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

Posted in Uncategorized at 6:56 am by admin

Day 101

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 post will continue with the Common Core Standards for Grade 1, listed in blue and followed by their ambient counterparts.

But first, I just need to say at 101 days I am a bit blog-weary.  So today’s and tomorrow’s posts will be brief.  (It does help that these two standards are quite straightforward and simple.)  Next, on to the briar patch of place value.  All the more so because both Waldorf and Math By Hand do not teach place value directly until Grade 2.  But it may be possible to lay the foundation with age-appropriate lessons and activities, we’ll see.  On to #7.

Operations and Algebraic Thinking   1.OA
Add and subtract within 20.
7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.  For example, which of the following equations are true and which are false?   6 = 6,  7 = 8 – 1,  5 + 2 = 2 + 5,  4 + 1 = 5 + 2.

This sort of activity is best not done on paper at first.  Worksheets are not appropriate either.  And you may want to substitute “is” for “equals” at this stage, since it’s more direct.  Here are a few ambient, alternative suggestions:

Using a play clay equal sign, glass gems or other manipulatives, and the 4 processes color-coded strips, set up equations like those above and ask if they’re correct.  Allowing enough time for counting and exploration to discover the answer, have your child use special markers to say yes or no.  For example, shells = yes, small stones = no.  After checking the markers, change those that need to be changed, while reviewing them.  Then say that all the stones need to be changed to shells, providing guidance where needed.  As always, making the abstract concrete is key.

Using a beanbag or a soft ball, play a game of math catch while saying the equations.  As the child catches the ball, s/he says yes or no, providing the correct equations where needed.  If you say “6 is 6″ the child says yes as s/he catches the ball.  If you say “5 + 2 is 8″ the child says no as s/he catches the ball and says ” 5 + 2 is 7″ as s/he tosses it back to you.  Be slow and deliberate with this, and bring it sparingly because of the child’s deep respect for you as the teacher to always tell the truth.  Time enough for critical thinking later on!

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 Operations and Algebraic Thinking Standards along with their ambient counterparts tomorrow!

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

Posted in Uncategorized at 7:23 am by admin

Day 100 (yay, triple digits!)

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 post will continue with the Common Core Standards for Grade 1, listed in blue and followed by their ambient counterparts.

Operations and Algebraic Thinking   1.OA
Add and subtract within 20.
6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.  Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 13).

Wow.  This is the basis of one of the Common Core controversies that’s all over the internet, posted by confused parents defending children stressed over too much homework and testing.  And coming to the aid of teachers who are stressed over possible censure and/or threats of job loss.  All that aside, it’s a mistaken “improvement” on existing math methods.  This sort of juggling numbers requires a good deal of abstract thinking which is developmentally beyond the ken of first graders.  Manipulatives are most likely being used in some cases to teach this, but it still demands more than a 7 year old is able to handle.

Some of the Common Core demonstration lessons I’ve seen use snap-together plastic cubes as manipulatives to solve these equations.  And there are many colorful, imaginative plastic sets available as classroom kits, like 20 plastic strawberries in a plastic basket, or whole picnic-themed sets with several different items, used for counting, addition and subtraction.  These are specifically aligned to Common Core coordinated worksheets, either consumable pencil and paper or laminated to be used with wipe off crayons/markers.  All of this is pre-fab and truly deadens the creative impulse every child is born with.

Using simple, natural materials as manipulatives and creating child/parent-made learning tools is far superior.  The Math By Hand glass gems and colored paper strips are a good example of this.   Re making ten, decomposing to lead to a ten, or creating equivalent, easier known sums, these may all be seen as mental shortcuts and could be valuable at the right age and place, which Grade 1 is decidedly not.  Basic and straightforward beginnings are best, such as learning addition, subtraction, multiplication, and division side by side with simple visual aids.  This is a good way to practice the section of the standard that asks for juxtaposing addition and subtraction.

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 Operations and Algebraic Thinking Standards along with their ambient counterparts tomorrow!

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

Posted in Uncategorized at 6:32 am by admin

Day 99

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 post will continue with the Common Core Standards for Grade 1, listed in blue and followed by their ambient counterparts.

Operations and Algebraic Thinking   1.OA
Add and subtract within 20.
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

In the Waldorf Grade 1, rhythmic counting is used for more than one purpose: to learn to count up to 100 by 1′s and to learn some of the times tables.  Counting by 2 happens this way: the children whisper and step lightly on the 1, 3, 5, 7, etc. while stamping and speaking the 2, 4, 6, etc.  The odd numbers are later dropped and just the 2′s are spoken and rhythmically stepped.  The 2′s can be further enhanced by clapping, skipping, waving a small flag, or playing a tambourine.  Rhythm and meter are close friends of math, and they provide the best means to effectively teach and learn it.

Counting to add and subtract is made clearer and more accessible by using manipulatives like the glass gems.  This can be done with equations or composing and decomposing numbers, and can also later translate to factoring composite numbers.  The 4 processes should always be viewed as comparable and parallel.  Just as counting on 2 can be used for multiplication as well as addition, it can also be used for division along with subtraction.  The glass gems, then the real numbers, and finally pencil and paper can be used for this.

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 Operations and Algebraic Thinking Standards along with their ambient counterparts tomorrow!

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

Posted in Uncategorized at 3:31 am by admin

Day 98

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 post will continue with the Common Core Standards for Grade 1, listed in blue and followed by their ambient counterparts.

Operations and Algebraic Thinking   1.OA
Understand and apply properties of operations and the relationship between addition and subtraction.
4. Understand subtraction as an unknown addend problem.  For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

The unknown number concept works well with the color strips.  Using this example, place the two strips, green (addition) and blue (subtraction) next to each other.

For addition (green):
Place 8 gems or counters in the first square and 2 in the second.
The white square is third for the answer: 10 gems.

For subtraction (blue):
Place 10 gems in the first square and 8 gems in the second.
The white square is third for the answer: 2 gems.

Move the white square around to different positions for finding the unknown number:
___ +  2  =  10,  8  + ___  = 10.  Or  ___  -  8  =  2,  10  -  ___  =  2.

Continue with many similar equations, all under 20, using the gems or counters.  Transition to working with the real numbers before writing the equations.

Using the color strips, expand this practice to include multiplication (yellow) and division (red) placed side by side.  (Note that in Grade 1, the 2, 5, and 10 tables are learned.)

For  6  x  2  =  12  multiplication (yellow):
Place 6 gems in the first square and 2 gems in the second.
The white square is third with the answer: 12 gems.

For division (red):
Place 12 gems in the first square and 6 gems in the second.
The white square is third with the answer: 2 gems.

Move the white square around to different positions for finding the unknown number:
___  x  2  =  12,  6  x  ___  =  12.  Or  ___  /  6  =  2,  12  /  ___  =  2.

Continue with many similar equations, all under 20, using the gems or counters.  Transition to working with the real numbers before writing the equations.

Work with equivalent and inverse equations by comparing addition with multiplication and subtraction with division.  Using the colored strips to see all 4 processes together in various relationships is an example of working from the whole to the parts.  This is most beneficial, as is using the colored strips and white squares in place of the operation signs at first.  This helps to not only simplify the process but also to work with the pictorial or visual aspect, so much more effective than abstract symbols at this age.

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 Operations and Algebraic Thinking Standards along with their ambient counterparts tomorrow!

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

Posted in Uncategorized at 7:38 pm by admin

Day 97

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 post will continue with the Common Core Standards for Grade 1, listed in blue and followed by their ambient counterparts.

Operations and Algebraic Thinking   1.OA
Understand and apply properties of operations and the relationship between addition and subtraction.
3. Apply properties of operations as strategies to add and subtract.  Examples: if 8 + 3 = 11 is known, then 3 + 8 = 11 is also known.  (Commutative property of addition.)  To add 2 + 6 + 4, the second two numbers can be added to make a 10, so 2 + 6 + 4 = 2 + 10 = 12.  (Associative property of addition.)

Math By Hand  introduces all three properties: associative, commutative, and distributive, in the Grade 2 Daily Lesson Plans book, in a thoroughly concrete and playful way.  When concepts are presented too abstractly too soon, it results in confusion, anger, and loss of confidence.  This is what we see happening with the protest by parents trying to understand and help their children with Common Core homework.  And with parents deciding to opt their children out of testing because it will likely fail them, even those who were high math achievers until the Common Core came along.

The “high stakes” nature of the testing has put teachers and children at risk: failure, mislabeling as learning challenged, and losing a grade for children, censure and job loss for teachers.  High stakes then becomes high stress, for teachers, children, and parents alike.  It would be immeasurably better for Grades 1-3 (Kindergarten is intentionally omitted) to learn the basics first, before tackling the more abstract properties.  There certainly can be some flexibility within this though, as mentioned yesterday.  When using the color strips, the blank, white square can be placed anywhere to represent either the answer or a missing number.  This allows a flexibility of thinking that can later translate to algebraic concepts.

As previously stated, working with all 4 processes at once allows a broader, more effective comparison.  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.  Tune in tomorrow for more on the Common Core Grade 1 Operations and Algebraic Thinking Standards along with their ambient counterparts.

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