We just finished our final week of SAI for summer 2013, and we had a nice comment from the mother of one of our students. These remarks are pretty typical of parent responses to our SAI program.

She wrote, I wanted to tell you what Day #1 has done for my son. For a super sports-oriented guy, he rated the camp very highly. The next morning he woke up early, busted out of bed, did his SAI homework, and declared himself finished before I could properly wake up! He had everything right. He went on to show me all of his work for the day. I cannot believe that he got so much work done. I am delighted at the exercise his math brain is getting! What an advantage for going back to school! 

   He usually feels like he is competing with peers and time. I believe this is a symptom of his athletic and competitive focus. So he makes many errors. I usually see many erase marks. I saw none on your worksheets.  

   This morning after completing his homework and reviewing his classwork, he ran excitedly to the game closet and pulled out one of the games that you sell, insisting that I play with him. Then he accosted me with our multiplication index cards. He is exhausting me with his enthusiasm. That is a good thing. I promised to play both with him this morning before class.

    He is happy, excited, confident and enthusiastic. What more can I ask for! Thank you for day 1! I cannot wait to see what the rest of the week brings!

Shelly

I’m convinced that listening to skip counting songs is vitally helpful – not the only factor, of course, but a huge help in acquiring number sense and basic facts. I’ll give an example after I explain what skip count songs are.

Skip counting is counting in multiples of a number – 7, 14, 21, 28, 35, etc., or 4, 8, 12, 16, 20, 24, etc.

Skip count songs are catchy tunes containing the skip count sequence of a number. The skip count songs I recommend have different melodies and lyrics for each of the numbers from 2 through 10.

The power of music and songs is well known. Most of us can still remember the A-B-C’s song for the alphabet. Many adults have songs they have memorized from singing or hearing – and can sing along (or recite!) at the drop of a hat. How many of us can hear just the first few opening notes of a song after years of absence – and have the entire song come flooding back into memory? This power of songs on memory is a dynamic connection that can be used to help kids become good at math from an early age. Here are some examples…

Case 1: Our younger son (now 25yo) started listening to the skip count songs at the age of about 15 months. Before he was two years old, he could count to 20 by 2’s. Did this 2-year-old know what he was doing? Of course not – but at least the number-words for this sequencing were in place at an early age. And these skip count sequences were associated with fun and mastery. Several years later when he was ready for school, he had the usual check-up with the kindergarten nurse. After checking his ears and throat, she asked him, “Can you count?”

He said, “How do you want me to count?”

The nurse didn’t understand what he meant (!), and so he said, “Do you want me to count by 7’s or 9’s or 4’s or 8’s or 3’s…?” The nurse’s jaw dropped and she said in amazement, “You can count by 4’s?” He flawlessly sang the 4’s chorus from one of our skip count CD’s – and the nurse said, “I’ll record that he can count.”

Because of the skip count songs, my son associated numbers with fun and music and order and sensibility. From the very beginning for him, math simply made sense – because of the early exposure to skip count songs.

Case 2: Quite a few years ago, I was working with a 7-year-old whose family had both of our skip counting CD’s (The Skip Count Kid and Skip Count Bible Heroes). One day this 7yo told me something – he didn’t ask me – and here follows our dialogue:

• 7yo: 9 times 2 is the same as 2 times 9.

• me: How do you know that?

• 7yo: Because 18 is on both choruses – it’s on the chorus for the 9’s song, and it’s on the chorus for the 2’s song.

• me [holding back my impulse to give a mini-lecture on the commutative property of multiplication]: Does that work for any other pairs of numbers?

• 7yo [after a few moments of thought]: Yes, 7 times 3 is the same as 3 times 7, because 21 is on both choruses.

• me: Are there any other pairs of numbers that works for?

• 7yo: [silence… while pondering the question]

Our session ended before he could say anything else. The next day when I saw him, the very first thing out of his mouth was, “It works for all the pairs of numbers I can think of.” Apparently he had spent a significant amount of time running the skip counting songs through his mind to confirm his own discovery that when multiplying two numbers, it doesn’t matter what order they’re in.

Case 3: My older son (now in his late 20’s) was not even six years old when he started listening to the skip count CD songs. After listening to them for a few weeks, he said to me, “Daddy, I noticed that 12 is on the chorus for the 2’s song, and for the 3’s song, and for the 4’s song. Then it isn’t on the chorus for the 5’s song, but then it is on the chorus for the 6’s song. Then it’s not on any other choruses after that. Daddy, should I be noticing things like that?”

I was floored – here a six year-old was talking in meaningful ways about what he would many years later realize involved vocabulary like factor, divisor, multiples, common multiples,and divisibility. When he told me that the number 12 was on the chorus for the 2’s song and 3’s song, for example, that was a setup for later learning each of the following:

• 12 is a multiple of both 2 and 3

• 12 is a common multiple of 2 and 3

• 12 is divisible by both 2 and 3

• 2 and 3 are factors of 12

• 2 and 3 are divisors of 12

When my young son mentioned this to me, I was thrilled. Rather than laying on him a math lecture about factors and divisors, I instead just grinned a huge smile and said, “I’m proud of you that you notice things like that. Keep it up.” He did, and he went on to do well in math throughout his schooling.

 

Time for a multiple-choice quiz! Which of the following is the most fully true about the connection between arithmetic and algebra?

A. There is no connection – they’re different branches of mathematics.

B. Sometimes arithmetic is used in algebra, like to solve equations.

C. Six or seven years of arithmetic is necessary in order to do algebra.

D. Algebra is generalized arithmetic.

Let’s discuss each of these:

A. No connection

While arithmetic and algebra are different branches of math, this statement is too strong. There are several significant connections between algebra and arithmetic, which will be discussed below.

B. Arithmetic used sometimes in algebra

Obviously, arithmetic does get used in algebra (and in almost every branch of math). So this statement is true – but it’s not the most fully true statement, because there’s more to this algebra-arithmetic connection.

C. Six years of arithmetic before algebra

This statement is how most people think of algebra and arithmetic – and this is what their own experience has been for most adults. But this is precisely what we have found at Algebra For Kids for the past 20 years: kids DON’T need six years of arithmetic before algebra.

D. Algebra = generalized arithmetic

In arithmetic, we deal with specific exercises like 5 + 3 or 7 • 9. In algebra, we deal with general principles like a + b or 7•x.

In arithmetic, we learn the mental math of 7•(10 + 2). In algebra, we study the general idea of 7•(x + 2).

In arithmetic, students learn about base 10 operations involving “borrowing” and “carrying.” In algebra, students learn how in some ways algebra is easier simply because there is no “borrowing” or “carrying” in the algebra of base x.

In arithmetic, kids learn the relatively complicated two-digit multiplication procedure of (10 + 2)•(10 + 4). In algebra, students practice the comparativelyuncomplicated procedure of (x + 2)•(x + 4).

Are you starting to see how connected algebra and arithmetic are?

This connectedness is why in AFK, for 20 years we have been able to teach substantive algebra to students as young as 3rd grade.

Get the 3rd graders in your life into our summer camp – SAI: the Summer Algebra Institute for Kids.

Is basic fact mastery a goal to achieve as an end in itself? Or is it a goal to achieve as a means to something else? In other words, why do we want kids to master basic arithmetic facts?

Adults don’t go through their day thinking, “Hmm – yes, do the laundry; go to the game Thursday night; 8 times 7 is 56; gotta get the oil changed next week; oh yes – 54 divided by 9 is 6…” If adults did do that, then basic fact mastery would be an end in itself.

But we adults don’t utilize basic math facts like that – we use them as a way to get other things accomplished. In my opinion, basic fact mastery is an extremely important goal – but is not an end in itself. Basic fact mastery is a goal to reach for as a means to other things, such as…

• adding up a series of numbers from a grocery trip or a card game;

• determining commissions on sales;

• figuring or estimating tax and tip on the purchase of a meal, a shirt, or a car;

• calculating the area or perimeter of a room or house for purposes of painting or wallpapering;

• shopping for items that involve discounts;

• estimating gas mileage and the cost of a trip.

In general then, basic fact mastery empowers a person to be more savvy, more alert, more on top of one’s dealings with the surrounding world.

So, if basic fact mastery is a means to an end, then our kids need experiences where they can practice basic math skills as a means to an end – as a way to achieve other goals! What are some of those experiences where kids can practice basic math facts as a means to other ends? Here are some:

• playing board games: think about the practice of basic addition in almost any board game involving two dice –Monopoly, Chutes & Ladders, Parcheesi, etc. Sometimes there is other math involved in these games, such as in Monopoly with the purchase price of a property and making change with the bank;

• playing card games: card games involving counting are great – like Cribbage and Rummy 500;

• making things: anything involving a tape measure (whether sewing or working with wood);

• making fun food stuff:

• participating in sports (keeping score): adding up in point totals in basketball from the number of 3-point field goals and/or 2-point field goals and/or 1-point free throws; keeping track in football the number of 6-point touchdowns and/or 3-point field goals and/or 2-point safeties; estimating batting average in baseball or softball.

Yes, there should be times when those same skills are practiced in isolation – just not all the time! Achieving basic fact mastery only through flash cards and drill sheets would be like learning to play piano by only playing scales and never getting to play lighthearted melodies, practice beautiful hymns, listen to excellent music, or lead a family sing-along at the piano. Mathematical games can provide the variety and motivation to engage the three key elements of memorization: repetition, repetition, and repetition. If you are not convinced, try putting this book down, and playing a few rounds of Chutes & Ladders or Cribbage or Parcheesi. You’ll see how much basic fact practice is actually used in just one game – and it’s fun.

So why have a child study algebra as early as 3rd grade? – or as early as 1st grade in a longer program of study (as I did in the 1993-94 school year)? There are several sound educational and mathematical reasons. 

   First, arithmetic too often gets focused on the many different individual procedures. This detail-oriented emphasis on the myriad of different arithmetic procedures reinforces the notion that arithmetic is merely and only a collection of a whole lot of particular procedures. Too often, all these procedures seem to be just a collection of disconnected, unrelated rules to memorize. With year after year of elementary math focusing on arithmetic procedures, too many students have trouble seeing the wider forest of mathematics because all the trees (of individual arithmetic procedures) are in the way. What is needed here? Precisely the very thing in SAI that ties directly to the second reason for studying algebra at an early age. 

   This second reason is this: it is so helpful to also study what unifies and connects all those different arithmetic procedures. This is precisely what algebra reveals – especially if taught in the way we teach algebra in SAI. 

   In arithmetic, the focus is on individual facts like 8+1, 9+7, 5+3, 2+10, and 4×6, 3×9, 6×5, etc. In algebra, the focus is on the general case of x + y and a•b. What’s needed for math mastery is both the individual particulars of arithmetic as well as the broad generalities of algebra.

   This unity between arithmetic and algebra is especially revealed by how we teach algebra in SAI. One very exciting aspect of SAI is that we have identified a handful of broad, practical principles that are kid-tested and adult-friendly. These principles – these “grand ideas” of math – turn out to be extremely helpful in assisting kids from an early age to see and experience the unity between arithmetic and algebra. Experiencing this unity at an earlier age will help your child have a smoother transition from arithmetic in grade school to algebra in middle school – and higher math in high school. 

   This is why we make a point of saying that SAI is not just for the gifted and talented. Learning the connectedness between arithmetic and algebra is helpful for students of a wide range of abilities. 

   We hope to see you and your child at SAI this summer! 

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