2 Math Manipulatives That Will Solve Your Math Problems

April 30, 2022

"What the hand does, the mind remembers." - Maria Montessori

Research shows over 7% of students experience severe learning disabilities in math. Countless more struggle because of the way the language of math is spoken to them. You might as well be speaking Greek. Our goal at ShillerLearning is to make math understandable for every child with hands-on, exploratory learning – learning by doing.

Is your child struggling with math or would you like to avoid your child having math problems in the first place? The decimal material (also known as base ten blocks) paired with Montessori number cards are a game-changer for students young and old. I wonder how many parents, who themselves struggled with math as children, would have had an easier time learning (and even loving) math if they had worked with these tools.

As Dr. Maria Montessori so eloquently stated, "What the hand does, the mind remembers."

Together these hands-on tools make the decimal system and place value visually obvious and number sense easy to acquire – something with which many students struggle. These easy-to-use manipulatives and accompanying activities set your child up for math success and banish the tears and frustration. They provide a concrete introduction to the decimal system in an amazingly understandable way and can be added to any math instruction.

What are they?

The decimal material physically represent each place value category. There are unit cubes (1 cm x 1 cm x 1 cm), ten rods (1 cm x 10 cm x 1 cm), hundred flats (10 cm x 10 cm x 1 cm), and thousand cubes (10 cm cubed). Used in sufficient quantity, these math manipulates not only help children visually quantify each place value amount, but instruct in and provide learning by doing for inequalities, multi-digit operations (addition, subtraction, multiplication, and division), exchanges within operations (borrowing and carrying), skip counting, and more.

The decimal material is introduced using the Montessori three-period lesson – a simple, but effective gap preventer that ensures the student understands the concepts and materials being used. The ShillerLearning YouTube channel features videos on the three-period lesson, as well as doing operations with the decimal material and number cards.

First the units are used to introduce and strengthen the recognition of quantities one through nine. The materials are then paired with number cards to learn the symbols – the numerals – 1 through 9.

The number cards help children recognize the numerical symbols for each place value category of the decimal system. One of the fringe benefits of the number cards for students with special needs is that they empower students to work with numbers without having to be able to write numbers. The number cards consist of four sets of cards for the first four different categories of numbers in the decimal system (base ten). The smallest sized cards are numerals 1 – 9 (units/ones), increasing in size algorithmically for place value with numerals 10 to 90 (tens), 100 to 900 (hundreds), and 1000 to 9000 (thousands). The cards are color-coded for place value in green, blue, and red with the color pattern repeating for each new place value family - everywhere a comma goes. If we were to create cards in higher place values, the color pattern would continue. Ten thousands would be blue and hundred thousands would be red.

With success achieved in numbers one through nine, students are ready to learn place value into the thousands! As they work with the material, they also discover that if we count beyond the number nine in any decimal (place value) category, we continue counting into the next highest category.

Abstract, sometimes difficult concepts become hands-on, concrete, interactive. This introduction and exploration of numbers is unparalleled in creating a solid number sense and understanding of place value – the cornerstone of a solid math foundation.

The magic continues as you work with operations: addition, subtraction, multiplication, and division. Students learns 4-digit static addition (addition without an exchange, commonly referred to as carrying), doing in the concrete with the materials and number cards what they will later do on pencil and paper. Once that skill is mastered, they learn to play the exchange game where various quantities of a place value are combined; when ten is reached, it is traded in the materials bank for the next place value, adding that to whatever is left over. Now the student is ready for dynamic addition – addition with an exchange (addition with carrying).

Here is a peek at 4-digit addition with exchange. The following is a summary of activities that are scripted in the ShillerLearning Math Kit I, Lesson Books 1 and 2.

◼ First, the problem is constructed on the work mat with the number cards and addition sign.

◼ Then, the decimal material (base ten blocks) for the first addend are laid out with the number cards.

◼ Now we do in the concrete exactly what the student will do later abstractly with pencil and paper. First, bring down the units (ones) and lay the 9 number card below them.

◼ Next, we bring down the tens. In this case, we have 13.

◼ We trade 10 of those tens in for 1 hundred flat, keeping the remaining 3 tens.

◼ We show that the hundred flat goes above the hundreds column, just like they will do abstractly with pencil and paper.

That floating one on the paper which is confusing to many students is no longer mysterious. They have experienced exactly what it represents.

◼ Then, you guessed it, bring down the hundreds, lay out the 500 number card, bring down your thousands, and lay out the 3000 number card.

Finally, the problem is read as a whole, overlapping the cards as you read it: 2192 plus 1347 equals 3539.

A ruler may be used as the equal sign bar as shown on the right.

Here is a video that pulls it all together with a different example.

You will be amazed at how quickly young children achieve conceptual understanding as well as success in performing operations – preventing math struggles from occurring. The activities are also a powerful remediation for older students struggling with math for a variety of reasons and crucial support for students with special needs. These are two math manipulatives that will promote the success of every student in homeschool math.

The decimal material and number cards are available separately and can be added to any math program. They are both included in Math Kit I (covers pre-k, kindergarten, 1st grade, 2nd grade, and 3rd grade).

Antoinette LaGrossa

Veteran homeschool mom of five, Antoinette LaGrossa has been homeschooling since 2001. Having experienced her family’s frustration with multiple math programs, Antoinette understands the struggle that can come with teaching math. Everything changed in 2004 when she tried ShillerMath. She quickly joined the ShillerLearning team and has been supporting home educators for almost two decades - sharing hands-on learning tips, encouragement, and practical experience from homeschooling five very different children. Antoinette speaks at conventions across the country and is host of ShillerLearning’s Tuesdays@2.

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Fun Pi Day Inspired Homeschool Math and Language Arts Activities

March 11, 2022

Most writers know that pi is always written lowercase. But even at the beginning of a sentence? Even then, pi is not capitalized. And a preposition is still a word never to end a sentence with.

pi (seen in math as the symbol “π”) is an important number representing the ratio between the circumference of a circle to its diameter. Although it is an irrational number that goes on forever, the first three digits are 3.14.

March 14th (3/14) is Pi Day. But that does not satisfy all pi lovers: Some also celebrate July 22 (the reciprocal of 22/7, a common pi approximation).

There is another reason to celebrate Pi Day: March 14th is also the birthday of Albert Einstein and Apollo 8 Commander Frank Borman.

We have creative ideas to celebrate with all ages.

Fun Pi Day Inspired Homeschool Math and Language Arts Activities

Math Kit I (ages 4 to 8):

Math Lesson Book 1

Lesson 2- Shapes

Lesson 10- Shapes Matching

Lesson 15- Comparisons Among 2

Math Lesson Book 2-

Lesson 42- Subtraction with Circles

Math Kit II (ages 9 to 13):

Math Lesson Book 5

Lesson 10- Diameter and the Chord

Lesson 113- Angle Degrees

Math Lesson Book 6

Lesson 84- Planes

Lesson 85- Circles and Spheres

Lesson 118- Measuring a Circle’s Circumference

Lesson 119- Circle Circumference and pi

Pi Day Inspiration for Your Homeschool:

• Explore math in literature with the books Sir Cumference or the Dragon of Pi

• Measure out 3.14 inches or feet of string

• Practice making the pi symbol in your quinoa tracing tray

• Select 10 different colored beads; assign each color to a digit from 0-9; string the beads to reflect the order of pi’s digits to create a colorful, math-inspired statement necklace.

• Make the pi symbol with playdough or clay

• Bake a pie - reading and math skills, plus snack all rolled into one! (What more could a homeschooler want??)

• Use a drawing compass to create circles of various sizes

• Montessori matching activity using pi digits

• Have pot pie for dinner

• Practice pennmanship writing the digits of pi

• Read about famous mathematicians (Einstein is a perfect choice considering it's his birthday)

• Research how it got the name “pi”

• Make a skyline landscape by making a series of bar graphs to represent each digit of pi

• Take a field trip to a local pizza place and get a lesson on spinning the perfect pizza pie

• Find pi with a variety of circular objects - like that pie you baked or a wall clock (using a sewing measuring tape, measure the distance around the circle to find the circumference, divide the measured circumference by the diameter, and you should get the value of pi)

• Using a roll of paper or strips of paper taped together about the length of the perimeter of your room, write out the digits of pi to use in your homeschool learning space as a wall border

• Host a pi digit memorization contest with family and friends (makes a fun challenge for homeschool co-op too)

• Create a pop art inspired pi wall decoration

• Write a poem, joke, or limerick about pi (an acrostic could be short but fun)

As noted above Pi Day is not only celebrated on March 14th. Here are alternative Pi Days and Pi Approximation Days:

• July 22nd: 22 divided by 7 equals 3.14

• March 4th: 14% of the 3rd month

• April 5th: 3.14 months of the year have passed

• April 26th: Earth has traveled two radians of its orbit (April 25 in leap year), celebrated exactly on the 41st second of the 23rd minute of the 4th hour on April 26 or the 116th day (In leap year, it is celebrated exactly on the 3rd second of the 2nd minute of the 12th hour on April 25 or the 116th day.)

• November 10th: 314th day of the year (November 9 in leap year)

• December 21st at 1:13 pm: The 355th day of the year (December 20 in leap year), celebrated at 1:13 for the Chinese approximation 355/113

Challenge your child can find another date to tie into pi!

When you post pictures of your Pi Day activities, tag @ShillerLearning. We'd love to see you in your pi glory.

Antoinette LaGrossa

Veteran homeschool mom of five, Antoinette LaGrossa has been homeschooling since 2001. Having experienced her family’s frustration with multiple math programs, Antoinette understands the struggle that can come with teaching math. Everything changed in 2004 when she tried ShillerMath. She quickly joined the ShillerLearning team and has been supporting home educators for almost two decades - sharing hands-on learning tips, encouragement, and practical experience from homeschooling five very different children. Antoinette speaks at conventions across the country and is host of ShillerLearning’s Tuesdays@2.

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How Long Should a Math Lesson Take? A Montessori Homeschool Approach

February 4, 2022

A question we get at ShillerLearning is, “How long should this math lesson take my child to complete?”

With a personalized approach to education, there is no “right answer” to that question. For example, some children get a concept right away; others need lots of repetition over several days, even weeks.

“How Long Should a Math Lesson Take?” A Montessori Homeschool Approach

Every child and activity is different. Some children get an activity’s concept right away and need no further practice to have it stored in their brains forever. Yet they insist on repeating the activity anyway. These children may not need further practice, but they enjoy the play associated with an activity they are good at. So let them explore, for five minutes or for two hours, until they are finished. If they can work without you right there, that is OK too.

Such active interest on the part of a child is a good sign; the child wants to repeat the satisfaction that comes from achieving a successful completion of an activity. What may seem like repetitive actions to a parent may actually be the child putting closure on the activity, and a child may become anxious if the activity is taken away prior to closure. Closure may also occur when the child – or you - becomes frustrated. When frustration occurs, you may say, “let’s put this activity aside for now, we’ll come back to it later.”

When a child asks to repeat an activity from last week, last month, or even last year, happily comply with the request. Your child will let you know when he or she is ready to move on.

On the other hand, there could be times when you wish the child would be more enthusiastic about repeating an activity that he or she did not fully grasp. When the child (or parent) becomes frustrated or restless with a particular activity, just mark that activity as one that needs more attention next time; we recommend circling the activity number and marking it in your Completed Work Tracking Sheet – and move on. (You can read more about the tracking sheet and other learning management tools.)

For a child to grasp principles fully, the decimal system activities in particular need to be repeated often. For example, one decimal system activity may be repeated once or twice each week for several weeks. Addition and multiplication tables also require several weeks to complete.

Among ShillerLearning's innovations is the Incremental Spiral Method™, which ensures that bite-size activities cover syllabus topics clearly and thoroughly at each age. Because topics need to be revisited regularly - as in a spiral with increasing levels of difficulty and abstraction - children don't learn addition or all aspects of a square or circle all at once. As a result children learn both math and self-confidence at the same time.

This technique was inspired from a study in which two groups of children each had five minutes to work on the same memorization problem. One group had one five minute session; the other group had five one-minute sessions interspersed with one minute sessions of a simple and unrelated activity. The latter group had significantly better results – despite having spent exactly the same amount of time on the problem. That’s why activities on related subject matter tend to be spread throughout the lesson books.

ShillerLearning’s goal is for students to speak and write mathematics clearly, accurately, and concisely based on a solid understanding of mathematical concepts. We find that this personalized approach to the “How long should this math lesson take?” achieves that goal.

Here at ShillerLearning, we are available to support you and your child in your homeschool math journey. We invite you to call 888-556-6284 or message/email with your questions. We are passionate about helping your kids learn - and love - math!

Larry Shiller

Larry Shiller is President of ShillerLearning, whose mission is to help every student fulfill their potential. Shiller has degrees from Massachusetts Institute of Technology and the Harvard Business School and is the author of Software Excellence (Prentice-Hall).

A father of three, Shiller is active in non-profits and his hobbies include working with local startups, music (Shiller is an accomplished violinist who - when not helping children learn math and language arts - performs in the NYC tri-state area), tennis (Shiller's team made it to the USTA national finals in his skill bracket), Quoridor (Shiller is a former USA Champion), backgammon (Shiller is the Voice of Backgammon, doing commentary on backgammon tournaments worldwide), table tennis, and flying (Shiller holds a private pilot's license).

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The US Math Crisis

December 29, 2021

We have a math crisis in the US.

Maybe that's the least of our problems. But maybe not. Consider this:

• 80% of US 8th grade students cannot calculate decimals, fractions, and percentages

• 40% of US 4th grade students cannot tell SW from NE on a map

The US Math Crisis

It's really just us. The world does better.

• In Germany, 35% of teens take and pass advanced placement exams; it's 4% in the US

• In Japan, kids start algebra two years ahead of those in the US

What are the repercussions?

• 40% to 60% of freshman are required to take remedial math classes, trapped in classes that only cover high school content
• 75% of students required to take remedial college classes fail to graduate
• One in five adult Americans cannot:

- Calculate the total of a purchase w/tax & tip

- Locate an intersection on a road map

- Enter background information correctly on a form

• MIT economist Lester Thurow says that only 20% of Americans have the work skills and education to be competitive in the global marketplace

And no one believes it! In a recent study 71% of high school parents say that they are satisfied with the math education their children are getting. Ouch.

The US has a math crisis. And the repercussions are serious. Join ShillerLearning and the Rising Stars Foundation in creating better outcomes for our children and country.

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Language Arts A - PreK to 1st Grade

Larry Shiller

Larry Shiller is President of ShillerLearning, whose mission is to help kids learn and fulfill their potential. Shiller has degrees from Massachusetts Institute of Technology and the Harvard Business School and is the author of Software Excellence (Prentice-Hall).

A father of three, Shiller is active in non-profits and his hobbies include working with local startups, music (Shiller is an accomplished violinist who - when not helping children learn math and language arts - performs in the NYC tri-state area), tennis (Shiller's team made it to the USTA national finals in his skill bracket), Quoridor (Shiller is a former USA Champion), backgammon (Shiller is the Voice of Backgammon, doing commentary on backgammon tournaments worldwide), table tennis, and flying (Shiller holds a private pilot's license).

3 Key Solutions to Math Success

October 22, 2021

"True understanding of the child's needs requires that we focus on the uniqueness of each child rather than group them under broad categories of common behavior." - Larry Shiller

3 Key Solutions to Math Success

Society values math skills.

According to the Bureau of Labor Statistics, a solid math education can mean a 350% increase in pay.

Yet even with unfilled jobs in our economy, NASA estimates that two million jobs in technology are going unfilled by American workers because they are unqualified.

The U.S. has an ongoing math crisis. Fortunately, there are solutions that do work!

Here are a few key solutions to the US math crisis:

1. Get rid of grade levels

Huh? How could we do that? But think about it: students don't learn in grade chunks; they learn - and master material - at their own pace. Segregation is bad - any kind of segregation. We have grades because once we grew beyond the one-room schoolhouse, it made life easier for bureaucrats. That's not good enough: Let's get rid of grade segregation. Ideally, education should nurture a child's intrinsic motivation to learn, create, and do satisfying work. An educator should guide the child to progress through the curriculum as concepts are mastered (and can be built on), rather than, “Time to move on!” Knowing the 3 possible outcomes of a lesson will guarantee that a student successfully moves on without regard to grade. Montessori schools the world over have been liberated from grade levels, focusing instead on personalized education, for over a hundred years. That's also happening now at the Ridgeview Charter School in Colorado and hundreds of other schools across the US - and it's a success.

2. Don’t rely on labels and categories

In a well-meaning attempt to help more students, we create categories such as LD, ADD, and ADHD, and define processes for each.

In fact, every student should have a label: His or her name.

Wait, don't we need those labels in order to better serve our kids? Yes, diagnostic labels serve an important role in bringing support services to students. However, they also can have unintended side effects. In trying to group children under categories of diagnosis, we group together children who are actually quite different from one another. And in doing so, we underemphasize their important differences. Parents can become frustrated when their child is not fitting into the diagnosis mold.

True understanding of the child's needs requires that we focus on the uniqueness of each child rather than group them under broad categories of common behavior.

Let's use our great American marketing muscle to target each student individually - and to let students fulfill the American dream by providing an environment that lets them reach their own potential, not some bureaucrat's idea of their potential.

3. Multisensory math education

A typical homeschool curriculum engages about 30% of the brain. By and large there are no kinesthetic or auditory lessons, and the use of manipulatives is limited. Unless a student takes to abstract thinking right away, the odds are good he or she won't be good at – or like – math.

Your child completes a lesson correctly - say it was a visual lesson that requires looking at a picture and doing a simple calculation. The child then successfully completes several drills on the same material. Does that mean your child has mastery and a full foundational understanding of the topic covered?

The problem is that this lesson and drill only reached 30% of the child's brain. The rest? Not getting this concept! Now, imagine four lessons on the same concept - one each for visual, tactile, kinesthetic, and auditory neurons. Now the child's brain is being engaged 100% - to its fullest potential - and connections are rampant. Only then will a child truly have mastery.

Without a complete multisensorial experience children lose the richness that comes from absorbing the same material from all learning styles: visual, auditory, tactile, and kinesthetic. Only then will a student form a rock-solid web and foundation of knowledge and ability. The sensorial materials enable the child not just to memorize, but to gain conceptual understanding.

Whether your child is average, gifted, ASD, a combination of those, is a current or former Montessori math student, or not, is a pre-k math student or in junior high, make sure that the math program for your child includes a multisensorial approach like the one used by ShillerLearning.

Need help with multisensory math ideas? The 20-page Fun with Fractions Activity Pack homeschool freebie is full of multisensory fractions games and math activities. Complete the form below to download yours today!

Larry Shiller

Larry Shiller is President of ShillerLearning, whose mission is to help every student fulfill their potential. Shiller has degrees from Massachusetts Institute of Technology and the Harvard Business School and is the author of Software Excellence (Prentice-Hall).

A father of three, Shiller is active in non-profits and his hobbies include working with local startups, music (Shiller is an accomplished violinist who - when not helping children learn math and language arts - performs in the NYC tri-state area), tennis (Shiller's team made it to the USTA national finals in his skill bracket), Quoridor (Shiller is a former USA Champion), backgammon (Shiller is the Voice of Backgammon, doing commentary on backgammon tournaments worldwide), table tennis, and flying (Shiller holds a private pilot's license).

Multisensory Math Hacks

Multisensory math is something we love here at ShillerLearning. Counting, learning place value, exploring geometric shapes, and all types of mathematics become more fun when partnered with interactive activities. Not only do kids learn math more effectively with a multisensory approach, they enjoy learning and become lifelong learners. Multisensory learning engages with visual, tactile, kinesthetic, and auditory lessons. Here are some ways to incorporate multisensory math into your homeschool classroom:

• Building: By using blocks and other building materials, children can make clear visual representations of the problems they are trying to solve, create shapes, and more. The classic children’s building blocks are some of a child’s first encounters with math and some of the best ways to make math fun and more concrete.
• Touch: Handling manipulatives (fun, colorful objects for learning), tapping out numbers, and creating tactile points for numbers are all great ways to incorporate the sense of touch. It’s also fun to create manipulatives out of things that have interesting textures such as modeling dough, quinoa, or putty.
• Counters: Unit cubes (included in Math Kit I and Math Kit II), small pieces of food, blocks, beads, or any other small object can be used as counters. This is especially beneficial with younger children learning 1:1 correspondence, solving math problems, and counting. We use the unit cubes as support for math facts in the Math Facts Boot Camp live classes. When learning place value, the unit cubes are paired with ten rods, hundred flats, and thousand cubes. As students advance in their math skills, unit cubes can be used in geometric modeling and creating numeric modeling plans. Never underestimate the benefit of a counter!

• Music: Some children learn incredibly fast through song. Music like the 50 ShillerLearning math songs helps reinforce concepts through catchy songs kids will enjoy. (And music isn't just for math; there are over 50 language arts songs - including a song for EVERY letter of the alphabet and fun activities to go with them.) Learning to play and read music can also help children with math skills when they’re older!
• Movement: Incorporating math into outdoor games, using dice, dominos, or a ball to practice math facts, dancing to math music, and counting steps or objects while walking are all ways to get the body moving and grasp math concepts. Sometimes even just a simple movement break in between lessons helps reinforce a concept and improve focus.
• Drawing: Drawing out a problem (such as 5 stars + 2 stars = 7 stars), making shapes and illustrating word problems engage different parts of the brain and build different neural connections.
• Images: Using visual materials such as pictures, graphs, tables, and images will reach visual learners as well as help reinforce concepts in another way.

If you find yourself trying to present a concept to a child that they seem to be struggling to grasp, try approaching it from a different angle using one of these multisensory techniques. If you need help generating ideas or tips feel free to call or email us or ask in the ShillerLearning Customers  Facebook group. We also post multisensory math ideas and tips on the ShillerLearning Facebook page.

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Language Arts A - PreK to 1st Grade

Amanda Osenga

Amanda is a former Montessori teacher, now homeschooling her dear son - an only child. Amanda and her family reside in an Airstream parked in Washington State. She loves Washington's outdoor opportunities. When not homeschooling, she also blogs, works as a virtual assistant, and loves creating hand-lettering pieces.