The Scaffold Method

Mathematic Definition:
Scaffold method--
When you solve the division problem A/B, you start with the number A, and repeatedly subtract multiples of B until a number remains that is less than B.  Since you subtracted as many Bs as possible, when you add the total number of Bs that were subtracted, that is the largest whole number of Bs that are in A.  What's leftover is the remainder.

Concept Meaning:

The scaffold method of division is basically the same set up as basic long division. The number you are dividing is placed underneath the division bar with the number you are dividing it by to the left of the division bar. For example, if you were dividing 440 by 4, you would place the 440 under the division bar and the 4 to the left. You would then divide the largest place-value number by the division number. Write the answer down above the division bar. Move to the next place value and divide that by the number. Place this result above the original number. Keep working until all numbers have been divided. Add up all the results to find your answer.

Example

• Divide 440 by 4 by dividing the hundreds place first. The hundreds place is represented by 400. Divide it by 4 to get the result of 100. Write 100 above the division bar, lining the one up with the four underneath and the zeroes above the zeroes underneath. Move to the next place value, the tens. The tens are presented by the 40. Divide the 40 by the 4 to end up with 10. Write the ten above the 100, placing the one in the tens place value and the zero in the ones place value. You cannot divide the zero in 440 by four so stop your division. Add up the 100 and the 10 to come up with 110.

Advantages

• The scaffold method is a visual method that helps break down the numbers in a way that some of your students may understand more fully than the normal method of long division. It breaks the problem down into its root values. It also helps simplify the division process. Instead of thinking of a problem like 1684 divided by 6 in terms of dividing 6 by the whole number, students can think of it in terms of dividing 1,000 by 6, 600 by 6, 80 by 6 and 4 by 6. It basically breaks the problem down into simpler steps.

Real-World Application

• The scaffold method of long division can be used in many real-world situations. For example, you work at a bank and you have $1,682 that you have to split up four different ways. You have to keep track of all of the different bills you use for each individual division. Using the scaffold method, you find the result of $420.50. The scaffold method would show that you had $400, $20 and $0.50 to give out. As a result, you now know that you have four $100 bills to give out, two $10 bills and two quarters to give out to each person. You can find this using normal division but the scaffold method can help some people better visualize the division.

Teaching Resources:

CONSTRUCTING TASK: SCAFFOLDING DIVISION THROUGH STRIP MODEL DIAGRAMMING  (Lesson plan)
http://gfletchy.com/2014/09/18/scaffolding-division-through-strip-model-diagramming/

Click the above link.  There is a short article.  In the article you will see a sentence that says "click here to change your life forever"  click the link and a new tab will open with the lesson plan.
It takes a few steps to get there, but I think it is worth it. 

Some information on the scaffold method.
Article by Heather Coffey found at http://www.learnnc.org/lp/pages/5074

Scaffolding in the classroom

When using scaffolding as an instructional technique, the teacher provides tasks that enable the learner to build on prior knowledge and internalize new concepts. According to Judy Olson and Jennifer Platt, the teacher must provide assisted activities that are just one level beyond that of what the learner can do in order to assist the learner through the zone of proximal development.¹ Once learners demonstrate task mastery, the support is decreased and learners gain responsibility for their own growth.
In order to provide young learners with an understanding of how to link old information or familiar situations with new knowledge, the instructor must guide learners through verbal and nonverbal communication and model behaviors. Research on the practice of using scaffolding in early childhood development shows that parents and teachers can facilitate this advancement through the zone of proximal development by providing activities and tasks that:

• Motivate or enlist the child’s interest related to the task.
• Simplify the task to make it more manageable and achievable for a child.
• Provide some direction in order to help the child focus on achieving the goal.
• Clearly indicate differences between the child’s work and the standard or desired solution.
• Reduce frustration and risk.
• Model and clearly define the expectations of the activity to be performed.²

In the educational setting, scaffolds may include models, cues, prompts, hints, partial solutions, think-aloud modeling, and direct instruction.

Eight characteristics of scaffolding

Jamie McKenzie suggests that there are eight characteristics of scaffolding instruction. In order to engage in scaffolding effectively, teachers:

• Provide clear direction and reduce students’ confusion. Prior to assigning instruction that involves scaffolding, a teacher must try to anticipate any problems that might arise and write step-by-step instructions for how learners must complete tasks.
• Clarify purpose. Scaffolding does not leave the learner wondering why they are engaging in activities. The teacher explains the purpose of the lesson and why this is important. This type of guided instruction allows learners to understand how they are building on prior knowledge.
• Keep students on task. Students are aware of the direction in which the lesson is heading, and they can make choices about how to proceed with the learning process.
• Offer assessment to clarify expectations. Teachers who create scaffolded lessons set forth clear expectations from the beginning of the activity using exemplars, rubrics.
• Point students to worthy sources. Teachers supply resources for research and learning to decrease confusion, frustration, and wasted time.
• Reduce uncertainty, surprise, and disappointment. A well-prepared activity or lesson is tested or evaluated completely before implementation to reduce problems and maximize learning potential.
• Deliver efficiency. Little time is wasted in the scaffolded lesson, and all learning goals are achieved efficiently.
• Create momentum. The goal of scaffolding is to inspire learners to want to learn more and increase their knowledge and understanding.³

Martha Larkin suggests that there are eight guidelines that teachers most commonly follow when developing scaffolded lessons.⁴ According to research in the area of scaffolding, teachers often:

• Focus on curriculum goals to develop appropriate tasks.
• Define a shared goal for all students to achieve through engagement in specific tasks.
• Identify individual student needs and monitor growth based on those abilities.
• Provide instruction that is modified or adapted to each student’s ability.
• Encourage students to remain focused throughout the tasks and activities.
• Provide clear feedback in order for students to monitor their own progress.
• Create an environment where students feel safe taking risks.
• Promote responsibility for independent learning.

Advantages and disadvantages of scaffolding

This type of instruction has been praised for its ability to engage most learners because they are constantly building on prior knowledge and forming associations between new information and concepts. Additionally, scaffolding presents opportunities for students to be successful before moving into unfamiliar territory. This type of instruction minimizes failure, which decreases frustration, especially for students with special learning needs.⁵
Although scaffolding can be modified to meet the learning needs of all students, this is also disadvantageous because this technique, when used correctly, is incredibly time-consuming for teachers. Scaffolding also necessitates that the teacher give up some control in the classroom in order for learners to move at their own pace. Teachers who engage in scaffolding as a teaching strategy must be well-trained in order to create effective activities and tasks for all students.⁶

Notes

1. Olson, J. and Platt, J. (2000). “The Instructional Cycle.” Teaching Children and Adolescents with Special Needs (pp. 170-197). Upper Saddle River, NJ: Prentice-Hall, Inc. [return]
2. Bransford, J., Brown, A., & Cocking, R. (2000). How People Learn: Brain, Mind, and Experience & School. Washington, DC: National Academy Press. [return]
3. McKenzie, J. (2000). “Scaffolding for Success.” [Electronic version] Beyond Technology, Questioning, Research and the Information Literate School Community. Date accessed: 21 February, 2009 from http://fno.org/dec99/scaffold.html [return]
4. Larkin, M. (2002). “Using Scaffolded Instruction to Optimize Learning.” ERIC Clearinghouse. ED 474 301. [return]
5. Van Der Stuyf, R. (2002). "Scaffolding as a Teaching Strategy." Date accessed: 21 February, 2009. http://condor.admin.ccny.cuny.edu/~group4/ [return]
6. Van Der Stuyf, R. (2002). [return]

One way to show the process

First, we begin by dividing 8 hundred by 2, which gives us 4 hundred. Since we have used 8 hundred, we subtract 860  800 = 60. In the model, we still have $60 to share.

Next, we share the $60.  Since 2 × 30 = 60, we need to add another 30 to our quotient.  Notice we place the 30 in the proper place value above the 400.   We have also subtracted 60  60 = 0. In the model, we have no money left to share.

The last step is to sum the two partial quotients to obtain the final quotient.  This method of division is called the scaffold algorithm for long division.

May and Jay would each receive $430.

        Here are two different examples that use the scaffold algorithm to divide 976 by 2. The first example uses the most efficient partial quotients. The second example uses more partial quotients but they are in smaller pieces; this is like passing out a large number of items by giving each person a few at a time. Since we may pass out any number of items at a time, the number of partial quotients we use does not matter. We do need to pay attention to the place values.

Documentation of Sources:
Defintion--Math for Teachers Textbook
Concept Meaning--Ehow.education.com
Lesson Plan--gfletchy.com
Two different examples--mnstate.edu


Common Core Standards:
MCC4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 
MCC5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.  
MCC6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.