Montessori Math in Preschool and Kindergarten
Fredrick L. “Rick” Silverman
Faculty of Elementary Education
School of Teacher Education
University of Northern Colorado
Greeley, CO 80639
Fredrick L. “Rick” Silverman
Surface Address: See Above
Montessori Math in Preschool and Kindergarten
About a hundred years ago, the first female Italian physician, Dr. Maria Montessori, skilled as a scientific observer, discovered how effectively young children learned when they were engaged in purposeful activity. In 2005, cognitive psychologist, Dr. Angeline Stoll Lillard, published a book titled Montessori: The Science behind the Genius. Lillard illustrates that current scientific research has supported the main ideas of Dr. Montessori’s approach to education. Lillard comments in the preface to her book that Dr. Montessori’s practical approach to education is “less ‘debunkable’ today than Piaget” (P. viii).
David Elkind (2003), scholar and professor in the field of child development, describes three epistemological perspectives that underlie Montessori education: empiricism, nativism, and constructivism. For the empiricist, knowledge is outside the learner. The human mind is a blank slate or a tabula rasa. For the nativist, the innate capacity for obtaining knowledge is entirely within the learner’s mind at birth and knowledge manifests itself during critical periods of life.
For the constructivist, knowledge arises in the interaction between the learners and the world in which they live. Hence, rich environment plays a substantively important role in knowledge acquisition. In elaborating these three positions, and particularly in comparing Montessori methods and contemporary constructivist approaches to teaching and learning, Elkind (2003) indicates that Montessori programs incorporate elements of all three perspectives, concluding that “… what children learn is socially constructed and cannot be arrived at by the child on his/her own. It is to Montessori’s credit that she recognized this and made room in her pedagogy for all three modes of learning (Elkind, 2003).”
In recent years, the popularity of Montessori education has gained momentum. Some public schools have adopted the Montessori system in the kindergarten and primary grades, sometimes supporting professional development for selected classroom teachers to implement the Montessori approach to curriculum and teaching. The purpose of this article is to introduce Montessori math education for children 3-6 years of age. The authors do not intend to convert every early childhood teacher, or even every reader of this article, to the Montessori system, but rather to provide ideas that might add to the repertoire of the early childhood and primary grades educators who are not so familiar with the system.
A Glance at the Classroom
This section presents brief statements about young children engaged with various Montessori materials, and at the end of the article are links to images and descriptions of those materials. In this realistic account of life in a Montessori classroom or school, Sara, a three year old, just finished the Pink Tower activity-- stacking 10 pink wooden blocks from the largest to smallest. She looks at the Pink Tower in front of her with a big smile on her face. Four year old Leo is screwing a nut and bolt combo together; there are four sets of various sizes of nut and bolt combos and he has successfully screwed one set together already. His eyes are on the combos and he quietly continues his work. Four-year old Lily is working on the Positive Snake activity; six year old Jim sits next to Lily watching her intently. Jim just finished working on the Binomial Cubes and has decided to take a break. Lily is trying to replace a combination of 2 sets of beads, 7 and 3 (a total of 10), in her “snake” with a golden bead bar with 10 beads in it. Since Lily didn’t set it right in the first place, she is struggling, concentrating hard, and she still trying to accomplish the task. After a while she stops, and Jim begins to help Lily reset some of the bead combinations, and the two of them do the activity together. Four-year old Kyle is working on Classified Picture Cards. Across the room, two five-year olds, Ming and Salina, are stacking cubes each representing 1000 to complete the Forty-Five Layout activity which requires them to place numeric symbols and corresponding manipulatives side by side. These two children have repeatedly chosen the forty-five layout many times and have enjoyed the activity. While Ming and Salina are working on this activity, 3-year old Sam is watching intently. Nearby, the teacher is introducing 3-year old Henry to Geometric Solids which includes 10 wooden geometric shapes-- sphere, cone, cylinder, cube, etc. His teacher asks Henry to feel the solids. Close to the entrance to the classroom, six-year old Sue is using the shapes in the Large Hexagon Box and Small Hexagon Box to create her own geometric design. Each of the other 19 children in this Montessori kindergarten classroom in a public school is independently working on hands-on activities of their choice. It is evident that there is only one set of materials for each activity, and none of the children uses the Montessori hands-on materials frivilously. In keeping with Montessori principles, the children have learned to use the materials in the purposeful ways for which they are intended. These examples show the children in the roles of young mathematicians engaging in mathematical thinking.
Characteristics and Purpose of Montessori Education
A self-paced, choice-based, hands-on, and movement oriented approach to learning is characteristic of Montessori education. Montessori curricula consists of 5 strands: Practical Life, Sensorial, Language, Geography and Culture, and Mathematics. The Montessori method reflects its founder’s assumption that the child has the potential for self-development in a prepared environment in which the child is free to select from a variety of tasks that have first been taught by the teacher and that meet her developmental needs. The child is seen not only as capable of facilitating her own development, but also of educating herself (Orem, 1971). Eric Erickson (1963) indicated that during the preschool years, the most important task for a child is to learn to take initiative, a perspective consistent with Montessori education. Similarly, Kamii (2000) supports autonomy as a highly desirable outcome of education.
Control of error is a feature of many Montessori materials. It enables the learner to discover when errors are present in his interactions with them. The terminology “self-correcting material” is virtually equivalent in meaning to “Control of Error.” Some Montessori materials are constructed so that learners easily perceive or notice discrepancies in expected patterns when reflecting on their own work. Another type of control of error does not lie in the materials themselves; instead, it is a process that scaffolds the child through the use of physical or pictorial clues to identify errors.
Spindle Boxes, in which the child places 0-9 spindles (wood sticks) in the 10 compartments, provides an example of control of error—a self correcting material. When a child completes the work by putting 9 spindles in the last compartment and has spindles left over, the child will realize that there is a counting error to be found. The Hundred Board offers an example of control of error by scaffolding through the use of pictorial clues (Please see section titled “Introduction to numbers 1-100”). Control of error prompts self-correction and self-assessment, without relying on teachers or other adults, which leads to independence and autonomy.
Fundamental Mathematics in Montessori Materials
Montessori math is not an isolated subject; instead, it is supported by other subject areas. One-to-one correspondence, seriation, and classifying appear in the sensorial, language, and practical life curricula. The concepts of part-part-whole relationship and algebra are embedded in Binomial and Trinomial Cubes in the Sensorial curriculum. Likewise, the Geometric Solids (sensorial activity) and the Picture Cards (sorting) in Language are examples of activities in other subject areas that support mathematical reasoning.
Lecturing to children as a means of instruction does not occur in Montessori education for mathematics or any of the other school curricula. Children learn, instead, through involvement with the diverse materials available for them to explore. Teachers work individually or else with small groups of children. Each child progresses at his/her own pace, consistent with differentiated instruction. The teacher is flexible in timing the instruction for each child. When the teacher gives the Spindle Boxes lesson to a child, for example, if that child is not able to detect her own mistakes, it signifies to the teacher that this task is not at the child’s developmental level. Consequently the teacher will delay teaching this activity until a later date (East Baton Rouge Parish School System, 2008).
When the teacher presents an activity, she must be precise in modeling correct manipulation of the materials step by step. The teacher does not intervene when the child is concentrating on the materials, nor does she intervene when the child is reflecting to find his own mistakes. Concentration and independence are hallmarks of Montessori education.
In Montessori’s view, children use movement to extend their understanding and to acquire abstract ideas (Montessori, 1966, 1967a). “The mental effort should be accompanied by a physical one because the latter is needed to stimulate the action of the brain. Hence it is unnatural to use one without the other” (Wentworth, 1999, p. 33). This view is supported by numerous research studies that indicate that movement leads cognition and cognition is imbedded in action. (Lillard, 2005). Piaget (1970) stated as much in these words: “to know an object is to act upon it.”
The Montessori math manipulatives are often large enough that the child needs to incorporate movement of the body to complete the task. The materials for a given task are laid out on a rug on the floor in front of the child, who is also seated on the floor.
The Montessori Mathematics curriculum, as with each of the other four subject areas - Practical Life, Sensorial, Language, and Geography and Culture - is embodied in a prescribed set of hands-on activities, each of which includes a set of specific manipulatives that appropriately challenge the child at his developmental level of comprehension (Orem, 1971). Montessori teachers work directly with each child to teach them intended use of each of the manipulatives, thus minimizing the possibility that the child would treat the Montessori manipulatives as toys. Some educators criticize this limitation. However, research by DeLoache (2000, cited in Lillard, 2005) shows likelihood that when a child has difficulty perceiving the manipulatives in their symbolic role, he has difficulty recognizing the underlying, abstract concept the manipulatives are intended to communicate.
The Mathematics Curriculum Content
The math curriculum for learners 3-6 years of age is numeration. Concepts taught include but are not limited to: 1) Introduction to numbers 1-10, 2) Introduction numbers 1-100, 3) introduction to the base 10 numeration system and the concept of place value, 4) construction of numbers from 1 to 1000, 5) addition, subtraction, multiplication and division, up to 4 digit numbers, 6) fraction concepts, and 7) memorization. The following are typical of hands-on activities that constitute the mathematics curriculum.
Introduction to number 1-10: number rods, short bead stair.
Ten wood rods of increasing incremental length, the shortest of which is 10cm and the longest of which is 100cm. The rods are painted in alternating 10cm color segments of red and blue. The child perceives the difference in length of the rods in terms of how far apart he must hold his hands to lift and place the rods on the rug. Next, the child is working to place the rods in sequence. This activity captures the child’s interest and sustains it .
Control of Error:
The child knows when the activity is complete because all of the rods will be in place; a regular stair pattern will be evident to the child and the color segments (red and blue) of the rods will be aligned.
Short Beads Stair
A set of 9 wired bead bars, each bar a different color, and containing a different number of beads from 1 to 9. This set of manipulatives is similar to the Cuisenaire Rods that were introduced by Georges Cuisenaire, a Belgian elementary school teacher in 1931, and popularized beginning in 1953 with Caleb Gattegno (Cuisenaire, 2008).
Control of Error: The materials themselves provide the control of error for this activity. An error in the placement of one or more of the bead bars will result in an irregularity in the pyramidal pattern.
Introduction to the base 10 numeration system and the concept of place value,
The Forty-five Layout
The materials for this activity consist of forty-five unit beads, forty-five 10 bars, forty-five 100 squares, forty-five 1000 cubes, and cards containing the numerals 1-9, 10-90 (by 10’s), 100-900 (by 100’s), and 1000-9000 (by 1000’s). The items are laid out in alternating columns of beads or bars or cubes and the corresponding numeral cards. The unit beads are on the far right and the 1000-cubes are on the far left. The child must place the correct number of manipulatives next to the numeric symbol for that number.
Control of Error: Miscounting or mispositioning any of the manipulatives causes materials to be left over or to run out before the activity has been completed, thus controlling for error. Another control of error lies in the visual appearance of the distribution of the manipulatives: If completed correctly, the number of beads, bars, or cubes increases from the least to the greatest number and from the tops of the columns to the bottoms.
Introduction to numbers 1-100
The Hundred Board
The activity materials consist of a framed board and 100 small squares numbered 1-100.
First the child lays the 1 square in the upper left corner of the board and then places the other squares in numerical sequence ending with 100, moving left to right, top to bottom.
Control of Error:
The child is scaffolded with an exact pictorial representation, i.e., a chart to which she can refer.
The Snake Game is structured to ensure that children will use and remember all the possible combinations of 2 counting numbers that equal 10. The child makes a snake of 9 segments using the combinations of bars containing 1 and 9 beads, 2 and 8 beads, 3 and 7 beads, and so on. To complete the activity, the child counts and replaces each 10 bead segment with a golden 10 bead bar until the entire snake is a golden snake.
Control of Error: The materials control for error in that they must be set in segments correctly so that each segment (of the snake) consists of 10 beads. The child knows that the activity has been completed correctly when the entire snake is golden.
Addition, subtraction, multiplication, and division
The Stamp Game
In the process of learning from the manipulation of Montessori math materials, the child gradually learns to depend less upon the sensorial cues as he or she internalizes the abstraction embedded in the materials. The Stamp Game is an example of an activity in which the size of the objects does not provide a visual cue pertaining to their relative value. The materials and the activity assume that the child understands the differences in value between 1’s and 10’s, 10’s and 100’s, etc.
The color coded materials for this activity consist of 1 (unit) tiles, 10 (unit) tiles, 100 (unit) tiles, and 1000 (unit) tiles and a box with compartments for each denomination. Each of the tiles is identical to the others in size. Consequently, there is no visual size clue given. The color coded tiles can be laid out in columns right to left by 1’s, 10’s, 100’s and 1000’s to represent numbers from 1 to 9999. In addition, the bead bars and small numeral cards serve as a review for the concept of base 10.
Control of Error: Because the tiles are color coded and marked with 1, 10, 100, or 1000, the child may notice very quickly the misplacement of a tile.
This activity involves the manipulation of fractional circles representing the whole, halves, thirds, etc. Children are introduced to the visual representation of fractions equaling 1.
Control of Error: Only the correct number of the fractional pieces with a common denominator will fit together and fill the frame.
Montessori education generally, and in particular for mathematics education, implements mathematics instruction for young children that affords them abundant opportunities to explore hands-on materials that engage them in mathematical thinking. They build towers, put blocks of graduated lengths in order from small to large, classify picture cards according to content, compose units to tens and ten to hundreds and the like, use geometric shapes to compose other geometric shapes, and so on. The approach is consistent with Principles and Standards for School Mathematics (NCTM, 2000) that heavily influences contemporary reform mathematics curriculum, pedagogy, and content.
The materials have proved their interest to children over the decades of their use in Montessori classrooms around the world. Central to Montessori education is that children acquire competence and confidence in making choices, exercising autonomy and cultivating their independence in the course of their educational experiences, the Montessori teacher’s role being that of facilitator of learning, not that of pounder of knowledge into the empty their heads. Moreover, the teacher differentiates instruction based on coming to know as nearly as possible the readiness of each child for interacting with the wide variety of materials that are available. While group instruction, which is typical of most public school programs even for the youngest children, is rare in Montessori education, the children do interact quite a bit with one another through their pure social time together, as well as in their collaboration sometimes with each other in their focused explorations of the materials, as indicated earlier with Ming and Salina in the section, A Glance at the Classroom.
The environment is rich and prepared for children’s inquiry, designed to invite their exploration. When children’s interest levels are high for their learning experiences, their wrangling with one another and likelihood of causing causing disturbances diminishes. Intrinsic motivation for learning and for behaving in socially acceptable fashion is usually high in Montessori education (Lillard, 2005). Certainly these insights are important for all teachers to keep in mind who worry over classroom management and child discipline matters.
For those readers who want to bring constructivist mathematics education to their classrooms, for those who already have and are seeking to sustain it, and for those who are skeptical and who’d like to try some approaches that have stood the test of time, Lillard (2005) states, “In contrast to other constructivists, Dr. Montessori left the legacy of a broad, field-tested curriculum covering all the major subject areas -- math, music, art, grammar, science, history, and so on -- for children ages 3 - 12.” (p. 18). Indeed, for teachers who might not have considered Montessori education as a resource for implementing reform-minded mathematics education, including content, pedagogy, a humanistic view of child development, and some integration of mathematics with other disciplines, this article says, “Take a look at Montessori. There is a wealth of material and a long history of desirable impact on children’s learning in the implementation of the program whose founder is Maria Montessori.”
Links to websites at which to view specific Montessori materials:
Pink Tower- http://www.fmployola.com/materials.htm
Nuts and Bolts- http://www.kidadvance.com/productdetails.asp?pid=259
Positive Snake- http://www.infomontessori.com/mathematics/tables-of-arithmetic-addition-snake-game.htm
Binomial Cube- http://www.fmployola.com/materials.htm
Trinomial Cube- http://www.fmployola.com/materials.htm
Golden Bead Materials- http://www.fmployola.com/materials2.htm
Classified Picture Cards- http://www.montessori-matters.com/catalog_descriptions/description46.htm
Forty-Five Layout Activity- http://www.ehow.com/video_4403337_the-45-math-montessori-materials.html
Geometric Solids- http://www.fmployola.com/materials.htm
Large Hexagon Box- http://www.expertvillage.com/video/118082_hexagon-box-montessori-activity.htm
Small Hexagon Box- http://www.expertvillage.com/video/118084_small-hexagon-box-montessori-material.htm
Spindle Boxes-- http://www.fmployola.com/materials2.htm
Hundred Board- http://www.fmployola.com/materials2.htm
Cuisenaire (2008). Cuisenaire and gattegno. Retrieved December 25, 2008, from http://www.cuisenaire.co.uk/background.asp .
East Baton Rouge Parish School System. (2008). Instruction: Montessori Program. Retrieved December 30, 2008, from http://instruction.ebrschools.org/explore.cfm/montessoriprogram/ .
Elkind, D. (2003). Montessori and constructivism. Montessori Life, Winter. Retrieved December 30, 2008, from http://findarticles.com/p/articles/mi_qa4097/is_/ai_n9199626?tag=artBody;col1.
Erikson, E. H. (1963). Childhood and society (rev. ed.). New York: Norton.
Kamii, C. (2000). Young children continue to reinvent arithmetic--2nd grade (2nd ed.) . New York: Teachers College Press.
Lillard, A. S. (2005). Montessori: the science behind the genius. Oxford/New York: Oxford University Press.
Lillard, P. P. (1972). Montessori: a Modern approach. New York: Schocken.
Montessori, M. (1966). The secret of childhood. Notre Dame: Fides Publishers. (P. 99)
Montessori, M. (1967a). The absorbent mind. New York: Henry Holt.
Orem, R. C. (1971). Montessori today. New York: G. P. Putman’s Sons.
Piaget. J. ( 1970 ). Science of education and psychology of the child. New York: Orion Press.
Wentworth, R. A. (1999). Montessori for the new millennium. Mahwah, NJ: Lawrence Erlbaum Associates.