A Year in the Life: Ambient Math Wins the Race to the Top!
Day 195
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 3.OA
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Patterns! are what Math By Hand and Waldorf Education and math itself is all about! This post will focus on patterns taken from the Math By Hand Grade 3 curriculum, with illustrations. Patterns abound here because they are the true heart of math, of life itself. When a math curriculum aligns with and reflects the basic, underlying principles that structure virtually everything around us, that deep relevance is what carries the day and makes for a happily successful math student.
Now for some Math By Hand Grade 3 goodies, taken from the supplemental booklet called, “Tricks & Patterns.” Note that these are just a few of the many fascinating, motivating patterns found in the Math By Hand / Grade 3 / Kit 2: Rhythms & Stories & Patterns.
NUMBER PALINDROMES
Palindromes are words, phrases, or numbers that read the same backwards and forwards.
PALINDROME WORDS: /madam /kayak /level /net-ten /gel-leg /trap-part /step-pets
PALINDROME PHRASES: / don’t nod / never odd or even / no lemon, no melon / bird rib
NUMBER PALINDROMES: / 2772 / 33 / 6776 / 112211 / 11 / 535 / 779977 / 4884 / 22
Here’s how to make a number palindrome:
1)Start with any whole number.
2)Reverse it.
3)Add it to the original number.
4)Reverse the answer and add it.
5)Continue this process until you see a number palindrome. Does it always work? Try it!
Here’s an example: 3782 + 2873 = 6655 + 5566 = 12221
THE 9′S PYRAMID
0×9 + 1 = 1
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1,111
1234 x 9 + 5 = 11,111
12345 x 9 + 6 = 111,111
123456 x 9 + 7 = 1,111,111
1234567 x 9 + 8 = 11,111,111
12345678 x 9 + 9 = 111,111,111
123456789 x 9 + 10 = 1,111,111,111
See below for the absolute jewel of all number patterns, Pascal’s Triangle. Note the patterns! The first row is 1′s, the second row is the counting numbers, the third row is the overlapping, alternating square and triangle numbers, and the very best, the hockey stick pattern, goes like this: go down any number of numbers on a diagonal, add them, then make a sharp turn right or left for the answer. Again, these are just a few of many, many Math By Hand patterns.
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 3 math CCSS and their ambient counterparts.
The post 3.OA 9: Patterns! Make Math Meaningful & Exciting (#195) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 194
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 3.OA
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
8. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing in for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
An online search for Grade 3 Common Core word problems turned up the usual suspects. Anonymous boys and girls juggling quantities of various objects between and among themselves. “If John has 28 pieces of candy and Jane has 12 less, how many pieces of candy do John and Jane have altogether?” Hmmmmm. What do these detached scenarios accomplish, really? They may be able to teach the bare facts successfully, but at what price? Boredom and alienation?
The Waldorf Grade 3 curriculum is rich in opportunities to practice life rather than word problems. “If John is building a chicken coop and he needs eight 12 foot boards for two walls, and eight 6 foot boards for one wall, how many board feet does he need altogether? If the boards cost 76 cents per linear foot, how much will it cost to build the three walls?” Now the crucial difference here is that John really is building this chicken coop with his teacher/parents/class and they all really do need to know if the chicken coop cost is within their means. Guarantee: no third grader will fall asleep over this kind of math practice.
A letter standing in for the unknown quantity. This was accomplished in essence in Grade 1, only without the letter. A white square is used to represent the unknown quantity in equations practice with the color-coded strips, in all four operations. This will translate nicely to its algebraic equivalent when the time is right. My online search also turned up the use of parentheses. It may be too soon for this level of abstraction. Algebraic thinking could happen now, but without the formal language.
Drawings are a big part of Common Core math. But the drawings tend to be crude and somewhat scribbled. It’s recommended that students draw elements of their problem-solving as they go. One example I found online involved Jack and Jill, the number of buckets they each owned to begin with, and the number of buckets each one of them left on top of the hill before bringing the remaining buckets down the hill. Then it asked how many buckets remained (down at the bottom of the hill) altogether? Now not only does this chop up a perfectly good nursery rhyme, but it also leaves something to be desired in its execution, as the buckets are drawn as counters, repetitively and crudely.
This sort of scribbling does not meet the young child where s/he needs to be met, creating work that’s beautiful, full of integrity and purpose. Assess the reasonableness of answers. Reasoning is not quite ripe and ready at 9. Math facts should be just that, facts. Analysis will only produce confusion at this point. As for mental computation, this can happen beautifully and successfully in the context of the morning circle and other math skills practice with movement. A wonderful example of thinking on their feet! Rounding happens often in the context of skills practice, both oral and written. Once the rules are established, rounding is simple and as always, practice makes perfect.
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 3 math CCSS and their ambient counterparts.
The post 3.OA 8: Life-Sourced Word Problems, Too Soon for Algebra? (#194) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 193
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. For today, a break from the Common Core, because honestly, though I’m sure creative teachers everywhere are coming up with dynamic and interesting lessons based on the CC, standards, any standards tend to be on the dry and boring side.
So here’s a Grade 3 break, hopefully a refreshingly inspiring one. This craft activity is taken from the Math By Hand Grade 3 Form Drawing/Stories book. The Grade 3 stories are a collection of worldwide creation myths and North American Indian housebuilding arts and tools. The mix of creation and housebuilding stories supports the 9 year old’s need for reassurance that as s/he leaves the garden of childhood, survival tools and resources will be there.
California tribes were most diverse, reflecting its many geologically and ecologically different regions. The Yurok depended on fishing skills for survival, and to this end, wove complex nets that were attached to poles secured on either side of a river or stream. A version of this knot-tied net can be fairly easily replicated by a third grader with some adult assistance. See below for weaving/tying instructions and a finished net, tied and woven with rainbow yarn.
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 3 math CCSS and their ambient counterparts.
The post CC Break: A Yurok Carrying Net (#193) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 193
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. For today, a break from the Common Core, because honestly, though I’m sure creative teachers everywhere are coming up with dynamic and interesting lessons based on the CC, standards, any standards tend to be on the dry and boring side.
So here’s a Grade 3 break, hopefully a refreshingly inspiring one. This craft activity is taken from the Math By Hand Grade 3 Form Drawing/Stories book. The Grade 3 stories are a collection of worldwide creation myths and North American Indian housebuilding arts and tools. The mix of creation and housebuilding stories supports the 9 year old’s need for reassurance that as s/he leaves the garden of childhood, survival tools and resources will be there.
California tribes were most diverse, reflecting its many geologically and ecologically different regions. The Yurok depended on fishing skills for survival, and to this end, wove complex nets that were attached to poles secured on either side of a river or stream. A version of this knot-tied net can be fairly easily replicated by a third grader with some adult assistance. See below for weaving/tying instructions and a finished net, tied and woven with rainbow yarn.
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 3 math CCSS and their ambient counterparts.
The post CC Break: A Yurok Carrying Net (#193) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 192
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 3.OA
Multiply and divide within 100.
7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division(e.g., knowing that 8 x 5 = 40, one knows that 40 / 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
This post will be short and sweet. Short because it’s a bit repetitive to be consistently singing the praises of Waldorf education, and sweet because both Waldorf and Math By Hand’s goals are fluency in addition, subtraction, multiplication and division within 1000+ and knowing all times tables by heart, both by the middle to end of Grade 3.
“By heart.” This is quite literal because learning and practicing times tables should always happen with heartfelt joy. Yes, there needs to be an element of rote memorization, but for the most part, fun is had, colorfully creative learning tools and books are made, and a joyful noise is sung, recited, marched, and rhythmically moved to the beat of the times tables.
I call it “back door learning” because as Yeats said, “Education is not the filling of a pail but the lighting of a fire.” And not many fires can be lit with the boring statement, “six times four is twenty four.” Rather, the creative fire is sparked with the elements of mystery and surprise. Setting up a lesson with 10 river rocks placed in a large, perfect circle on the floor, with a chunky, colorful rope in the center, accompanied with the unlikely question, “What pattern does the 4 table make on this circle?” will pique any child’s curiosity and interest.
When I taught this lesson in a Waldorf Grade 2 class, it was met with gobs of enthusiasm. The next day, one very excited boy brought in a wonderful rendering of all the patterns superimposed on one another in different colors. See below for these patterns on a circle of 10, noticing that 1′s and 9′s, 2′s and 8′s, 3′s and 7′s, 4′s and 6′s are all mirrored.
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 3 math CCSS and their ambient counterparts.
The post 3.OA 7: Fluency In All 4 Processes & Times Tables By Heart (#192) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 191
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 3.OA
Understand properties of multiplication and the relationship between multiplication and division.
6. Understand division as an unknown factor problem. For example, find 32 / 8 by finding the number that makes 32 when multiplied by 8.
As stated in earlier posts, this concept is solidified very early on since division is worked with in Grade 1, alongside the other 3 processes. The white “unknown number” square is placed anywhere on the color practice strips to simulate the unknown factor in today’s standard. When solid foundations like this are built early, familiarity wins the day as the same concept spirals back later on with greater and greater complexity. So, the Math By Hand or Waldorf student will fully comprehend this concept in this standard, from Grade 1 on through the grades.
A really essential piece for teaching the 4 processes is conveying them with personality. And this is beautifully accomplished in Grade 1 by applying one of the four temperaments to each of the 4 processes. Division, as you may have guessed, is choleric, colored red! The nature of each of the processes becomes second nature to the child, and more importantly, math becomes humanized and friendly, approachable. All fears and phobias fall away in the face of such “personalizing.”
The 4 processes stories told in Grade 1 are simple, and aligned with that year’s story theme, the fairy tales. The King might organize a counting house where the farmers bring their wares to distribute to the townsfolk in exchange for gems. The glass gems shown above can be used as the first calculating tools, and many imaginative scenarios are created in which the 4 characters engage in calculations.
Here is Divida’s story, taken from the Math By Hand Grade 1 Daily Lesson Plans book.
Once there was a kingdom where 4 special children lived. Each one was different from the others, but in spite of their differences they got along well. All good friends. they always helped each other. More importantly, the kingdom itself could not get along without them. The King gave them honorary names and decreed that each wear an especially bright color so everyone would know who each one was at first sight.
DIVIDA was dressed all in red because she was so decisive! Whenever the King needed a strong, fair person to take charge and keep things in order, he called on DIVIDA. When a disagreement broke out among her friends or others in the kingdom, she was a wonderfully impartial judge, helping to settle matters fair and square.
DIVIDA was respected because she could be counted on to step in and make all the tough decisions, but all that decision making sometimes made her feel that she alone was right. Sometimes the King had to step in and help her to see others’ points of view. At times like these, DIVIDE readily listened because she was so very fair after all. DIVIDA, all in red, was a treasure. Without her, the kingdom would often be stuck in confusion and indecision.
And here is Divida herself! You can see that attaching personalities to each of the 4 processes helps to make them more friendly and less daunting, lifelong! Do invest the time and energy it would take to personalize division as well as the other 3 processes, at any age or stage! The Grade 3 personalities could coordinate to the stories that are told in that grade.
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 3 math CCSS and their ambient counterparts.
The post 3.OA 6: What Is the Essence of Division? First Things First (#191) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 191
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 3.OA
Understand properties of multiplication and the relationship between multiplication and division.
6. Understand division as an unknown factor problem. For example, find 32 / 8 by finding the number that makes 32 when multiplied by 8.
As stated in earlier posts, this concept is solidified very early on since division is worked with in Grade 1, alongside the other 3 processes. The white “unknown number” square is placed anywhere on the color practice strips to simulate the unknown factor in today’s standard. When solid foundations like this are built early, familiarity wins the day as the same concept spirals back later on with greater and greater complexity. So, the Math By Hand or Waldorf student will fully comprehend this concept in this standard, from Grade 1 on through the grades.
A really essential piece for teaching the 4 processes is conveying them with personality. And this is beautifully accomplished in Grade 1 by applying one of the four temperaments to each of the 4 processes. Division, as you may have guessed, is choleric, colored red! The nature of each of the processes becomes second nature to the child, and more importantly, math becomes humanized and friendly, approachable. All fears and phobias fall away in the face of such “personalizing.”
The 4 processes stories told in Grade 1 are simple, and aligned with that year’s story theme, the fairy tales. The King might organize a counting house where the farmers bring their wares to distribute to the townsfolk in exchange for gems. The glass gems shown above can be used as the first calculating tools, and many imaginative scenarios are created in which the 4 characters engage in calculations.
Here is Divida’s story, taken from the Math By Hand Grade 1 Daily Lesson Plans book.
Once there was a kingdom where 4 special children lived. Each one was different from the others, but in spite of their differences they got along well. All good friends. they always helped each other. More importantly, the kingdom itself could not get along without them. The King gave them honorary names and decreed that each wear an especially bright color so everyone would know who each one was at first sight.
DIVIDA was dressed all in red because she was so decisive! Whenever the King needed a strong, fair person to take charge and keep things in order, he called on DIVIDA. When a disagreement broke out among her friends or others in the kingdom, she was a wonderfully impartial judge, helping to settle matters fair and square.
DIVIDA was respected because she could be counted on to step in and make all the tough decisions, but all that decision making sometimes made her feel that she alone was right. Sometimes the King had to step in and help her to see others’ points of view. At times like these, DIVIDE readily listened because she was so very fair after all. DIVIDA, all in red, was a treasure. Without her, the kingdom would often be stuck in confusion and indecision.
And here is Divida herself! You can see that attaching personalities to each of the 4 processes helps to make them more friendly and less daunting, lifelong! Do invest the time and energy it would take to personalize division as well as the other 3 processes, at any age or stage! The Grade 3 personalities could coordinate to the stories that are told in that grade.
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 3 math CCSS and their ambient counterparts.
The post 3.OA 6: What Is the Essence of Division? First Things First (#191) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 190
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 3.OA
Understand properties of multiplication and the relationship between multiplication and division.
5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 24 – 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property of multiplication.)
Wow. This is very complex for Grade 3, and should be concretized with stories, art, and movement. Math By Hand teaches these three properties in Grade 2, using these methods. Here is an excerpt from the Grade 2 Daily Lesson Plans book.
Tell the three stories on the next page, one each day for the next three days. The stories will help to show these three properties in a simplified, fundamental way. Note that they will be revisited later on, with more advanced math and with algebra. The terms used here are concrete, informal versions of the more abstract terms: commutative, associative, and distributive. Parentheses will be given the informal name, “huggers” previewing their later use in algebra.
Commuting Numbers / Tell This Story for Day 8
When numbers swap and switch places on either side of the equals sign, they’re having fun commuting from one side to the other! It’s like being on a seesaw when you are balanced with your partner. Numbers on the seesaw say, “Commute! Commute!” as they hop from one side to the other. Both Plus and Times like to play this game.
Associating Numbers / Tell This Story for Day 9
When numbers are friendly they like to associate, with hugs. Sometimes they will want to change partners for hugs, but they cooperate no matter how they get together, to be sure they always have the same answer. They say “my associate,” meaning “my friend” whenever they hug. This is a favorite game for Plus, but Times likes it too.
Distributing Numbers / Tell This Story for Day 10
Remember how busy Times can be? Well he loves to rearrange numbers, sometimes asking them to hug and sometimes not. He can be heard calling, “Distribute!” loud and cheery as he hops and jumps into the room. He busily puts numbers in hugging pairs, then takes them apart again. This is a favorite game for Times, but Plus (and sometimes Minus) like it too.
Each story is told on Day 1 and retold by the children on Day 2, after which they do a guided illustration then practice, with many examples. This practice should include other forms of visual, auditory or kinesthetic memory aids. For example, an actual seesaw could be set up for the commutative equations, balancing numbers or even children-as-numbers (wearing them as tags) on either side.
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 3 math CCSS and their ambient counterparts.
The post 3.OA 5: Have Fun First & Foremost With New Concepts (#190) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 189
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 3.OA
Represent and solve problems involving multiplication and division.
4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 x ? = 48, 5 = __ / 3, 6 x 6 = ? .
Back to Math By Hand Grade 1 to fulfill this standard! Much practice occurred with color coded strips in all 4 processes, alone and side-by-side to show equivalency. A little white square is the color strips’ magic ingredient. Here’s how it works. The color strips are divided into 6 sections: they are 18″ long and folded into 3″ squares. They’re used with manipulatives, the glass gems shown above or counting beans, to represent the numbers in equations.
And here’s the magic part. A 3″ white square can be placed anywhere on the strip, in any position in the equation, perfectly fulfilling this Grade 3 standard in Grade 1. And even though the equations never total more than 20, the concept is learned and carried on later, with higher order equations in all 4 processes. The next step is to use the sets of “real numbers,” lovingly crafted in Grade 1, in place of the gems/beans.
Transitions are made slowly and solidly. Slow and steady wins the race to the top, after all. The next step is writing the equations on paper, but this is always paired with using the manipulatives so the numbers and their functions remain grounded and tangible. This is foundational and can form a strong basis for learning new concepts and all skills practice.
Skills practice needs to be more than merely 2-D with pencil and paper. For today’s standard, oral practice with a ball or bean bag works well. Here’s how. For the equation, 8 x ? = 48 say, “eight times,” throwing the ball for the student to catch on the blank. Then say, “equals forty eight.” The student throws the ball back saying, “six.”
For the equation, 5 = __ / 3, have tambourines or other percussive instruments on hand (preferably made by you and the student(s) and say, “five equals.” Strike the hand drum or tambourine for the blank then say, “divided by three.” The student accompanies the answer, “fifteen” with his/her tambourine or drum.
For the equation, 6 x 6 = ?, you might modify the game “Simon Says” by using a gesture for the blank, like jumping up and down, making a funny face, or clapping three times after saying, “six times six equals.” The student then accompanies the answer, “thirty six” with the same gesture. Have fun with it, 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 3 math CCSS and their ambient counterparts.
The post 3.OA 4: Keeping It Real While Filling In the Blanks! (#189) appeared first on Math By Hand.
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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 188
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 3.OA
Represent and solve problems involving multiplication and division.
3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Much creative teacher energy has been expended on creating worksheets to address this standard. One of the problems with worksheets (or computer apps for that matter) is that they are 2-dimensional in nature, and largely divorced from reality or anything familiar that could act as an anchor in the learning process. Computer, iPad, or other digital media apps require very little physical activity, and in fact require much rigidity, in that the hands are poised on the keyboard, the eyes are riveted on the screen, and the child’s body which is built and meant to move, is glued to a chair.
In the Waldorf and Math By Hand systems, this standard is addressed early on, from the beginning, since multiplication and division are taught along with addition and subtraction in Grade 1. And this is for the most part accomplished 3-dimensionally and with much movement, as word problems are practiced orally in the morning circle or skills practice time. Gems (like those pictured above) are used as counting manipulatives to accompany situations that are play acted. And although the numbers did not total up to 100, the concepts were clearly taught and seen through groups and arrays made with real objects.
Because this concept was solidified in Grade 1, it can be applied now in Grade 3 to higher order problems. The 4 processes are reviewed at the beginning of Grade 3, and since place value was mastered in Grade 2, students are ready to take on problems totaling far greater amounts than 100! Long multiplication and division are learned in the middle of Grade 3, and then incorporated into 4 processes practice.
Learning basic algorithms is sometimes shortchanged with the Common Core standards and materials, in favor of innovative and alternative methods. Teaching the classic, basic algorithms alongside alternative methods is a much sounder, more sensible approach. Math By Hand explores the alternatives like Lattice or Window Frame multiplication as fun and novel approaches that are motivating and interesting, but are never allowed to replace the basics.
See below for an example of Window Frame and another alternative called Russian Peasant Times. (Divide all numbers on the left in half (rounding down where needed), double all numbers on the right until reaching number 1 on the left. Cross out all numbers on the right that line up with even numbers on the left. Add the remaining numbers on the right for the answer. Try it on your calculator, it’s correct!)
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 3 math CCSS and their ambient counterparts.
The post 3.OA 3: Moving in 3-D vs. Worksheets & Apps (#188) appeared first on Math By Hand.
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