A Balance of Active and Teacher-Directed Learning in Mathematics: Grounded Theory to Inform Kindergarten Curriculum Design

A Dissertation Proposal

 Introduction

Early mathematics education is an important concern in discussions of curriculum, especially given the fact that the United States is ranked lower than several other developed nations in the areas of mathematics and science (DeSilver, 2015).  Some researchers have suggested the need for more intensive mathematics curriculum in Kindergarten (Chard et al., 2008) because early learning experiences are considered among the most significant factors determining later educational performance (de Haan, Elbers, & Leseman, 2014). It seems appropriate for teachers to focus on maximizing student’s potential by directing them so they can build a strong foundation for mathematics education they will receive in grades K-12.

However, when educators use a teacher-managed approach and direct the learning process they may take away some of the student’s ability to take active roles in their learning processes; active learning boosts motivation and improves outcomes (Antonetti & Garver, 2015), so there seems to be a trade-off between the necessary direction from teachers and the active engagement of students. Early educators must reconcile their interest in promoting lifelong learning with their interest in promoting equal educational opportunity by using a directive approach that attends to the areas in which students need the most improvement but also maximizes the opportunity for student-directed, active learning. This project is intended to generate theory grounded in self-reported data from experienced early educations, which can be used to inform curriculum design for early mathematics education.   

Background of the Study

Research by Haan et al (2014) suggests that mathematics activities are best directed by a teacher, and the reason may be that they are more abstract and perhaps less relevant to the interests of early learners, so in this area, teacher-managed activities are more appropriate. This would be consistent with Vygotsky’s theory of the Zone of Proximal Development (ZPD). Vygotsky rejected Piaget’s approach and argued that mental development in children comes from interaction and collaboration with more mature people who can stimulate their intellectual development (Morgan, 1999). The work of de Haan and colleagues, although they did not present it as such, can be understood as a way to reconcile and balance the conflicting approaches of Piaget and Vygotsky.

            What implications does such a balance have for curriculum design in early education? In the example discussed above, the different approaches to mathematics and language arts demonstrate a principle of effective practice which could be discussed concerning Vygotsky’s ZPD for use with preschool and kindergarten students. The findings by de Haan and colleagues (2014) suggest that more scaffolding is necessary for mathematics activities than for language activities.  Going beyond the insight that scaffolding is necessary to help students move toward the ability to apply their skills independently, an important point to notice may be that students can make greater gains in all areas of competency if teachers are vigilant about transferring responsibility to them so they can indeed work independently as soon as they can do so.  If a teacher continues to manage activities after students are already able to work independently, it may be detrimental to their intrinsic motivation and the ideal of lifelong learning.  

            The use of a child-managed approach is conducive to active learning, which has many benefits for early learners. Yet, using a more teacher-centred approach, other recent studies have led researchers to conclusions involving reliance on the theory of the Zone of Proximal Development (ZPD), and the issues they raise are useful for informing the design of the present study. In one case, ZPD became central to the recommendations of researchers who were concerned with learning the best ways to even the playing field for students of culturally and economically diverse backgrounds (Sylvester & Kragler, 2013). Working to promote the ideal of equal educational opportunity, Sylvester and Kragler observed classroom activities with a focus on identifying areas where early education can correct misconceptions that give rise to unequal circumstances. 

            Two main insights are expressed in the results, and they are interesting to the present study.  The first is that early learners need culturally relevant content in their early learning experiences; the notion of cultural relevance being of particular importance is consistent with Piaget’s conception of active learning.  The second is the importance of situating that culturally relevant content within each student’s Zone of Proximal Development; this is consistent with Vygotsky’s more directive approach.  The fact that Sylvester and Kragler (2013) reached a conclusion that consists of these two concepts supports the premise of the present study, which involves an approach to fine-tuning early education with a construct based on the a spectrum with the seemingly contradictory approaches adopted by Piaget and Vygotsky at each end. 

            On the other hand, although this freedom is the ideal for constructive, active learning, it is also true that teachers can direct what students do with that freedom, slightly prompting them in the direction of meeting the early learning standards.  In this regard, and in keeping with the use of Vygotsky and Piaget to structure the discussion, this balance of freedom and direction is reflected in play-based instruction.  Although play seems more aligned with Piaget’s notion of children making their meanings, it is also discussed by Vygotsky who was enthusiastic about play as an opportunity for intervention and direction.  Symbolic play creates an imaginary situation in which children can gain important skills, such as sharing toys. Vygotsky observed that during play children always seem to act noticeably more mature than they act under ordinary circumstances.

            Another important concept for consideration when creating such a model is “executive function.” This term refers to the self-regulation that enables students to interact properly with others and achieve positive academic outcomes. Play can be combined with the theory of the Zone of Proximal Development (ZPD) to effectively enhance executive function among early learners (Berk & Meyers, 2013).  In their discussion, Berk and Meyers emphasize the importance of facilitating play but not controlling it. Their insight is useful for guiding educators as they balance child-managed learning experiences with the scaffolding and guidance that can lead to improved academic outcomes, as long as they are done with the kind of finesse that does not inhibit student’s ability to control the learning experience. 

            The importance of maximizing the extent to which students can control their learning processes is also emphasized by Cubukcu (2012), who conducted a research study focused on teachers’ evaluations of the student-managed learning that took place in their classrooms.  Cubukcu notes that participants ranked the psychosocial dimension of student-directed learning the highest, and that they expressed the importance of giving students as much autonomy as possible.   

            A recent study from educational researchers in the Netherlands focused on assessing the extent to which preschool and kindergarten students benefited from teacher-managed and child-managed classroom activities. De Haan, Elbers, & Leseman (2014) were inspired by other research which shows that competencies among early learners are strong predictors of their outcomes in subsequent years, so they designed their project to examine the effects of two different approaches on these competencies. Among their findings was the observation that language development and literacy were improved more by child-managed activities, whereas teacher-managed mathematics activities were more effective for improving mathematics skills.

            The results of this study are a source of great insight into the way to refine the early learning curriculum and improve outcomes. Student-managed language activities may be more effective. They are better prepared to manage such activities when they begin their formal education because they have been developing language skill through an ongoing process of what might be viewed as problem-based learning. This notion is consistent with the aspect of Piaget’s theory, as explained by Morgan (1999), affirms that children, “through their egocentrism, are the primary agents in enabling themselves to make their meaning” (p. 216). They are accustomed to engaging in authentic learning experiences using language to solve the “problem” of satisfying their curiosity or expressing themselves. 

Problem Statement

            Early learning competencies are important predictors of future academic outcomes (de Haan, Elbers, & Leseman, 2014). In the past, some researchers have attempted to utilize this insight by emphasizing the need for a more rigorous curriculum in early learning classrooms (Chard et al., 2008). Yet, a more directive, the teacher-managed approach may be detrimental to motivation, as researchers suggest that it is important to allow students to control their learning experiences as much as possible (Berk & Meyers, 2013; Cubukcu, 2012).  This is consistent with the idea of active learning, rooted in the work of Piaget. To improve learning outcomes throughout a child’s education, early educators need to find the subtle balance between active learning and directed learning. 

            This challenge is urgent for all aspects of early education, but this project focuses on mathematics activities because mathematics and science are areas in which American education is in particular need of improvement.  Educators see a great deal of room for improvement especially in the areas of mathematics and science.  DeSilver (2015) of the Pew Research Center writes; “While U.S. students are scoring higher on national mathematics assessments than they did two decades ago (data from science tests are sketchier), they still rank around the middle of the pack in international comparisons, and behind many other advanced industrial nations” (paragraph 3). The problem addressed by this paper is a need for a research-based conceptual model to help early educators make decisions about how to most effectively balance the need for direction (which requires students to be passive as they follow directions) with the need for students to take an active role in their learning activities related to mathematics. 

Purpose of the Study

            To contribute to the efforts associated with improving mathematics education in American schools, the author of this paper hopes to generate theory that can inform early education curriculum design and is grounded in self-reported data from experienced teachers. Specifically, this study is intended to provide insight into how teachers can give early learners the direction they need to have a strong foundation for future study of mathematics while simultaneously leaving students as free as possible to direct their learning processes. Through the use of open-ended questions about early mathematics instruction, it is possible to learn about the activities and concepts most useful for achieving an optimal balance of direction by the teacher and active engagement by the student.   

Research Questions

What concepts do experienced kindergarten teachers consider to be the most effective for inspiring curiosity about mathematics concepts?

What activities do experienced kindergarten teachers consider to be the most effective for achieving common core standards while maintaining high levels of interest and active participation?

Theoretical Basis

            The theories of Piaget and Vygotsky are both celebrated among educational theorists, and the present study is situated directly on the point at which they seem to emphasize different values.  On one hand, Piaget expresses the importance of the learner as the main actor in her or his education.  While this is certainly a profound insight at the heart of the constructivist theory, it could also be argued that the teacher is the main actor in the process of instruction from her/his point of view.  The emphasis of the teacher’s direction is aligned with Vygotsky’s theory of the Zone of Proximal Development (ZPD).  Although early educators must stay mindful of letting students take an active role whenever possible, research suggests that mathematics is an area in which a more teacher-managed approach might be important (de Haan et al 2014). This is accordant with Vygotsky’s ZPD theory of the Zone of Proximal Development (ZPD). These two extremes – focus on the action of the teacher, and focus on the active role of the learner – represent two ends of a spectrum that forms the theoretical basis for this project. 

Definition of Terms

Active learning: Refers to “a process whereby students engage in activities, such as reading, writing, discussion, or problem-solving that promote analysis, synthesis, and evaluation of class content” (Center for Research on Learning and Teaching, 2015).

Curriculum: A specific set of instructional materials that order content used to support pre-K–grade 12 classroom instruction (Clements, 2007).

Zone of Proximal Development: “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem-solving under adult guidance or in collaboration with more capable peers” (Vygotsky, 1978, p. 86).

Assumptions and Limitations

            The design of this research study is based on an assumption that participants will share their most useful insights in ways that are accurate and forthcoming. It is also assumed that their ideas about concepts and activities most effective for maintaining active learning while giving the necessary direction will be generalizable enough that they are useful for other early educators.

            This study is limited by the constraints that are common to all research involving self-reported data that must also be interpreted by the researcher.  For example, reporting bias by the subjects may result from a discrepancy between the methods they find to be most effective and the methods they have in mind but have not necessarily put to the test.  Additionally, researcher bias has the potential to distort the results; if the researcher approaches the analysis of data without perfect objectivity, preconceived ideas may cause misunderstanding.  

            Importantly, the results are limited by the fact that the concepts and activities considered by teachers to be most useful will reflect the circumstances in their classrooms – the interests and cultures of the particular students with whom those concepts and activities have been used. 

Significance of the study

            This study is important because of its potential to make a meaningful contribution to curriculum research. In light of Clements’ (2007) ideal of research-based curriculum, it is important to inquire with experienced teachers to learn about the specific concepts and activities that have proven to be most effective for maintaining motivation while also giving the necessary direction during kindergarten mathematics lessons. By using grounded theory to systematically analyze self-reported data from experienced teachers, it is possible to identify their best insights for informing decisions about curriculum design.  

Background on the Proposal and Design

            Grounded theory is “a qualitative strategy of inquiry in which the researcher derives a general, abstract theory of process, action, or interaction grounded in the views of participants in a study” (Creswell, 2009, p. 13). This method is particularly useful in the present study because it involves a methodical analysis of the key concepts and themes that emerge in self-reported data. In pursuit of research-based curriculum, it is necessary to use a methodical approach when analysing qualitative data so that the most important insights will be accurately reflected in the research findings.           

Research Design and Approach

            The research design is quantitative, and the process will include the use of a survey distributed to experienced kindergarten teachers. The following ten items consist of mathematics concepts and activities, and participating teachers will rank each item according to interesting and motivational they are to the children. Although teachers are interested in making all aspects of the curriculum interesting and motivational, they are being asked to express their ideas about which among the common standards are inherently most motivational. Insight about their relative levels of interest/motivation to students will inform instructional practice.  

            The activities and concepts listed below are taken from the Massachusetts Curriculum Framework for Mathematics Grades Pre-Kindergarten to 12, incorporating the Common Core State Standards for Mathematics. For each of the following concepts and activities, participants are asked to select a rating from 1 to 3.  The number 1 represents the least motivational, and the number 3 represents the most motivational. 

Counting and Cardinality

  1. Know number names and the count sequence.
  2. Count to tell the number of objects.
  3. Compare numbers.

Operations and Algebraic Thinking

  1. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Number and Operations in Base Ten

  1. Work with numbers 11–19 to gain foundations for place value.

Measurement and Data

  1. Describe and compare measurable attributes
  2. Classify objects and count the number of objects in each category.

Geometry

  1. Identify and describe shapes (squares,circles, triangles, rectangles, hexagons,cubes, cones, cylinders, and spheres).
  2. Identify relative positions of objects in space.  
  3. Analyze, compare, create, and compose shapes.

Setting and Sample

            The researcher will use social media, especially Facebook Groups, to find kindergarten teachers willing to respond to the questions.  It is hoped that response rates will be high because only two questions are used, leaving respondents plenty of time to participate and elaborate on their answers. The participant criteria include only two requirements: Participants must have at least five years of experience teaching at the kindergarten level, and they must feel confident that they have useful insights to contribute about keeping students motivated while providing the necessary direction in early mathematics instruction.  Convenience sampling will be used so that the data collection instrument can be distributed to as many teachers as possible for maximizing response rates.  

Data Collection and Analysis

            Data collection will consist of distributing the qualitative questionnaire via email and social media.  Facebook groups are particularly useful for connecting with a large number of kindergarten teachers who can contribute valuable ideas about the most useful mathematics concepts and activities for optimizing motivation among early learners and enabling teachers to teach the standards while also enabling students to stay actively engaged in the learning process.   

            Data analysis will consist of the constant comparative method from grounded theory (Strauss and Corbin, 1998). This method involves three stages of analysis, including open coding, axial coding, and selective coding.  Respectively, these states require a line-by-line review of the qualitative data applying labels (codes) to identify key concepts; the formation of categories based on these concepts; and systematic relating of emergent categories with one “core” category identified as most significant for the research study.  This method will lead to a theory grounded in real data from experienced teachers, and it can be used to inform decisions about curriculum design. 

Data Analysis – Chapter 4

Question 1

Teacher-Directed Learning in Mathematics
What is your gender?
Answer OptionsResponse PercentResponse Count
Female81.8%9
Male18.2%2
answered question11
skipped question1
ETD Tutors Dissertation Proposal 1

A large number of kindergarten teachers in the United States consist of females. This is explained by the high number of female respondents as compared to their male counterparts.

Question 2

Teacher-Directed Learning in Mathematics
What is your age?
Answer OptionsResponse PercentResponse Count
18 to 246.7%1
25 to 3413.3%2
35 to 4433.3%5
45 to 5433.3%5
55 to 6413.3%2
65 to 740.0%0
75 or older0.0%0
answered question15
skipped question1
ETD Tutors Dissertation Proposal 2

The above results indicate that a high number of respondents came from the ages of between

35 – 55. Majorly, only this portion contributed valuable ideas for this study, which will be used in developing useful mathematics concepts and activities for optimizing motivation among early learners and enabling teachers to teach the standards while also enabling students to stay actively engaged in the learning process.   

Question 3

Teacher-Directed Learning in Mathematics
What is your approximate average household income?
Answer OptionsResponse PercentResponse Count
$0-$24,9990.0%0
$25,000-$49,99926.7%4
$50,000-$74,99926.7%4
$75,000-$99,99926.7%4
$100,000-$124,9990.0%0
$125,000-$149,9996.7%1
$150,000-$174,9990.0%0
$175,000-$199,99913.3%2
$200,000 and up0.0%0
answered question15
skipped question1
ETD Tutors Dissertation Proposal 3

The level of household income, as a factor, came close to explaining the variation in teacher response. Following this assumption, we can as well deduce that the level of income is also a key contributing factor for performance.

                Question 4 Teacher Directed Learning in Mathematics What is your ethnicity? (Please select all that apply.) Answer Options Response Percent Response Count American Indian or Alaskan Native 0.0% 0 Asian or Pacific Islander 0.0% 0 Black or African American 86.7% 13 Hispanic or Latino 20.0% 3 White / Caucasian 0.0% 0 Prefer not to answer 0.0% 0 Other (please specify) 6.7% 1 answered question 15 skipped question 1

The above data presents that over 96% of kindergarten teachers in America consist of black/African Americans and Hispanics/Latinos.  This is mainly attributed by the low level of income for kindergarten teachers due to education budget cuts at local and state levels.

Question 5

Teacher Directed Learning in Mathematics Counting and Cardinality Answer Options 1 2 3 Rating Average Response Count Know number names and the count sequence. 5 1 3 1.78 9 Count to tell the number of objects 2 7 2 2.00 11 Compare numbers 4 2 8 2.29 14 answered question 16 skipped question 0      
0
In counting and cardinality, the concept of comparing numbers seems to be more popular among teachers in promoting active learning, which is considered to have many benefits for early learners. In the whole, we can say that its classroom activities like this which help in identifying areas where early education can correct the misconceptions that give rise to unequal circumstances. 

Question 6

Teacher Directed Learning in Mathematics
Operations and Algebraic Thinking
Answer Options12Rating AverageResponse Count
Understands addition as putting together and adding to451.569
Understands subtraction as taking apart or taking from681.5714
answered question16
skipped question0
ETD Tutors Dissertation Proposal 3

In operations and algebraic thinking generally, understanding subtraction is harder for children to learn than addition. The above data proposes the adoption of strong subtraction skills among teachers, which they can pass on to the children since subtraction facts could pose a big challenge.

Question 7

Teacher Directed Learning in Mathematics
Number and Operations in Base Ten
Answer Options12Rating AverageResponse Count
Works with numbers 1- 10 to gain foundation for place value441.508
Works with numbers 10 -20 to gain foundation for place value671.5413
answered question14
skipped question2
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Most of the above respondents preferred working with numbers 10 – 20 to gain a foundation for value. Numbers in the base often are more kinder-friendly; since kindergarten kids are generally interested in learning big numbers.

Question 8

Teacher Directed Learning in Mathematics
Measurement and Data
Answer Options12Rating AverageResponse Count
Describe and compare measurable attributes371.7010
Classify objects and count the number of objects in each category771.5014
answered question15
skipped question1
ETD Tutors Dissertation Proposal 5

Through classifying objects and counting the number of objects in each category, children begin to understand how to devise questions that can be answered using data. This helps them in developing skills for statistical understanding at an early age.

Question 9

Teacher Directed Learning in Mathematics
Geometry
Answer Options123Rating AverageResponse Count
Identifies and describe shapes (squares, circles, triangles, rectangles, and cubes).5131.789
Identifies relative positions of objects in space.1562.4212
Analyze, compare, create and compose shapes3542.0812
answered question15
skipped question1
ETD Tutors Dissertation Proposal 6

Pre-school teachers are mandated to create an environment in which kids are excited to learn and explore math. According to the above data, a huge number of respondents preferred using shapes to identify relative positions of objects in space.

Question 10

Teacher Directed Learning in Mathematics
Recognizing Numerals
Answer Options12Rating AverageResponse Count
Recognizes numbers 1 – 20551.5010
Compare/Contrasting or More/Less671.5413
answered question15
skipped question1

Recognizing numerals helps in understanding the relationship between quantities and numbers. From the above data, most respondents preferred comparing/contrasting or more/less as they are beneficial in developing a positive learning attitude, enhancing and developing physical skills, developing basic skills and concepts and lastly in enhancing communication skills.

References

Antonetti, J. V., & Garver, J. R. (2015). 17,000 Classroom Visits Can’t Be Wrong: Strategies That Engage Students, Promote Active Learning, and Boost Achievement. Alexandria, VA: ASCD.

Berk, L. E., & Meyers, A. B. (2013). The Role of Make-Believe Play in the Development of Executive Function: Status of Research and Future Directions. American Journal of Play, 6(1), 98. 

Center for Research on Learning and Teaching. (2015). Active learning. University of Michigan. Available:  http://www.crlt.umich.edu/tstrategies/tsal

Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33(2), 234-248.

Clements, D. H. (2004). Major themes and recommendations. In D. H. Clements & J. Sarama (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education. Mahwah, NJ: Erlbaum.

Clements, D. H. (2007). Curriculum research: Toward a framework for ‘research-based curricula.’ Journal for Research in Mathematics Education, 38(1), 35-70.

Creswell. (2008). Educational Research: Planning, Conducting, and Evaluating Quantitative and Qualitative Research (4th Edition). Pearson.

Creswell, J.W. (2009). Research Design: Qualitative, Quantitative, and Mixed Approaches. Thousand Oaks, CA: Sage.

Cubukcu, Z. (2012). Teachers’ Evaluation of Student-Centered Learning Environments. Education, 133(1), 49. 

De Haan, A. K., Elbers, E., & Leseman, P. P. (2014). Teacher- and child-managed academic activities in preschool and kindergarten and their influence on children’s gains in emergent academic skills. Journal of Research in Childhood Education, 28(1), 43. 

DeSilver. (2015). U.S. students improving – slowly – in math and science, but still lagging internationally. Pew Research Center. Available: http://www.pewresearch.org/fact-tank/2015/02/02/u-s-students-improving-slowly-in-math-and-science-but-still-lagging-internationally/

Massachusetts Department of Elementary and Secondary Education. (2011). Massachusetts Curriculum Framework for Mathematics Grades Pre-Kindergarten to 12. Available: http://www.doe.mass.edu/frameworks/math/0311.pdf

Morgan, H. (1999). The imagination of early childhood education. Westport, CT: Bergin & Garvey.

Piaget J. 1963. The origins of intelligence in children. New York: Nonon.

Strauss and Corbin. (1998) Basics of qualitative research: Grounded theory procedures and techniques. 2nd ed. Sage.

Sylvester, R., & Kragler, S. (2013). “Banana Starts with Yellow”: Curricular Implications of Preschoolers Missing a Concept. Childhood Education, 89(1), 19.

Vygotsky, L. S. (1978). Interaction between learning and development (M. Lopez-Morillas, Trans.). In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.), Mind in society: The development of higher psychological processes (pp. 79-91). Cambridge, MA: Harvard University Press.