Tuesday, February 1, 2011

Math Progression Matrix Steps 1 thru 3

NOT COMPLETE!! As many of you know I've been working on this for a couple of years. Basically, I teach my boys math & science more like the Eastern world and less like the American world. I've been trying to put together into words the flow of how I teach,  when most of it I don't think about..lol.... So, here's what I have so far, and I'll keep adding to it as I work things out....

All parents want to ensure their children have the best educations available to them. Many believe you can nurture the mind beginning in the very young – reading to your baby helps prepare them for reading, scribbling unrecognizably helps develop the fine motor skills later needed to learn to write, etc. For many parents one aspect of both home educating and helping your child’s mind bloom terrifies: Math. Can you, growing up, remember thinking or hearing, “Why do I have to learn this stupid stuff, it’s not like I’ll ever need it.” Well, really every single day you use math – from setting your alarm clock, to planning out your chores, to balancing your checkbook. Math is the one subject that every person WILL use throughout their life. The beauty of math is that it’s a step by step process, and elements of it can be seen in the world around us. If you follow the progression then not only will your child excel at math but you’ll have little trouble in understanding it yourself. I firmly believe so many dread math today simply because of the way it was presented to them. If your child sees you’re excited and having fun, then they will too.

How do infants learn? Well if you smile at your baby enough times then they’ll mimic the motion. Mimicking becomes prevalent at around the 6-12 month mark. So if you’re repeating concepts, like math, then your baby will repeat you. This is how babies learn Sign Language at such young ages, they’re just mimicking motions they’ve seen repeatedly. Between 6-12 months a baby will begin to associate motion/sound to action. If you say cookie, show a baby a cookie, say cookie, then give them the cookie, every time they have snack-time, then eventually they’ll associate the word/sound ‘cookie’ with getting a yummy treat. This is the basis for teaching any pre-preschooler. Repetition and Consistency are the bread and butter of early education – and they are the cornerstone of learning mathematics. The thing is – if you look at learning math by imagining an advanced logarithm the you’ll be way to intimidated to get excited about deciphering the language. Start with the everyday and work toward the arcane.

A bit about 1-to-1 Ratio – this is associating the word for a number with the symbol for the number with the group of that number. For instance, say ‘One’ Show the child the number 1 and then show them 1 cookie. BUT - and here's the first turn from American Matrix to more Western teaching, don't make a big deal about the Name of a number or it's value....for instance a German Sheppard can be called Spot, Bob, Rex, etc....but they are all German Sheppard's. This is one of the first stumbles for Americans. We get so attached to the definition - this symbol "2" means there are 1 and another 1 but we could call it X we could call it anything. The symbol for numbers is just a name, a label to make early mathematics more tangible....but then when we try to move into more advanced math like Algebra, and start calling these numbers simple integers and then mix it up with variables so that we stop writing 1 + 1 = 2 and start writing 1 + 1 = (x + x) = 2x we get very messed up. Honestly all this equation is saying is that we will now call all groupings of 1 item by the name of x that's it...so the answer is still 2 it is saying we have 2 x's.....don't worry you're probably trying to complicate things if you get frustrated with math..lol..But this 1-to-1 Ratio of teaching the definite definition of a symbol for the integers 1, 2, 3 really messes us up when later we, the teacher, are all "Ha!" fooled ya! There is no true definition we're just giving them one and we can change it anytime we want! So, just to clarify- a 1-to-1 Ratio is all about associating 2 things, the most basic way of understanding 1-to-1 Ratios is that you are showing how 2 things are equal. The integer 1 is the same thing as 1 cookie, they are equal in quantity. In more advanced math we use the 1-to-1 ration with unknown quantities....we don't know the volume of this shape but it has a 1-to-1 Ratio (or equal value) to the volume of this cup. So anything can be made to have equal quanities - 2 different equations, amounts of food, geometric shapes, etc. Now, moving along, Ratio simply shows how something relates to something else. If you look at the design '1-to-1' the same number resides on both sides of the -to-. This shows it is equal (so technically you could use 4-to-4 because if you have 4 of this and it equals 4 of that then they are exactly equal.) But you could have un-even Ratios to show how much larger or smaller something is when compared to something else....ie. a 1-to-2 Ratio shows you have 1 of this and 2 of that to show that the first something is larger because it takes double the quantity of the second to be equal to the first...see? (Think of it for a while)

It is also important to remember that every child is different. Some children will ‘get it’ much younger than others. This is fine! There is no competition, as long as kids learn it eventually, then no problem. But you should focus – in my opinion – on mastering one skill before attempting the next in the matrix. Otherwise, you’ll find yourself bouncing back and forth confusing both yourself and your child. So, instead of listing ages for the matrix I’ve listed steps. It doesn’t matter at what age one masters the concept, like I stated, some children might be ready to master step 1 as toddlers – others might be in elementary grades. Don’t stress about fitting into a preconceived box, let your child move comfortably from one step to the next.

Don’t push too hard! I think one of the hardest things about teaching is knowing when to push and when to let it lay. My personal theory is to push a topic for several days, if they don’t catch on the first day (are still confused) then try a different approach the second day. If after several days – thus several different approaches – the child is still lost, let it go for a month, then come back. There are several different points to make in each step so you can move to a different idea within the step and come back to the confusing ones later before moving onto the next step. You can also reward the child – make a production – out of moving onto the next step, this is especially true for homeschoolers who don’t have that feedback at moving to a new class when finished with another.

Stages of each Step
Number Sense - this is working with basic numbers -integers, fractions, and decimals. The whole point of this stage is to understand that 1-to-1 Ratio of what each symbol means.
Algebra - is working with variables. All addition/subtraction, multiplication, etc falls under the umbrella of Algebra.
Geometry - working with shapes, and their coordinates in space.
Trigonometry - In this case we're focusing mostly on Triangles and Circle Trig, basically trigonometry is the study of these two shapes - their angles, sides, etc.  {so the difference between Geometry and Trigonometry? Geometry is the study of how shapes relate to the environment, how they fit into a situation, whereas Trig focus' within the shape...in other words Geometry is an open set working - infinite, whereas Trig is a closed set}
Statistics & Probability- this is the study of the likelihood of an occurrence/recurrence of an event
Calculus - This is another open set type of math. Honestly we're dealing with a lot of infinities in life (a line goes on forever after all) Calculus works to take a piece of the infinite and use it to find solutions. For instance calculus works with the slope of a hill, the rate of change (like speed), working with volume by parallel cross-sections (like taking core ice info out of the glaciers), etc....so Calc very much marries Math and Science - especially Physics, Geology and Chemistry.
Economics - the dealings with money and the economy, arguably the most used form of mathematics in adults.
Critical Thinking - I like to think of this as thinking outside of the box - outside of that ratio. It's allowing the child to come up with as many or varied solutions to a problem. Even though there may be one right answer, in math there are hundreds of means of achieving that answer.

Step 1:

Number Sense-

  • Learning to count - start small and work your way up. Use manipulative tools like snacks, shoes, flash cards, number line, etc. At first it’s ok to not be able to say numbers from 1 - 100 (they’ll often skip or reverse numbers) it’s much easier to have an actual item to count. “How many Cheerio are in your cup? 1, 2, 3, 4, 5? 5 Cheerio’s!” This is almost always the first step on the road to mastering mathematics. For this step to be mastered a child should be able to count to 25. 
  • Same & Different - as relates to ‘how many’ this teaches grouping. “Is this your hand? No, it’s your Tummy!” I know when my kids were as young as toddlers I would have them help me with things like laundry or putting away groceries. That’s an excellent start in determining what fits into a pre-defined grouping. Child’s socks-with-child’s socks, mommy’s pants go with mommy’s cloths….soup cans go on one shelf whereas rice goes on another. Believe it or not this is an exceedingly important tool to learning mathematics. Basically you are learning what defines a group – it’s parameters – then you are learning to recognize if something fits into a group. In this step you are using very broad and obvious groups. Food, clothes, body parts, toys. Start talking about these things with your infant early, then have your toddler help you group them together. 
  • Irregular Measurements Length and Width- how many left feet does it take to measure the width of the room? If we laid pencils end to end how many would it take? Find fun things to measure length and width.
Step 2:

Number Sense

  • Counting Integers– continue building on your child’s counting skills. This point will encompass the entire time you are in step 2 – meaning, even while working on the other points, you should be also working on counting. This step is mastered when the child can count to 50. We are not looking for perfection, but the understanding that the 30’s come after the 20’s etc. (definition: integer is any whole number - it can be positive or negative but no fractions or decimals here!) 
  • Reverse Counting Integers - help your child learn to count backward from 10 and 20.
  • Writing Integers – this is probably one of the more tricky stages in Step 2 – if only because writing is dependant on fine motor control, which may not be equal to cognitive abilities. Start off with tracing. Don’t worry about them being able to remember what the number 2 looks like in this stage. It’s the motor memory we’re working on.
  • Number line = Real Numbers – long before using a calculator or computer a child’s mathematical best friend is the number line. Once the start working on equations the number line gives them the answer! Have your child verbalize filling out the line with you doing the writing, have them trace the numbers to fill out the line, have them find you the number 6, etc. [definition: Real Number - any number, integer, fraction, etc that is found on a number line. If it has a spot on the line then it's a real number]
  • Skip Counting By 2's & 5's - count to 20 by 2's and 50 by 5's. FOr thise Step we're just exposing the child to the idea of skip counting, like a set. The set includes {5, 10, 15, etc} in Step 3 we become familiar with different methods of finding the solution to what comes next in skip counting.
Algebra -
  • Variables & Sets - I love to introduce Algebraic language with a simple set of playing cards. Each of us write on a piece of paper the name of our variable - "t", "a" "z" whatever...this is also nice for reinforcing the alphabet that the child is learning at the same time...then each of us selects a card from the deck. That is what our variable is worth. Now we each try to pick up the matching card from a new deck - the one that matches first gets a pass on chores for the day! definition - a variable is just a symbol used to identify a quantity that changes....for instance if I were writing about the temperature in my area I might want to do a form of shorthand by calling it "w" as I live in the North East and the weather is ever changing. So a variable can be a number that is always changing or one that you just don't know the quantity of as of yet. A set is a series of numbers separated by a comma - 1,2,3 or 23,78,92.
  • Early Addition & Subtraction - if you have an apple and I have an apple, how many apples are there? Addition is easiest taught through manipulative tools like physical objects, number lines, etc. While the student will not master this concept at this stage, they should have an understanding of what addition & subtraction does. Use the correct vocabulary – addition, subtraction, sum, difference, solution. This step is all about either using manipulative tools or using a worksheet with clear picture graphs to demonstrate the equation – not using numbers for the equation or resulting answer.
  • Extending Patterns of Sets- using a variety of pictographs or materials have a child continue the pattern. They should also be encouraged to create patterns within a set. (ie. when using toy cars they can create a set of  red cars then the pattern can be 2-door, 4-door...etc)
Geometry –

  • Basic Shapes - circle is round, square has 4 angles and it’s sides are all the same, etc.
  • 3-D Shapes - identify, name and describe 3-d images (sphere, cube, cone adn cylinder).
  • Seeing Shapes – learning to recognize simple shapes by sight. Triangle, square, circle, star, diamond. I really like to do this during art projects, “do you notice the roof of the house looks like a triangle?” as well as while driving. I usually post a Shape of the day and the kids have a contest, putting a sticker next to their name for every time they are the first to point out that shape throughout the day. The winner at the end of the day get’s to pick the game we play that night, or what to have for dinner.
  • Primary Colors - red, blue and yellow, this is often done in grouping, but you can also sing about the rainbow or colors in general. I like working with a triangle. One a big sheet of art paper I make a big triangle (might as well get used to one now..lol) and have the child make a smear of paint on each corner. I then make a dot in the exact center of the triangle and have the child swoosh the primary colors toward the dot - letting them overlap and make more colors and such.
  • Basic Grouping - organize items by shape, quantity, color, etc. “Where’s your yellow blanket? Here it is!” Some of this also overlaps the Critical Thinking lessons – where are the Big Lego’s, where are the Small cars, etc.
  • Volume - Begin familiarizing the child with different ways of measureing liquids - cups, quarts, gallons, etc.
Statistics -
  • Basic Data Accumulation - use something the child can relate to and finds amusing; this will keep their attention over time. My kids were always fascinated with the weather. They would wake up early to watch our local weather man, and when my DH & I came stumbling in for coffee they were the first to tell us the prediction for the day. So we set up a simple blank month calendar. I found some stickers that matched weather - clouds, sun, rain, lightening, etc...and when we started school for the day the child would read the thermometer to me and I would write the temperature for each day. (also see Critical Thinking - Measurements)
  • Basic Graphing - I think bar graphs are the easiest to make for these youngsters. Using our month of accumulated data on weather, we would set out to see how many days were a certain temp, or type of weather. The kids love coloring each of the bars differently then looking at the colorful graphs.
Economics –

  • Identify Money – identify coins and bills. Which is paper money and change. Be able to tell the difference between the types of change and the types of paper money. 
  • Worth – begin working with Pennies and $1 bills. Help the child to understand that it takes 100 pennies to equal $1. 
  • Saving vs. Spending – take the child to the store and show them how each item costs a different amount (we tied in social studies on how items got to the store). Make a big sign for savings. Help the child to see their money building as they put money into their account – even if it’s only pennies.
Critical Thinking -

  • Directions/Opposites - over/under, side-side, top/bottom, full/empty, in/out, etc.
  • Sizes - small, medium, big, bigger, biggest 
  • Measurement Weight & Temperature – we’ve already been practicing reading a thermometer for the graphing and data parts of this step, now we also learn to read a ruler and scale
  • Time – learning the days of the week and months of the year. Become familiar with the passage of time with vocabulary like before, after, soon. My kids would change our daily board to say the day and date each morning (we have a Velcro board). While the student won’t master telling time at this level, they should begin to understand that time moves in increments - seconds-to-minutes-to-hours, days-to-weeks-to-months-to-years.
  • Ruler Measurement of distance - getting comfortable with American measurements (inches, feet,etc)
Step 3:

Number Sense –

  • Counting – continue building on your child’s counting skills, a continuation from step 2. Again this step will encompass Step 3. This step is mastered when the child can comfortably count to 100, in order. 
  • Number Line Skip Counting – continue using the number line. Skip counting is a predominant part of step 3 as it’s the easiest way to deal with answering equations later on. So, you’ll notice there are several different ways of skip counting in Step 3, all of which need to be mastered before step 4. In number line skip counting first show the child how to skip by 2’s by scribbling out every other number. In this step focus on counting by 2’s to 40, 5’s to 50 and 10’s to 100.
  • Skip Counting Grouping – by grouping things like toys by number (groups of 2 cars each, for example) your child can count, this also helps remind them of the addition for the next stage of Step 3.
  • 100 Unit Cards - this is the first start in understanding place value. I basically took some sheets of foam paper and marked 1 sheet with 100 squares or units. I cut another sheet into Vertical rectangles and marked it with 10 squares, and the final sheet I cut into 100 little squares. These were used to demonstrate how a number breaks down. Although for this stage we just used integers below 50, so no 100 blocks as of yet (those are used in Step 4). Then I would write a number and my son would get the right amounts of units - 10's and 1's - to show the number, or he would give me the 10's and 1's and I would have to figure out the number he wants. For instance, for the number 48 he would show 4 10-units and 8-single units.
  • Even Odd - determine whether a number is odd or even by pairing pictographs or objects - if all the objects have a partner and there are none left over then it is an even number.
Algebra -
  • Addition & Subtraction to 50 – start with the pictographs and move up to numbers. 
  • Writing Integers– to master this step your child should be comfortable writing to 50. 
  • Ordinal-Cardinal Relationships - This is learning about places – who’s in first, second, third, etc. Have fun with it. Play Simon Says, etc.  
  • Number-Numeral Relationships – understanding number order and the symbol we give it….so you can rename a group of 2 items A. This is a great way to start the concepts of Algebra.
  • Sequences - this is also an important foundation. I taught it through stories, first a happens then b, etc. Without this the student could become lost when deciphering word problems.
  • Commutative Property - For addition demonstrate no matter the order the answer is the same - 1 + 2 = 2 + 1, be sure to write it in this fashion not just looking at 1 + 2 = 3 and 2 + 1 = 3 show them that it's ok to have another equation as the solution, it's not wrong. This is the first step toward true Algebraic equations where many times there is an equation as a solution, and it's helping them learn to show their work - their thought process.
  • Greater/Less Than - Equal To - help child determine if an integer is greater/less/equal to another integer, or if a set is greater/less/equal to another set.
  • Inverse Relation - use the inverse relationship between addition and subtraction to check your solutions....if 7 + 6 = 13 then 13 - 7 = 6 and 13 - 6 = 7.
  • Find the Unknown Variable - (once they have been reasonably exposed to Inverse Relations) use simple equations for addition and subtraction to find the value of x or y or z or whatever letter you choose. (ie. 5 + __x_ = 8, to solve you subtract 5 from both sides to get 8-5 = 3 therefore x = 3)
Geometry –

  • More on Shapes – look into what makes each shape – the number of corners, the number of lines. Are the lines straight or curved, etc.
  • Symmetry - You can fold many shapes in half and they will be identical on each side.
  • Shapes Working Together – this is a fun part of step 3 – where many shapes come together to make a different shape. For instance, two triangles can come together to make a square. 
  • The Face - help the child to recognize that a 2-D image is the face (or one plane) of a 3-D image.
  • Additional Geometric Shapes – Yup, I suggest learning the more complex shapes this early – pentagon, octagonal, etc. You’re just learning to count the sides/angles and what it’s called.
  • Secondary Colors - we did a lapbook on Rainbows (which you can find on my YouTube site) where we learned to mix primary colors to make secondary ones, and we learned about prisms.
  • Fractions 1/2, 1/3 and 1/4 - expose your child to separating a whole item into equal parts, then writing that part in its fraction.
Trigonometry -
  • Is it a Polygon? - become comfortable with the definition of a polygon. Polygon: any 2 dimentional closed shape made up of 3 or more lines connected end to end....so it must have 3 or more straight lines, as well as an equal number of joints or angles.
  • Begining work with a Circle - taking any size circle the child should draw 2 perpendicular lines through it. The point where those lines intersect is the Center Point. Each of those lines is also considered the Diameter (or width of the circle). From the centerline to the edge is the Radius. Children should become comfortable labeling a circle and should be exposed to measureing each of these important factors.
  • Congruent? - This is easiest working with a triangle, but work toward using many different polygons. If the length of the sides are all the same then the triangle is congruent. Basically it's a fancy word for saying these two items are exactly the same shape and size.

Statistics -
  • Estimation - guest mating, and learning how to estimate in the most basic sense. I have a bag of gumballs, how many do you think I have?
  • More Basic Graphing - Moving onto the Pie Chart Graph. Again we used things like their Matchbox Cars and graphed them by their colors and such.
Economics -
  • Using a Register - In Step 2 we used a big poster board (like charities often do) to measure the growth of our savings. In Step 3 we move onto using a simple register - just like balancing your checkbook (but with less entries..lol).
  • Worth - Nickel, Dime, $5 and $10. Become comfortable using these forms of money and their worth.
Critical Thinking –

  • More/Less - this is terribly important to learning addition/subtraction
  • Time Measurement - compare units of time seconds/minutes/hours or days/weeks/months/years. As well as begin to read the analog clock to 5 minute interval (skip counting by 5!). Work with time elapsed problems - to further compare passage of time.
  • Digital Time – reading a digital clock.
  • Measurement Relationships - this is also a fun project for kids - learning how inches make a foot and a foot makes a yard, as well as cups are in a quart and quarts are in a gallon. My kids mostly loved it because they could make a mess by scooping up various things.

1 comment:

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