Comprehension Strategies in Math

Dr. Maria Montessori emphasized the value of active learning for children—experiences involving the use of their hands for practical purposes and for the acquisition of knowledge. Today’s educators would call this experiential learning.

She wrote in her book The Secret of Childhood that the child’s capacity to distinguish similarities and differences impacts his or her capacity to “mold [their] own intellect.” As a result, in the Montessori classroom, we place much more importance on the process of understanding how a concept is formed and implemented, than upon the accuracy of the end product.

 

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Accuracy is not necessarily the aim when teaching math to young children.

 

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Perhaps no subject area in the classroom is more challenging than math to adhere to this idea of the supremacy of ideas and procedure over product. After all, what could be more straightforward than 1+2=3?

How about…

When the child writes a number backward.

When the concrete materials representing a quantity are short a few units, but the answer is correct.

When the written answer is correct according to the concrete materials but the equation is written inaccurately.

When the child clearly understands the procedure but executes it incorrectly.

Accuracy is Not Always the Aim

In the Montessori classroom, the young child learns mathematical concepts through experiencing progressive stages of work where the tangible, direct associations with real objects are slowly eliminated. He or she is building over time a knowledge base—a foundation of understanding—to reference when problem solving now and in the future. This is the most important task at hand.

 

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Solid grounding in the theoretical always trumps accuracy for the immediate and the particular.

 

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Dr. Montessori described this “construction of the mind’s content” in The Absorbent Mind as a process of abstracting; the process serves to simplify and unite with the end goal of expression. Many of the early childhood Math works are in actuality indirect preparation for much more conceptually complex operations, such as extracting square and cube roots. The young child is subconsciously absorbing many complicated mathematical concepts at an early stage of development. Whether or not he or she has a solid grounding in the theoretical may ultimately affect the acquisition of knowledge in later years.

Many of the Montessori materials serve as preparatory exercises for later, more complex math, including algebra, geometry, and eventually calculus.

Active Observation

In the Montessori classroom, one of our primary objectives is to nurture the capacity for active observation. A child receives images and data through the senses but eventually becomes more selective in his or her choices and conclusions. Experience is being integrated into a larger, more general understanding of the world. Trusting your child’s inner sensibility is trusting their capacity to function on a higher, rational level. Unnecessary correcting can plant seeds of self-doubt.

The ability to reason is gradually developed and significantly influenced by a child’s access to environmental stimuli. Increasing a child’s comfort level with math materials more often than not outweighs correcting isolated instances of inaccuracy.

Also, why diminish a child’s enthusiasm for math by needlessly correcting him? This is especially true with longer math works that methodically take a child through a series of steps, with the goal of reinforcing a particular concept.

The Bottom Line

Central to the Montessori Method is the idea that a child should feel fulfilled, content, successful, etc., as a result of manipulating materials. Outside intervention, positive or negative, has the potential to lessen feelings of confidence and self-sufficiency. Correcting (or praising) a child becomes superfluous.

In simple terms, the outcome of completing or failing to complete a work is just more information for the educator or parent. Observations of a child’s progress, ideally, lead to responses correctly reflecting where he or she is at developmentally.

For a child, the end product of his or her efforts is more data about the world—more information to absorb and more feedback that he or she will correctly apply later.

Final Note about Writing Numbers (or Letters) Backward

Current research indicates that recognizing mirror images comes naturally to children. According to an article in The Economist (10 July 2010, 77-78), the Federation of European Neuroscience Societies (FENS) concluded that learning to read requires the brain to “undergo profound changes, including unlearning the ancient ability to recognize an object and its mirror image as identical.” Montessori teachers have known for some time that it’s simply a matter of time; the child will eventually correct himself. So, don’t worry. It’s normal for a child to initially write his numbers backward.

Understanding the Transition from Non-reader to Reader

Your child is progressing, as he or she should, at his or her own pace, which is the right pace.

There is no reason to rush through the important period of valuable skill building for reading and writing. Consider how critical skills are developed over time in all areas of your child’s life, in school and in your home. The early childhood years can be an exciting, joyful time—for the child to experience and for the parent, too, to observe and even participate.

Process of Building Pre-reading Skills

For now, our primary task is to together (teacher and parent) instill in your child a great curiosity and love for language. In the classroom, we aim to provide access to the materials and opportunities needed to practice and perfect the necessary skills leading to reading.

 

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Start with the Practical Life area in your child’s classroom. What skills are being refined and how? Try viewing this activity as preparation for reading and writing.

 

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Skills learned in the classroom’s Practical Life area reflect fundamental Montessori principles of order, self-discipline, and concentration. A child masters his or her movements through a series of isolated steps and through repetition. He or she eventually learns to analyze and to respond with rational, orderly actions; intellect becomes the primary guide for navigating a challenging scenario.

Role of Practical Life and Sensorial Works

Foundation skills, such as the capacity to organize one’s workspace and to use reason to problem-solve, are needed for accomplishing work throughout the classroom. (Proficiency here is a strong indicator of future academic success, too.) However, many of the Practical Life works are specifically about preparing your child for reading and writing.

 

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The presentation of Practical Life materials often adheres to a left-to-right, top-to-bottom order, which is consistent with other works in the Montessori classroom that help a child prepare for reading and writing the English language.

 

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Teaching strategies associated with the Practical Life area often focus on a child’s fine motor skills—specific to hand and eye coordination. Grasping, pouring, and spooning works support the child in refining skills for gripping and delicately maneuvering a pencil. Similarly, variations in textures, shapes, and sizes of manipulatives help a child to refine the sense of touch and to increase visual discrimination. The former directly relates to Montessori sandpaper letters and the latter aids in distinguishing the differences between letters through sight.

A similar approach for cultivating reading readiness is apparent in the Montessori Sensorial works. For example, tracing forms in the Geometric Cabinet is a form of training for both the eyes (exactness of shape) and for the hands (muscular development).

Early works, such as described here, are necessary preparation for visually distinguishing and eventually forming letters. Your child must first master this kind of work before learning to read and write.

Practice Makes Perfect

A less obvious, yet, no less important category of work in the Montessori classroom involves oral language activities. These teaching techniques involve allowing the child to see the juxtaposition of spoken and written language. The child must discover (on his own!) the value of using labels, categorizing and ordering his or her environment, and following written instructions. From an understanding and then application of these concepts eventually comes the knowledge necessary for reading.

Be aware that your child practices the implementation of pre-reading skills. This prep work takes place on the playground and in the home, in addition to school. Socializing with young peers is a critical component of a child’s development leading to reading, for the child is learning how to mentally organize and communicate his or her thoughts, as well as how to anticipate and decode information received.

Understand that your child’s recount of his or her day is also about refining the pre-reading skills learned earlier in the classroom. Consider the intellectual agility involved in recalling, organizing, and then narrating your day!

Your Role is Important

As a parent, you play a critical role in helping your child to build communication skills by modeling accurate syntax in all verbal interactions and by continuously introducing new vocabulary. Sharing should coincide with your child’s actual experiences; you want to help him or her to make the connection between communication (whether verbal, written or read) and real-life happenings.

How special it is to be a part and to witness those first steps toward understanding language’s use and its ultimate purpose of self-expression. Savor each and every moment.

The Montessori Method and Memorization

This second article in a two-part discussion describes additional areas of the Montessori math curriculum as practiced by our teachers at The Montessori Schools. However, the following explanation delves deeper into how Dr. Montessori’s teaching methods help bridge concrete applications for computation to a more nuanced understanding of mathematics. Skills developed here will serve the child well in elementary grades.

Progression Away from Concrete Aids

In a Montessori classroom, young children often begin by counting beads to solve simple addition, subtraction, and multiplication problems. More advanced mathematical concepts are also comprehensible through direct manipulation of classroom materials.

In the final years of Primary, the goal is to successfully solve a problem on paper with pencil, independent of the Montessori materials. However, it typically takes several years (usually into the elementary years) for a child to completely wean himself or herself from such concrete aids.

In response, our early childhood math materials include numerous works designed to help a child gradually memorize the basic math data contained in more traditional memorization tables. These works include the Snake Games, the Strip Boards, the Finger Boards, and the Tile Boards. For multiplication memorization, we also have the Bead Box and Board.

Memorization in Practice

Addition with the Bead Stairs is an early memorization work, as it still relies heavily upon concrete representations of numeration, or individual beads as units. The work performs an important role at a critical juncture in a child’s cognitive development: practice adding while building equations using the symbols + and =.

 

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Practice and more practice with equations is necessary, for young children don’t readily comprehend higher numbers without one-to-one correspondence.

 

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The Short Bead Stairs reinforce the abstract idea of quantities from two through nine. Addition with the Bead Stairs uses a second Bead Stair to initiate the child into the process of memorizing simple addition facts.

fig. 1. The Montessori Bead Stair. Each quantity (1-9) is represented by the corresponding number of beads, wired together as a bar.

Each bar in the Bead Stair is a specific color, which enables a child to visualize combining one quantity of beads with another quantity of beads. The answer is secured from the second bead stair, which provides a kind of control of error, as the child can match the combined beads with the final sum.

Subtraction Memorization

Dr. Montessori designed a series of memorization charts to assist the child in more advanced (and more abstract) stages of working with the operations. The Subtraction Strip Board prepares the child to work on memorizing subtraction facts through exploring all the combinations when the numbers are subtracted from one another.

fig. 2. The Montessori Subtraction Strip Board. Red and blue sets of wooden strips increase in size: the shortest corresponds to one square on the board; the longest is nine times the length of the first.

Three sets of wooden strips (red, blue, plain) enable a child to “build” a problem on the board. He or she uses a plain strip to cover numbers greater than the largest number, which has a quantity being subtracted from it (represented by a blue strip). The remaining red strip represents the solution to the problem. The child can continue to exchange components, therefore, exploring all the ways that numbers 1 to 18 can be subtracted from one another.

Multiplication Memorization

The Multiplication Finger Boards are simple visual aids to assist a child in memorizing multiplication tables. Consistent with an important tenet of Dr. Montessori’s philosophy, movement is employed for cognition—or, more simply put, a young child’s comprehension is increased with (literally) hands-on learning.

After placing the right index finger on the smaller number and the left index finger on the larger number, a child moves one finger at a time, slowly inching his or her way toward the solution. The two fingers meet where the select column and row naturally convene.

Closing Thoughts

All of the memorization boards have a control chart, which permits the child to check his or her own answers. The control charts are also indicative of the Montessori Method’s emphasis on nurturing within children, even within very young children, a sense of autonomy and personal responsibility for their own education.

Our teachers also encourage children using the boards to write the answers to problems on paper, further reinforcing the concepts embodied in the work. The boards are tools to aid children in their abstract memorization of math equations. Although the materials are somewhat abstract, the process is still concrete.

Journey from Concrete Reality to Abstract Mathematical Concept

The following two-part discussion describes key areas of the Montessori math curriculum as practiced by our teachers at The Montessori Schools.

We are a community committed to establishing the intellectual, emotional, and physical foundation that will develop the skills for your child to become a self-directed learner, flexible thinker, and creative problem solver. Our math curriculum follows a traditional Montessori education, as developed by world-renowned doctor and educator Maria Montessori.

 

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Our classrooms contain a range of teaching aids for math, but the common factor is their origin: concepts are taught using materials originally designed by Dr. Montessori more than a century ago.

 

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Children begin by counting beads to solve simple addition, subtraction, and multiplication problems. Boards with separate cards demonstrate the base ten system for teen and double-digit numbers. Still more advanced mathematical concepts are made real—comprehensible—through direct manipulation of classroom materials.

Numeration 1-10

This initial series of math works includes some of the most poignant examples of Dr. Montessori’s concept of “materialized abstraction,” as realized in her didactic materials. Works, such as the Number Rods, Sandpaper Numbers, Number Rods with Cards, Spindle Boxes, Cards with Counters, Memory Game, are to be experienced in an established progression.

The math works involving numbers 1-10 have a child physically associate symbols with concrete representations of quantities. The quantity can be fixed—for instance, as per the number rods—but the goal is always to help him or her express abstract mathematical concepts.

 

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A child moves from an understanding of quantity to a meaningful comprehension of symbol.

 

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The Number Rods are one of the first math works to be introduced to a child. Readiness for this work can be measured by the level of interest in numbers as expressed in everyday interactions, as well as previous success with select works in the Sensorial area of the classroom. Ten wooden rods vary in length, with the shortest one at 10 cm and the longest one at one meter.

fig. 1. Child working with the Number Rods. Each succeeding rod increases in length by the length of the first. All other dimensions remain consistent.

In fig. 1., the student is learning her numbers (1-10) in conjunction with the concept of quantity and within the context of an established sequential order. Through working with the Number Rods, the child grows to understand that each number is a separate object in itself.

Decimal System

More advanced works, those involving the use of beads, cards, and “fetching” activities, build on an understanding of how quantity is associated with a symbol. Children are prepped to perform operations using the decimal system. Exercises are repeated over and over again, providing significant exposure and practice and laying the groundwork for an understanding of place value. A child also becomes skilled in reading complex numbers.

The Formation of Numbers, or “fetching,” is a foundation work for a child. With mastery, a child will have success with more complex “banking games,” involving the concepts of static and dynamic addition, multiplication, static and dynamic subtraction, and division.

As a child progresses through a series of steps, he or she experiences and eventually comprehends multiplication, for example, in concrete terms. Similarly, the process of exchanging is introduced using tangible elements. As a child counts quantities in each column, he or she may have to (literally) trade groups of ten, exchanging ten units for one ten bar, 10 ten bars for one hundred, or ten hundreds for one thousand, in order to determine the final addition/multiplication sum.

 

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“Ten units makes one ten,” says the child who has grasped the mathematical concept of exchanging."

 

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fig. 2. Using the Golden Bead materials, children “build” numbers, physically manipulating the materials to arrive at the sums.

Linear Counting

Still another series of Montessori math works involve children again working with the concepts of quantity (beads) and symbol (boards) but, this time, in pursuit of developing a sense of sequential order. Part of the genius behind the Montessori Method is establishing a correlation between different concepts.

A child builds on concepts internalized earlier with works that teach the names of numbers, such as the Teens/Tens Beads and Boards (11-19, 10-90). Sensibly, he or she will eventually move on to the next work, the 100 Board, which sequences the numerals from 1 to 100 in their totality.

fig. 3. The Montessori Bead Cabinet teaches linear counting while providing a concrete representation of abstract mathematical concepts.

The Short and Long Chains, or the Bead Cabinet, is also sometimes referred to as the “Cabinet of Powers,” due, in part, to its remarkable flexibility as a teaching tool. For the 4 to 6-year-old child, the work’s primary objective is to practice linear counting and reading numbers, as well as skip counting, from 1 to 100—and 1 to 1,000 with the long chains. However, significantly, the Bead Cabinet also indirectly teaches multiplication, squaring, cubing, and preparation (for the elementary years) for extracting square and cube roots.

Operations of the Decimal System

When a child is ready to work with the operations, he or she has a series of works that progress from a more concrete understanding of adding, multiplying, subtracting, or dividing quantities to a more abstract reinforcement exercise. The Stamp Games represent a shift toward a more theoretical understanding of math.

fig. 4. In the Stamp Game, counter tiles are used to layout the initial quantities. Additional units can be secured by “cashing-in” a tile from the next column to the left.

In the Stamp Games, a child no longer has such tangible, three-dimensional representations of quantities. Counter tiles are identical in size. Differences in color and in labeling are standardized to represent individual units (green “1”), a set of ten units (blue “10”), a set of 100 units (red “100”), or a set of 1,000 units (green “1000”). Consequently, the material’s direct aim, or purpose, is to reinforce the operation with the decimal system at a higher level of abstraction.

At this point in a child’s progress through the math works, he or she is writing work—copying the problem and final sum onto paper. The color-quantity correspondence (place value) is reinforced through using green, blue, and red colored pencils.

Why Understand the “Why”

In a Montessori classroom, children use tangible materials to solve simple and long math problems. Consider how many of us, as children in traditional classrooms, learned how to perform math operations. We likely memorized “tricks,” for example, for long addition: work from right to left, placing the right digit of a multi-digit answer under the line, and then carry the left digit into the next line of numbers to add.

Math tricks may ultimately work, but the process leaves the young child feeling disconnected from the mechanics of the operation. On the road to comprehension, it’s much better to initially connect on more solid terms.

Note on the Next Article in this Series:

Part 2 of 2 delves deeper into how Dr. Montessori’s teaching methods help bridge concrete applications for computation to a more nuanced understanding of mathematics.

Early Literacy and Emergent Reading in the Montessori Classroom

When early childhood educators talk about sensitive periods, they are referencing a young child’s inner development, something that is not always readily apparent to the adult. You might observe for a time in your 3 to 6-year old child’s behavior a special sensibility—a sudden responsiveness to mastering particular skillsets, including the building blocks for reading: concentration, vocabulary development, visual and sound discrimination, and more. Your child is focusing attention on certain elements in his or her surrounds to the exclusion of others.

What might feel like tunnel vision is in actuality a specific stage of development.  

Critical Role of Concentration
As a result of their extensive training, teachers at The Montessori Schools are acutely aware of each child’s stage of development as it relates to the timing of these special, yet transitory, periods of sensibility. The ability to concentrate for long periods of time is an early indicator of reading readiness, as is repetition with a chosen classroom material.

 

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Our teachers zero in on the quality of concentration and take advantage of a sensitive period in your child’s development.

 

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Your child is learning how to concentrate through interacting with teaching materials. Our teachers look for the general meaningfulness of the material to the child, which tells them whether it is the time to introduce something new or to modify an existing strategy.

Vocabulary Development

During the first 3 to 6 years, your child is capitalizing on a period of sustained proficiency in the spoken language. A child entering into the sensitive period for language needs vocabulary infused into his or her world whenever and wherever possible. In truth, he or she will be very vocal and very insistent about this need. “Mommy, what is this? And, what is this called? And, this and this and this…” and so forth.

Your child is adding new words to his or her vocabulary at a rate not to be experienced again.

Children desire exact and sophisticated nomenclature. During this sensitive period, language is ideally acquired in a contextually meaningful way. When your child uses vocabulary in context, inside and outside the classroom, reading is likely just around the corner.

Learning Sounds

During the sensitive period for auditory discrimination, your child will spend significant time vocalizing (sounding out) the names of objects. He or she has likely grasped the concepts behind word construction when the sounding out is no longer vocalized. During this sensitive period, your child enjoys works, games, and songs that develop the ability to hear the initial, medial, and ending sounds of words. This may help explain why “I spy” and rhyming songs are so popular at this time.

The Montessori Bells contribute to improved auditory perception—an acuity needed for “sounding out” words. 

Before mastering initial/ending sound discrimination and (later) word composition, the young Montessori child has experienced significant success with auditory teaching materials elsewhere in the classroom. The Montessori Bells and the Sensorial sound boxes have already, at this time, contributed to his or her ability to “decode” and, therefore, to write words.

In the classroom, our teachers take great care to arrange materials in an orderly manner, which encourages classification of items, a critical component of language development. The various matching works (picture-picture, object-picture, etc.), sequencing works, and categorization works (go-togethers, opposites, etc.) help your child to mentally organize the onslaught of auditory and visual information received during the course of the day.

We pay close attention to the degree of organization in a child’s work, because it reflects the mind’s order and relative capacity to perceive the underlying structure of language.

Writing and Tactile Discrimination

The Montessori child learns the shapes of letters that correspond with speech sounds. Initially, your child may be preoccupied with the sensorial aspects of writing, including the shape of the sandpaper letters and each letter’s corresponding sound. Touch is still the dominant sense in this sensitive period and, as a result, the child will not necessarily at first be able to recall the phonetic sound of a letter by looking at it; he will remember, though, once he has traced the tips of his fingers over the letter.

A child initially uses sensorial input for written communication. The sandpaper letters allow him or her to feel the contours of a letter using two writing fingers.

When a child has had sufficient time to explore the sensorial aspects of language, it is common to see a sudden and dramatic move into writing.

During this sensitive period for touch, the phonetic learner can also use writing for expression. Young children have an intense need to label and otherwise assign descriptors to objects. Both the sandpaper letters and the moveable alphabet enable him or her to do this without having, yet, perfected muscular hand control.

Signs You have a Reader in Your Near Future

Multiple areas within the Montessori classroom prepare your child for reading. Works in the Practical Life and Sensorial areas help your child to refine skillsets considered the foundation for lifelong readers and writers: concentration, left-right sequencing, pencil control, visual and auditory discrimination, and more.

Your child may be on the verge of reading when he or she exhibits a general mastery of “pre-reading” skills, in addition to other behaviors, such as comprehending the words he or she has “built” (composed).

The first words a child reads may be the exact same words he or she originally composed using the moveable alphabet.

Much of this new interest in reading comprehension is satisfied through classroom works in the content areas, such as those on the geography, science, and cultural shelves. In addition, a child learns that there is a correlation between words on a page and the spoken language. This is typically first manifested in “pretend reading,” where a child acts out reading behavior—for example, “reading” a favorite, very familiar story to a younger child or to a toy animal/doll.

Have Fun!

A child’s comprehension of a good number of sight words is also a strong indication that he or she is about to truly read. In effect, the child’s extensive oral vocabulary is eventually translated into reading vocabulary. This is an exciting time for your child and for your family.

The sensitive periods leading to reading proficiency are a joyful time filled with much discovery, awe, and respect for your child’s enthusiasm for learning. There is nothing humdrum and routine about learning how to express oneself. Rather, your child is entering into a keen period of cognitive growth in his or her young life. Enjoy the ride.