Montessori Math Why is it successful? Dilek Buchholz, Ph.D. Belfair Montessori Magnet Elementary Pre-K/K Montessori Teacher dbuchholz@ebrschools.org Pamela Autrey, Ph.D. Belfair Montessori Magnet Elementary Pre-K/K Montessori teacher pautrey@ebrschools.org Presented at the EBR Flip Flop Professional Development, June 13, 2012 1

Montessori Environment The Montessori prepared environment is holistic; it all works together, through both direct and indirect preparation. 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 writing 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 Sensorial materials are designed to isolate one sensation, teach similarities and differences and the correct language for each gradation of the sensation. They offer the subconscious experience with a base ten system.

Montessori math: The Search for Ten Red Rods: Order based on length (sensorial-precursor 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 NUMBER SENSE

Colored Bead Stair: Fixed Quantity, Loose Order Hanging Bead Stair Table Rods

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Montessori math: Ten and some more (11 – 19) Ten Bars and Colored Bead Stair Teen Boards Hanging Teens Number sense

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In a base ten system, there can never be ten of anything. Montessori math In a base ten system, there can never be ten of anything.

Montessori math: 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 Cube Leads to four-digit addition (static addition) Dynamic addition (carrying) Subtraction and other operations

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Young children love order Give out standards cut in strips. Prepare 35 sets. Let’s look at the Montessori math materials. Can you identify the materials with common core standards? Young children love order

CCSS AND MONTESSORI MATERIALS 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; 9-lay-out; 100 board; bead cabinet (1-10 squared) and long bead chains 1-10 cubed).

Counting & cardinality continued… K.CC.2: Count forward beginning from a given number within the known sequence 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 objects). All of the above in K.CC.1 and addition strip board; 45 layout; snake game; handful exchanging; bank game. 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 Chain

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. K.CC.7: Compare two numbers between 1 and 10 presented as written numerals. Red & blue rods; table rods; bead bar stair; graphing activity; making charts; cards & counters. Sandpaper numerals, numeral cards; written extensions; graphing activity; making charts.

Montessori math 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 taught. The three periods are (1) “This is …,” (2) “Show me …,” and (3) “What is this?” Used for teaching vocabulary, materials, and everything. Integral part of the Montessori Way 21

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. 22

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 equations. 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 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

Geometry materials

Montessori math There must be conversations about math and geometry.

GEOMETRY: identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders and spheres). 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 to. 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 shapes 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 rectangle?” Same as above Clay, playdough, toothpicks, art activities using various media

Measurement & Data Describe and compare measurable attributes 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.1.: Describe measurable attributes of objects, such as length, or weight. Describe several measurable attributes of a single object.

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 findings.

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

The Case for Montessori education 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 intervention. 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 3.) 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

Useful Websites

Neuroscience http://www.learner.org/courses/neuroscience/common_includes/si_flowplayer.html?pid=2448 http://www.ted.com/talks/lang/en/rebecca_saxe_how_brains_make_moral_judgments.html   http://www.ted.com/talks/lang/en/patricia_kuhl_the_linguistic_genius_of_babies.html http://www.deakgroup.com/pdocs/brainbasics2.pdf