Making Math Facts Memorable 2016 AMS Conference Chicago, Illinois Betsy Lockhart Montessori Education Center of the Rockies lockhart-learning.com

Handouts 1. Procedure for teaching skip-counting by 6 Demonstrated method by skip-counting by 17 during workshop How to generalize the process for skip counting by 6, 7, 8 in your classroom

2. Sample chart for addition facts Demonstrated use of multiplication chart during workshop Sample provided here to enable teachers to create all four operations at home

3. Resources and key concepts



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Material: Red and Blue Rod (10 only) • Lesson Notes: Direct instruction – Skip Counting the Trickier Multiples This is a group activity for harder to remember multiplication tables (usually 6, 7, 8, or maybe 12) that are really hanging up the children after significant practice with bead bars and long chains, and might be a great preparation for the mini tramp / jump rope. Think of it as a way to unstick children who can skip count with partial accuracy for a given number and are weary of the process used for all of the tables that have come before. Do not use this routinely! The initial presentation can be done in 10 minutes. Choose a times table that many children are finding slippery. Here, we will illustrate teaching skip counting by 6. Materials: Red and Blue 10-rod, two colors of post-it-notes, sharpie. • Start by labeling the Red and Blue Rod with pink post-its at the leftmost position (0) and the rightmost position (60). • At the first position, ask what times tables we are learning today . Label the first position with a yellow post-it labeled 6. • Ask what double the first position value is. Mark the second position with a yellow post-it labeled 12. • “What is double 12.” . “Am I going to put this on the next position? Why not? Where should it go?” • Now double the 24. . “Yes, and I am going to put it there.” (place a yellow postit labeled 48 at the 8th position. When working with adults, ask where it should go) • “Can you think of a way that I can get this number?” (pointing to position 3) We can add 6 to 12 or subtract 6 from 24. Which is easier? Place a yellow post-it labeled 18 at position 3. • “What is double 18?” OPTIONAL: Ask where to place it on the number line. Place a yellow post-it labeled 36 at position 6. • “What is 6 more than 36?” Place a yellow post-it labeled 42 at position 7. • “There are three ways of getting the number in the middle. What are they?” ! add 6 to 24 ! subtract 6 from 36 ! take half of the largest number, 60 " may omit this for children Place 30 (on a pink post-it) at the center (position 5) • “There is just one left. There are three ways of getting this number. What are they?” ! add 6 to 48 ! subtract 6 from 60 ! triple the number in position 3 (18) " point this out to the children (a cue). • Place a yellow post-it marked 54 in position 9. • “My challenge to you is to be able to say all of these multiples in order (skip count by 6) in about 6-7 minutes, without any of the stickers on the rod!” • “Shall we try saying them together to start with?” • “OK, the rules are that you say what you think out loud. It is ok to make a mistake; we are all learning. In fact, we will all be concentrating so hard, we won’t even

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notice. If you goof, join back in as soon as you can. If you can only do some, join in on the ones that you can.” “Now we are going to remove some of the tickets. Ready? Do we need the 0? And we know that we are doing the 6 times tables, so we don’t need that number. There’s 2 gone already. Do we need 60? We know that 6 x 10 is 60.” Children skip count from 0 – 60. Pointing at each space in turn. “Now the times table that we are doing is? And double that is? . And you know that.” Remove the 12 ticket. Pointing, “Tell me what times table we are doing. And double it? What goes at this end? And at this end? Fantastic! Are you ready?” Children skip count from 0 – 60. “Fantastic! Tell me the times table. Double that. Double that. ” Remove the 24. Pointing, “OK, what have I just taken off? What is half of that? What is half of that . OK, what have I just taken off? What is half of that? What is half of that? . (Note: if children struggle with “half of that”, simply go back to building up ! “What is the times table that we are doing?, Double that. Double that. Just be sure that your cues remain the same throughout the activity) “Fantastic! Are you ready?” Children skip count from 0 – 60. “Brilliant. Now this one (3rd position) is extremely important. It is so important that we call it the key. 18 is the key. Tell your neighbor what the key is.” Remove the ticket marked 18. Pointing, “What is the key?” Pointing to the 4th position, “What did I take off here?” What is half of that? What is half of that? . Tell me the key again. Wonderful! Are you ready? Children skip count from 0 – 60. Pointing throughout: “Tell me the key. Double it. .” Remove 36. It’s so easy! We know that double 18 is 36! Tell me the key. Double it , “What did I take off here?” What is half of that? What is half of that? . Wonderful! Are you ready?” Children skip count from 0 – 60. “Tell me the key. Double it. . Add 6. “ Remove 42. “This is just so easy! We know that 36 + 6 is 42! Tell me the key. Double it , add 6 What did I take off here?” What is half of that? What is half of that? . Wonderful! Are you ready?” Children skip count from 0 – 60. “Put your fingers on your noses. What goes in the middle? ” Remove 30. Placing finger on nose. “What goes in the middle? What’s the key. Double it , add 6 What did I take off here?” What is half of that? What is half of that? . And what goes in the middle? Wonderful! Are you ready?“ Children skip count from 0 – 60. “Just 2 more to go! You are doing so well!“ Pointing, “What’s the key. Double it , Triple it . It’s so easy!” 2

Remove 54. “Tell me the key. Double it , Triple it . What goes in the middle? What did I take off here?” What is half of that? What is half of that? . “Tell me the key. Double it , Add 6 it . Are we ready again?” • Children skip count from 0 – 60. • “Now I have a confession to make. I can’t remember this one, so that is your job to remember this one! What is the one I can’t remember? . Tell me again, what is the one I can’t remember? Now what goes in the middle? What did I take off here?” What is half of that? What is half of that? . “Tell me the key. Double it , Triple it What’s the one I can’t remember? Back to the key double it Add 6 it . OK, this is your big moment for glory! Are you ready for this? • Children skip count from 0 – 60. To keep this from being nothing more than a parlor trick, these numbers must go from short-term to long-term memory. Repeat an abbreviated version of this (from whatever point the children start to slow down / struggle to the end) every day for a few days – perhaps a week. Encourage children to practice skip counting this set of numbers on the mini-tramp with a friend verifying that the numbers are correct. To take it to true multiplication, label the red and blue rod with the numbers from 0 to 10 rather than the multiples of 6. Repeat the same routine with the same words, “What is the key?” Repeat this with the children on a closely spaced interval (daily?) until they are responding automatically. Then change your language to, “6x3”, etc. Have fun!



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Why do we care?

Why do we care?

Not a frivolous question!

Why do Montessorians care? •  Correct answers when working abstractly

•  Facilitates transition from materials

•  Vital to child’s comprehension of the gestalt

How do we define “automaticity”? How fast is fast enough? •  30-50 correct responses per minute – oral - 2 seconds per fact if oral

•  30 facts in 3 minutes - written -  3.6-6 seconds per fact if written

My definition of “automaticity”

A fact is “automatic” if you can retrieve it without researching it.

Goals •  Insure that children understand why automatic fact recall is important for them, engage in goal-setting, measuring and celebrating •  Understand how the brain learns ready-recall facts and the modalities used to process /store information •  Adapt OUR strategies/practices to engage multiple modalities and optimize duration/frequency of practice •  Individualize activities so that the majority of time is spent on facts still being mastered •  Have FUN!

“We Value Math Fact Mastery” •  Why should I want to master math facts? (a.k.a. What’s in it for me?) •  Establish a mentor system •  Create goals; celebrate success

“Why should children learn math facts?” •  How would your children respond?

“Why should children learn math facts?” •  How would your children respond? •  Reasons that will resonate with some: -  You feel super-smart! - It makes the math operations work easier and more fun, like knowing sounds of individual letters makes reading fun! -  When new lessons are presented, it is easier to get the new concepts! -  It helps you see patterns, like solving a mystery! •  You can amaze your family and friends by doing math in the real world! •  (Last resort) It will help you abstract more quickly.

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Establish a Mentor System •  It is the Montessori way!

The “Math Fact Master” Process •  Child self-identifies as knowing all + facts •  Child shows proficiency •  Celebrate new Addition Fact Master –  Acknowledge achievement at line –  “How’d you do it?” –  Present a token –  Identify the child as a resource –  “What is your next goal?” –  Acknowledge all Addition Fact Masters –  Take a photo

•  Elementary children are social beings!

•  Elementary children love to be philanthropic!

Anatomy of the Brain

•  Special Celebration when all 4 operations mastered

Each Lobe has Two Hemispheres

Frontal lobe—movement, planning, reasoning, responses to our surroundings, emotions, word meanings Parietal lobe—sense of touch, focusing on and understanding our surroundings, naming things, writing words, drawing, reading Occipital lobe—vision Temporal lobes—hearing, memory, understanding speech, recognizing faces, behavior

Montessori Math and Brain Function

Anatomy of the Brain Brain stem — (grey) connects the brain to the spinal cord. It controls breathing, heart rate, swallowing, alertness, sleep, balance, blood pressure, digestion, temperature, and sweating. Cerebellum — (purple) behind the brain stem and below the occipital lobe. It controls coordination of movements, balance, equilibrium Cerebrum — outer layers of the brain, largest and most complex part of the brain made up of lobes. The lobes have different jobs…

Brain-Based Learning

Instruction that engages all four lobes, integrates both hemispheres, and activates the pre-frontal cortex is a complex of neural connections acting like a net, catching and retaining more concepts.

The beauty of Montessori Math is that it links the analytical thought processes inherent in calculations with the spatial awareness of geometry. It not only engages all 4 lobes, but also both hemispheres.

Learning vs. Practicing

Stages of Math Fact Mastery •  •  •  • 

Understand the Concept: Addition

Understanding the Concept Math Fact Exploration Math Fact Research Reach for Automaticity –  Assess Known Facts –  Narrow the Focus –  Targeted Practice

•  Achieve “Master” Status •  Maintenance

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Explore: Addition Combinations to 10

Explore: Addition Combinations to 10

Research: Addition Combinations to 10

Exploration: Addition Combinations to 0!18

Exploration: Addition Combinations to 0!18

Exploration: Addition Combinations to 0!18

Research: Addition Combinations to 0!18

Research: Addition Combinations to 0!18

Reach for Automaticity: Assess Known Facts

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Reach for Automaticity: Narrowing the Unknown • Revisit Commutative Law (Research Essential Combinations) • Direct instruction on fact families as needed •  +0 •  +1 •  +2 •  +4 •  +3 •  +5 •  +6 •  +9 •  +8 •  +7 • No pattern in holes – proceed to “Targeted Practice”

Target Practice

Practicing math facts

8 + 5 ==

–  Basketball –  Jump rope –  Mini-tramp –  Hand-claps alone or with partner

6 + 8 = 14 8 + 5 = 13

5 + 7 = 12 9 + 7 = 16

9 + 6 = 15 8 + 7 = 15

Target Practice –  Basketball –  Jump rope –  Mini-tramp –  Hand-claps with partner

–  Basketball –  Jump rope –  Mini-tramp –  Hand-claps with partner

5+7= 9+7=

•  Kinesthetic Activities

•  Kinesthetic Activities

•  Kinesthetic Activities

6+8= 8+5=

Target Practice

•  Slippery Facts –  Personal flashcards –  Rhymes –  Songs

9+6= 8+7=

Maintenance

Reassessing

•  Serve as an Addition Fact Master

•  Weekly assessment of + while learning x

•  Software practice with + for speed 8+5=

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xtramath.org

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Stages of Math Fact Mastery

Skip Counting in the Classroom

Understanding the Concept Math Fact Exploration Math Fact Research Reach for Automaticity

•  Morning line

–  Assess Known Facts –  Narrow the Focus –  Targeted Practice

•  Achieve “Master” Status •  Maintenance

–  Song / chant https://www.youtube.com/watch? v=uIwjuKcWsfQ –  Multiples Slap

•  Lesson gathering •  When lined up to leave for an activity

Goals •  Insure that children understand why automatic fact recall is important for them, engage in goal-setting, measuring and celebrating •  Understand how the brain learns ready-recall facts and the modalities used to process /store information •  Adapt strategies/practices to engage multiple modalities and optimize duration/frequency of practice •  Individualize activities so that the majority of time is spent on facts still being mastered •  Have FUN!

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