Number sense is a person’s ability to have flexibility and fluency with numbers – to use and understand numbers, know approximate values, and know how to use numbers to make decisions.

So, for example, a person with strong number sense would know that 6×3 is a relatively small amount, whereas 7×8 is a relatively larger amount.

A student with poor number sense might not know or realize this.

To paint a visual picture, think of the positive whole numbers as a forest of numbers in some sort of rectangular arrangement – say a 10×10 chart, or a 20×20 chart. Each whole number is a tree in this forest. The trees of the smaller whole numbers are closer to us, and the trees of the larger whole numbers are further away. Picking larger and larger numbers means we get further and further away from where we’re standing in this forest.

Moreover, the trees in this forest are highly organized. There are lots of diagonals that line up as well (depending on where we’re standing).

Students with poor number sense seem to see the trees and the forest as simply an enormous number of trees that are scattered all over the place in front of them. They might know that the value of 4×2 (its tree) is pretty close by, that the value of 9×8 (its tree) is further away, and that the tree for 15×17 is a lot further away. But they usually don’t see the number forest as a highly organized chart of numbers.

So how can we help our kids develop strong number sense? Especially, how can we help young kids (ages 2-7) develop strong number sense from early on?

By having kids listen to skip count songs. Skip count songs have catchy melodies whose choruses are a skip counting sequence. For example, the chorus for the 4’s song is…

four, eight, twelve, sixteen,

twenty, twenty-four,

twenty-eight, thirty-two,

thirty-six, that’s fun – yahoo!”

Skip counting songs help kids memorize skip count sequences – 3, 6, 9, 12, 15, 18, 21, 24, etc., and 8, 16, 24, 32, 40, 48, 56, etc.

Skip counting songs implant number sense into a child’s mind. For example, notice that the number 24 is on both of the above lists. That is, 24 is a multiple of 3, and 24 is also a multiple of 8.

Kids who know these skip count sequences have an experiential sense of the following:
• we get to 24 pretty quickly when skip counting by 8, but…

• it takes us longer to get to 24 when skip counting by 3.
That in a nutshell is number sense.

There are so many success stories we have with kids who have learned how to skip count from the songs. Skip count songs are so powerful, I tell parents and teachers this: if there were only one math resource you could get for your child from ages 2 to 7, get the skip count songs.

There are two great sets that we provide. Contact us at info@AlgebraForKids.com about acquiring one or both of them.

• Because Algebra focuses on the general, while Arithmetic focuses on the particulars. Both are needed. Each provides insight to the other. But kids are already overloaded with the many particulars of Arithmetic – they can benefit from seeing and learning about the big picture generalities of Algebra.

• Because with Arithmetic, kids often lose sight of the forest because the focus is on all the many trees.

• Because the Algebra focus on the forest shows how similar trees are related to each other.

• Because the Algebra focus on the forest shows how apparently very un-similar trees are also related to each other.

• Because Algebra requires Arithmetic skills, thus putting Arithmetic in context and showing one answer to “Why do we need to learn this?”

• Because kids can learn both Algebra and Arithmetic at the same time.

• Because the early experience with and study of Algebra precludes the 6-years-of-only-Arithmetic approach. This Arithmetic-only path leads too many kids to complacency and boredom before getting blown away with this new creature called “Algebra.”

• Because the teaching of Arithmetic is too often rule driven, rather than concept driven. This means that many kids up through 5th-6th grades can memorize all the rules without really understanding what they’re doing – and then they hit a wall with the many new concepts in Algebra.

• Because how many times have you heard an adult say, “I was good at math through 6th grade, until I got to Algebra…”?

In a recent blog, I mentioned the value of games involving lots of counting and calculation. This blog lists some commercial games, available in most toy/game stores.

• Chutes & Ladders (sometimes titled Snakes & Ladders): the classic dice game of moving your piece to the 100th square, hoping to avoid the downward chutes and trying to land on the upward ladders.

– variation on Chutes & Ladders: as your child gets older and learns how to subtract and multiply, after the dice are rolled allow each player to choose whether to add, subtract, or multiply the two numbers. A player rolling a 5 and a 3 could move either 8 (from 5 + 3), or 15 (from 5 x 3), or 2 (from 5 – 2), or even -2 (from 3 – 5). This variation helps kids realize there are more relationships between two numbers than always just adding.

• Mancala: remember this? – moving stones around from your side of the board to your end zone – counting, estimating, visualizing moves.

• Monopoly: lots of practice both with the counting of your piece across the board as well as use of play money, making change, buying and paying and renting.

• ParcheesiSorry: these traditional games involving lots of counting and moving pieces. From sheer experience of lots of counting, players learn to look for shortcuts, counting on, counting back, etc.

• Ring-A-Round: another game of three regular six-sided dice to find expressions equal to number goals from 2 to 18, in a ring around your own colored post. Players can use any combination of +, –, x, or ÷ to find a number goal – for example, rolling a 3, 4, 5 could results in 12 (from 3 + 4 + 5), or 17 (from 5 + 3 x 4, or from 5 x 4 – 3), or 11 (from 3 x 5 – 4), or 3 (from[5 + 4] ÷ 3), or 6 (from 5 + 4 – 3), etc.

• Backgammon: the ancient dice game allowing players to decompose sums into desired parts in order to move all pieces to their own goal.

• Yahtzee: there’s no better dice game allowing so many intriguing combinations and arrangements of numbers.

• Cribbage: the pegging game of counting points with pairs, three of a kind, cards adding to 15, strategizing to reach 31.

• Kenken: this game is available as a board game but is more commonly found in newspapers, magazines, and booklets. Kind of like Sudoku but with arithmetic thrown in.

• The Game of 24: use each of the four numbers on a card exactly once each to create an expression equal to 24 – for example, a card with the numbers 2, 2, 6, 8 would have a solution of (6 + 8 – 2) x 2. Each box has dozens of such cards, with each card marked with degree of difficulty.

There certainly are other games of computation that I haven’t included here. Write me at info@AlgebraForKids.com with your favorite counting/calculating game for kids.

Recently I had an aha! moment that connected, I believe, with why so many students in middle school and high school have difficulty getting meaning from written directions.

My challenge in this blog is to connect that aha moment with algebra. Here goes.

My aha was partially birthed in several elementary classrooms in which I was a substitute teacher recently. In the early elementary grades (1st through 3rd or even 4th grade), it’s understandable that kids have some degree of challenge understanding what the written directions both say and mean.

During several substitute teaching assignments, quite a few 2nd or 3rd graders kept saying to me about the written directions on a worksheet (or quiz or test), “I don’t get it.”

I found myself defaulting to the tried and true strategy of re-wording what the directions meant – “Kids, what it means is…”

No doubt my re-worded explanation was brilliant. No doubt it helped some to many kids, of course.

But as I pondered what had just happened, I began to think about some things that were really going on here when I orally re-worded the written directions.

One element that was present was I was subtly teaching kids that they didn’t have to pay attention to the written directions! Why pay attention to what the words say – why learn to decode the meaning of the written directions at all – when the teacher is just going to step in (sooner or later) and say, “What this means is…”

That inference is not a good thing for kids to learn.

So I tried taking a different approach the next time students told me they didn’t understand the written directions: I was going to “drag them through” the exact wording of the written directions.

What this meant was examining how the directions were structured:

• We examined the verbs, especially command verbs like underline, draw, explain, solve, simplify, group, match, etc. Along the way we happened upon the distinction between an action verb like explain and a stative verb like is or were.

• We pinpointed the subject and object of the verbs – which often were the nouns and pronouns of the directions. Write a fraction that is equivalent to the percentDescribe the similarities and differences between… Simplify the improper fraction and then change it to a mixed number.

• I pointed out how common it is to see prepositional phrases that contain nouns (like the capital of the country) – but how that noun country isn’t the subject of the sentence.

In other words, I gave a grammar lesson from the directions. Whether the subject matter was English or history or math or science – it didn’t matter.

We focused on getting meaning from written directions by “dragging ourselves through the English of the written directions” – without re-wording those written directions (at least at first).

Of course, providing synonyms for certain words is helpful at times. It’s unavoidable and part of language. But I didn’t end at the synonym. I always went back to the exact wording of the directions to ensure that students were getting meaning from that exact wording.

Several students summarized an important truth about certain kinds of problems in saying, “The hardest part about problem solving in math isn’t the numbers. It’s the English.”

Getting meaning from written directions – a lifelong skill. A skill we can help our kids develop. Such kids become better learners, more self-reliant learners, more capable problem-solvers.

Drag kids through the exact wording of written directions. It’s the educational equivalent of a well-known adage that I’m revising here: Give a child the decoded meaning of the written directions and you provide help for today. Teach a child how to decode written directions and you provide help for a lifetime.

Consider the following examples of details versus big picture:

Detail: What’s the answer? Big Picture: What’s the question?

Detail: How much is 9 x 6? Big Picture: 9 x 6 is somewhere less than 10 x 6.

Detail: Do I add or multiply? Big Picture: What does add mean? What does multiply mean?

Detail: Is this inches or square inches? Big Picture: How are perimeter and area different? Similar?

Detail: How do I add 2/7 + 3/7 ? Big Picture: What does 2/7 mean?

Detail: What is 3/4 as a percent? Big Picture: What does percent mean?

Detail: Is it true that 5 + 3 = 2 x 4? Big Picture: What does the equal sign mean?

Detail: Is 11 x 13 equal to 93? Big Picture: Since 10 x 10 = 100, then 11 x 13 should be more than 100.

Detail: (5 x 2 – 10) x 172 is a long problem. Big Picture: (5 x 2 – 10) is 0, and 0 times anything is 0.

Too often, kids learning math can be sidetracked to focus primarily on the details of math. Such details include the exact values of basic facts, the placement of the decimal point in a multiplication exercise, or converting fractions to percents.

Such details are important. As important as they are, however, details are not the only thing that’s important in math.

It’s also important that kids learn how to see the big picture, the larger perspective. Seeing the big picture requires non-math skills, such as overfamiliarity, impulse control, knowing definitions, practicing patience, knowing multiple ways to think about how much a number is or how to solve a problem, etc.

So what helps kids learn how to think with the big picture in mind? Here are some suggestions:

Teach definitions of things, but keep definitions brief. Add means put together with. Equal means is the same value as. Percent means out of 100 or for every 100.

Use those definitions. Ask your child regularly and repeatedly for the meaning of key terms like add, divide, equal,

Aim for overfamiliarity – shoot for your child becoming overly familiar with, not just acquainted with, basic facts. Teach and play lots of math-type games – games involving lots of counting and calculation (more on those kinds of games in the next blog).

Ask kids questions that go beyond details. See all of the Big Picture questions above. Encourage them to ask questions too.

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