Sacramento Homeschool Math By Hand

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 203

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.

Measurement and Data 3.MD
Solve problems involving measurement and estimations of intervals of time, liquid volumes, and masses of objects.
2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).  Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

Two words for teaching this standard: cooking and baking.  Metric measurement may be a bit of a problem here in the US, but that’s another opportunity to practice math!  Have a conversion chart handy to translate US standard to metric.  Both liquid and dry (masses of objects) are used in just about any recipe.

Halving or doubling recipes will afford a lot of 4 processes practice while cooking or baking.  And once again, real life trumps word problems.  How much better than a drawing of a beaker with a measurement scale is a real-live measuring cup and set of measuring spoons?

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.MD 2: Real Measurement: Cooking & Baking! (#203) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 202

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.

Measurement and Data 3.MD
Solve problems involving measurement and estimations of intervals of time, liquid volumes, and masses of objects.
1. Tell and write time to the nearest minute and measure time intervals in minutes.  Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

There’s that number line again.  From the last blog post, “As for the number line, waiting until the full picture (negative numbers, the concept of zero, and the 4 processes on the number line) can be introduced is optimal.”  Math By Hand introduces the number line with all of its complex concepts at the end of Grade 4.  Partial work with it before then is counter-productive since it cannot be correctly understood without zero holding the middle, positive numbers to the right, negative numbers to the left, and a basic outline of how the numbers interact on it, within the 4 processes.

Time is one of our most engaging and pervasive every-day mysteries.  Einstein certainly thought so, and pondering its nature resulted in some of his best work.  Time should not be taught as minutes on a clock face or digital dial without first honoring its biography: how did it come to be and what words and elements form its essence?  This can and should be related in an anecdotal manner, with generous doses of humor and wonderment.  Here is an excerpt from the Math By Hand Grade 3 binder:

A good place to start is with the history of timekeeping and timepieces. Simply noting natural units of time measurement, made apparent by the sun and moon, could be the first reference. Then, going on from there, note the many different ways of measuring these units. In ancient Rome, priests announced the first day of the month with the appearance of the new moon. The Romans called this first day of the month Kalends from their word calare or “to proclaim.” This is where our word calendar comes from. The word clock is more direct, taken from the Latin word clocca or “bell,” since bells were the first means of announcing the time or marking the hours. And storytelling, as always, is the best cloak to wrap around these facts or anecdotes.

The teacher should research and learn these facts then present them in story form.  Rather than going directly to the concept of the modern clock, which can be complex and dry, time should be presented as it evolved.  Constructing a sun dial would be a wonderful hands-on project, tying time to the sun where it originated and making it more concrete than abstract.  Examples of and instructions for making sun dials can be found at the library or online.

This one would be fairly easy to construct, using any variety of materials, and illustrates the morning and afternoon path of the sun.  The hours are 6 to noon in the morning on the left, and noon to 6 in the afternoon on the right.  Increments for the quarter and half hour could be indicated as well.  No better foundation for learning about time exists than a child making a sundial like this, before moving on to navigating time in minutes.

Problems taken from real life are far better than abstract word problems, which can be boring at best and even worse, alienating and ineffective.  Some examples of real-life time problems would be, “Let’s time how long it takes us to plant one row of carrots, then multiply that by the 5 rows we’re planting.  Then let’s compare that to how long it takes us to mulch one row of strawberries times the 4 rows we’ve planted.  We can then record all that information on the garden graphs we’re keeping.”

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.MD 1: Teach Time By Telling Stories First! (#202) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 202

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.

Measurement and Data 3.MD
Solve problems involving measurement and estimations of intervals of time, liquid volumes, and masses of objects.
1. Tell and write time to the nearest minute and measure time intervals in minutes.  Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

There’s that number line again.  From the last blog post, “As for the number line, waiting until the full picture (negative numbers, the concept of zero, and the 4 processes on the number line) can be introduced is optimal.”  Math By Hand introduces the number line with all of its complex concepts at the end of Grade 4.  Partial work with it before then is counter-productive since it cannot be correctly understood without zero holding the middle, positive numbers to the right, negative numbers to the left, and a basic outline of how the numbers interact on it, within the 4 processes.

Time is one of our most engaging and pervasive every-day mysteries.  Einstein certainly thought so, and pondering its nature resulted in some of his best work.  Time should not be taught as minutes on a clock face or digital dial without first honoring its biography: how did it come to be and what words and elements form its essence?  This can and should be related in an anecdotal manner, with generous doses of humor and wonderment.  Here is an excerpt from the Math By Hand Grade 3 binder:

A good place to start is with the history of timekeeping and timepieces. Simply noting natural units of time measurement, made apparent by the sun and moon, could be the first reference. Then, going on from there, note the many different ways of measuring these units. In ancient Rome, priests announced the first day of the month with the appearance of the new moon. The Romans called this first day of the month Kalends from their word calare or “to proclaim.” This is where our word calendar comes from. The word clock is more direct, taken from the Latin word clocca or “bell,” since bells were the first means of announcing the time or marking the hours. And storytelling, as always, is the best cloak to wrap around these facts or anecdotes.

The teacher should research and learn these facts then present them in story form.  Rather than going directly to the concept of the modern clock, which can be complex and dry, time should be presented as it evolved.  Constructing a sun dial would be a wonderful hands-on project, tying time to the sun where it originated and making it more concrete than abstract.  Examples of and instructions for making sun dials can be found at the library or online.

This one would be fairly easy to construct, using any variety of materials, and illustrates the morning and afternoon path of the sun.  The hours are 6 to noon in the morning on the left, and noon to 6 in the afternoon on the right.  Increments for the quarter and half hour could be indicated as well.  No better foundation for learning about time exists than a child making a sundial like this, before moving on to navigating time in minutes.

Problems taken from real life are far better than abstract word problems, which can be boring at best and even worse, alienating and ineffective.  Some examples of real-life time problems would be, “Let’s time how long it takes us to plant one row of carrots, then multiply that by the 5 rows we’re planting.  Then let’s compare that to how long it takes us to mulch one row of strawberries times the 4 rows we’ve planted.  We can then record all that information on the garden graphs we’re keeping.”

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.MD 1: Teach Time By Telling Stories First! (#202) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 201

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.

Number and Operations – Fractions 3.NF
Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a) Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
b) Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a) Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b) Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. 
c) Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. 
d) Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Fractions.  The word fraction is taken from the Latin frangere, meaning “to break” and since the goal of Waldorf Education is to keep the world of the child whole for as long as needed, I’ll attempt to explain why Grade 4 is the optimal time to introduce the concept of fractions.

All of the lower school grades and ages are a progression from innocence to knowing.  It’s terribly essential that this process not be rushed.  The graph of brain development pictured below indicates that sensory pathways peak early and are complete at age 7.  Language development also peaks early and is complete at age 7.  In contrast, higher cognitive function peaks later and is not complete until age 16!

Note the slow, curving decline of higher cognitive function development.  This is a clear indicator that human child development needs to be a long, slow process.  There is a necessary letting go of innocence at each stage, and we rush it at our, our children’s, and the world-at-large’s peril.  We must be vigilant and closely observant of the readiness signs, and meet all of these severances with stories and art.

At age 7, the fairy tales with their sometimes frightening images serve as a wake-up call to the child’s awakening psyche.  At age 8, polarities become more apparent as the sometimes harsh reality of the world dawns on the child’s horizon.  Justice is not always served!  Reynard Fox and other unscrupulous, wily fables’ characters win out over the just and the innocent when they shouldn’t.  But the child finds support and sustenance in the deep goodness found in the saints’ and heroes’ tales.

At the very beginning of Grade 3, the 9 year old is evicted from the garden via the Old Testament or other Creation stories.  Again, polarities hold sway as the drama of the early human journey unfolds.  And here, sustenance is found in the comforting home arts: gardening, building, and crafting.

In Grade 4, polarities become ever stronger with the Norse Mythology stories or trickster tales from other cultures.  Here, there’s an echo of Grade 2 as the wily and unscrupulous win out over the innocent.  Loki kills Baldur after all, and the whole world is sunk into oblivion, as Fenris and Ragnarock destroy every living thing.  In handwork, the cross-stitch helps to keep the world at bay with every little “x” that’s sewn.  The child’s world widens with geography that starts at the center and radiates outward to the immediate community, though overall, the brokenness presented in story form presents a window for first introducing fractions.

As for the number line, waiting until the full picture (negative numbers, the concept of zero, and the 4 processes on the number line) can be introduced is optimal.  The 9 year old is in a fragile state as s/he crosses the Rubicon, out of childhood.  Time and measurement, long division and multiplication, and fully mastering the times tables, along with refining skills in all 4 processes, are all quite enough to take on at this time.  For all these reasons, fractions can wait!

Unfortunately, this also means that your third grader will not be standardized-test-ready.  With the firestorm surrounding the new high-stakes testing though, that may not be a bad thing.  Healthy, timely, proper development is a priority after all.  In these times, with technology hurrying them along, children can more and more seem like little adults.  But for their sake, for all of our sakes, please recognize that they are not, that they need your protection as they slowly grow to maturity.  Recognize this as a sacred trust, and abide by it.

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.NF 1-3: Fractions Wait Until Grade 4 (#201) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 200

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 features string stars and how to make them.

The 6, 8, and 12-point stars shown below are an offshoot of form drawing, since all of these forms will have been drawn by now.  It’s a very different experience however, to construct the forms with string.  More hands-on, and that’s always good!  It can also be a prelude to the much more complex string forms created in Grade 6 when formal geometry is introduced, like this amazing masterpiece:

The stars shown below can be constructed with rainbow string (available at most craft stores) and map pins or push pins.

All of these patterns begin with a circle drawn in yellow.

The best base for this project is white foamcore, which can be obtained in scrap pieces for free (or inexpensively) at most craft or frame stores

Here’s how:

1) Draw the circle with a yellow crayon, by exploring many circles lightly until the perfect one is found, darkening it in the process.

2) Distribute the pins evenly on the circle for the 6, 8, or 12 pointed star.

3) Begin the pattern by tying the string to the top pin and tracing triangles for the 6 and 12 pointed stars or squares for the 8 pointed star.

4) After completing each triangle or square, bring the string to the next point by following the circle, winding the string around the pins as it goes.

5) When the triangles or squares are finished, continue winding the string around the circle till the top is reached. Tie it off.

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 Grade 3 Fun Before Plunging Into Fractions! (#200) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 199

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.

Number and Operations in Base Ten 3.NBT
Use place-value understanding and properties of operations to perform multi-digit arithmetic.
3. Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.

Be playful with this.  Play and math go together like strawberries and cream, soap and water, or _______________ and _____________ (favorite pairs of your choice).  Einstein himself said so!  So be playful.  Here’s an example, a page taken from the Math By Hand Grade 2 Fables and Tables booklet.  But first the fable!

The Crow and the Pitcher

A Crow perishing with thirst saw a pitcher, and hoping to find water, flew to it with delight. When he reached it, he discovered to his grief that it contained so little water that he could not possibly get at it. He tried everything he could think of to reach the water, but all his efforts were in vain. At last he collected as many stones as he could carry and dropped them one by one with his beak into the pitcher, until he brought the water within his reach and thus saved his life.
Necessity is the mother of invention.

The idea here is that the crow started with 12 stones and when he found they weren’t nearly enough to raise the water level to where he needed it to be, he collected many more stones until he had 120, and then was finally able to drink.  The activity, which teaches/reviews the 10 times table, goes like this:

The post 3.NBT 3: Pebble Math Reinforces Place Value (#199) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 198

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.

Number and Operations in Base Ten 3.NBT
Use place-value understanding and properties of operations to perform multi-digit arithmetic.
2. Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Math By Hand students add and subtract within 1,000 and more when place value is taught, in the middle of Grade 2.  Four oversized, color-coded columns representing 1′s, 10′s, 100′s, and 1,000′s are placed on the floor for both addition and subtraction.  Students replicate a vertically stacked problem from the board by placing matching numbers (on large index cards) on their respective columns.  The numbers are then added or subtracted, and regrouping happens with small numbers that are walked or jumped to the tops of the columns.

After practicing this for some time, the problems are transposed to workbooks, and at that point addition and subtraction relationships are explored.  This has an excellent foundation since all 4 processes were taught together in Grade 1, along with exploring reciprocal relationships among and between all 4.  Fluency is achieved early on, since lots of practice occurs with exploring intriguing math tricks and patterns.  Here are a few examples from the Grade 3 booklet, “Number Tricks and Patterns.”

THE MYSTERY NUMBER: 1089

1)Subtract the reverse of any 3 numbers from the number.
2)Add the reverse of the answer.
3)The final answer is always 1089! Here ‘s just one example, do try more!

632
-236
396
+693
1089

MAGIC WITH THE NUMBER 9

1)Add the digits of any number.
2)Subtract the answer from the number.
3)Add the digits of the answer.
4)Continue adding until you get 9.
You always will. Here’s just one example, do try more!

                                                                             7586312 =(32)                                 

                                                                              -       32                                                                  

                                                                             7586280 =(36) =(9)             

As you can see, very large numbers can be used for this fun version of skills practice.  This is no drill and kill . . . rather it presents interesting and compelling patterns as irresistible incentives.  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 3 math CCSS and their ambient counterparts.

The post 3.NBT 2: Addition & Subtraction Within 1000+ (#198) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 197

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.

Number and Operations in Base Ten 3.NBT
Use place-value understanding and properties of operations to perform multi-digit arithmetic.
1. Use place value understanding to round whole numbers to the nearest 10 or 100. 

Rounding is really a very simple concept, but it can be overly abstracted and thus made confusing at this age.  An online search for suggested activities to meet this Common Core standard turned up mostly (surprise!) worksheets!  Very repetitive and boring, drill and kill.  One suggested activity involved rounding on a number line.  Math By Hand teaches the number line in Grade 4, along with the concept of using the 4 processes with positive and negative numbers on it.  Very complex, too complex for Grade 3.  

Place value is taught mid Grade 2, with beans and large, color-coded columns.  The first step is to correlate the beans with the values: 1′s are black beans, 10′s are kidney beans, and 100′s are lima beans.  This “bean bank” is used to illustrate how 10 ones are cashed in to make 10, and 10 tens are cashed in to make 100.

If this is learned pictorially and imaginatively once, that’s enough.  There’s no need to belabor the point with insisting on having all work shown with endless dots, bars, and squares.  Watch this youtube for a perfect example of this enforced, unnecessary busywork.

So.  The beans could be adapted to show rounding as well.  I found piles of worksheets with example after example of rounding.  Doesn’t this communicate a lack of trust by conveying the message, “You will have to do this over and over to prove to me that you understand it.”  Drill and kill.  Shouldn’t a gentle reminder when needed suffice?

Math By Hand brings long division and multiplication mid Grade 3, with the same large, color coded columns that were used to teach place value.  Remainders in long division are another opportunity to teach rounding.  

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.NBT 1: Place Value & Rounding In 3-D (#197) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 196

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 features an article from a Canadian newspaper, The National Post, on the merits of slowing down and exploring the joys of tried-and-true algorithms and times tables! (Graphic by Mike Faille / NP Graphics)

Here are a few excerpts, but do read the whole thing, it captures the crux of the Common Core dilemma:

University of Winnipeg math professor Anna Stokke and two of her colleagues knew there was “a huge problem,” when they started hearing about Manitoba grade school students not being taught how to do vertical addition, carry or borrow numbers, or knowing their times tables.

Then, two years ago, she and Robert Craigen, a fellow U of W professor, and Fernando Szechtman a math professor at the University of Regina, formed WISE Math — the Western Initiative for Strengthening Math Education. They set up a website with a blog, gave lots of media interviews and started meeting with government officials to push for changes in the way math was being taught.

“Then we started hearing from a lot of parents, from all across Canada,” said Ms. Stokke, whose group has collected nearly 1,000 signatures supporting its calls for reform. “It’s a lot of work and it’s a lot of trouble to advocate for things like this … but our kids are worth it, because in the end we really need our kids to learn math.”

The group is now seeing the fruit of its efforts this fall, as Manitoba rolls out a “back to basics,” revised curriculum for kindergarten to Grade 8, one explicitly requiring students to learn times tables; have automatic recall of answers to basic problems such as 30 – 5 = 25, known as math “facts”; and standard algorithms for key math operations — and perform them without using a calculator.

It marks a step back from “new math” and “inquiry-based” teaching approaches that emphasize such things as estimating and multiple “strategies” in basic calculations — complicated methods of solving math problems in a bid to develop students’ deeper understanding of how those calculations work. Such approaches are common across Canada and are part of the Western and Northern Canadian Protocol (WNCP), a common framework, initiated in 1995 and revised in 2006, used to develop curriculum in all western provinces, Canada’s three territories, as well as in Atlantic provinces including Newfoundland, Nova Scotia and Prince Edward Island.

“We were hearing concerns from parents and we were hearing concerns from some math professionals,” said Ms. Allan, who called her province “a leader” in math reform. Besides worrying about students not learning basic math skills, parents trying to pitch in with their children’s homework, “were having difficulty helping their young people because they weren’t able to understand it either.”

Sound familiar?  It seems Canada has been dealing with a scenario very much like the Common Core for a while now.  Note that both Math By Hand and Waldorf Education feature a lively, effective blend of the “basics” with the “deep understanding” that the new math standards are going for.  But the crucial difference is that both approaches are so holistic and child-centered that there’s never the danger of age-inappropriateness, over-abstraction, or alienating disconnection.

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 Grade 3 CC Break: Back to Basics! (#196) appeared first on Math By Hand.

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.

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