Sacramento Homeschool Math By Hand

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 260

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understand a fraction a/b with a> 1 as a sum of fractions 1/b.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation.  Justify decompositions, e.g., by using a fraction model.  Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

Like so much else in math, taking fractions apart and putting them together is best approached with a hands-on format.  Here is a fun and effective lesson plan from the HotChalk Lesson Plans page.

Objective: Students will compare fractions with unlike denominators.

Materials: Fraction game board, Playing Cards

Provide students with a fraction game board (a paper with two fraction bars that specify where the numerator and denominator are) and playing number cards. The game is played like war. Students place cards on the numerator and denominator spaces at the same time. The first person to identify the larger fraction by tapping or placing their hand on the fraction that is larger gets all four cards. As students develop their knowledge of fractions, you can write a denominator for students to use consistently. (For example, have students write 10 in the bottom of the fraction, so that all the fractions are consistently less than one).

This is one area in which Common Core, Waldorf, and Math By Hand agree: hands-on and interactive is the way to go with fractions!  The Math By Hand / Grade 4 / Fractions / Kit 2 includes a DIY set of playing cards with instructions for multiple familiar games that are adapted to learning about fractions.  Here are excerpts from the instructions and the fraction equivalency chart used with the playing cards.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NF 3b: Fraction Equivalency, Hands-On! (#260) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 259

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understand a fraction a/b with a> 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

Simple is best.  Searching online for lesson plans that fit each standard is my first task when writing these posts.  Invariably, I am confronted with multiple convoluted, complicated examples.  This voluminous collection is the result of much hard work on the part of teachers sharing their ideas with other teachers.  My conclusion, having found similar results each time, is twofold:

1) The standards, lessons, and worksheets are overly complex: the wording is obtuse, not clearly or plainly presented, there’s a lack of cohesiveness due to both repetition and omission, and the attempts at going deeper by using reasoning and logic to explain the process or the answers conversely results in more confusion rather than a greater understanding.

2) The standards, lessons, and worksheets are overly simplified: the piecemeal nature of the standards obscures the whole or the meaning, resulting in not seeing the forest for the trees.  In actuality, simplicity becomes complexity because if care is taken to present the whole picture in a series of clear, simple steps, the naturally complex nature of math will shine through in a language and form that is both comprehensible and dynamic.

Nothing is more alienating at this age than to be forced to perceive the world (or any new concept) in pieces.  Ideally (and ironically) fractions are best taught holistically and all-at-once.  Just such an approach is used in the Math By Hand fraction block.  All 4 processes are learned at the same time, with hands-on tools and a pocket reference guide that’s there whenever it’s needed.

For example, small black beans and red kidney beans are used with a small, 6-well plastic tray to represent the numerator, denominator, and whole numbers.  A clear picture forms as the black beans are stacked in the tray, reduced where need be, or traded for the whole number kidney beans.  After working it all out with the beans, the problems are written in a workbook.  The reference guide mentioned above is called Fractions Rules and Keys and is a concise summary of the 4 processes’ rules (as referenced in #4 below).

The post 4.NF 3a: Fractions That Are Full Of Beans! (#259) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 258

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models with attention to how the number and size of the parts differ even though the two fractions themselves are the same size.  Use this principle to recognize and generate equivalent fractions.

Note that fractions are first introduced in Grade 4, after an immersion in factors and factoring.  The Waldorf and Math By Hand approach is visual and hands-on, and although this standard recommends using visual fraction models, they are usually of the flat, 2-dimensional variety, either as an overhead projector slide or in a worksheet format.  And although it is helpful to have these comparative graphic images of 1/2, 2/4, 4/8, etc., it still falls short of using 3 dimensional models to convey equivalency.  Here’s an excerpt taken from the Grade 4 Math By Hand binder: introduction to fractions.

The post 4.NF 1: Real Fractions Teach Fractions Best (#258) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 257

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations in Base Ten 4.NBT
Generalize place value understanding for multi-digit whole numbers.
3. Use place value understanding to round multi-digit numbers to any place.
Use place value understanding and properties of operations to perform multi-digit arithmetic.l
4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Are these repetitive and boring?  I thought so, therefore decided to include all of them in one post rather than stretching them out.  The Math By Hand curriculum covers rounding and estimation in regular, daily skills practice, both oral and written.  Place value is taught with large, colorful columns and manipulatives, mid to late second grade.  Long division and multiplication are also taught with the same methods, mid third grade.

As for the rectangular arrays and area models, Math By Hand covers this concept first thing, mid first grade.  Once it’s “gotten,” there is no need to endlessly repeat it, especially with boring worksheets like this one from www.commoncoresheets.com:

Paraphrasing  John Holt, don’t subject children to lessons and activities that adults would never sit still for.  This worksheet certainly qualifies as that!  We need to consider that this kind of teaching wrings the very life out of math.  This youtube beautifully illustrates the same mistaken notion of labor-intensive math “explanations.”

Now back to some Grade 4 fun, like this local geography main lesson book page from Waldorf Today: a map of the route to school, surrounded by local birds and squirrels.  Starting from home at the center, geography radiates outward slowly, until it embraces the entire world in Grade 8.

Or this form drawing pinned by Janet Langley as a Grade 5 drawing, but also suitable as a Grade 4 Human and Animal study, integrated with the weaving forms that are typical in Grade 4.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NBT 3-6: Oh Those Common Core Rectangular Arrays! (#257) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 256

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. 

So much has been said about the Common Core, with its detractors seeming to outnumber its proponents.  The position taken in this blog has primarily been its developmental inappropriateness, especially for the youngest students, from K-3.  Moving into Grade 4, a shift is apparent, perhaps one that could accommodate a more academic focus.  Having said that however, it remains true that until the “age of reason” at 12-14, the primary focus should remain artful and heart-based.  All academics until then need to be cloaked in story, art, and movement.

Here is an image from Catie Johnson’s wonderful page, Chalkboard Drawings in the Waldorf Classroom.  Note how the babies are cared for.  Remember, human childhood is the longest of that of all species!

Please read this Forbes article by Alice Walton, The Science of the Common Core: Experts Weigh In On Its Developmental Appropriateness, for an in-depth look at the Common Core’s short and long range effects.  Here is an excerpt, one with a philosophy that closely parallel’s that of Waldorf Education:

“David Elkind, long-time child development expert at Tufts University and author of The Hurried Child, says that a related problem with the Common Core standards is that “children are not standardized.” Between ages 4 to 7, he says, kids are undergoing especially rapid changes in cognitive ability, but this neurological and psychological development occurs at all different rates. “Some children attain these abilities—which enable them to learn verbal rules, the essence of formal instruction—at different ages. With the exception of those with special needs, all children attain them eventually. That is why many Scandinavian countries do not introduce formal instruction, the three R’s until the age of seven. In these countries children encounter few learning difficulties. Basically, you cannot standardize growth, particularly in young children and young adolescents. When growth is most rapid, standardization is the most destructive of motivation to learn. To use a biological analogy, you don’t prune during the growing season.”

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post The Common Core: A Long, Thoughtful Look (#256) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 255

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations in Base Ten 4.NBT
Generalize place value understanding for multi-digit whole numbers.
2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.  Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Well, I just spent too much time looking for a Facebook Grammarly post and didn’t find it.  It has a sad face and it says, “I just spelled a word so wrong, autocorrect was like, “I got nothin’ man.”  My sentiments exactly re this standard.  All of these things were taught and learned in the Math By Hand Grade 2 curriculum, by writing out answers to 4 processes problems in words and in expanded form.  Also in Grade 2, the symbols > and < were taught using very pictorial images, with lots of subsequent practice.

Once it’s “gotten” theres no need for repetitive drill and endless worksheets!  I just have nothing further to say about this.  Instead, I will jump ahead to seventh grade with this lovely image from the Bright Water School.

Representative of all we’re missing because we have cut the heART out of education.  Right brain thinking is responsible for most of the creative endeavors that grace our world.  How short-sighted of us to have gone so far to the left.  Left-brain thinking is sickeningly dominant in how we teach our children now.  All from the mistaken notion that more is more, i.e., more work = better scores.

Really, we need to take a breath and take a step back.  To how we used to teach, with breathing room.  We need to have a little faith, in ourselves as teachers, in our children, and in the things of the world that we need to teach them.  As always, knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NBT 2: Too Much Left Brain? (#255) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 254

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. CCSS math standards are listed here in blue

Number and Operations in Base Ten 4.NBT
Generalize place value understanding for multi-digit whole numbers.
1. Recognize that in a multi-digit whole number, a digit in one place represents ten time what it represents in the place to its right.  For example, recognize that 700 / 70 = 10 by applying concepts of place value and division.

Place value, in the Waldorf and Math By Hand methods, is introduced with lots of tactile manipulatives and a variety of materials that support a pictorial grasp of the concept, so the place values of 1’s, 10’s, 100’s, and 1,000’s are thereby firmly established.  If initial impressions are given and taken in the appropriate language to each specific age, they have staying power and are there for good. And from then on, those concepts do not need to be constantly and repetitively reinforced and drilled.

There are bigger fish to fry, so to speak, in Grade 4.  Math lessons should be devoted (in the true sense of the word) to the complexity of fractions, equivalency, mixed numbers, decimals, and a beginning look at positive and negative numbers on the number line, as well as basic, introductory algebra.  In addition, the well-rounded fourth grader nurtures a nascent consciousness of the surrounding world, through local geography and comparative studies of the human and animal.

Along those lines, here is a wonderful idea from the Atelier Pour Enfants‘ blog: creating animal figures with leaves!  This one quite nicely captures the essence of elephant.

The main focus in the Grade 4 human and animal study is to find traits in both that are reflective of the other.  For instance, the dexterity of the elephant’s trunk might be compared to the human’s opposable thumb. Many opportunities abound to build bridges between the two worlds and therefore nurture a love of and caring for nature and the environment.  Bigger fish to fry indeed, that preempts completing reams of busy-work worksheets or teaching and learning the focuses on standardized testing prep.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NBT 1: Place Value Redux (#254) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 253

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. 

The trickster tale encompasses the essence of the Grade 4 year: brash and brave with a will to take on the world, and the panoply of the Norse Gods and Goddesses fill this bill quite nicely.  But there are many tales from other times and places that also beautifully fit this genre.  American Indian tales are full of mythical trickster figures, with Raven and Coyote as just two of the towering animal totems who are both heroic and renegade, locked in a reciprocal and magical dance with their human counterparts.  This Coyote tale from the Karok Indians can be found in the Math By Hand / Grade 4 / Form Drawing/Stories book.

HOW COYOTE STOLE FIRE
(Karok) 

Long ago, when humans first appeared upon the earth, they were the happiest creatures of all during the long, warm days of summer and autumn, when all the fruits and grains ripened and their children happily played in the sun. They were the best times. But when the days grew shorter and evenings more chilled, the people knew winter was near and they became fearful and unhappy. Knowing that many of the very young and very old would die in the snows and bitter cold brought them great sorrow. Yes, every winter so many beautiful babies and children as well as the revered elders who kept the precious and sacred tales of the tribe were forever lost. Now Coyote, like the rest of the animal people, had no knowledge of or need for fire because they always wore warm fur coats. So he never thought about fire until one spring day as he was passing a human village. There, the women were singing a song of mourning for the lost babies and old ones who had died in the winter. Their voices, like the moaning west wind through a buffalo skull, prickled the hairs at the back of Coyote’s neck. One of the men wondered aloud why the sun that now warmed the earth and made the rocks hot to the touch could not be captured in small pieces and taken into their teepees in the winter. Coyote felt sorry for the men and women of the village, and thought there must be something he could do to help. He remembered the three Fire Beings who lived at the top of a tall mountain and jealously guarded their fire, fearing that if humans got hold of it, they would one day become as strong as the Fire Beings were. Coyote hated their selfishness and thought that he might do a good turn for the humans while teaching the Fire Beings a lesson. So Coyote stealthily crept to the top of the Fire Beings’ mountain and watched the way they guarded their fire. While he crouched hidden among the trees, one the Beings leapt to her feet and gazed piercingly around the camp, eyes glistening like hot, red coals and claws clenched like great, black vulture talons. She screeched to the others that there must be a thief lurking about in the camp, and then she spied Coyote. But because he was going about on all fours, she thought he must be an ordinary coyote, slinking as coyotes do, among the trees. The lookout shrieked that it was nothing and no one, but merely a lowly, gray coyote. As the other two looked, they agreed and all sat down again. So, as he was safe and unnoticed, Coyote was able to watch unhindered the rest of that day and all that night while the Fire Beings guarded their fire. He saw them feeding it with pinecones and dry twigs and branches, and he saw them angrily stamp out any runaway flames that escaped from the confines of the fire, as it gobbled up the dry grass around it. All night, the Beings took turns sitting by the fire, sending two off to sleep while one continuously watched and kept the fire going. But Coyote noticed that there was one part of the day when the Beings were not so closely watching. Early in the morning, when the first chilly winds of dawn arose, the Being who was watching the fire would crawl off shivering into the teepee, calling for one of the others to go out and sit and watch. But the next guard, still fuzzy and groggy with early morning sleep, would always be slow to come out and take the post by the fire. Coyote went to the animal people and told them of all this, and how the furless humans shivered so in winter and lost many of their young and old to the cold. The animal people agreed that it was for the good of all if the warm, bright flames were stolen from the selfish, greedy Fire Beings and shared equally among the humans. They promised to help him in any way they could, and encouraged Coyote to go back and steal the fire. Again, when Coyote came close, the Beings loudly screeched that there was a thief nearby, and again paid him no attention when they saw that he was nothing more than a harmless, ordinary gray coyote. Coyote waited all day and night and watched as they changed guards at intervals through the long, dark hours, until the dawn winds rose again and the next Fire Being was slow in coming out of her teepee saying that yes, she was indeed on her way and do not shout for her so! In those brief moments before she came out of the teepee, Coyote lunged out and grabbed a burning coal, running as quickly as he could down the mountain. The Fire Beings flew screaming in pursuit of him, and quick though Coyote was, one of the Beings caught up with him enough to grab just the tip of his tail, turning the hairs there pure white. That’s why a coyote’s tail tip is white to this day. Coyote shouted as he flung the glowing coal away from him. The animal people, who had promised to help, were waiting at the foot of the mountain. Squirrel caught the coal, putting it on her back as she fled through the treetops. It so painfully burned her back that her tail curled up and back, as it still is, to this day. Squirrel then threw it to Chipmunk who held it, frozen with fear, as the Fire Beings came terribly close. She finally turned to run, but not before one of the Fire Beings reached her, clawing at her back as she escaped. The three stripes that we still see on a chipmunk’s back today are the marks that the Fire Beings’ claws left. Chipmunk then threw the fire to Frog who took a mighty leap as one of the Fire Beings grabbed him by the tail. He did get away but without his tail, and that’s why frogs today have no tail. As the Fire Beings pursued him, Frog flung the fire to Wood who swallowed it. All three of the Fire Beings gathered around Wood and frantically watched, but did not know how to get the fire out of Wood. They threatened and pleaded and tore at Wood with their sharp claws, but to no avail. They finally gave up and went back to their mountaintop, defeated. Coyote, having watched carefully, knew how to get the fire out of wood, and he taught the humans in the village. He showed them the trick of rubbing two dry sticks together, and the trick of spinning a sharpened stick in a small hole made in another piece of wood. Thanks to Coyote, the humans now enjoyed bright fires in their teepees, keeping them safe and warm through the winter’s cruel, killing cold.

Winter can be cruel, but it can also be a time of coming together around the warmth of the fire.  Math By Hand is offering a special Winter Sale for the month of December: 15% off your entire shopping cart!  Hop on over to your grade level of choice on the Shop page and explore.  What better time to try this hands-on, creative, and experiential approach than now, mid-year, when interest in math may be waning.  Give the gift of joyful math!

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post Grade 4 Trickster Tales: How Coyote Stole Fire (#253) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 252

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. CCSS math standards are listed here in blue, followed by ambient math suggestions.

Operations and Algebraic Thinking 4.OA
Generate and analyze patterns.
5. Generate a number or shape pattern that follows a given rule.  Identify apparent features in the pattern that were not explicit in the rule itself.  For example, given the “Add 3″ and the starting number 1, generate terms in the resulting sequence and observe that the terms paperer to alternate between odd and even numbers.  Explain informally why the numbers will continue to alternate in this way.

As usual, I searched the net for lesson plans and work samples for this standard, rounding up all the usual suspects: labor-intensive, intricate, mostly worksheet oriented content.  Some handy tricks and patterns as well, but the overall impression was one of random pieces cobbled together.  “Doubling and Halving” was one such trick.  Here’s how it works: to multiply 25 x 8, double the 25 to 50 and halve the 8 to 4, so 50 x 4 = 200.  It’s a streamlined approach, one that makes mental math easier.

But it’s a only a piece, one of many disjointed pieces that make up the Common Core.  Both the Waldorf and Math By Hand Grade 4 curriculum follow a whole-to-parts rather than a parts-to-whole approach, that’s more methodical and consistent.  An unbroken thread connects factors to fractions, then decimals are followed by the number line and a brief introduction to algebra to end the year.

The year begins with a thorough times tables review to insure that they’re in place before moving on.  The sorts of number patterns mentioned above are in constant focus beginning in Grade 2, as patterns within the times tables are worked with in many ways.  For instance, in the large Grade 2 times tables chart, it can easily be seen that the square numbers form a diagonal line from the upper left to the lower right corner, from 1 to 144.  Find the square numbers’ diagonal line in this chart from Algebra One Blog.

Many, many patterns are found within Pascal’s triangle, shown below.  When this is presented along with a brief biographical sketch of its creator Blaise Pascal, a broader picture of mathematics prevails, one that reveals its inherently elegant properties.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.OA 5: Patterns & Math? Always! (#252) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 251

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. 

The Art of Learning, a favorite education page on Facebook, posted this quote by Eugene Herrigel, Zen and the Art of Archery:

“The right art,” said the Master, “is purposeless, aimless! The more obstinately you try to learn how to shoot the arrow for the sake of hitting the goal, the less you will succeed in the one and the further the other will recede. What stands in your way is that you have a much too willful will. You think that what you do not do yourself does not happen.”

“Don’t think of what you have to do, don’t consider how to carry it out!” he exclaimed. “The shot will only go smoothly when it takes the archer himself by surprise.”

The post CCSS & Testing: A Much Too Willful Will? (#251) appeared first on Math By Hand.