4.23.14 | A Year in the Life: Ambient Math Wins the Race to the Top!

Posted in Uncategorized at 6:15 am by admin

Day 108

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.  Today’s standard will be listed in blue, followed by its ambient counterpart.

Number and Operations in Base Ten   1.NBT
Use place value understanding and properties of operations to add and subtract.
6.  Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Today’s standard is similar to yesterday’s, and it seems both are meant as preparation for working with the tens place in place value.  The “concrete models or drawings” portion has resulted in students having to draw “hash marks and polka dots” to represent numbers and groups of numbers.  This is cumbersome at best and deadening at worst.  Math begs to be enlivened and made part of life.  This is what engages students and enables math success.

Generally speaking, I find the Common Core language to be convoluted and unnecessarily obscure and difficult.  The California State Math Standards by comparison, are clear and straightforward.  I have retained these and the National Math Standards as references for the Math By Hand curriculum.  Here are the California standards that are comparable to the Common Core Operations and Algebraic Thinking and Number and Operations in Base Ten.

Number Sense
1.0) Students understand and use numbers up to 100.

1.1) Count, read, and write whole numbers to 100.

1.2) Compare and order whole numbers to 100 by using the symbols for less than, equal to, or greater than (<, =, >).

1.3) Represent equivalent forms of the same number through the use of physical models, diagrams, and number expressions (to 20) (e.g., 8 may be represented as 4+4, 5+3, 2+2+2+2, 10-2, 11-3).

1.4) Count and group objects in ones and tens (e.g., 3 groups of 10 and 4 = 34, or 30+4).

1.5) Identify and know the value of coins and show different combinations of coins that equal the same value.

2.0) Students demonstrate the meaning of addition and subtraction and use these operations to solve problems.

2.1) Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory.

2.2) Use the inverse relationship between addition/subtraction to solve problems.

2.3) Identify 1 more than, 1 less than, 10 more than, 10 less than a given number.

2.4) Count by 2s, 5s, and 10s to 100.

2.5) Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference).

2.6) Solve addition and subtraction problems with one- and two-digit numbers.

2.7) Find the sum of 3 one-digit numbers.

3.0) Students use estimation strategies in computation and problem solving that involve numbers that use the ones, tens, and hundreds places.

3.1) Make reasonable estimates when comparing larger or smaller numbers.

Algebra and Functions
1.0) Students use number sentences with operational symbols and expressions to solve problems.

1.1) Write and solve number sentences from problem situations that express relationships involving addition and subtraction.

1.2) Understand the meaning of the symbols +,-,=.

1.3) Create problem situations that might lead to given number sentences involving addition and subtraction.

Aren’t these standards clearer and more straightforward?  ”If it ain’t broke, don’t fix it.” comes to mind here.  There does seem to be a grass roots movement afoot to repeal/replace the Common Core with more workable standards.  And this may very well happen in more states than not, in light of the fact that the Common Core may have been hastily conceived then rolled out without proper vetting and viability.  The fact remains that math is a much-beleagered subject and sorely in need of a lively and sensible teaching approach.  Waldorf and Math By Hand methods may be just the ticket!

As always, it’s the movement, story, and art that win the day with math!  Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal.

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4.22.14 | A Year in the Life: Ambient Math Wins the Race to the Top!

Posted in Uncategorized at 5:52 am by admin

Day 107

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.  Today’s standard will be listed in blue, followed by its ambient counterpart.

Number and Operations in Base Ten   1.NBT
Use place value understanding and properties of operations to add and subtract.
5.  Given a two-digit number, mentally find ten more or ten less than the number, without having to count; explain the reasoning used.

Having explored the -teens and -ty’s through stories, pictures, and hands-on activities, it’s now possible to find the next ten, up or down, of any number.  The -ty’s lesson showed that the multiples of ten are basically the same as counting in ones.  It was seen that some of the ‘ty’s change their names and some don’t.  For example, two becomes twenty, three becomes thirty, four becomes forty, five becomes fifty, but six stays sixty, seven: seventy, eight: eighty, and nine: ninety.

Armed with this inside information the first grader can confidently proceed, knowing that it’s all just counting, using the same numbers!  So then it’s just simple mental calculation to go up and down the tens ladder.  Take ten away from seventy two = sixty two.  Being able to count from 1-10 forward and backward makes all of this possible. Here’s a foundational place value activity/lesson.

Gather twenty or so palm-sized flat, smooth stones.  (We’re living in rock heaven here, with four major rivers nearby.)  If none are readily available outdoors, your local garden supply should have a nice selection of stones for sale.  Have your child write the numbers 0 – 9, one number per stone, two stones for each number.  Line up the number stones 1-9 in a vertical column, then take the other stones, randomly one at a time, and place each one next to the numbers in the first column, having the child say each number while doing this.  For example, with the 3 stone it would be thirteen, twenty three, thirty three, etc.  Physically moving the numbers up and down are a good hands-on means of exploring ten more and ten less.

This could inspire morning circle activities as well: 1) Toss a ball or bean bag while saying, “sixty six, ten less” and have the child toss it back with the answer, “fifty six.”  2) Have the child toss a ball or bean bag from hand to hand / toss it up and catch it / pass it from front to back and back to front while saying several tens in a row, “forty three, fifty three, sixty three.”  3) Marching in a circle with your child, say a number as the second digit and have the child say the -ty’s with it while marching: you say, seven s/he says, seventeen, twenty seven, thirty seven . . . and so on, up to ninety seven.  Add some lively fun to this last one with a tambourine, rhythm sticks, a small drum, etc.  ”Explaining the reasoning used” can wait until there’s the capacity for abstract thinking at 11 or 12.

As always, it’s the movement, story, and art that win the day with math!  Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal.  More Common Core Grade 1 Number and Operations in Base Ten Standards with their ambient counterparts tomorrow!

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4.21.14 | A Year in the Life: Ambient Math Wins the Race to the Top!

Posted in Uncategorized at 6:16 am by admin

Day 106

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.  Today’s standard will be listed in blue, followed by its ambient counterpart.

Number and Operations in Base Ten   1.NBT
Use place value understanding and properties of operations to add and subtract.
4.  Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of ten, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.  Understand that in adding two-digit numbers, one adds tens and tens, ones and ones, and sometimes it is necessary to compose a ten.

Ah, the Common Core.  Again, unfamiliar territory tends to be introduced with interpretive lessons and homework that seem unnecessarily unwieldy and complex.  I’m afraid that the concrete models or drawings and strategies have taken the form of the polka dots and hash marks that have puzzled parents and children alike and appeared so frequently online and in the media.  The rushed implementation that’s resulted in inadequate orientation for teachers may be responsible for some of the confusion.  But complicating a simple 2-step equation with a mega multi-step solution that includes complex visuals is questionable at best.

“Explaining the reasoning used” is really beyond the innate logical ability of a 6-7 year old.  Or, considering Piaget’s theory of child development recommending concrete vs abstract thinking from ages 7-12, any child under 12 for that matter.  The now famous example of a second grader’s “constructed response” illustrates the fact that the capacity to reason abstractly has not yet dawned.  See the actual second grade Q&A below.

Place value per se is not brought in the Waldorf or Math By Hand method until Grade 2, when it’s taught with large, colorful manipulatives.  Four color-coded columns are placed on the floor with large numbers for the 1′s, 10′s, 100′s, and 1,000′s place values, and small numbers that are physically jumped all the way to the tops of the columns.  This will be explored in depth in the Grade 2 posts.  For now in Grade 1, the 4 processes continue to be practiced side-by side, so that the relationship, not only of addition to subtraction, but of all 4 to each other, is continually experienced.  Equations are kept in a horizontal format until the end of Grade 1 or the beginning of Grade 2, when they are switched to a vertical format.

As always, it’s the movement, story, and art that win the day with math!  Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal.  More Common Core Grade 1 Number and Operations in Base Ten Standards with their ambient counterparts tomorrow!

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4.20.14 | A Year in the Life: Ambient Math Wins the Race to the Top!

Posted in Uncategorized at 6:50 am by admin

Day 105

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.  Today’s standard will be listed in blue, followed by its ambient counterpart.

Number and Operations in Base Ten   1.NBT
Understand place value.
3.  Compare two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

After becoming familiar with the -teens and -ty’s along with regular practice counting to 100 or 120, the above exercise can be attempted with lots of support, if needed.  As always, any new concept should be introduced with a story!  Here’s the equations and equivalency story from the Grade 1 Daily Lesson Plans book.

Equations / Equivalency
The numbers love to play, alone and in teams.  Often their games are tied, with both sides scoring the same.  The equals sign is a good score keeper who sees both sides. (Hold your hands in front, palms down facing different directions, with index fingers slightly pointed.)   The numbers also like to compete one on one, to show whose value is greater or less.  Then they use a special sign, one that cheers the greater number, turning away from the number that’s less.  (A “talking” gesture with your fingertips shows the “greater-less than” sign.)

The hand gestures that accompany the story clearly show not only the operations signs for equal and greater-less than, but show their functions and meanings as well.  This story could be told after the two-digit numbers are well established using story, art, and movement.  The day after the story is told, the children retell it and then draw a colorful picture of it.   The concept is practiced after this, with manipulatives and real numbers first, then on paper.

The morning circle is an excellent time to practice this as well.  The teacher could call out two numbers and have the child repeat the numbers along with using the hand gestures mentioned in the story.  Here are some examples with the accompanying gestures:  18 and 18 / equal sign (Hold hands in front, palms down facing different directions, with index fingers slightly pointed.)  27 and 30 / greater-less than sign (A “talking” gesture with fingertips, the open end (fingers) toward the 30 and closed end (wrist) toward the 27.)  The two numbers could be represented by a closed fist held up in the proper position as each number is said.

As always, it’s the movement, story, and art that win the day with math!  Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal.  More Common Core Grade 1 Number and Operations in Base Ten Standards with their ambient counterparts tomorrow!

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4.19.14 | A Year in the Life: Ambient Math Wins the Race to the Top!

Posted in Uncategorized at 4:59 am by admin

Day 104

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.  Today’s standard will be listed in blue, followed by its ambient counterpart.

Number and Operations in Base Ten   1.NBT
Understand place value.
2.  Understand that the two digits of a two-digit number represent amounts of tens and ones.  Understand the following as special cases:
a) 10 can be thought of as a bundle of ten ones — called a “ten.”
b) The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c) The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

Yesterday’s post fits today’s standard perfectly!  So I will copy and paste it with a few additions, along with an illustration of the -teens and -ty’s in the story at the bottom of this post.  This is also a good example of how to work in a main lesson book.  Ideally, the pages of the book should be at least 9″ x 12″ and blank rather than lined.  Organizing content on a blank page is a wonderfully effective exercise!  (Note that guidelines for printing can be created with a yellow crayon.)

Waldorf Ed and Math By Hand do not teach place value until Grade 2.  In Grade 1 however, there needs to be a bridge or transition from the numbers 1-12 after they’ve been slowly and carefully taught (each one accompanied by a story, a geometric form, and lots of movement) to the -teens and -ty’s (20, 30, 40, etc) up to 100.  In the Math By Hand Grade 1 Binder, lessons are aligned to California State and National Math Standards.  Before the inception of the Common Core, the requirement for Grade 1 was counting up to 100.  (At this time I am not planning on revising the standards since Common Core is so much in flux.  Besides, the old standards are more user-friendly, child-friendly and developmentally appropriate.)  Here is an excerpt from the Math By Hand Grade 1 Binder:

Count, read, and write whole numbers to 100.  Know 2s, 5s, 10s to 100.

Move beyond the number 12 to teach the -teens, with glass gems as counters.  Using a piece of newsprint paper, draw 10 large ovals (about the size of an egg) in a vertical column on the left, spacing them evenly.  Then add gems to each one, 1 for 11, 2 for 12, etc.  Replace the gems with color dots, and write each number and its word out on the right side.  Have the child(ren) copy yours as you model it on paper or the board.  Note that the words for eleven and twelve are different.  Their origins go back to the early use of Arabic numerals.  The word “eleven” comes from an Old English word meaning, “one left over.”  And “twelve,” from a word meaning “two left over.”  Except for thirteen and fifteen, the rest just add the suffix -teen to the number.  Both suffixes -teen and -ty derive from the word “ten.”  Try telling a story like this while showing it visually.

Once upon a time, the numbers were happy to be a small family, from 1 – 10.  They knew and understood each other very well, though each one was so different.  Then one day, they were asked to count 2 more things than 10.  They called the first one “eleven” or “one left over,” and the second one, “twelve,” or “two left over.”  Then, they were called upon to count so much more that they needed to put the numbers together with 10.  That’s how the -teens came about, with all the numbers having to stand next to 10.  The numbers stood SO close together, that they squeezed the “0” out.  Then the things that had to be counted grew and grew to be so many that the 10’s had to stand together, asking all the other numbers to stand there with them.  These were called the -ty’s (say “tees”).  So, twenty is two 10’s, and thirty is three.  First, all of the “1’s” were squeezed out by the other numbers, then the other “0’s” were.   So, for example,  3 10s became 30, and 3 10s + 3 became 33!

Include all the numbers to 90, then say that one hundred is ten 10’s.  No need to show all the numbers visually, just a sampling’s enough.  Be sure to use the Real Numbers as props, then go on to illustrate the -teens and -ty’s on separate pages.

So the child is brought along slowly and carefully, in a very pictorial way.  Lots of illustration and color make up the heart of this lesson and activity.  The image below illustrates the story, using real numbers as props.  The real numbers can either be placed as manipulatives above the written numbers or worked with separately, then drawn on the top of the page.  (The three sets of real numbers shown can be found in the Math By Hand / Grade 1 / Kit 2 / Real Numbers.

When teaching from the whole to the parts, there is no need to rush into abstraction.  If the numbers are known in sequence and counting order, taking this “whole” apart can come a little later.  But the lesson on -teens and -ty’s does build an excellent foundation for being able to do this, because of the fact that within any group of -ty’s numbers follow the known sequence (1-2-3-4-5-6-7-8-9).

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal.  More on the Common Core Grade 1 Number and Operations in Base Ten Standards along with their ambient counterparts tomorrow!

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