Math Fact Memorization: Montessori & Current Research

When someone starts talking about memorization of math facts, people tend to have strong opinions.  We all had a variety of experiences as children ourselves, and those experiences coupled with notions of best practices in education can cause for heated debates.  In today’s post we would like to share the results of recent research on the topic.  At first glance, the results may seem at odds with Montessori theory, but upon further examination this couldn’t be further from the truth.  Read on to find out why our materials and methods can provide children with exactly what this “new” information suggests they need.

The Findings

This summer Paul L. Morgan, Ph.D. published an interesting article for Psychology Today.  Morgan works at Penn State as a professor in the education department as well as Director of the Center for Educational Disparities Research.  He and several colleagues (George Farkas and Steve Maczuga) conducted research to investigate instructional practices and their effects on student achievement.  Their specific focus was on first grade classrooms in math.  

The researchers observed a variety of students and classrooms and determined that the only teaching method that had a positive effect on student achievement was teacher-directed instruction, as opposed to student-centered.  [This is the part where those of us who adore Montessori will audibly gasp, but keep reading!]

Morgan, Farkas, and Maczuga define teacher-directed as the following: “Teachers initially demonstrate specific procedures for solving problems, and then provide students with repeated opportunities (e.g., worksheets, routine practice and drills) to independently practice these procedures. Teacher-directed practices should help students increase their procedural fluency in applying explicitly taught and repeatedly practiced sets of procedures to solve mathematics problems, which should result in more effective use of higher order thinking and problem-solving skills.”

What Do Montessori Teachers Do?

According to the researchers’ definition of teacher-directed learning, this is exactly the approach that is taken in regards to students learning their basic math facts in Montessori classrooms.  Our initial reaction is simply a result of semantics and misconceptions.  Do Montessori teachers utilize worksheets and drills?  Well, not exactly, but we still meet the definition in other ways.

One of the most common misconceptions about Montessori education is that the children are let to do whatever they want all the time.  Some people think that choice is the driving force (it is, at least in part) and the students run amok.  Anyone who has spent any time learning about Montessori or observing in a classroom knows the opposite to be true.  Montessori is really all about choice within limits.  Teachers create an environment rich with materials that call to the children, and while they do have lots of decision-making opportunities, they are only provided with options that will lead them to meet desired outcomes.  The same is true for math fact instruction and practice.  

Let’s get down to the facts: in Montessori classrooms, the teachers provide direct fact memorization instruction.  At the lower elementary level in particular, one will find shelves stocked with materials that were designed to aid the process of math fact memorization.  Typically a teacher will give a child a lesson on a material to explain how it is to be used, and then detail their expectations to the child.  Children are generally going to be practicing and recording their facts on a daily basis.  Montessorians believe that while understanding the concept of why we manipulate numbers and having a visual representation helps children in the long run, we agree that when it comes down to it those basic facts really need to be memorized with an emphasis on speed and accuracy.  This is one of the reasons our children are able to solve larger complex operations problems at a younger age than children in many traditional settings.

Bead Cabinet

Taking a closer look at specific materials, how they are used, and their intended outcomes may help to refine these points.

• The bead chains/bead cabinet: Children begin using this material as early as 3 years old.  While the initial purpose is for children to learn to count, during the elementary years that skip counting translates into speedy memorization of multiplication tables.  The transition tends to be fairly smooth, as they’ve already been practicing for years!  Bonus: the same material will help them understand squares and cubes just a little further down the road in their school career.

• Addition and subtraction strip boards: While a student initially uses these materials to explore the concept of adding and subtracting, they quickly notice patterns and build speed as they gain confidence.  These materials are typically introduced at age 5 or 6.

• Multiplication and division bead boards: Like the strip boards, these are initially used by students to gain a basic understanding of the concept.  As they master individual facts, they naturally start to create shortcuts for themselves and a trained Montessori teacher will observe that they are ready to move on to more challenging materials.

• Finger boards: Created for each of the four operations, these materials are essentially wooden versions of the classic fact charts.  Children are slow to fill them in the beginning, but after repeated practice they build speed and accuracy.  Control charts are readily available for immediate feedback, ensuring that even when a child is practicing independently they will be able to know whether they are answering correctly or not.

• Blackline masters: Montessori teachers have blackline masters, which are essentially paper copies of the finger boards, that are available for children.  Children are expected to complete the material and then record the information on the paper version, thus providing another layer of repetition.  

Room for Improvement?

If there is one area that many math teachers (Montessori and otherwise) could stand to improve upon when it comes to math facts, it would be taking the time to target math fact instruction.  Making sure our students are practicing daily, quickly, and accurately is critical, but might we also help them practice smarter?

Sitting down with each individual child to gather a quick assessment periodically can make a huge difference in progress.  While gathering data in this fashion is time-consuming, we may find it well worth the effort.  Teachers can sit with a child and quickly run through a chart of facts, asking the child for answers orally.  Highlight the facts the child can answer quickly and confidently, leaving the others blank.  Teachers make a copy of this sheet so that the student can keep it to reference.  When it’s time to practice math facts, children can focus on the ones they haven’t yet memorized, rather than eating up time going over the facts they’ve already mastered.

If you are interested in reading more about Dr. Morgan’s work, follow the links to his article and the research findings:

https://www.psychologytoday.com/us/blog/children-who-struggle/201808/should-us-students-do-more-math-practice-and-drilling

http://journals.sagepub.com/stoken/rbtfl/J2BxFXoAWRPSo/full

The Keys to Handwriting Success

This is not a news flash:
Handwriting instruction is disappearing in schools across the United States.

You’ve probably already heard this sad revelation, and while it’s certainly not true for all schools, more and more are eschewing handwriting instruction to make more time for other, standards-based skills.  The result is a generation of children who are not gaining a sense of how important it is to be able to write beautifully and they are simply not learning cursive – period.  

If this makes you cringe, here’s the good news: people are noticing and speaking up, and some schools are finding ways to fit handwriting back into the schedule.  Even better news?  Montessori schools never dropped it in the first place.  Read on to learn more about how this 100+ year-old educational approach guides children in the art of writing beautifully.

Indirect Preparation

If you walk into a Montessori toddler or primary classroom, you will see very young children working with materials that develop fine motor skills.  While fine motor proficiency can serve children in a wide variety of ways, Montessori intentionally created materials that strengthen the hand as indirect preparation for handwriting.

Each time a three-year-old lifts a knobbed cylinder they are developing proper pincer grip.  This same action is repeated in many other materials.  The child may be working to joyfully refine a sensorial skill, but at the very same time their tiny fingers are slowly working their way toward being able to hold a pencil correctly.

Many Montessori materials are designed to be used working from left to right in order to prepare the child to move in that direction while writing.  Even the materials themselves are organized in a left to right fashion on the shelves.

Manipulating a Pencil

Long before they are ready to write a story (or even a word!), Montessori children begin learning how to carefully manipulate a pencil.  The metal insets are a beautiful material that were designed specifically to prepare the hand for writing.  While the shapes in the material are reminiscent of a geometry lesson, that is not the primary intention.  What’s meant to be the focus is the teaching of a variety of handwriting skills, including pencil grip, applying appropriate pressure, moving the pencil left to right, and further strengthening the muscles of the hand to build stamina.

Early Letter Formation

Montessori primary classrooms are equipped with a special material that helps children learn how to form letters.  The sandpaper letters are wooden tiles with letters made out of a sand-textured surface.  The children use their fingers to trace the shape of each letter, and later use the tiles as a reference while learning to write for the first time.  

Another option for children to practice letter formation is to use their finger and ‘draw’ the letters in a small tray of sand.  Both sand writing and using the sandpaper letters appeals to the sensorial nature of the primary child, making these activities fun.  

Cursive or Print?

By the time a Montessori student is 4 or 5 years old they begin writing joyfully because they are well prepared.  Montessori schools typically focus on teaching children to write in cursive, even in the primary classroom.  We have found that there are many benefits to emphasizing this style over manuscript/print writing.

Learning to write in cursive has many advantages:

• It’s nearly impossible to reverse letters in cursive.

• Cursive writers can read print, but the reverse is not always true.

• The ligatures in cursive may help early readers see groups of letters (oa, ing, th, and so on).

• The flow of cursive words allows the writer to focus on the ideas of the writing rather than the formation of individual letters in isolation.

A Continuation

When children enter a Montessori elementary program, their teacher will emphasize the mastery of cursive writing and take the time to review any letters or skill gaps they may have.  From here on, children practice constantly.  They have notebooks they are expected to record their daily work in, and that work is expected to be written beautifully and neatly.  Not only that, but the children themselves take great pride in the beauty of their own writing.

As time goes on, students do eventually learn skills such as keyboarding.  Fortunately, they have been given a foundation that emphasizes the power of neat handwriting.  In our fast-paced, shortcut-filled world, it’s nice to think that our children will grow up to enjoy sitting down to craft a thoughtful letter, using a pen, some paper, and their own hand.

Can Montessori Boost Your Child’s Success?

Many parents choose Montessori for their kids because they appreciate how the approach respects their children as people and as learners.  We love how the structure, materials, and lessons appeal to the developmental nature of our children.  But how does it affect kids in the long-term?  

Plenty of people are noticing the positive effects of a Montessori education and researchers are paying closer attention in recent years.  For anyone who has been involved with the philosophy for any length of time, the results are not surprising.  If you are interested in reading more about current research findings, check out this website: https://www.montessori.org/research/ .

The big question that remains is, will Montessori children grow up to become successful later in their lives?  We may want to start by examining our definition of success.  Is success the ability to score well on tests?  To be empathetic?  To be an innovator?  Regardless of how we define the word, research indicates that Montessori does seem to give kids a leg up.  

Did you know that many creative, innovative, and successful people that have and continue to shape our world attended Montessori schools?  Read on for a list of just some of those folks.

• Julia Child - Where would Americans have found their fondness for French cooking if not for Julia Child?  Her books, television show, and the realness of her delivery made a huge impact on households across the country.  Child accredited Montessori with teaching her to love working with her hands

• Jeff Bezos - Most of us recognize Bezos’ name as he is the founder of Amazon, the world’s largest online retailer.  

• Prince William and Prince Harry - The sons of Prince Charles and Princess Diana both attended Montessori schools as children.

• Prince George - Perhaps unsurprisingly, Prince George, the son of Prince William, also attended a Montessori primary school.

• Gabriel Garcia Marquez - The Nobel Laureate who wrote One Hundred Years of Solitude and Love in the Time of Cholera has said, “I do not believe there is a method better than Montessori for making children sensitive to the beauties of the world and awakening their curiosity regarding the secrets of life.

• Anne Frank - Famous for the penning of her diaries written while she was in hiding during World War II, Frank was a Montessori student.

• Larry Page and Sergey Brin - The co-founders of Google have been vocal advocates for Montessori.  In 2012 there was a Google doodle dedicated to Montessori.  Page has said, “I think it was part of that training of not following rules or orders, and being self motivated, questioning what’s going on in the world in and doing things a little bit differently.” 

• Sean “Diddy” Combs - Best known as a rapper, Combs has also worked as a talent director, songwriter, record producer, actor, and entrepreneur.  He once said, “I feel like I was nurtured into wanting to be somebody special.”

• Dakota Fanning - Fanning is an actress credited as being the youngest ever nominated for a Screen Actors Guild Award for her performance in the film I Am Sam at age seven.  She has starred in many celebrated films since.  

• Will Wright - Wright is a well-known video game designer, and creator of games such as The Sims, SimCity, and Spore.  He says that children “can become interested in pretty complex theories, like Pythagorean theory, say, by playing with blocks.  It’s all about learning on your own terms, rather than having a teacher explain stuff to you.  And when kids discover these things on their own, what they learn sticks with them so much more.”

Check out this short video of Barbara Walters interviewing Larry Page and Sergey Brin:

https://youtu.be/0C_DQxpX-Kw

Montessori Basics: Geometry from the Start

Perhaps it happens one day when your four-year-old comes home from school one day, excited to show you their work for the day.  They proudly show you a perfectly traced pentagon with elaborate, colorful patterns inside that they have created.

Maybe it’s when your eight-year-old casually references acute-angled scalene triangles.

Regardless of when it happens, as Montessori parents, there comes a moment when we become acutely aware (pun intended) of our children’s interesting knowledge of geometry.  We may recall our own study of the subject beginning much later - likely sometime during our high school years and typically not as exciting as our own children depict!  We notice that our children seem to be really ready for the information, which can feel surprising.  Not only are they ready, but the work seems to fill them with joy and satisfaction.

What, exactly, is going on?

As with so many things, Montessori discovered that young children are fully capable, and in fact developmentally primed, to learn about subjects that have traditionally been reserved for much older children.  Geometry is a perfect example.  Read on to discover what this portion of a Montessori education can offer your child.

The Primary Years

From ages 3-6 much of children’s geometry instruction in Montessori classrooms is indirect.  That is to say that while they are practicing crucial developmental skills, they are often doing so through the lens of geometry preparation.  One obvious example, as mentioned above, is with the metal insets.  Children trace a variety of geometric figures including squares, triangles, circles, curvilinear triangles, and quatrefoils, among others.  The main objective of this work is to prepare the child’s muscles for proper pencil grasp and handwriting.  When they have mastered tracing they work to create intricate designs within the figure.  

Primary children are also given a number of simple geometry lessons that allow them to begin naming figures and exploring shapes.  Wooden geometric solids are held and named by the children (cube, sphere, square-based pyramid, etc.).  The geometry cabinet is composed of drawers of related figures; small wooden insets are organized into a polygon drawer, curvilinear figure drawer, triangle drawer, and so on.  Children also use constructive triangle boxes to manipulate triangles in order to form larger triangles and other geometric figures.  The key during these early years is to give children early exposure to geometry and allow them to use their hands to explore these concepts.

The Elementary Years

During the elementary years the Montessori geometry curriculum expands significantly.  Teachers often begin by reviewing content taught during the primary years, but 6-year-olds are ready and eager for more.  This begins with a detailed study of nomenclature.  Using a series of cards and booklets that correspond with lessons given by the teacher, children explore and create their own nomenclature sets.  Topics include basic concepts such as point, line, surface, and solid, but go on to teach more in-depth studies of lines, angles, plane figures, triangles, quadrilaterals, regular polygons, and circles.  For example, when children learn about lines they begin by differentiating between straight and curved lines, but go on to learn concepts such as rays and line segments, positions (horizontal and vertical), relational positions of lines (parallel, divergent, perpendicular, etc.)

Throughout the second plane of development (ages 6-12) the study of geometry continues to spiral and go into more and more depth.  Children as young as seven learn about types of angles and how to measure them.  Eight-year-olds explore regular and irregular polygons, as well as congruency, similarity, and equivalency.  In lower elementary children begin learning about perimeter, area, and volume.

In upper elementary, children begin to learn about the connections between the visual aspects of geometry and numerical expressions.  They apply what they’ve learned about perimeter, area, and volume to measuring real-life objects - including Montessori materials they’ve seen in their classrooms since they were three years old.  They learn about things like Fibonacci numbers and Pythagoras which appeal to their sense of number order and geometric patterns.

Now, when your child comes home with surprising knowledge about geometry content, we hope you have a better idea of where they’re coming from.  If you have any questions or would like to see this type of work in action, please give us a call.

Montessori: What’s in a Name?

A common question among parents is, “What, exactly, makes a school ‘Montessori’?”  The answer is more layered than you may think.  The truth is, any school can call themselves ‘Montessori’ but the interpretation of the approach can vary greatly.  Read on to better understand the differences...

Humble Beginnings

As you may already know, Montessori education had its start in the slums of Rome, Italy.  Dr. Maria Montessori was a physician who had been studying child development.  She already found some success with institutionalized children who had been deemed uneducable.  Her first school, Casa dei Bambini, was created to serve the children of poor families while their parents worked during the day.  It was here that Dr. Montessori worked to create more materials, observe the children, and further develop her ideas and methods.  

Dr. Montessori’s successes quickly gained attention of the international community and schools began to open across the globe.  

Organization: AMI

Dr. Montessori soon realized the importance for standardization among Montessori schools.  She felt it critical to preserve the integrity of the method, ensure teachers were well-trained, distribute publications, and manufacture materials.  In 1929 she created AMI, Association Montessori Internationale, to meet these goals.

Today AMI has its headquarters in the Netherlands and supports affiliated societies in thirty-five countries around the world, including the United States.  AMI works to provide high-quality teacher training, materials, consulting services, publications, materials, and much more to Montessori schools.  AMI is the original Montessori organization and is regarded as having high standards and preserving Montessori’s original ideas, methodology, and work.

You can learn more about AMI here: https://montessori-ami.org/

Information about AMI USA can be found here: https://amiusa.org/

New Ideas: AMS

While Montessori had come to the United States much earlier, it wasn’t until the 1960s that its popularity really began to spread.  Nancy McCormick Rambusch was a young American teacher who trained at an AMI center in London.  She was appointed by Mario Montessori (Maria Montessori’s son) to be AMI’s United States representative.  Rambusch opened the Whitby School in Greenwich, Connecticut, and worked to support the spread of Montessori education in the United States.

Over time, Rambusch and her colleagues began to advocate for certain changes within Montessori.  They felt that for Montessori to be successful in the United States certain elements of the curriculum needed to remain flexible.  Leaders at AMI disagreed, arguing for preservation of Montessori’s original ideas in their entirety.  Representatives from both perspectives worked together toward a solution, but eventually parted ways and the American Montessori Society was created.

Rambusch established AMS at Whitby in 1960, and it continues to be the most prevalent Montessori organization in the United States today.  AMS functions similarly to AMI, in that it provides teacher training, publications, and resources to Montessori schools across the country, as well as to a number of international schools.

More information about AMS can be found here: http://amshq.org/

Montessori Schools Today

As mentioned earlier in this article, any school may call themselves a Montessori school.  Montessori can mean different things to different people, and it can be helpful for parents to understand the differences.  Montessori schools can be public, private, or charter schools.  They may be affiliated with a church, but most are non-denominational.  Beyond those basic definitions, the delivery of a Montessori program can vary widely.  Some of the many possibilities include:

• Montessori Inspired Schools

Educators around the world have learned from the important work of Maria Montessori.  Her texts and lectures are often regarded as some of the most respected guides to non-traditional education.  The strong emphasis on child development, individualized education, and a beautiful environment appeal to educational facilities across the nation.  Montessori materials are now much more readily available than they were even a decade ago, so preschools or even homeschool families implement them in various ways.  The creation of online programs has increased access to basic teacher preparation.  

• Montessori Member/Affiliated/Associated Schools

For a school to become an AMS full member school, the school must meet specific requirements.  Most importantly, every lead teacher at the school must be certified through an approved teacher education program (including those affiliated with AMS, AMI, and several other well-respected organizations).  Starting in 2020 there will be additional requirements for heads of school as well.

AMI requires specific standards to be met in all its schools including AMI trained teachers, a full complement of AMI approved materials, and specific requirements concerning class sizes, ratios, and organization of the work period.  Schools that meet a certain percentage of these requirements or are committed to meeting all requirements within three years may be considered affiliated or associated schools.

• Montessori Recognized/Accredited Schools

Schools who wish to be formally recognized at the highest level by either AMI or AMS must adhere to the strictest of standards.  

If a school meets all the requirements of an AMI school, they may receive an AMI Certificate of Recognition.  Schools must reapply annually.  Details on those requirements can be found here: https://amiusa.org/school-standards/

For those wishing to be accredited by AMS, the process is typically about eighteen months long and includes the writing of a self-study report, a site visit by a specially trained team of evaluators, and a commitment to ongoing evaluation and improvement.  After initial accreditation, schools must apply for reaccreditation every four years.  More information on the process can be found here: http://amshq.org/School-Resources/AMS-Member-Schools/AMS-Accredited-Schools

Still have questions about what it means to be a Montessori school?  We would be happy to chat with you.  Contact us today!

Assessment in Montessori Schools

Testing is a hot-button topic for many families.  When making decisions regarding your child’s education, it’s important to know a school’s stance on assessments.  Read on to get an idea of where Montessori schools stand.

Let’s Define Assessment…

Merriam-Webster defines assessment as:

“the action or an instance of making a judgment about something : the act of assessing something”.

If we take a look at the evolution of the word itself, we find that assess comes from the Latin word assidere, which means ‘to sit beside’.

While any type of assessment is a means of judging progress, Montessori teachers take the Latin root to heart.  We literally sit beside the child, observing and assessing as we go.

While there are many different forms assessment can take, most of them can fit into two main categories: formative and summative assessment.  Formative assessment happens while the teaching and learning is taking place.  This is the type that Montessori teachers rely heavily on.  It allows teachers to shift gears mid-lesson and to get an instant record of how a child is doing with a particular skill at any given moment.  Summative assessment is more like your traditional test at the end of a unit, or a major standardized test at the end of the year.  These tests are typically data collection points and are often used mostly by the adults and not to give feedback to the student.  

How Do Montessori Teachers Track Progress?

Notes.  Notebooks full of thoughtful and detailed handwritten notes.  At least that’s the traditional way of recording progress.  Many schools are now shifting over to digital platforms that are created for and cater specifically to Montessori schools and their goals and values.  Still, many Montessori teachers continue to keep their own detailed records by hand.

Montessori teachers are masters of observation.  They think like scientists and spend lots of time sitting back and quietly watching the children at work.  When they’re not giving lessons, they’re observing.  They write all these observations down and then review them later to help decide what lessons to revisit, what new materials to present, or even what parts of the classroom environment need attention or change. 

How is Mastery Evaluated?

Often, mastery is evaluated while the teacher is giving a lesson.  Montessori developed a fascinating tool called the ‘three-period lesson’.  When a teacher is presenting new material to a child, they may only present the first period, or the first two, depending upon how the child reacts to the work during the lesson.  When the teacher suspects mastery, the third period portion will be given.  There is a certain amount of variation depending on the subject matter, but the general pattern is as follows:

• First Period: “This is ____.” The teacher introduces the skill.  If the child is to learn the parts of a mountain, the teacher may say, “This is the summit.  The summit is the highest point of a mountain.”  A visual will be presented along with any other supporting materials.
• Second Period: “Where is ____?” The teacher provides part of the information and asks the child to identify the rest.  For example the cards highlighting the various parts of the mountain may be laid out and the teacher asks the child to point to each defined part in turn.  “Where is the summit?”
• Third Period: “What is ____?” The teacher is determining whether the child can independently recall the information.  The mountain cards are now laid out without any labels, and the child must identify the parts without any cues.  “What is this part?”

The best part?  Because of the beauty of the materials and the tone of the classroom, the child perceives this as a sort of game rather than a test to be dreaded.

What About When Children Get Older?

Parents often wonder how their children will make the transition into their local public schools or other more traditional private schools once they age out of their Montessori school.  This is where there’s a little more variability.  Different schools take different approaches, but some give the option of offering some form of standardized testing for their oldest students.  This could be in the form of state testing, or something similar.  This testing is typically not a requirement, but is sometimes an option for students or families who are interested.  Contact us to learn more about how our school handles this transition.

A Note About Self-Assessment:

Montessori classrooms are not just designed for teachers to assess the students, but also for the students to assess themselves.  This is done in two main ways.

Most Montessori materials are autodidactic, that is they are self-teaching.  They have been intentionally developed in such a way that the child can not complete the work incorrectly, or there is a built-in means for them to check their own work.  This looks different at the different levels and is best understood by visiting a classroom to observe, which we always encourage parents and prospective parents to do when they are curious.  When given a lesson on how to use a material correctly, the children learn about these built-in tools and how they can use them to guide their work.

Secondly, Montessori students are taught to be reflective.  As they get older (typically elementary and above), individual meetings with their teacher give them the opportunity to be an active participant in planning their own education.  They are not told what they must do, but they are asked how they plan to accomplish specific goals.  Some of these goals are set by the teacher but others are set by the child.  When needed, teachers will give strategies and suggestions, but the hope is that eventually the child will develop more of these on their own.  

We want our children to be able to take a look at their work and evaluate it with a critical eye, while still basking in the joy of accomplishment and learning.  By not passing obvious judgement in the form of grades or other traditional feedback methods, Montessori children come to see their learning as a constantly fluctuating process that they are empowered by.  If we can instill those values in them as children just imagine what they will be capable of as adults.