Not long after encountering the possibility that number lines have a privileged place in math learning, I read H. Wu's revelatory definition of a fraction as a point on the number line:
The following is a new approach to the teaching of fractions. It is not new in the sense of introducing new concepts; the subject is too old for that. Rather, it is new in the way the various skills and concepts are introduced and woven together. Whereas it is traditional to ask you to believe that the concept of a fraction is so profound that you have to be willing to accept multiple meanings for it at the outset, we merely ask you to accept one clear-cut deļ¬nition of a fraction (as a point on the number line), and use reasoning to deduce as logical consequences all other meanings of this concept.I've been relying heavily on number lines for self teaching and reteaching for several years now.
On the Teaching of Fractions (pdf file)
H. Wu
David Geary's new longitudinal study seems to add further evidence that number lines are important:
The researchers also found that first-graders who understood the number line and how to place numbers on the line and who knew some basic facts showed faster growth in math skills than their counterparts during the next five years.
MU Psychology Study Finds Key Early Skills for Later Math Learning
Long-term study shows students must know about numbers at beginning of first grade
July 11, 2011
I have a friend who tutors, among other subjects, math.
ReplyDeleteHe made a BIG point of asking me to formally introduce my child to the number line at an early age.
He claims to know multiple kids who have problems with math because of a deeply rooted MIS-understanding of this. His " poster child" example is a kid who thinks of numbers as being on a 'clock thermomter' ... 0 - 12 work differently than other numbers...
-Mark Roulo
Anecdata: I very clearly remember my first grade teacher showing us the number line drawn on the wide chalkboard at the front of the room. The zero was in the center, and she explained that the number line went on forever in both directions, but we would only need to worry about the right.
ReplyDeleteI remember we used the number line in first grade, too. Each student had a adhesive-backed number line stuck on the top of the desk, right under the grooved pencil holder.
ReplyDeleteOf course, that was back in the day, when students had individual desks.
As a parent, I can't tell you how important the number line is, especially for children who are experiencing early math problems. The ironic part, my child who has dyslexia, also experience problems in math early on. I taught my child the number line, not only to acquire a deeper understanding of numbers, but as a more efficient way of counting, since she did have sequencing and short-term memory processing problems. More efficient, than the current fuzzy math instruction, but she was not allow to used it. So she had to do it in secret, and than erase her work, and than put the require picture in it. My child over the years have developed some novel strategies with number lines over the next few years, and once again she was chewed out for teaching grade 1 students who were doing poorly in math, how to add, subtract and multiply using the number line. All one needs to know are your numbers up to 100. No need for fancy pictures, just a straight number line will do. By the way, the grade 1 kids were passing their quizzes, doing their work, and more importantly they developed a deep understanding of numbers and counting. And my dyslexia child got chewed out, for teaching the kids such advance methods. I burst out laughing when I heard it, but the parents thank my child and she was out of the running for the award of best lunch monitor for the primary grades.
ReplyDeleteMath is done so badly in today's schools, I often wonder how many potential geniuses in math are lost with the current fuzzy math curriculum?
Why not teach it earlier? In preschool, perhaps? At least to 3-4 year olds who have a rudimentary understanding of ordinal numbers and their symbols?
ReplyDeleteI can only talk anecdotally about one child (my own), and he's not representative at all. But when I showed him a number line at age 4 and used it to demonstrate addition and subtraction, something "clicked" for him, and even though he'd been confused about subtraction before, he never misunderstood it again. I've since used the number line to talk about simple fractions (with a ruler standing in for the number line) and even negative numbers. He understands them both now, enough to answer novel arithmetic problems with very high accuracy.
On a different subtopic: wasn't there a post here some months ago about some people simply not having a mental representation of a number line? Could someone refresh my memory of that?
Buzzmath.com and MRI's Blueprint both emphasize the number line extensively. I've found Buzzmath to be the best of the programs I use to build students into being capable for Algebra 1.
ReplyDeleteDoing it earlier is limited by a couple things, but the obvious being the question of who would teach it to preschoolers. The number line is not well enough understood by adults that you can assume a preschool teacher understands it.
ReplyDeleteAnother reason not to start too early is the fence post problem. How do you explain to most 4 yr old that you are counting jumps, not the ticks where the numbers are? How many times will you have to state this before they follow it? Most will count their starting place as "1", because they count their thumb as 1, too.
our prior conversation
ReplyDeletehttp://kitchentablemath.blogspot.com/2011/03/number-sense.html
Ah, thanks for the link.
ReplyDeleteThe fence post problem was a big stumbling block for me! (as an adult, re-teaching myself math -- )
ReplyDeleteorangemath - thank you for those links!
ReplyDeleteBack in my day, we did number line in 2nd and 3rd grade, which is way too late IMHO. As a kid, I thought it was trivial and easy, so I paid little attention. I think it should be taught in kindergarten, the age at which kids are actually trying to understand numbers and quantities.
ReplyDelete