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A Closer Look at Montessori Math

The Montessori math curriculum is quite unlike the traditional approach that each of us experienced.  It is based on developing a strong foundation through concrete experience and manipulation until the time a child reaches the age of abstraction, typically around nine years old. As they engage in the Cosmic Curriculum, children are given a basis for the interconnectedness of all things and encouraged to engage in the wonder and magic of mathematical concepts.  Various activities and materials develop the mathematical mind, preparing the child for their inevitable explosion in to abstraction and connection to the power of relationships. 



The following was written by Lower Elementary Spanish teacher, Diana Haro Reynolds.

Mathematics is the study of quantity, form, and magnitude. We live among it. It is in the position of the sun and in the shell of a snail. We carry math in our pockets, in our devices. It is what makes our communication possible. We touch and live math, whether we know it or not. It is our responsibility as Montessori guides to help the child discover this framework of mathematical order that makes up our world. This rationale proposes several reasons for teaching math in the Montessori classroom. It will explain the journey the child will take from concrete concepts through to abstraction.

Human beings have a tendency for order. Since the beginning of human origin, math has been used to unlock the mysteries of the world. It began with a man’s need to
keep track of his belongings. Then came early techniques that created the experience
with numbers of counting. After a long time, came comprehension, which led to
improvements and shortcuts. Finally, humankind reached abstraction. This same
process is seen in a child. (Doer, 2012)

Mathematical order leads to a mathematical mind. As the mathematical mind unfolds, it develops capacities such as sensorial interaction among objects, observation
of patterns, and awareness of the physical world, mental classification, abstract thinking, and knowledge of the power of relationships. Math supports understanding by encouraging order, concentration, independence, special relations, patterning, one to one correspondence, combination, difference, and similarity. Additional goals supported by math are predictability, exactness or sense of accurateness, concreteness, logic and reasoning, problem solving, and decision making skills, as well as refinement of the mind and thinking. (Stockton-Moreno, 2015)

Why the need to teach math in a specific Montessori way? There are a lot of aspects that prepare the child for math. These include the prepared environment, giving the child the power of choice. Practical life works build concentration and confidence. The child enjoys practicing a task over and over for the pure pleasure of it. The joy is in the process. This mentality is preparing him for the academic areas. Sensitive periods serve as the specific times in a child’s development where part of their needs include an insatiable thirst for specific tasks. There is a sensitive period for concrete tools of precision. We must capitalize on these sensitive periods. (Stockton-Moreno, 2015)

The aims of Montessori math are to make the child aware that math is a part of her life; to build confidence and prepare the child for life. Confidence comes from the sequential growth in which the materials are presented. It starts with the importance of the Three Period Lesson. The first period being the presentation of the concept. The
second period is where the child practices and shows that which has been presented. In math, this second period is much longer than in other areas of study. This is the time in which the child is practicing, exploring and making discoveries, day in and day out, about the concept presented. The third period is that in which the child shows understanding of the concept through teacher observation or helping someone else.

The main goal of Montessori math is to move the child from concrete to abstraction and helping him form a mathematical mind. In the book, A Way of Learning, Ann Burke Nerbert explains that “the mathematical mind derives from experience” (Stockton-Moreno, 2015). We must not rob the child from forming her mathematical mind. She must have ample time to experience the joy of working with the materials and for understanding and internalizing the processes and concepts. The materials are
multimodal in that they appeal to multiple senses. This aids in the "permanent wiring of the brain that will be available as your child gets older and uses her brain for analytical thinking and problem-solving" (Duffy, 2008). Knowing is not understanding. Montessori math provides the path toward understanding.

According to Michael Doer, the passage towards abstraction is done in four stages. The first is the Concrete stage. This is where the child works purely with the material. No works is shown on paper until the child is nearing the end of this stage. The second stage is Concrete Materials lead to Symbols. This is when the child works with
the materials and records the process in writing. This is the longest stage and requires
that the focus be on the process, not the end result. This is the time in which the child is “internalizing the algorithm” (Doer, 2012). Towards the end of this stage the child may begin to work with charts rather than manipulatives. The third stage is often overlooked perhaps because it is the shortest. This is the stage when Symbols connect to Concrete Material. Essentially it is the reversal of stage two. The child does the work on paper then uses the materials to check their answer. The Symbolic stage is the fourth and final stage. This is where the emphasis is on showing the written work. (Doer, 2012)

Doer also emphasizes mental calculations and mental carrying as the two key elements in reaching abstraction. Mental calculation or memorization requires that the
child know math facts with accuracy and speed. The child should take no longer than
three second to recall a fact, otherwise, memorization has not been reached and the
child is calculating. Accuracy should be no less than 98%. It should be recall only. The
second key, mental carrying, requires that the child be able to keep track of the carrying without making a mark on paper. Having the child work on other forms of memorization, such as poems or definitions, will greatly help achieve this goal.

Math is part of our society. We need it in order to function. But there is also a math phobia. Math in Montessori makes it more than accessible, it makes it real. Whenever possible, real life problems should be presented to the child so as to give her
the context for these new skills. Among with word problems, research in the area of
math is a great way to expose the child to the practicality of math. We must cultivate a love and understanding of mathematics in our children by proving the keys and allowing them to make their own discoveries.

Diana Haro Reynolds - Lower Elementary Teacher/Intern



References

Doer, M. (2012). Numbers: Montessori arithmetic for lower elementary.

Duffy, M. (2008). Math works: Montessori math and the developing brain. Hollidaysburg,

PA: Parent Child Press.

Stockton-Moreno, L. (2015). MONT. 633*01, week 1 notes [PowerPoint slides].
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Parent Education Night








Sign-up outside of your student's classroom.

Childcare will be provided, however, you must sign up in advance.

This is a really great night full of insight regarding the education of your child in relation to Montessori Philosophy. Don't miss out!

(Your attendance can go toward Parent Volunteer Hours).

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Your Daily Dose of Resilience-Building by Melissa DeVries, Ph.D., Licensed Psychologist

b2ap3_thumbnail_Raising-Resilient-Children.jpgRaising children in the twenty-first century is a most rewarding challenge. In modern society we have increased access to mass media and greater sprawl within families. Youth are increasingly influenced by sources of information beyond parental control. Thus, our task as parents is to figure out how to balance sheltering our children while still preparing them for the future.

Research has identified many key elements that predict better quality of life in adulthood; academic achievement, absence of medical and mental health problems, financial stability, and rewarding social connections with others. Yet most of us at one point or another face situations that create vulnerabilities in these areas. So this begs the question, how do we bounce back? And more importantly, how do we teach our children to demonstrate the same perseverance when faced with stressors?

Everyday I work with families who are striving to bolster the skills and abilities of their children. They seek to help them to adapt to current stressors and challenges, and to acquire characteristics likely to help them lead a successful life in the future. My method of teaching is based on building resilience.

...
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Elementary Curriculum - Language

The elementary reading curriculum is designed to incorporate phonics, whole word and phonetic exceptions. Lower elementary students progress through a leveled reading program using the Pink, Blue and Green Montessori reading exercises while additional materials and experiences allow them to perfect their reading skills, develop their fluency and comprehension. The Grammar and Vocabulary materials allow the students to assimilate an understanding of the structural rules that govern the English language. Literary elements are explored during Group Literature. Lower and upper elementary students practice writing on a daily basis in classroom journals that cover a variety of writing forms. In lower elementary, Writer’s Workshops are held throughout the year to target specific writing skills. In upper elementary the different varieties of writing and writing skills are integrated into their cultural, science and literature studies. Our goal is to help the students become comfortable using writing as a communication skill. Students learn to think clearly, to research, and to express themselves with confidence and clarity in writing and speech.

  Lower Elementary Upper Elementary
Reading Reading readiness, phonic skills, guided reading, sight words, contextual clues, S.S.R. (Silent Sustained Reading), vocabulary Shared reading, dictionary skills, fluency, expression
Comprehension Responding to questions regarding Story-time book (sequencing events, recapping & summarizing, identification of character, plot & setting) context clues & main ideas Continued study of main ideas, sequencing & context clues, assumptions/inferences, following written directions & instructions
Penmanship Metal inset exercises, D’Nelian print & cursive, spacing, left justification, neatness Mastery of cursive
Spelling Unconventional to conventional, leveled spelling works Conventional spelling lists, spelling demons, vocabulary, spelling rules
Mechanics Ending punctuation, capitalization, commas Apostrophes, commas, quotation marks
Composition Complete sentences, journaling, picture prompted stories, modeled writing, editing Journaling, character & plot development, proofreading, revising, publishing
Study Skills Categorizing, table of contents, index, beginning reports Outlining, note taking, organizing information, skimming, advanced reports, paraphrasing
Grammar Parts of speech, parsing Sentence analysis, verb tenses
Speaking Poetry presentations, in-class reports, drama, story-telling Poetry presentations, in-class reports, drama,   story-telling

Elementary Curriculum - Mathematics

The elementary Montessori math curriculum takes the students through a series of precise exercises using specifically designed materials that support the students’ emergent abilities to abstract. Using hands-on manipulative materials the students in 1st – 3rd grade are given tools by which to do their math work and so acquire a concrete understanding of math skills and knowledge. This solid foundation allows a smooth transition to abstract understanding and application of math skills during the 4th – 6th grades.

  Lower Elementary Upper Elementary
Numbers Linear counting, sequencing, place value through millions, before & after numbers, <, =, or >, skip counting, ordinal & Roman numbers, one-step word problems, patterns & relationships Factors & multiples, rounding numbers to nearest 10s & 100s, prime numbers, squaring and cubing, estimating, multiple-step word problems
Operations + - x / of whole numbers, regrouping, missing values, inverse operations, memorization of numerical patterns Large operations in all 4 operations (including long division, multi digit multipliers), operations involving decimals, memorization of tables, percentages, averages
Functions Identification of fractions, addition & subtraction with common denominators, multiplication & division of fractions by whole numbers, equivalencies Mixed numbers, + and – of fractions with unlike denominators, simplifying fractions
Measurement Standard and metric units of measurement for length, mass & volume Perimeter, area, capacity, word problems
Time Telling time to the minute Elapsed time, 24 hour clock, word problems involving time
Statistics Interpreting data, block and bar graphs Line graphs
Geometry Classification of solids, quadrilaterals, triangles and polygons, study of lines & triangles Study of circles, congruency & symmetry, use of protractor and compass
Money Coin value, totaling amounts Making change, word problems involving money

The Elementary Curriculum Overview

Different from a traditional school setting where teacher-directed curricula determines the daily lessons, the Elementary Montessori students choose their own work. Under the guidance of the teachers, the Montessori students select activities that reflect their ability levels yet present opportunities to practice and perfect skills. Students and teachers work together for large blocks of uninterrupted time within a classroom that is rich in resources. The students work at their own pace while the Montessori teachers observe and facilitate the learning process. The curriculum’s goal is to encourage students to become active learners rather than passive participants in education.

The elementary Montessori curriculum is designed to meet the needs of students between the ages of six and twelve. Elementary students have an increasing ability to abstract and to imagine; the curriculum engages the students in activities that utilize these affinities. While the curriculum builds upon the student's early childhood classroom practice, it expands to include experiences, opportunities and instruction that are appropriate for the students’ developing minds. The Montessori materials continue to play an important role as the students transition from the concrete to the abstract. The teachers’ lessons involve exploration, research and hands-on experiences that guide the students in developing their reasoning minds.

Elementary studies include geography, biology, history, language, mathematics, science, music, movement and art. Studies are enriched through field trips, visitors and workshops that support the curriculum and expand the learning outside of the classroom into the community.

The Montessori Teacher

“Follow the child” – as Dr. Montessori asserted, the Montessori teacher focuses on the whole child and not on the daily lesson plan. Dr. Montessori wanted to create a clear distinction between the role of the Montessori teacher and that of a traditional teacher. She coined the new title “director” or “directress” for the adults in her classrooms and as the name implies their role is that of a director of activities. Nowadays the term “guide” is more commonly used. The Montessori classroom is a student-centered environment rather than teacher-centered. The teacher is rarely the center of attention. She spends the majority of her time in individual or small group activity or observing the progress of the students.

The Montessori guide:

  • tailors lessons and activities to suit the student’s learning style and abilities.
  • prepares the classroom environment to promote autonomy amongst the students.
  • maintains an investigation and discovery approach when presenting topics rather than giving facts and figures.
  • is trained to assess knowledge and achievement through observation of the student.

The Elementary Montessori Teaching Method

Elementary Montessori programs emphasize active learning rather than passive reception of information.

  • The elementary Montessori curriculum builds upon the student’s early childhood experience. The Montessori materials continue to play an important role as the student transitions from the concrete to the abstract.
  • Lessons involve exploration and hands-on experiences. The student in the elementary classroom learns by doing hence the classroom is rich in materials, resources, movement and conversation.
  • The curriculum is individualized. The needs, ability, interests and skills of each student are taken into consideration when lessons are planned and knowledge assessed. The Montessori student will receive extra help or direction on areas where she needs it and move rapidly through other areas where she excels.
  • The elementary program teaches the student how to think clearly, how to research, how to express herself in writing and speech.
  • The program fosters independent work as well as group effort.
  • The multi-age classroom creates an atmosphere of non-competition making it possible for the student to work at her own pace, unrestricted by traditional grade standards.
  • The program supports a variety of learning styles.
  • Elementary Montessori education integrates all the different areas of study rather than compartmentalizing them.

The Elementary Classroom

Students learn best within an environment prepared to nurture and enhance each student’s unique development.

  • Multi-Age Groupings – Elementary Montessori classrooms are comprised of multi-age groupings. This is the practice of teaching students of different ages and abilities together without organizing either the curriculum or the classroom by age or grade designations. The students remain in the same classroom, with the same instructors, for several years.
  • Classroom Areas – The elementary Montessori classroom is divided into distinct curriculum areas: Science, Geography, History, Art, Math and Language. Many of these subjects are then organized into separate skill areas. There is a large floor area for spreading out work and gathering in for community meetings and lessons. There are tables for individual and group activities. The students have notebooks for recording their work and folders to store ongoing projects. Group supplies are located in a central area. A message board displays the day’s schedule as well as reminders and announcements. A wide variety of plants and animals are located throughout the classroom. Arrangements of cut flowers often decorate the tables and music is almost always playing in the background. Replicas of artists’ work adorn the walls. Cleaning materials are accessible to the students since they are custodians of their classroom. A library is located nearby and available for the students to visit in order to support their research and interests.
  • Materials – The wealth of materials in each area allows the students to follow their own interests. Materials are arranged so as to allow sequential progress in skills. Usually there is only one example of each material to encourage turn taking and patience. Materials and their activities vary from individual work to partner work to group activities. The materials are aesthetically pleasing with a great many being teacher-made. Many of the materials employ an internal control of error so as to encourage self-monitoring and foster independence in the elementary student.

The Elementary Student

The elementary Montessori program and curriculum is structured around the very specific needs and characteristics of students between the ages of six and twelve years.

  • Reason & Imagination - The inquisitive nature of the elementary student provides the fuel for the research and exploration focus of elementary Montessori. The elementary student wants to know the “why?” and “how?” The six- to twelve-year-old is able to use both reason and imagination to explore and understand increasingly abstract concepts.
  • Exploring Society – While the early childhood student was primarily focused on the construction of the individual, the elementary student begins to explore his place in society. Opportunities continuously present themselves for the student to observe or participate, moments in which to lead or follow.
  • A Need for Togetherness – This is the age of clubs and groups. The elementary student explores friendship and cooperation; he learns how to be a leader, a partner and a follower. While collaboration is encouraged, individual contribution and strength is also valued.
  • Exploring Right and Wrong – The six- to twelve-year-old student is actively developing his moral conscience; “That’s not fair!” is heard over and over again in the elementary classroom. Every student may know the rules but keeping them is another matter. Problem solving techniques are modeled and fostered in the Montessori program. Community brainstorming for solutions and rules helps form the elementary Montessori classroom’s code of conduct.
  • Freedom & Discipline – Independence and inner discipline continue to develop in the elementary years. The six- to twelve-year-old student is capable of increasingly complex and numerous responsibilities and needs opportunities to exercise judgment and demonstrate self-conduct. Everything from classroom management to the student’s work stems from the student’s freedom to choose and think. Mistakes and failures are viewed as learning opportunities.

Why Spanish?

The United States has approximately 50 million Spanish speakers.  The Western United States including Utah have the highest percentage of Spanish speakers.  Demographic and economic trends, including greater purchasing power among Latinos and Spanish speakers and interconnectedness in the Americas from Canada to Chile, suggest Spanish will grow even more important throughout the century. Additional reasons for learning languages including better access to other cultures and communication possibilities.

Why should my child be enrolled in dual language?

  • The dual language program offers your child the chance to develop communication ability in Spanish and English. This could be difficult at first, but being bilingual may enhance your child’s potential opportunities (jobs, culture, etc).
  • Acquisition of language by children has been studied widely. Children have advantages over adults in language learning. Natural curiosity and a willingness to make mistakes help the child. Adults often hesitate when speaking another language for fear of erroneous pronunciation or grammar. Children, especially young children, tend not to have that fear.
  • Children easily pick up and model accents. Few adults who learn other languages pick up the correct accents. Many children do to the point at which they pronounce as well as a native speaker. Of course, those children generally start to learn the language before their teens with the help of native speakers.

Does it affect or help my child if I have a language background?

Every child that is enrolled in the Dual Program goes through the same process of learning. Children who hear other languages spoken at home tend to learn languages faster.  Some ideas for home are:
  • Interact with your child.
  • Learn about dual language education.
  • Encourage your child to speak the second language.
  • Give your child the benefit of the doubt.
  • Do not ask your child to translate. This requires advanced skills that could frustrate your child.
  • Try to avoid comparing your child’s progress to that of other children. Rates of progress differ.
  • Be willing to participate in opportunities to expose your child to Spanish and culture(s)outside school.
  • Praise your child for his or her progress.

How does the Dual Language Program work at Montessori Community School?

In the dual language class, children are encouraged to evenly divide their time between Montessori works in English and Spanish. One teacher speaks and presents lessons to the children in English while the other speaks and presents lessons in Spanish. Teachers generally speak their native language though the Spanish teachers are bilingual in case a child needs assistance in English. With time, every child in the class will learn Spanish. Children in the classroom have the opportunity to learn how to communicate in both languages. They will experience speaking, listening, reading and writing. Reading and speaking in both languages is necessary to become bilingual, bi-literate and bicultural.

Montessori: Your Daily Dose of Resilience-Building

Raising children in the twenty-first century is a most rewarding challenge. In modern society we have increased access to mass media and greater sprawl within families. Youth are increasingly influenced by sources of information beyond parental control. Thus, our task as parents is to figure out how to balance sheltering our children while still preparing them for the future.

Research has identified many key elements that predict better quality of life in adulthood; academic achievement, absence of medical and mental health problems, financial stability, and rewarding social connections with others. Yet most of us at one point or another face situations that create vulnerabilities in these areas. So this begs the question, how do we bounce back? And more importantly, how do we teach our children to demonstrate the same perseverance when faced with stressors?

b2ap3_thumbnail_playing-working-together.jpg

Everyday I work with families who are striving to bolster the skills and abilities of their children. They seek to help them to adapt to current stressors and challenges, and to acquire characteristics likely to help them lead a successful life in the future. My method of teaching is based on building resilience.

...
More Info