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!      

Math Triangle of Meaning

I've been on sabbatical this semester researching ways to help students who are English Learners learn math in US classrooms. Part of the project has involved interviewing both adults and children who have settled in the US having grown  up in other countries, just like I did. This week I have interviewed a couple of third graders who were born in Burma and came to the US via Thailand.

Both students spoke English well so communication was not an issue. In the first interview the student did something quite remarkable in that it was so exactly in line with a particular theory I use in my math education courses. It is always so neat when this happens because it reconfirms the value of a theory-based education. When we can make an observation and relate it directly to a particular theory, or piece of a theory, as teachers we are in such a better position to know where to go next in the instructional process.

The particular observation happened as I was interviewing the student on his understanding of basic numeracy; more specifically being able to read, write and understand 3 and 4 digit numbers. The triangle of meaning is a piece of theory that identifies the  three component of a piece of math understanding; the idea, the word and the symbol. For example 5 is "five" and *****. Put one of these at each  corner of a triangle and there are 6 relationships (e.g hold up five fingers and ask how many, hold up 5 fingers and ask student to show the correct numeral). Students may  frequently have some of the relationships but not all 6.

Yesterday the student kept reading 4 digit number incorrectly. For 4,582 he would say "forty-five thousand and eighty two". So I asked him to write numbers such as 4,275. He did this correctly every time. I then asked him to read back to me the numbers he had correctly written and he read them incorrectly using tens of thousands as in the example above.

He could model the numbers using base ten blocks and he could write them when given them orally  but he could not read them even when he had written them. The important lesson here is to remember that just because a student can write a number doesn't mean to say she/he can read it, or knows what it means.   
  

Place Value and Reader's Theatre

In a previous life and many years ago I taught a language arts and social studies course as part of a teacher education program. My favorite part of the language arts part of the course was Reader's Theatre (and I intentionally put the r before the e). I think I enjoyed it so much because it was an opportunity for students to come to terms with the most incredible teaching tool we all have, our voices. I used to tell them that they had to imagine they were performing a play on the radio so no-one could see any of their actions or facial expressions; everything had to be communicated through the voice.

Yesterday evening I spent half an hour with Stephanie, a graduate education student to help her develop her project which was to teach place value through reader's theatre. Since I believe math should always have an aesthetic component this seemed like a wonderful way of developing the fundamental concepts of place value in an artistic and motivating context.

We talked primarily about place value, how it is groups of tens of tens and how we have ten numerals with which to make every conceivable number possible. We also talked about the misconceptions caused by phrases such as "0 is a place holder" and how 0 really means "none of". For example, in 103 the 0 means there are no tens. We also talked about the reason for putting a comma every three digits to help us read large numbers. Most people remember being taught this but few people ever remember being taught why. If you think about a large number such as 21,487,439 the first 3 digits from the right, 439 refer to ones, the next 3 digits, 487,  refer to thousands while the 21 refers to millions. Between each set of commas, from right to left are ones, tens and hundreds. 439 is ones, tens and hundreds of ones. 487 is ones, tens and hundreds of thousands and 21 is ones and tens of millions. This pattern of ones, tens and hundreds repeats itself between each comma for ever.

Thinking about Zero

Just think, without the numeral 0 there would be no place value system, 10 would be 1, and we wouldn't have a clue how to show a million. Traditionally, in our base 10 place value system of numeration, 0 has been called a "place holder" which is pretty meaningless when you stop to think about it. In the number 307, the zero means there are no tens in just the same way that the 3 means we have three hundreds and the 7 signifies seven ones. To call it a "place holder" reduces the real meaning of zero to a valueless grapheme that in no way contributes to young children's understanding of our place value system. Zero, or nought,  means nothing, none of, the starting place, neither positive nor negative,

Whether we should teach children to start counting at 1 or 0 has been a point of discussion among academics for some time and has recently been complicated by the addition of subitizing, the instant  identification of numerousness without counting. It has recently seemed to me that we need to start at 0; here's why. All children know what it's like to have no ice-cream or no M&Ms or no Hotwheels so why not begin counting with this idea that all children are familiar with. We still need to teach number naming, the sequence of the number names similar to learning the alphabet, and we can still teach subitizing skills. If we start with zero, however, the introduction of fractions and negative numbers later will make so much more sense.

If children see 0 as the point of reference they will better be able to see how fractions fit between 0 and 1. Fractions are initially taught as being parts of wholes, which on a number line, for example, is the space between 0 and 1. All the models used to teach fractions rely on students being able to take one apart into its fractional parts.

It's a similar issue with developing the idea of negative numbers. Without 0 as a reference point many children develop the misconception that anything less than one, such as a fraction, is a negative number.

It's as if 0 is a state of equilibrium, a fulcrum, an origin, a place at which all life begins.


  

The Value of a Place

I can remember being told, as a student in primary school, to put a comma every three numbers when writing large numbers. I was told it would separate the hundreds, thousands, millions etc. I remember dutifully doing this with absolutely no sense of why or feeling any nearer to being able to read very large numbers.

Today, we still use the commas when writing large numbers but
unlike the instruction of the last century, we now teach why we put a comma every three numbers. It certainly does separate things but, more importantly, it shows the repetition of the ones, tens and hundreds every three numbers. First we have ones, tens and hundreds of ones, then we have ones, tens and hundreds of thousands and then we have ones, tens, and  hundreds of millions, and so on. If we learn this repetition of the place value referents it makes reading and understanding large numbers much easier.

It also helps students develop a far more realistic idea of what zero, or nought, is.  I remember learning 0 as a "place holder", a bit like a pot holder, perhaps. What did that really mean? The 0 holds a place when there is no number there, perhaps. What nonsense!  The 0 means there are none of that particular referent in this number as in 23,405, where the 0 means there are 0 tens. In this number, 305, 877  the 0 means there are no ten-thousands; there are 3 hundred-thousands and 5 thousands but 0 ten-thousands.

'Nought', by the way is the British word for zero and became quite popular during the first decade  of the current  century with years like '03 and '06 which was referred to as the 'noughties'.