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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
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.

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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