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Developing Double-Digit Division Skills in Children

Learning Problem:

“Julia loves using math workbooks . We buy them at local bookstores whenever she gets the urge. She
studies math in spurts, yet moves fairly quickly through the books when inspired. When she came upon
long division, however, she asked for my help. I explained the mechanics and what it meant in real terms ,
but she was stumped.“

“When teaching long division, students often forget what to do next after they bring down a number.”

“We are doing some hard math--DIVISION! Single-digit division isn't too bad, but the double-digit
examples are a little bit harder. Please help by making sure your child knows the multiplication facts
instantly.“

As suggested by the large number of web pages addressing the issue, double-digit division is a
troublesome learning concept for many students. It appears that even students demonstrating an interest
or ability in math may have trouble with double-digit division. The number of steps involved in solving one
problem
often throws students for a loop when they first encounter these types of problems in 4th or 5th
grade. Halfway through the problem, the student may even forget what to do next. As the last quote
suggests, the biggest issue with learning double-digit division may be the fact that students have not
mastered the necessary component skills. This problem becomes even more readily apparent when
students are asked to solve word problems with double-digit division. Without the foundation and
prerequisite skills, students cannot be expected to solve problems in a complex situation .

Needs Assessment:

In order to gain a better understanding of the problem of teaching and learning double-digit division, a
study must be conducted in which the present learning environment is analyzed. In turn, this may provide
us with insight into how double-digit division could be better taught. Therefore, we must complete a
thorough needs assessment that examines the current practices in the classroom and evaluates their
effectiveness. Based on our hypothesis that the behavioral perspective on learning would best teach
them the necessary skills for double-digit division, we intend to model our needs assessment on the
design principles that reinforce this type of learning.

Design Principle Rationale for Principle Study How to Test
Information and
Procedures are
presented
systematically (taking
into account what
students already know)
In order for students to
learn about a difficult
concept, they must first
understand the
individual components
of the concept,
subtraction,
multiplication, simple
division, word problem
analysis and
organization skills.
Prior Knowledge: We
need to determine what
skills the students already
have in place in terms of
basic
mathematics. Can
they subtract? Can they
solve simple word
problems?

Components: An analysis
of the components of
double-digit division must
be undertaken in order to
create a learnable
sequence. Here, we must
identify all of the
necessary components
skills and rules for
teaching
double-digit
division.
We will conduct a
pre-test of math skills
for all students, in
order to identify their
strengths and
weaknesses.
We will also complete
a literature review, to
determine the
complete and most
effective sequence
for teaching double-digit
division.
Controlled Practice with
Feedback
Behaviorism supports
the learning of
knowledge through
controlled practice with
feedback. We need to
know if this is already
present in the learning
environment and if not,
what practices are
already in place. This
will give us a guideline
for what is working and
what is not.
Present Methods: The
current methods of
teaching
double-digit
division in the classroom
need to be analyzed. Are
teachers using repetition
with students? Are they
teaching under controlled
circumstances? Do
students know when they
are right, and when
they’ve made a mistake?
We will observe the
current math
classroom and
interview teachers to
find out their methods
and rationale. What
do they think is
working – what are
the current
deficiencies?
Promotion of Errorless
Learning
In addition, behaviorism
supports learning
environments where
errors are not made.
The environment we
design should identify
errors immediately and
not allow students to
continue until they have
identified and corrected
their mistakes.
Present Methods: Are
students presently
learning in an environment
that allows them to create
links between the stimulus
and the wrong response?
Are they allowed to make
mistakes and continue on
with their work? How
much flexibility is built into
their learning
environment?
During our classroom
observations, we will
note whether
students are making
errors, particularly
consistent ones. How
much feedback is
available in the
classroom now to
identify and remedy
students’ errors?
Practice should occur in
simple contexts
Practice of stimulus-response
situations
should not complicate
learning in any way.
How are students being
asked to learn double-digit
division now? Are
they using complicated
word problems or
intricate games? If so,
this environment may
be too complicated to
build the links desired.
Present Methods: We
must examine the context
in which double-digit
division is being taught.
Are teachers using the
simplest methods, or are
they cluttering practice by
introducing more difficult
or irrelevant concepts.
We will interview
students and
teachers here, asking
them to describe how
they practice this
skill. Perhaps the
students will view
their work differently
than their teachers
who assigned it. This
method will give us
both perspectives.
Sequence of
Component-to-
Composite Skills
Learning is most
effective when it
teaches simple skills,
creates links between
them that constitute
rules, and then builds
on those rules to create
coherent knowledge.
Present Methods: The
order of presentation of
double-digit division
components in the current
learning environment
should be studied. Are
teachers incorrectly
introducing the concept of
division before securing
the component concepts
of subtraction and
multiplication
? Are they
testing to ensure that
students have mastered
basic concepts?
We need to work with
teachers and observe
classes in order to
outline the steps that
are currently used for
teaching double-digit
division in the
classroom. Are they
consistent between
classes? Are they
slowly progressive?
Is there testing before
moving to new skills?
Individualization In behavioral learning,
teaching is most
effective when it is
individualized for the
strengths and
deficiencies of each
learner. Assumptions
about where students
should be learning are
dangerous to the
development of rule
learning.
Present Methods: We
must examine whether
students are being taught
according to their
individual abilities. Are
there tutors (such as
computers) that can tailor
skill building to the
particular learner? Do the
teachers have the time
and inclination to focus on
individual students, or are
they taught as a class or
in clusters?
We will take an
inventory of the
resources available
to teachers. Are there
enough teachers and
aides in the school to
individualize
learning? Are there
enough computers
and corresponding
software to address
specific needs?

Interviews with
teachers can also
help determine
whether this is a
possibility in the
current structure of
the school. Interviews
with students will
elicit their feelings on
how their individual
needs are being
addressed.
Effective Use of
Reinforcement
In order to incorporate
effective use of
reinforcement, the
students’ motivations
must be understood.
Therefore, we can
ensure that
reinforcement is
productive and
appropriate
to the
individual.
Reinforcement: We must
gather input from parents
on the likes and desires of
the children. In that way,
reinforcements that are
offered can then be
individualized and serve
as true motivation for
students.

Present Method: We
must also determine
whether reinforcement,
particularly appropriate
reinforcement at
appropriate times, is now
being used in the
classroom.
Interviews with
parents and students
will determine a small
number of
reinforcements that
can be offered for
successes in
learning.

As part of our
classroom
observation, we will
learn how
reinforcement tools
are now being used
in the classroom. Are
students adequately
motivated? Are they
reinforced after
appropriate learning
achievements?

With this important background knowledge, we can now begin to design a new learning environment that
uses behaviorist principles to improve the teaching of double-digit division. We will use the literature
review to plan our new design, and the observation, testing and interview results to understand how the
present teaching methods can inform our design suggestions. Perhaps most importantly, we will
understand what concepts are already in place, creating a jumping off point from which we can begin to
build new skills.

Planned Design:

The learning environment for acquiring the skills necessary for performing double-digit, long division word
problems could possibly be improved in our design by the learning activities defined in the following chart .
We have investigated the sequence of learning components as they pertain to the composite skill of
double-digit, long division problems. In conjunction, a learning procedure is suggested and defined. In
the end, chaining each of the learned concepts, procedures and rules should lead to a full understanding
of not only how to execute these complex division problems, but a foundational perception of the true
components of division problems.

It is important to note that arrangement and sequence of learning tasks must be based upon the
complexity of each component. The design will begin by enforcing mastery of small units of mathematics
(subtraction and multiplication); mastery of each subsequent unit must be completed before moving into
more complex units and chaining the procedures. The learning environment will begin with paper-based
drills and practice and then transition into individualized computer-learning programs, designed to better
shape students’ knowledge based on their personal progress and understanding.

Sequence of Component to
Composite Skill
Concept Recommended Procedure
Concept Learning of Information
and Procedures
Pre-test students’ knowledge of
basic underlying skills in math
including subtraction and
multiplication. Evaluate the
outcome to determine if
foundational concepts must be
reviewed or re-learned. Develop
concepts (basic skills) to
mastery.
Using a paper-based, pre-test
format, children will execute
basic drills for math problems
including subtraction and
multiplication and basic division
skills. The students will be
required to work through multiple
drills until mastery of these basic
mathematics components.
Teachers will evaluate their
progress based on educational
standards to be sure that
students are ready to progress to
the next sequence.
Rule Learning Practice of repetitive drill
problems using the following
“DMS ” technique for each
problem:
D
M
S
In working the division problems,
students will check each
sequence in their progression
against this chart. First the
student needs to divide (D) and
once the student writes quotient,
draw a check mark beside the
"D". Then (M) which tells the
student that the next step is to
multiply and check off the "M".
Subtraction (S) is the next step
in the division process and they
will check off that they have
subtracted. Finally, the down

arrow ( ) indicates bringing
down the next number from the
dividend. The student continues

following the D, M, S, .process
by checking off components in
each sequence of division
problem.
Rule Learning (errorless learning
in individualized, simple context
environments)
Once the former skill is
mastered, the child moves to a
computer-aided environment to
learn double-digit, long division
rules in a self-paced, errorless
learning environment. Errors in
division may negatively influence
progress since each calculation
in the problem requires distinct
accuracy of the former
calculation.
In using computer-aided learning
environment the ‘smart agent’
monitors student skill levels so
they can work at all different
levels (i.e.: children who have
mastered the double-digit
division concept and others who
are learning at slower rates).

Word problems are presented on
the computer screen. Children
work on ‘ graph paper ’ on the
computer and the system will
know whether the student
entered the right number in the
right space.

The computer system catches
students’ errors before they can
be performed and alerts them to
the mistakes that would occur if
that answer had been entered
incorrectly. Errors are noted and
explained before they are made.
If there is a mistake, the system
requires the student to correct
before they can proceed to the
next step in the problem.
Problem Solving Problem solving is the final stage
in our division learning design.
Once all sequences are fully
developed by the student, they
engage in longer problem
solving tasks that continue to
provide drills to practice long
division skills.
Students would begin working on
long word problems involving
complex, long division drills.
They should ultimately have
gained the chaining knowledge
to connect what parts of the
word problem will require their
long division skills and be able to
transfer the knowledge and
understanding of long division
into more complex situations.
Instead of simple, repetitive drill
and practice problems, these
word problems will involve
additional information and
students will have to properly
filter the components they need
to successfully perform the word
problem.
Reinforcement Reinforcement is a critical factor
throughout each sequence of the
learning design.
Children should be awarded
points based on the amount of
time they spend on problems or
how many they get right.
Rewards can also be delivered
to children for the number of
correct answers they’ve
achieved on homework, quizzes
and tests.

The programmed reinforcements
will be defined by teachers
(individualized so that each
students’ reward matches a
behavioral response to
accomplish the tasks).
Depending on points, the
teacher will decide how to award
each child.

Again, each stage in the
sequence should involve
reinforcement techniques. In
order to encourage the children
at each step, they will have to be
rewarded appropriately.

Assessment:

In order to determine how effective our new learning environment is for teaching double-digit division
problem-solving skills, we will conduct pre- and post-test on basic skills and a composite understanding.
We would use simple, paper-based testing that requires students to write out their work clearly, step-by-step.
The pre-test, conducted before we institute our new design, would test subtraction, multiplication,
simple division and problem-solving skills.

Then we will initiate our new behavioral learning environment, teaching skills, creating links that make
rules and building them into more complicated processes. Afterward, we will conduct a paper-based
posttest with similar questions to the pre-test, with basic skills and more complicated questions. Is it clear
from their work that they now have mastered the components of the larger problems? Are they working
through the proper steps to reach their conclusion? What are they still missing?

The final part of our assessment will look at the reinforcements that have been integrated into learning.
Are they working as motivation for students? Are they appropriate to the individuals? Do we need to
change them to make them more effective?

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