math

Math Fact Memorization: Montessori & Current Research

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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

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

Montessori Basics: Geometry from the Start

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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 Basics: How Math Progresses Through the Levels

Addition Strip Board

You know your four year old loves their classroom and their work.  You know their teachers are guiding them to learn early math skills.  But what, exactly, does that look like?  And how does it change as they get older?  Montessori math materials are nothing short of amazing.  While they look quite different than what we used growing up (pencil and paper?) there are intentional reasons for these methods.  Read on to learn more...

The Basics

Much of the Montessori curriculum is based on giving children exposure to concrete materials first, then giving them incremental opportunities to work to more abstract concepts.  This is no different when it comes to math.

What do we mean by concrete?  The children are able to hold a material in their hands.  The materials are symbolic or representative of something else (a number, perhaps), and that symbolism changes over time until children are ready to let go of the materials and find solutions on paper or even in their heads.  This idea of mastering a skill without the assistance of materials is what we refer to as abstraction. 

Number Rods

What Does Primary (Early Childhood) Math Look Like?

At the primary level math starts out simple, but you may be surprised at how much preschoolers are capable of.  

Even before a child is able to count, they experience the skill using materials like the number rods, a series of blue and red colored wooden rods that are arranged in a stair-like pattern.  Children learn how to count using a variety of materials.  The spindle box is an early material with which children place the correct amount of wooden spindles in compartments labeled 1-9.  Sandpaper numbers (just like their letter counterparts!) teach children how to correctly form each number to develop readiness for writing them on paper.

When a child is ready to learn about basic operations, there are plenty of materials to support them.  Montessori math uses the golden bead material; first to build numbers into the thousands.  For example a single golden bead represents 1, a group of 10 beads are strung together in a straight line for 10, and 100 beads are affixed into a flat square.  The thousand cube is as large as 1,000 of the original single ‘1’ bead.  Once a child is able to build a visual representation of a number, the beads are used to teach basic operations.  Young children are able to add, subtract, multiply, and divide numbers into the thousands using this material.  They first learn with static problems - that is, with no exchanges - and then move on to more complex, dynamic problems.  They quickly learn that ten 1s is equal to one 10, and they do this by holding those numbers in their hands.

Golden Beads

Montessori recognizes the importance of memorizing basic facts.  While when we were young we may have used flashcards to drill these facts into our heads, the Montessori approach begins by showing children why we manipulate numbers in different ways.  Young children appreciate the repetitive nature of the materials, which gives them plenty of opportunities to practice  (and memorize!) these facts.  The addition and subtraction strip boards show a child visually what is happening when we add numbers.  The same goes for the multiplication and division bead boards (which use small beads placed in divots on a wooden board to create an array).  

Division Board

A Period of Overlap

Somewhere between kindergarten and the first year of lower elementary, children are taught to use new math materials depending upon their individual readiness.  The stamp game is a classic example.  

The stamp game material is a sectioned box with small colored tiles sorted inside.  There are labeled green, ‘one’ tiles, blue ‘ten’ tiles, red ‘hundred’ tiles, and green ‘thousand’ tiles.  Instead of holding a large cube that actually shows the relative size of one thousand as they did with the golden beads, they are now representing series of tiles that are all the same size, but are differentiated only by their color and number label.  Like the golden beads, the stamp game material is used to teach all four operations, with children adding, subtracting, multiplying, and dividing into the thousands.  Some children begin this work in their primary classroom and continue when they reach elementary, while others begin once they enter their lower elementary classroom.

Stamp Game

It may be interesting to note that there are some Montessori materials that children spiral back to, over and over again, from ages 3 to 12!  The bead chains are a colorful, quintessential Montessori material.  In the primary classroom, children use them to learn how to count, and perhaps how to skip count.  In a lower elementary classroom they are used for skip counting and to help memorize multiplication facts.  In upper elementary children use them to solidify concepts like squaring and cubing, although they were indirectly preparing for that work for years previously. 

Bead Chains

What Does Elementary Math Look Like?

Remember the green, blue, and red tiles of the stamp game?  Montessori refers to those as the hierarchical colors, and they are used to teach children about number series.  They first appear in the stamp game, but they continue to follow the child through lower elementary and into upper elementary until they have a firm grasp on the idea of the simple family of numbers (ones, tens, hundreds), the thousand family (thousands, ten thousands, hundred thousands), and so on.

After a child masters operations with the stamp game, they move on to use a material called the bead frame, which can teach addition, subtraction, and multiplication.  It looks a bit like an abacus, but with ten beads on each rod in the hierarchical colors.  After a child masters the bead frame, they are typically ready to add and subtract into the thousands (and beyond!) using just pencil and paper.

To learn larger multiplication problems, children use a material called the checkerboard.  They begin small, but eventually work their way up to problems that have three or four digit multipliers.  For long division, children use a material that goes by different names at different schools: the racks and tubes, aka the test tube material.  Once children master the checkerboard and racks and tubes, they are able to multiply and divide large numbers without materials.

Decimal Checkerboard

During the elementary years fact memorization continues.  In early lower elementary, many children continue to use the strip boards and bead boards of their primary years, but eventually move on to using finger boards and tables in which they place numbered tiles.  Children notice the patterns numbers make, giving them more tools to memorize their facts.

There’s More!

Of course, math isn’t just about operations.  Montessori students learn about geometry and fractions from an early age. 

Geometric Solids

Did you know that primary children learn the names of geometric solids?  They can easily identify not just cubes and spheres, but square based pyramids, rectangular prisms, ellipsoids, and more.  As they move into elementary they learn about range of concepts, including studies of angles, triangles, polygons, and so much more.  A third grader can easily identify a right-angled, isosceles triangle.

Fraction Insets

When it comes to fractions, first graders start out simple with an impressionist lesson involving an apple and a definition of fractions that includes how they must always be fairly divided (the connections between fractions and division are impressed early on).  They next move on to using fraction insets, which look a lot like the metal insets they used for handwriting preparation in their primary classrooms.  Before you know it, many third graders are learning to multiply and divide fractions.

Still Curious?

The best way to really understand Montessori math is to see the materials in action.  Schedule a visit to watch children using them in the classroom, or join us for our upcoming parent education session on Monday, March 26, 2018, 6:00-7:30.  Contact us for more details!  

Check out the stamp game in action: