ASSESSMENT AND CORRECTION
MATHEMATICS EDUCATION:
ECED 4251
Rosalind Duplechain
University of West Georgia
College of Education
Decimal Numbers and Operations
Module 9
Basic Structure of PPt
• Opening Activity (slide 3)
• Lecture (slides 4-11)
– Decimal Numbers and Operations
– How the D&C Process works with Decimal Numbers and
Operations
• Application (slides 12-13)
– See textbook for error patterns associated with decimal
numbers and their operations (Tonya, Harold, Les, Marsha,
& Ted)
• Homework - (See Course Calendar).
Opening Activity: Module 9
Topics to consider
 Decimal Concepts and Operations
 Compare and contrast a decimal number
with a whole number. List as many ways
as you can find.
 Using base-10 blocks, how many ways can
you represent 2.5 using these place
values? Do not assume that you can only
use one of each place value. You can use
as many as needed.
 Explain why the following would not be a
correct representation of 2.5.
. .
 Use <, >, or = to show the relationship
between .001, .01, .1, and 1. Justify your
answers.
 .001 ____ .01 ____ .1 ____1
• Complete Homework Sheet #10.
What to do about topics?
 Self-Assessment
 Using the questions on the left side of this slide, what don’t you
“get” about decimal concepts and operations?
 Using what you don’t get, start a question log for decimals.
 Seeking Help
 Talk to two peers (at least) in hopes of finding answers to your
questions.
 Keep a log of other questions you have for me.
 Gallery Walk
 Take the Gallery Walk experience.
 Building on our past work, how might the D&C Process work
with decimals?


Using Tonya's error from the textbook,…

do the same diagnosing steps apply? If not, which
steps don’t apply? Why?

do the same correcting steps apply? If not, which steps
don’t apply? Why?
Compare and contrast Tonya’s correction strategies with
the correction steps provided in this course.
 Keep a log of questions you have for me.
Decimal Numbers and Operations
•
•
Decimals are fractions (and percents) written in a different symbol system. All are
merely different names for the same quantity or for the same point on a number line.
Despite the interrelatedness of these symbol systems, children see these systems as
being very distinct.
“The base-10 place-value system extends infinitely in two directions: to tiny values as
well as to large values. Between any two place values, the ten-to-one ratio remains the
same” (Van de Walle, 2004, p. 280).
– The decimal point is a convention that has been developed to indicate the units position to
the left of the decimal point that is being counted as singles or ones” (Van de Walle, 2004, p.
280).
– Although placing the decimal point is often viewed as a place value concern, it is clearly a
procedural concern as well.
– Finally, the further away a digit is from the right of a decimal, the smaller its value and vice
versa.
•
Like place value of whole numbers, with decimal place value, each place has its own
value. Thus, just as ones are added to ones and tens are added to tens, and so on, for
whole numbers, with decimal numbers, hundredths are added to hundredths, tenths
are added to tenths, ones are added to ones, and so on. Similarly with subtraction,
whether whole numbers or decimals, ones are subtracted from ones, hundredths from
hundredths, as so on.
Decimal Numbers and Operations
• Children tend to apply what they know about whole
number operations to decimal operations.
– In some cases, this application is good because it helps
students see that the meaning of the operations is always
the same.
– However, in other cases, this application contributes to a
variety of conceptual and procedural errors when
operating with decimals, namely, a misplacement of the
decimal.
• In which case, the role of estimating (rounding decimal numbers to
the nearest whole number) can help children determine where the
decimal point belongs.
Decimals and the D&C Process
• The same four D&C Sub-processes hold for decimal
numbers and operations as we used for whole
numbers and fractions.
–
–
–
–
Diagnose
Correct
Evaluate
Reflect
• The diagnosing checklist for decimal numbers and
operations is similar to the checklist used for
diagnosing whole number operations.
Sub-process #1: Diagnose
• Basic Facts Errors
– Interview Student
• Collect Data
– Analyze Data for Errors
– Final Diagnosis of Data
• Whole Number
Operations/Algorithm
Errors
– Collect Data
– Analyze Data for Errors
– Pre-diagnose Data
– Interview Student
– Final Diagnosis of Data
Sub-process #2: Correct
• Basic Facts Errors
– Teach meaning of
operation
– Teach and practice
number relationship
strategies
– Work on automaticity
• Whole Number
Operations/Algorithm
Errors
– Conceptual Only
– Intermediate
– Procedural Only
– Independent Practice
Correcting Decimal Errors…
Conceptual Only – Using manipulatives/drawings only, show
and talk aloud while solving problem. Emphasize ideas related to
student’s error. Repeat until student can do alone.
Intermediate – Using manipulatives/drawings, show and talk
aloud while solving problem. Teach and show that algorithm is the
step-by-step record of what is being done with manipulatives.
Emphasize ideas related to student’s error. Repeat until student
can do alone.
Procedural Only – Using only the algorithm, show and talk
aloud while solving problem. Emphasize ideas related to student’s
error. Repeat until student can do alone.
Independent Practice (procedural) –
Provide problems for
student to solve alone, using only the algorithms. Once practice is
completed, teacher checks work. If work earns <85%, teacher
repeats correction cycle beginning on either the intermediate level
or the procedural only level.
Teacher Guided
Experiences
Teacher Guided
Experiences
Teacher Guided
Experiences
Student-only practice
Teacher feedback
Sub-process #3: Evaluate
• Give a post-test and make sure that student
provides all of his/her work.
– Assuming you used the correct pre-test, use the same test
you used to collect your pre-data.
– Allow the same amount of time as you allowed for the pretest.
– Grade student’s work (Aim for at least 85%).
– Diagnose all errors and ask yourself:
• Are any of these errors like the original errors found on the pretest?
• Are any of these error new – unlike the original errors found on
the pre-test?
Sub-process #4: Reflect
• Use score from post-test to determine what to do next.
– If <85%, repeat correction cycle. Student has not mastered a sufficient amount of
concepts and skills associated with decimal numbers and their operations.
– If ≥85%, this student is on his/her way to mastery.
• Continue to work on number strategies and on automaticity in classroom through
learning centers (whenever students finished assigned tasks earlier than peers) and
through drill time (about 5 minutes of every math lesson).
• As another drill, create a timed drill that requires students to solve, in writing, no more
than 15 problems involving these operations (Similar to the way May’s tests are
designed, have students perform all operations with whole numbers, fractions, and
decimals – no more than 15 problems). Once time is up (no more than 10 minutes),
quickly model aloud how to solve each problem. Have students check answers as each
problem is modeled aloud. Collect these daily and keep an informal running record of
progress (entire drill about no more than 15 minutes of every math lesson).
• Aside from these learning opportunities (drills), move on to work with more needy
students or on other mathematical topics in mathematics curriculum.
Application
• Let’s apply what we’ve learned today about
the D&C Process to violations of algorithms,
and in particular to error patterns associated
with decimal concepts and operations.
– Tonya
– Harold
– Les
– Marsha
– Ted
Diagnosing Checklist:
Decimal Numbers and Operations
• The procedural error(s)
• Ask yourselves: What exactly is this student doing to get this problem
wrong?
– Basic Facts
– Violations of Algorithm
• The conceptual error(s)
• Ask yourselves: What mathematical misunderstandings might cause a
student to make this procedural error?
– Decimals
» Can identify and represent decimal numbers
– Place Value
– Meaning of the Operation in general
– Meaning of the Operation under specific conditions (i.e., larger quantities)
– Properties of Operations
– Number Sense
Homework
• See Course Calendar on CourseDen.
– Use Instructional Resource Item #29.
– Complete Homework Sheet #9.
– Read textbook chapter 4-6 (decimals only).
– Use overview sheet for Module 9 as checklist.
Complete all items on this checklist.