones place
on its left.
Any questions?
References
• “But what about the oneths?” by Amy
MacDonald in Australian Mathematics
Teacher, 2008.
• Place Value: Problem solving and written
assessment by Sharon Ross in Research,
Reflection, Practice
(Found online at AcademicOneFile via USA
library.)