This action research began with the definition of the problem to be studied and the formulation of the research questions. The problem statement focused on the analysis of the extent to which an authentic Montessori mathematics environment is compatible with the regular mathematics classes in order to fulfill the needs of all students. In this regard, the research involved comparing the Montessori system with the conventional education system in order to identify the model that best suits learners’ needs.
Several methods were used to collect data concerning learning activities in the class. The methods include the following. First, I used the results of formative assessments to collect data concerning students’ performance. The assessment results were retrieved from mathematics worksheets and students’ homework. Second, formal and informal observations were used to collect data. This method was used to collect data concerning students’ participation in class activities, as well as, the suitability of my instructional methods. In particular, videotaping was used to observe my performance in various lessons. Reviewing my teaching and student interactions during lessons helped me to evaluate my teaching strategies.
Furthermore, the aim of these observations was to enable me to identify the strengths and weaknesses of my instructional methods. The observations concerning students’ performance were recorded in teaching journals. Consequently, it was possible to evaluate students’ performance and the suitability of my instructional methods. Third, focus group discussions were used to collect data from the participants. The participants in these discussions included teachers and students. Through the focus group discussions, I was able to collect data concerning students’ knowledge, opinions, and class experiences. Concisely, the discussions gave me the opportunity to collect data concerning the instructional methods that the students were familiar with, as well as, the challenges that they faced in the process of learning. Moreover, the discussions enabled me to obtain feedback concerning my teaching methods.
The data collection methods that have been described in the foregoing paragraphs were used to collect information through learning activities that took place in the class. Verbally administered questionnaires and interviews were also used to collect data. Concisely, the questionnaires and the interviews were used in the evaluation of students’ understanding and performance in each activity. This involved asking students questions that were related to the learning activities. The learning activities or the lessons that have been completed and from which data has been collected include the following. The first exercise involved evaluating the students’ knowledge and skills in California mathematics standards 1.1 through 1.3 and Montessori mathematics curriculum pertaining to number sense. The California standard test assessments focused on skills such as comparing two or more sets of objects (up to ten and then up to 100 in each group); identifying equal sets; and recognizing greater than and less than number concepts. The evaluations for the California standards and the publisher-recommended assessments were conducted through written tests. Each student’s enduring understanding of the lessons was measured using a number grade, percentage, and a rubric (see Appendixes B and F). I applied the same rubrics to measure each student’s learning skill in all the Montessori curriculum lessons that pertain to California standard 1.1 through 1.3.
I presented Montessori lessons such as using number rods, spindle box counting, memory game, short bead stairs, the decimal system, the snake game, as well as, the teen and ten board quantities in order to compare learning outcomes in my class. Students were divided into groups in order to make the lessons enjoyable. Each group completed a reflective assessment after every major lesson (see Appendix C). Through the reflective assessment, I was able to gather more information about each student’s leaning experiences. By analyzing such information, I was able to reevaluate my teaching methods.
Having tested the students’ math skills, I proceeded to the next lesson, which involved teaching and evaluating the learners’ ability to perform simple additions and subtractions. The students’ ability to apply the concept of addition and subtraction was tested using California state standard curriculum 2.1 content, which employ colorful illustrations to attract students’ attention (McDonelle, Tourneau, Ackler, & Ford, 2001, pp. 10-150). The Montessori addition and subtraction materials that I presented to the students included golden beads, stamp game, short bead frame, strip boards, and snake game. All Montessori lessons were evaluated on an on- going basis. I used appendix B for formal assessments and appendix C for formative assessments.
Having tested the students’ math skills, I proceeded to the next exercise, which involved evaluating their sensorial skills. The California state standard 1.1 and the Montessori sensorial curriculum provide many opportunities for students to explore and sharpen their algebraic skills (Franker, 2007). I used textbook lessons to supplement the students’ homework. The formative evaluation was done with the aid of Montessori sensorial items such as flowers, as well as, red and blue rods. During the assessments, the students were expected to classify the flowers into different categories. The evaluation for this exercise was done through verbally administered questionnaires. Every questionnaire had ten questions with 10 points each. The students were assessed individually using questions, which focused on testing their ability to compare numbers between 1 and 10. Numeral counters were used to demonstrate the concept of odd and even numbers. The evaluation involved asking the students to place the counters under each number in order to illustrate the odd and even arrangement, as well as, to demonstrate the concept of less than or more than (see Appendix D).
The second phase of classifying objects in California state standard (statistics, data, and probability) involved the use of drawings, photographs, graphic organizers, picture graphs, and simple patterns. It also involved extending simple patterns through size and color variations. The lessons in the state recommended textbook helped the students to make connections to their environment. This was important because students are often motivated to learn with enthusiasm if they are able to relate lessons to their daily activities (Har, Segaran, & Aziz, 2008, pp. 8-50). The assessment involved a problem solving process in which the students asked themselves questions regarding the things they did and how they did them. The students reviewed their performance on a weekly basis (see Appendix C). Montessori sensorial activities such as constructing triangles and practical life – patterning and detecting triangles game helped the students to identify and to collect information about objects and events in their environment.
Initial lessons on measurements and geometry were taught within the California state standards 1.1 to 2.2. The state standards curriculum included the concepts of time, length, weight, capacity, and common geometrical shapes. After completing each unit, I applied the publisher recommended test units. Based on each child’s performance, I reevaluated my teaching strategies. Montessori hands-on materials and teacher generated circle activities, which were used to teach time, calendar, and clockwork, supplemented the textbook for California standard 1.2 (measurement and geometry). The students’ knowledge improved because they were able to identify and to describe geometric features using geometric solids and the geometric cabinet. Throughout these Montessori lessons, I used on-going assessments to measure learning outcomes (see Appendix E).
The fraction lesson began with an explanation of the importance of studying fractions. The California state standard (1.2) textbook on measurements focuses solely on line of symmetry. The lessons were brief and very abstract. I supplemented the students’ homework packet textbook pages as required by the school’s guidelines. I provided students with pieces of paper, which they were required to fold into half in order to understand the concept of dividing things equally. The use of hands-on materials such as paper often makes lessons more attractive (Rigg, 2000, pp. 3-45). The students were also expected to identify the shapes that they created by folding the papers. Additionally, they completed the dotted paper and shape-drawing exercises, as well as, dividing shapes equally. After the key introduction, I focused more on presenting the hands-on Montessori materials and supplemented the homework packet with the publisher-recommended texts. Using Montessori material involved dividing an apple into smaller parts in order to demonstrate the concept of the whole, half and quarter. I also made various shapes and divided them into equal parts. The students were expected to explain why they thought the parts were equal or unequal. The evaluation for this assignment was based on a standard formative matrix rubric (see Appendix E).
Using money to teach number concepts improves students’ math skills (California State Board of Education Sacramento, 1999, pp. 2-200). The money lesson was meant to enable students to use their math skills to count coins. The concept of one cent was presented with the aid of Montessori red and blue number rods, as well as, number cards. The classroom was setup as a shop so that children could practice the concept of buying. 1 dollar, half a dollar, and a quarter dollar were used to represent the concept of one whole, half and quarter. Assessments were done using scoring rubrics. Additionally, anecdotal evidence was measured using a scale of 1 to 4 (see Appendix E).
Having completed the state required standards and the textbook lessons, I focused on involving students so that they could learn beyond the state mandated curriculum. I introduced Montessori concepts such as the decimal system, addition, subtraction, division, and advanced geometry. The materials that were used to demonstrate the decimal system included introduction tray, building tray, large numerical cards, 45-tray, nine-tray, and the change game. The concept of addition was illustrated using bank materials, stamp game materials, and bead frame. These materials were used by students to create and solve mathematical problems. Subtraction exercises also involved the use of bank materials, stamp materials, and bead frame.
These materials were used to construct subtraction problems and answers. Multiplication exercises were presented with the aid of bank game, multiplication board, and bead box. Bank materials were used to illustrate multiplication operations. The concept of division was illustrated with the aid of bank game, division board, stamp game, bead frame and dot board. Using Bank materials was helpful to illustrate division operations. In order to understand the use of abstract methods to memorize the division facts, the students used their fingers to slide until they met the answers in the division board. Finally, geometry exercises were demonstrated using cylinder grid, geometric solids, and cards, as well as, constructing triangles and other shapes. Assessments were done with the aid of scoring rubrics, reflective assessments, observations and comments made by my fellow teachers (see Appendixes C, E, G, and H).
The collected data was prepared for analysis by ensuring that it was complete and accurate. Preparing quantitative data involved summarizing them with the aid of tables, graphs, and spreadsheets. The preparation of qualitative data, on the other hand, involved summarizing the participants’ responses and classifying them under common themes. These preparation procedures were also used for the data that has already been analyzed. The next section presents the results of the analyses that have already been done. Concisely, it presents the findings and conclusions concerning the extent to which an authentic Montessori learning environment improves students’ skills and knowledge in mathematics.
California State Board of Education Sacramento. (1999). Mathematcis Content Standard for California Public Schools. California: California State Board of Education.
Franker, K. (2007). Primary Grade Self-Evaluation Teamwork Rubric. Web.
Har, Y., Segaran, N., & Aziz, D. (2008). Early Bird Kindergarten Mathematics. New York: McGraw-Hill.
McDonelle, R., Tourneau, C., Ackler, K., & Ford, E. (2001). Progess in Mathematics. New York: McGraw-Hill.
Rigg, P. (2000). American Montessori 6-9 Mathematics Journal. San Leandro: Montssori Teacher.
Kindergarten Rubric for Number Sense
Enduring Understanding: The students will understand numbers and multiple ways of representing them
|Can count or draw a collection of specified amount of or specified amount up to and beyond 20||Can count out or draw a specified amount up to 20||Sometimes can count out or draw a specified amount with support||Needs more time to develop this skill|
|Verbally counts beyond 100 by 1’s and 10’s||Verbally counts by 1’s and 10’s up to 100||Verbally counts to 100 by 1’s and 10’s with support||Needs more time to develop this skill|
|Can write the number using proper formation legibly beyond 20||Can write the numbers 1to 20 using numbers formation legibly||Can write the numbers 1 to 20 using correct formation legibly with verbal support and a model||Needs more time to develop this skill|
|Can sequence numbers beyond 20 and determine more or less||Can sequence numbers up to 20 and determine more or less||Can sequence numbers up to 20 and determine more or less with support||Needs more time to develop this skill|
|Counts forward from any given number beyond 100||Count forward from any given number up to 100||Counts forward from any given number up to 100 with support (teacher or 100’s board)||Needs more time to develop this skill|
Statistics and Probability/ Sorting Assessment Rubric
Student Name —————————————
|Can you organize these objects by their color?||10 Points|
|Can you organize these objects by size?||10 Points|
|Can you organize these objects by shape?||10 Points|
|Which one of this is longer?||10 Points|
|Which one of this is shorter?||10 Points|
|Which group has more?||10 Points|
|Which group has less?||10 Points|
|Can you show me the even number group||10 Points|
|Can you show me the odd number group||10 Points|
|Can you show me the groups that have the same numbers?||10 Points|
State Standard Benchmark————————————–
|Knowledge: tell, repeat, name, relate||1 2 3 4|
|Comprehension: retell, show, describe||1 2 3 4|
|Application: convert, display, incorporate||1 2 3 4|
|Analysis: distinguish, question, categorize||1 2 3 4|
|Evaluation: judge, verify, choose, forecast, estimate||1 2 3 4|
Compose, create, produce, transform
|1 2 3 4|
|Materials||Test Strategy and questions|
|Positive Snake Game||Can the child make longer quantities using bead? (25points) |
Can the child count to 100 by 1’s? (25 points)
Can the child represent additions with the beads? (25 points)
Can the child represent subtractions with the beads? (25 points)
|The Hundred Board||Can the child place number tiles to show the order to 100? (25 points) |
Can the child count to 100 by 1’s? (25 points)
Can the child count to 100 by 10’s? (25 points)
Can the child count objects up to 100 correctly?
|The Hundred Chains||Can the child count beads and written numerals placed at 1- 10th? (25 points) |
Can the child count to 100 by 10’s’? (25 points)
Can the child correctly point up to 20? (25 points)
Does the child know that the last number name is the number of objects counted? (25 points)
|Teen Board (quantity, symbol, association)||Can the child add to base ten numerals to compose the teen numbers? (26 points) |
Can the child combine bead bars, 1- 9 to show teen quantities? (25 points)
Can the child count forward other than one? (25 points)
Can the child represent counters and numbers of object for written numbers 0 – 20? (25 points)
|Teen Board||Can the child use and show the quantities of 10 – 90? (25 points) |
Is the child able to count to 100 by 10’s? (25 points)
Can the child count forward from a number other than one? (25 points)
Regardless of arrangement, can the child identify any of the numbers? (25 points)
Enduring Understanding: Students will understand and be able to
- Describe characteristics and relationships of geometric objects
- Explore vertical and horizontal orientation of objects
- Manipulate two- and three dimensional shapes to explore symmetry
|Can independently name and identify additional shapes – example: rhombus, and trapezoid)||Can independently name and identify circle, square, triangle, rectangle and hexagon||Needs some support in identifying shapes||Needs more time to develop this skill|
|Student understand and can demonstrate special relationships between these additional shapes||Student understand and can demonstrate special relationship||Student has some understanding of special relationships||Needs more time to develop this skill|
Student Self- Evaluation Teamwork Rubric
My Team Work for the Day ———————————–
|As a team member I:||As a team member I:||As a team member I:|
|———–Let my partner do all of my work |
———-Did not help my partner
———-Did not listen to my partners’ ideas
———–Did not share my ideas
————Did not help the group solve problems
|————Let my partners do some of my work |
————–Only helped my partners when they asked me
————–Had trouble quietly listening to ideas
————-Shared no ideas
————-Waited for my group to solve most problems
|———–Did all my work |
———–Helped my partners
————Listened to my partners’ ideas
————Shared my ideas
————Helped my group solve problems