I'm Counting on You.

OK, before you read any further start counting by 5s. You can stop when you get to 50. Done that? Did you go 5, 10, 15, 20 etc? Yes you probably did.

Now, before you read any further count by 5s starting at 3.

OK, done that? What you probably did, if you are representative of 95 percent of the population is that you started off really slowly and hesitantly
3 + 5 is 8 + 5 is 13 + 5 is 18 and then, you probably went 23,28, 33,38,43,48 gathering speed as you went.

To check try counting by 5s starting at 7. Chances re you stumbled to begin with until you saw the pattern. Math is the Science of Pattern and once you see the pattern you are a long way to having a meaningful understanding of what you are doing.

Try the same thing with fractions. Count by halves starting at 0. Try it before reading any further.   Now start counting by halves beginning at a quarter. It's just the same, once you see the pattern everything gets easier.

So why not try it with decimals. Count by .3 starting at 0. Now count by .3 starting at .5. A little bit more tricky but it works. As soon a you see the pattern you can reel the numbers off as fast as you can speak.

This is the sort of thing we should be doing with students and adults to combat the paralyzing fear or hatred of maths that is so pervasive in the US. We are hampered, held back and disabled  by the debilitating phrase of "I'm no good at math" spoken aloud every day by so many people in so many places with large and small audiences by people of all ages and genders. Even the phrase "do the math" sounds like a  threat sometimes. No-one ever says "do the reading" and no one ever says, out loud, "I'm no good at reading" as described in this very eloquent Washington Post piece
sent to me recently by Erik, my wonderful engineer son-in-law.

Everyone can do maths if they focus on understand it and don't just memorize isolated facts to get scores on tests. Maths is made to be understood and played with; just like language.    

Arrow Cards Are The Best

This morning I watched one of my student teachers teach a wonderful lesson on place value to a group of first grade students using arrow cards. According to the new Common Core Standards for Maths first graders are required to be able to count, say and read numbers up to 120. The Arrow Cards are a perfect way of teaching both the procedural and conceptual knowledge associated with this skill. The really neat thing about this lesson was that Amy had made a set of Arrow Cards for each student in the class (sometimes, college room-mates can come in handy especially as there is a large amount of cutting out required). Amy used the Arrow Card model I developed which reduces significantly the amount of cutting required. (To make the "arrows" cut diagonally across the black square on each card).  There's also an interactive on-line arrow card activity but it doesn't seem as good as actually having the number cards in your hand to work with. Here's a paper on Arrow Cards I wrote several years ago

It was really neat to watch the students get their own set of Arrow Cards. The first thing many of them did was to count through them, first by ones, then by tens and then by 100s. The full set also include 1000s but 100s is enough for first grade.

There are two interesting discussion point about using arrow cards. One involves color coding the cards and the other has to do with including a 00 card that some people like to include. As far as color coding goes I would rather keep all the cards in one set the same color. I think this focuses the students' attention on the numbers and not on the colors. If all the ones were one color and all the tens were another color the students would focus on the color and not the numerals. I do like to make the sets in different colors so that children sitting next to each other will not get their sets mixed together.
As for the 00 card I don;t think it is necessary as it is not a real number and students seem to manage fine without it.

The moist important thing to remember when teaching place value is to treat 0 as a number and not use that terrible term "place holder". Every number is really a place holder!      

Counting from 2010

Learning to count in English is fraught with difficulty and non-intuitive sequences. The way our English counting words are constructed is probably one of the most significnt reasons why young children in English speaking countries tend to lag behind those, especially in Asian countries, in terms of early learning in Math. The counting systems in most Asian languages follow a very logical sequence, especially through 20, sequences such as ten and one, ten and two, ten and three where the confusing words of eleven, twelve and thirteen are used in the English language.

Once we get to twenty a logical sequence steps in where the sigle digits (1 - 9) are repeated  in each decade up to 100. Counting from 20 - 100 all seemed quite logical to me until yesterday when I suddenly realised we have been creating yet another dilemma for young children learning to count. One of the common errors young children make when they first learn to count in the decades is to say, for example,  "twenty-seven, twenty-eight, twenty-nine, twenty-ten (instead of thirty). In the past this has easily been corrected as it did not make any sense to say "twenty-ten". The problem is that the terms twenty-ten, twenty-eleven, and twenty-twelve have become legitimate terms of usage when refering to the date of the year. These terms seem to have replaced the more mathematically correct, for example, "two thousand twelve" (with or without the "and').

This is similar to the double use of the word "third", for example, as an ordinal number as in "I am third in line" and as a fractional number such as "I have a third of a cup of orange juice".