The implementation of the combination of Vygotzky's theorem of the zone of proximal development with the Singapore math program presents significant benefits in creating the critical thinking skills and developing the required cognitive functions for understanding complex mathematical concepts.

## Entry Narrative

The implementation of the combination of Vygotzky’s theorem of the zone of proximal development with the Singapore math program presents significant benefits in creating the critical thinking skills and developing the required cognitive functions for understanding complex mathematical concepts. This novel concept of combining these two powerful methods proved to be significantly more potent in elevating student’s levels in mathematics as well as projecting on other fields that require critical thinking skills. The implementation included an individualized program based on each student’s zone of proximal development, an on-going dynamic assessment based on the student’s learning potential rather than their immediate knowledge, and hands-on multi-disciplinary projects throughout the year.

In the past year I have been very successfully implementing a novel concept that I have come up with in classes in our school from 3^{rd} grade all the way to 8^{th} grade. The method has also been implemented by other teachers in our school for grades K-2^{nd}. I am a math teacher and serve as the math coordinator for Yavneh day School. I would like to tell you about the implementation of Vygotzky’s theorem of the zone of proximal development in combination with the Singapore Math program.

One of the major theoretical precursors of the current dynamic assessment (DA) approaches is Vygotsky’s notion of the zone of proximal development (ZPD). ZPD appears in Vygotsky’s texts in three different albeit related contexts. In the context of his developmental theory Vygotsky posed a question regarding those emerging but still “invisible” psychological functions that pave the way to child’s transition to a new developmental period. The process of their emergence is neither “natural” nor automatic. In the absence of certain sociocultural and learning conditions the maturing function may fail to play its designated role in the next period of development. ZPD is conceptualized in this context as those functions that need to be formed at a present moment so that the transition to a new developmental period could take place. Vygotsky’s answer to the question of how to make these functions visible follows his sociocultural theory that asserts that psychological functions of the child first appear in interactions with others and only later are internalized as inner mental functions. Thus the emerging functions can be detected if children are evaluated under conditions of joint activity with adults or more advanced peers.

The second context in which Vygotsky mentions ZPD is his critique of the static IQ tests. Vygotsky criticized IQ tests for their rigid focus on current performance level and their neglect of the learning potential of the students. He suggested to use the difference between the students’ performance under conditions of assisted problem solving and independent problem solving as a measure of their learning potential that coincides with their ZPD.

The third context connects the notion of ZPD with interaction between students’ spontaneous everyday concepts and systematic academic concepts presented to them in the school. Vygotsky identified ZPD as such a conceptual space where a dialogue between students’ spontaneous and teachers’ academic concepts can take place. If academic concepts are beyond students’ ZPD, the dialogue will be unsuccessful. If they are to close to everyday concepts they add little to students’ conceptual development. Vygotsky suggests that teaching should be done exactly within the students’ ZPD thus using their existent notions but simultaneously advancing and transforming them into systematic academic concepts.

The Singapore Math program focuses on the training and mastering of cognitive functions associated with mathematical and other critical thinking. The method involves hands-on conceptualization of various subjects in mathematics, advanced modeling and visualization mechanisms, and abstraction of mathematical problem solving. Students are initially engaged in an experience taken from real-life, tangible problem. Students, then, create a model that simplify the representation of the problem to enable use of already mastered skills. Eventually, students challenge themselves with a more abstract form of the problem used to generalize the desired cognitive function.

Using these descriptions of Vygotsky’s theorem and the Singapore math program in is quite clear to see why these would be a good match. The Singapore math program is inherently built to adapt to a student’s ZPD. It uses already mastered skills to build on new cognitive functions. It uses everyday problems to engage the students, increase motivation and situate the student in a comfortable thinking process within their ZPD. The visualization and modeling representation allow meta-cognitive processes to occur in the student’s mind without requiring them to consciously reflect or their method, skills and thought mechanisms. The abstraction then enables generalization and thus transference of complex mathematical concepts beyond the student’s initial ZDP.

My implementation of the combination of the Singapore math program with Vygotzky’s theorem manifests in multiple aspects of the learning process:

– An individualized program based on each student’s ZPD – each student is encouraged to work in their own pace based on an on-going dynamic assessment to evaluate their ZPD. Tasks are tailored for each student from a pool of tasks designed to create an incremental progress in the development of cognitive functions. Tasks are built to encourage deep understanding and mastery of critical thinking skills.

– on-going dynamic assessment based on the student’s learning potential rather than their immediate knowledge – students are assessed based on their learning skills rather then their knowledge of the subject. This enables each individual to work in their own ZPD which in turn increases their learning capability.

– hands-on projects throughout the year – Students are engaged in multiple motivation enhancing projects throughout the year. We tend to keep the projects to be on a Jewish nature (e.g. building a Menorah, or dressing up for Purim). These projects use hands-on experiences such as building, designing, art, engineering etc. A few examples are: Math Detective Investigation – children dressed up as detectives and were given a series of clues of mathematical nature. Clues led students on a path to discover a suspect in a police investigation. Students worked in pairs or groups to enhance the mediation based learning opportunity

– Work in collaborative mediation – many tasks are designed to be solved in groups by collaboration. This type of learning enables each student to completely learn in their ZPD while building on their own and their peers’ cognitive functions.

– work in stations – work is done in station to enable each student to experience different forms of learning to increase ZPD expansion efficiency.

I have been implementing this novel concept for over a year and it is extremely successful. Today , as opposed to a few years ago, many of the students attending Yavneh Day School are above grade level in math. Struggling students before have now been able to effectively learn advanced mathematical concepts, internalize critical thinking skills, master cognitive functions and improve learning strategies.

I am very proud of my students and believe that this method should be adopted by all math teachers as it tailors to every student’s needs.

Learning Fractions

Individualized learning experience

Collaborative hands-on learning of mathematical concepts (Geometry)

Weekly challenge problems

Math Detective Investigation – Purim costumes

Collaborative learning

Work Stations

Experimentation

Building a Menorah

Abstraction, generalization and strategies

## Entrant Bio(s)

Danielle was born and raised in Israel. She received her Bachelor's degree in Mathematic Education from the Levinski College in Tel-Aviv. She received her Masters in Cognitive Sciences, Summa Cum Laude from the Hebrew University in Jerusalem. She has over 15 years of teaching experience, teaching math for grades K-12. She constantly strives to improve methods for teaching math. She started working at Yavneh Day School in 2014 and serves as the School Math Coordinator since 2015.

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