## ZPD

The *zone of proximal development* (ZPD) is a simple and powerful concept widely accepted in the educational community. It was introduced by Lev Vygotsky about 90 years ago to refer to the area between what the learner can do independently and what he/she cannot do at all:

This model assumes that all three zones are related to the same learning objective. For example, mastering adding/subtracting within 20 while the learner’s competence is currently limited to adding/subtracting within 10. The aided activities in the transition zone result in expanding the learner’s competence to include a new skill.

The concept of ZPD turned out to be extremely stimulating for research and development in various directions, for example:

– Identifying the zone of proximal development for diverse learners

– Developing effective pedagogical strategies for aiding learning within the ZPD

– Incorporating technology as a means of support in a learner’s proximal development

Original works of Vygotsky as a psychologist were focused mostly on the first of these directions and also on the cultural-historical aspect of learning.

## Scaffolding

Applied interests of pedagogy are more related to the other two directions listed above. In that context, the term *scaffolding* is now widely used to refer to interactive support provided to learners that enables their progress through the ZPD and finally results in adding a new skill to the area in which the learner can work independently. The term scaffolding in education is a metaphor emphasizing that, like scaffolding in construction, it is a *temporary structure* removed upon attainment of the goal.

There is no doubt that currently the most effective scaffolding is one-to-one interaction of a highly qualified tutor with the learner. There is equally no doubt that this is unachievable on a large scale in mass math education, due to limited resources. However, progress in technology gives us hope that *computer-based scaffolding* will solve the problem to a certain extent. We are witnessing intensive development in that direction: within the framework of blended learning, advanced educational software provides differentiated support to students in many areas and also frees up teachers’ time for interacting with students in areas where it’s most needed.

### Example 1

For a student who has mastered counting up to 10 objects in aligned or arranged in ten-frame configurations, counting up to 10 scattered objects in highly likely to be in his/her ZPD. Happy Numbers provides interactive scaffolding that leads the student through the ZPD.

First, students are asked to tap one-by-one the objects they count. This highlights the counted objects, for example changing the color of a blue bead to pink, and creates a *counting path*.

The screenshot shows the beginning of the second tap.

The tasks start with sets of 4 or 5 beads. Upon mastering this content, the student progresses to tasks with sets of up to 10 beads, still with the same detailed scaffolding. When the student succeeds with these tasks, *the scaffolding he/she receives is reduced*: the counting path is no longer shown.

The reduced scaffolding is interactive — it still requires tapping the objects:

Then, the scaffolding becomes *optional*: students may tap the objects if they want. Finally, the scaffolding is removed *from the first attempt* to solve the task, and is returned only when correcting an error is necessary! It aids the student in solving the task.

Students leave the ZPD (which means that *the new counting skill is added to their area of independent work*) when they satisfy a certain criterion. The criterion varies for different subjects, but roughly it means that the student has a high percentage of success in solving the tasks.

Important features of computer-aided support can be seen in the scaffolding for the above example of a kindergarten-level objective:

– *Interactivity*. Each step requires student action. This is how *collaboration* in ZPD works on the learner’s side. In particular, this keeps students focused.

– *Adaptivity*. The system provides the student with tasks, hints, etc. based on the learner’s work and sometimes on their choice/request. (Happy Numbers provides the option to *request* context-sensitive support in some cases.) This is how *collaboration* in ZPD works on the system side.

– *Variety of manipulatives*. Colorful and easily available, due to technology, the manipulatives not only play their role in learning but also make it engaging.

### Example 2

Consider scaffolding for a topic in the Happy Numbers curriculum dealing with the standard algorithm for multiplication. The topic is limited to multiplying multi-digit by single-digit numbers and given to students who have already mastered such multiplication when it does not involve trading (a.k.a. regrouping or renaming), such as 2 x 314.

Without detailing prerequisites of the topic, it is worth mentioning that they include place values, addition algorithm based on them, and the standard algorithm for multiplication in the mentioned no-trade case. The case when trading is needed is the focus of the example at hand.

Let’s first look at four support areas of the scaffolding: conceptual understanding, operations, record, and structure of the algorithm.

★ *Conceptual understanding* is specifically supported by the place value disk model (discussed in detail in our “blog):

Students will trade the circled 10-disks for one 100-disk: an animation will make it happen.

★ *Operations* performed on each step are supported by hints and the corresponding numerical expressions, for example:

★ *Recording* of the multiplication in the standard algorithm format is supported by hints and explanation, for example:

★ Grasping the *algorithm structure* is supported by emphasizing that same procedure is applied place-by-place

Notice that the screenshots above show each scaffolding in its strongest form.

The scaffolding is *gradually removed* when students progress through the ZPD. The following chart shows how the scaffolding is fading and is finally terminated.

For example, the Operation scaffolding in the beginning includes hints and the corresponding numerical expressions (see the screenshot above). Then, the expressions are removed while hints are still available. Next, the scaffolding is completely removed, and students move into the zone of independent solving.

Similarly, the Algorithm Structure scaffolding in the beginning splits the multiplication into steps and for the current step specifies the place to work on. When the scaffolding is partially removed, students reflect on the sequence of steps and for the current step identify the place to work on. Finally, upon removing the scaffolding, students determine what and in what order to solve on their own.

The main takeaway from the above example is that scaffolding for an objective can include a number of parts with different sub-objectives, using different methods, and can be reduced/removed at different points.

## One Attempt of Independent Solving

In computer-based learning, there is an area “in between” aided and independent solving. From the practical viewpoint, it does not matter if it’s interpreted as a special zone, a part of ZPD, or a part of the independent solving zone. Happy Numbers finds it useful and often includes the corresponding tasks in its topics:

– Student has one attempt to solve a task independently, with no support

– When the answer is incorrect, the student receives an interactive support

– When the answer is correct, the student moves to the next task

An exercise, which is a set of such (quasi-randomized) tasks, is considered successfully completed if the student demonstrates a high percentage of the tasks solved on the first attempt.

Nobody is perfect, and moving from a student’s ZPD to the area of independent solving normally does not mean that the student solves 100% of the corresponding problems correctly. Actually, there is a transition area “in between” aided and independent solving. From the practical viewpoint, it does not matter if this area is interpreted as a special zone, a part of ZPD, or a part of the independent solving zone.

What matters practically is how to decide if the student has reached the required level of mastery, and on the other hand, what support is needed when he/she has not.

It is an important question, and the Happy Numbers answer to it is a “one-attempt-at-independent-solving” approach:

The student is considered working independently if he/she completes the set of tasks with a high percentage of tasks solved correctly on the first attempt.

The cutoff percentage can be specified differently for different skills: the more basic the skill, the higher the cutoff percentage should be. How detailed the support should be also depends on the skill. At the same time, the “one-attempt-at-independent-solving” approach seems to be an appropriate framework for transition from aided to independent solving.

## A Virtual Teacher Assistant

Through adaptive technology, Happy Numbers is able to guide students through their individual ZPD, creating optimal learning conditions. It acts as a one-on-one tutor, responding to student inputs with just the right amount of scaffolding, the way a teacher would. This not only provides students with individualized instruction in their ZPD, but it also frees up the teacher to differentiate instruction by working with small groups or individuals. The integration of technology and pedagogy results in an environment that optimizes the learning of every student.