Montessori Math - ebrprofessionaldevelopmentportal

• Why is it successful?
Dilek Buchholz, Ph.D.
Belfair Montessori Magnet Elementary
Pre-K/K Montessori Teacher
[email protected]
Pamela Autrey, Ph.D.
Belfair Montessori Magnet Elementary
Pre-K/K Montessori teacher
[email protected]
Presented at the EBR Flip Flop Professional Development, June 13, 2012
Montessori Environment
• The Montessori prepared environment is holistic; it all
works together, through both direct and indirect
• There are 4 major areas: Practical Life (aka
Everyday Living), Sensorial, Language , & Math.
• The purpose of Practical Life & Sensorial are:
• to prepare & train the hand,-the muscles of the mind- for
• to develop eye-hand coordination
• To increase concentration & independence.
• Materials are designed to train the senses and
teach vocabulary (e.g. long, short, wide, narrow)
precede the Math materials.
• Sensorial materials are designed to isolate one
sensation, teach similarities and differences and the
correct language for each gradation of the
• They offer the subconscious experience with a base
ten system.
• Red Rods: Order based on length (sensorialprecursor to red & blue rods)
• Red and Blue Rods (aka Number Rods): Fixed
Quantity, Loose Order
• Sandpaper Numerals
• Spindle Box: Fixed Order, Loose Quantity
• Cards and Counters: Loose Order, Loose Quantity
• Memory Game
• Colored Bead Stair: Fixed Quantity, Loose Order
• Hanging Bead Stair
• Table Rods
• Ten Bars and Colored Bead Stair
• Teen Boards
• Hanging Teens
20 - 100
Ten Boards
Squaring Chains – Skip Counting
One Hundred Board
Hundred Chain
1000 Chain
Decimal System
• Introductory Tray (1, 10, 100, 1000)
• Tray of Nines (9 units, 9 tens, 9 hundreds, Thousand
• Leads to four-digit addition (static addition)
• Dynamic addition (carrying)
• Subtraction and other operations
Let’s look at the Montessori math materials.
Can you identify the materials with common core standards?
Counting & Cardinality
• K.CC.1: count to100 by ones
and by tens.
• Pink Tower, red, rods, brown
prisms; red & blue rods;
table rods; sandpaper
numerals, spindle box; cards
& counters, memory game,
short bead stair; short bead
stair hanging rack; teen
boards; ten boards; ten
beads; teen beads hanging
rack; introduction tray; 9lay-out; 100 board; bead
cabinet (1-10 squared) and
long bead chains 1-10
Counting & cardinality continued…
• K.CC.2: Count forward
beginning from a given
number within the known
• K.CC.3: Write numbers from
0-20, represent a number of
objects within a written
numeral 0-20 (with 0
representing a count of no
• All of the above in K.CC.1
and addition strip board; 45
layout; snake game;
handful exchanging; bank
• Practical life activities; art
activities; chalkboard
exercises; metal insets;
sandpaper numerals, sand
tray; number work
extensions for various math
activities that include math
papers, spindle box
Counting & cardinality continued…
• K.CC.4: Count to tell the
number of the object:
Understand the relationship
between numbers and
quantities; connect
counting to cardinality
• K.CC.4a: When counting
objects, say the number
names in the standard
order, pairing each object
with one and only one
number name and each
number name with one and
only one object.
• Red Rods, Blue and Red
Rods with number cards,
table rods; Sandpaper
Numerals, Spindle Box,
Cards and Counters,
Memory Game, Short Bead
Stair, Hanging Rack, Teen
Beads, Teen Boards, Teen
Hanging Rack, Ten Boards,
Introduction Tray, Tray of
Nines, One Hundred Board,
Squaring Chains, Thousand
Counting & cardinality continued…
• K.CC.4b: Understand that
last number name said tells
the number of objects
counted. The number of
objects is the same
regardless of their
arrangement or the order in
which they were counted.
• K.CC.4c:Understand that
each successive number
name refers to quantity that
is one larger.
• Red & Blue Rods with
number cards, table rods;
Sandpaper Numerals,
Spindle Box, Cards and
Counters, Memory Game,
Short Bead Stair, Hanging
Rack, Teen Beads, Teen
Boards, Teen Hanging Rack,
Ten Boards, Introduction
Tray, Tray of Nines, One
Hundred Board, Squaring
Chains, Thousand Chain
Counting and cardinality
• K.CC.5: Count to answer
“how many?” questions
about as many as 20 things
arranged in a line, a
rectangular array, or a
circle, or as many as 10
things in a scattered
configuration, given a
number from 1-20, count
out that many objects.
• Practical life activities; pink
tower, brown prisms;
knobbed cylinders; red rods;
constructive triangle boxes;
geometric solids; geometric
cabinet; Montessori bells;
metal insets, the farm.
•What exist in the mind must first exist in the hand.
• K.CC.6: identify whether the
number of objects in one
group is greater than, less
than; or equal to the
number of objects in
another group, e.g., by
using matching and
counting strategies.
• Red & blue rods; table rods;
bead bar stair; graphing
activity; making charts;
cards & counters.
• K.CC.7: Compare two
numbers between 1 and 10
presented as written
• Sandpaper numerals,
numeral cards; written
extensions; graphing
activity; making charts.
• Parallel Work: A child can be working with the Teen
Boards while learning the colors of the short bead
stair; but mastery is required for the child to move
on. Clarity is a gift to the teacher as well as the
student. The teacher knows the child understands
when they have mastered a material. There is also
a time to challenge the child. Observation of the
child is the key to both. Mastery is checked through
the Three Period Lesson.
Three Period Lesson
• Montessori materials isolate the concept to be
• The three periods are (1) “This is …,” (2) “Show me
…,” and (3) “What is this?”
• Used for teaching vocabulary, materials, and
• Integral part of the Montessori Way
Response to Intervention
• With sequenced materials, you can always go back to isolate
what child is not getting.
• Go back to the most concrete material for that objective.
• Red and Blue Rods are used over and over again; until a child
is asked to do something with their knowledge, you cannot
be sure they have it.
• Golden Beads are always used to introduce operations.
• Squaring Chains teach skip counting; subconscious
preparation for addition and multiplication.
• Many kindergarteners realize, at some point, they do not
need materials; they have internalized the materials.
Operations and Algebraic Thinking
Understand addition as putting together and adding to, and understand
subtraction as taking apart and taking from.
• K.OA.1: Represent addition
and subtraction with
objects, fingers, mental
images, drawings, sounds
(e.g. claps), acting out
situations, verbal
explanations, expressions, or
• K.OA.2: Solve addition and
subtraction word problems,
and add and subtract
within 10, e.g. by using
objects, or drawings to
represent the problem.
• Addition Strip Board, Red
and Blue Rods, Addition with
Bead Bars (0 – 10), addition
finger chart with 5 control
charts, addition with Golden
Beads (0 to 9,999),
subtraction strip board,
Subtraction Charts,
Equation Boxes, Problem
Tickets, Equation Booklets,
Handful Exchanging,
Exchange Game, Stamp
Game, Dot game, positive
and negative snake game
All of the materials in K.OA1 to K.OA.5 apply to K.NBT.1: Work
with numbers 11-19 to gain foundations for place value.
• Stamp Game: Bridge to Abstraction: For each unit
tile, you say, “This stands for one unit.”
• Clarity
• K.OA.3: Decompose numbers less
than or equal to 10 into pairs in
more than one way, e.g. , by using
objects or drawings, and records
each decomposition by a drawing
or equation (e.g., 5=2+3 and 5=4+1).
• K.OA.4: For any number from 1 to 19,
find the number that makes 10
when added to the given number,
e.g. by using objects or drawings,
and record the answer with a
drawing or equation.
• K.OA5: Fluently add and subtract
within 5.
• Same as above
• Same as above
• Same as above
• There must be conversations about math and
GEOMETRY: identify and describe shapes (squares, circles,
triangles, rectangles, hexagons, cubes, cones, cylinders and
• K.G.1: Describe objects in the
environment using names of
shapes, and describe the relative
positions of these objects using
terms such as above, below,
beside, in front of, behind, and next
• K.G.2: Correctly name shapes
regardless of their orientations or
overall size.
• K.G.3: Identify shapes as two
dimensional (lying in a plane, “flat”)
or three dimensional “solid”).
• Pink tower, brown prisms,
red rods, knobbed cylinder
blocks, knobless cylinders,
geometric cabinet,
geometric solids, and all
appropriate extensions,
constructive triangles, metal
insets, botany cabinet, the
farm, preposition game
• Origami
• Same as above
Analyze, compare, create, and compose
• K.G.4: Analyze and compare two-and three
dimensional shapes in different sizes, and
orientations, using informal language to
describe their similarities, differences, parts
(e.g., number of sides and vertices/corners)
and other attributes (e.g., having sides of
equal length.
• K.G.5:Model shapes in the world by building
shapes from components (e.g. sticks and
clay balls) and drawing shapes.
• K.G.6: Compose simple shapes to form
larger shapes. E.g., “Can you join these two
triangles with full sides touching to make a
• Same as
• Clay,
art activities
Measurement & Data
Describe and compare measurable attributes
• K.MD.1.: Describe
measurable attributes of
objects, such as length, or
weight. Describe several
measurable attributes of a
single object.
• Dry & wet transferring
(pouring, spooning, tonging,
etc.) cooking with
measuring spoons, cups,
bowls, and ingredients, red
rods, pink tower, brown
prisms, knobbed cylinders,
knobless cylinders,
constructive triangles,
geometric cabinet, botany
cabinet, 3 various color
boxes, thermic tablets,
geometric solids, baric
tablets, pressure cylinders,
Montessori bells, red rods
with red & blue rods.
• K.MD.2: Directly compare
two objects with a
measurable attribute in
common, to see which
object has “more of/less of”
the attribute, and describe
the difference. E.g., directly
compare the heights of two
children and describe one
child as taller/shorter.
• Comparing contrasting; all
of the above; graphing
activities, written/drawn
Classify Objects and count the number of
objects in each category
• K.MD.3: Classify objects into
given categories; count the
numbers of objects in each
category and sort the
categories by count.
• All of the above
• However, Montessori materials are helpful at any
point in children’s development; these materials
can be useful assets in a traditional classroom.
Some children fail at math because they do not
have a clear concept of what a number means.
There are many extensions to provide for varied
• It is important that the child really has number
sense. Number sense is not only rational counting. It
is understanding numbers (quantity), numerals
(symbols) and relationships between numbers in a
number system.
So, why is Montessori math successful?
• Concrete materials
• Long time span (practical life & sensorial studies start at age
• Many materials for the same concepts
• Sequential scope & materials
• Mastery before new skill
• Explaining how they got the answer
• Individual attention
• Observation- of the child & the child observing other children
• Isolated skill
• Repetition

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