Base 10 blocks can make abstract ideas like place value and regrouping visible and tangible for your primary school students when the time comes to teach math.
In the struggle to teach math to first and second graders, the real enemy is Piaget's stages of development. The simple truth is that teachers have to find a way to present relatively abstract mathematical ideas to children who haven't yet developed much capacity for abstract thought. The solution is simple: find a way to represent those abstract ideas in a tangible, concrete manner that kids can see and touch. Base 10 blocks are a powerful tool for doing just that.
Overcoming Abstraction
It's no great trick to teach most elementary school kids to count. Many (probably most) can do that when they show up at school. And they certainly know the difference between one cookie and three cookies. Teaching them the names of our ten numerals usually isn't that complex a task, either. But when it comes time to cross the line from single digit numbers to using our numerals to represent more complex values, like eleven or two-fifths, mathematical ideas start to become a little muddier in the first and second grade brain...
In his or her mind a child may ask you, why is 11 worth more than 8, when eight is a bigger number than either of the two "ones" in eleven? It's even bigger than the two ones added together? Does that make sense? And you expect me to believe that 2/5 has less value than 1, even though both 2 and 5 have more value than 1...?
Of course, most first graders can't articulate their objections that clearly. They just know it has suddenly become (for some of them, at least) confusing.
Intro to Base 10 Block
Base 10 blocks (see image 1), like many other math manipulatives, allow children to see and touch the ideas they are being asked to cope with in math class. Base 10 blocks usually come in four sizes. There is the cube which represents a value of 1. The "long" is a block that looks like 10 of the cubes glued together; it represents a value of 10, and the kids can count 10 of the individual cube blocks on a long. The "flat" is a block that looks like 10 of the "longs" glued together; it represents a value of 100, and the kids can count the 10 "longs" on a "flat." Finally, there is the "block" - the size of 10 of the "flats" laid on top of each other. The "block" represents a value of 1,000.
The first time you use base 10 blocks it is a good idea to simply allow the children to examine them and compare them. Discuss them - their sizes and how they relate to each other. Practice counting them in order to allow the children to become comfortable with the fine motor skills required for using the blocks.
Representing Numbers with Blocks
When the children become comfortable with using the blocks, you can begin teaching them to build representations of numbers with the blocks. Many teachers create a place value mat that the kids can lay on their work surface. The mat should have columns that are four to six inches wide and are labeled "ones," "tens," "hundreds," and "thousands." Students learn to always put cubes in the "ones" column, longs in the "tens" column, flats in the "hundreds" column, and blocks in the "thousands" column. After that has been mastered, they are introduced to the basic rule of base 10 counting - you can't have more than nine pieces in any particular column. If you get more than that, you have to go back to the storage bin and trade your ten cubes in for a long, or your ten longs in for a flat...
Children should be allowed to use the blocks regularly. The initial goal is to develop the skill of representing numbers - like 37 (image 2), or even bigger numbers like 1,232 (image 3). Maybe the mats eventually disappear; maybe as a teacher you'll decide you want to keep using them. Once the children have mastered building numbers with their blocks, you're ready to use the base 10 blocks to teach operations - like addition and subtraction.
The copyright of the article Teaching Place Value in Curricula/Lesson Plans is owned by Greg Cruey. Permission to republish Teaching Place Value in print or online must be granted by the author in writing.
Image #3 (One Thousand Two Hundred and Thirty-Two) is not accurate. It
should say (One Thousand Two Hundred Thirty-Two) the (and) stands for a
decimal point.
Aug 28, 2008 4:53 PM
Greg Cruey :
You bring up an interesting point. I spoke to some teachers and I looked at
the math lessons on place value in the curriculum that my district uses -
the University of Chicago's Everyday Mathematics.
I understand
the convention that, when teaching decimals, we try to restrict the use of
the word "and" so that it serves only as a verbal representation
of the decimal. Everyday Math makes no statement about the convention, but
gives an example of a mixed number and says that the "and" in
"twelve and three-fourths" represents a decimal. Which is true.
But no prohibition is communicated against using "and" in other
ways. No "right" method for spelling out numbers is set forth.
My article doesn't deal with decimals.
In natural
spoken English, most fluent speakers place the word "and" between
the hundreds and the tens place: seven-hundred eighteen thousand, four
hundred AND eleven. In the English of the non-mathematical public, it is
even permissable to insert the "and" after EVERY hundred: four
hunderd AND thirty thousand, nine hundred AND sixteen. Few English speakers
leave out the "and" altogether from these numbers.
You're right in that it does violate the convention among math teachers.
I violated the convention here because, well, I taught college before I
taught grade school and I've never been exposed to the convention. I
managed to get certified in middle school math and to teach elementary math
for most of my four years in the classroom without ever having had the
convention pointed out to me.
The approach to math instruction
that Everyday Math promotes is one that acknowledges a variety of ways to
represent numbers and multiple approaches to most problems. The hope is
that children will THINK about math. Rigid conventions like the one you've
described are fading as a result, I think.
In thinking about the
issue I came upon the title of a study at the National Institute of Health:
A STUDY OF EIGHT HUNDRED AND FIFTY CASES OF SCARLET FEVER WITH A MORE
PARTICULAR CONSIDERATION OF SEVENTY-ONE FATAL CASES. Surely they know
numbers.
To say that the caption of image #3 is
"inaccurate" implies that people don't know what the number
means, or that the VALUE of the number is being misrepresented in some way.
While it may violate the convention, the caption is not innacurate; no one
who looks at the blocks in the picture and then reads the caption will
think that there are more than twelve but less than thirteen of the
blocks...
Feb 6, 2009 12:22 PM
Guest :
Just because some people say "and" when it should not be said
does not make it right for educated people to say it. What kind of research
have you done that validates your claim that "most fluent speakers
place the word "and" between the hundreds and the tens
place"? As educators we must teach our students to do what is correct,
not what is common. "And" does not belong between the hundreds
and ones place at any time. When writing a check for 102 dollars it is not
proper to write "one hundred and two dollars." It would be
written "one hundred two dollars." While what you wrote is not
inaccurate in your mind, it is misleading and not technically correct. When
you publish things you must be technically correct.
Feb 6, 2009 1:42 PM
Greg Cruey :
LOL. Saying "and" or not saying "and" is a matter of
right and wrong? Next you will be telling me that I can't end my sentences
with prepositions anymore. But I guess that sort of logic is what we're
headed towoard. ("But I guess that sort of logic is that toward which
we are heading?" Hmm, sounds awkward.) And while ending sentences in
prepositions is not permissible in Latin, the Little, Brown Handbook says
it's acceptable in English.
As educators we must teach our
children to think and understand, not just memorize facts and learn rules.
And if I write "one hundred and twelve dollars" or "two
hundred and eight dollars & 16/100" on a check, the bank will
certainly honor it and they'll pay out the correct amount.
You
are looking at convention as a cup that you can fill with values. It is
just a convention. No one suffers eternal punishment for violating it, and
it carries no mathematical information. No one misunderstands me when I say
"one hundred and four dollars" or construes it as anything but
$104.00.
What I wrote is not inaccurate PERIOD. It has nothing
to do with my mind...
May 13, 2009 6:08 AM
Guest :
As a true speaker of the English language, born and raised in England, I
feel I have a reasonable command of my mother tongue; I can assure you
there is absolutely nothing wrong in including the word 'and' when
stringing a series of figures together. In fact, when we write cheques here
in England (and no I haven't mispelled 'cheques' - that is what us English
people call them!) we always include the word 'and' when writing the figure
down. I am studying to be a primary teacher and found Greg's article about
place value most helpful in my studies, given that it is just about that -
place value; not a detailed study of written English. Let's all focus on
the mathematics, not the nit-picking.
Jun 18, 2009 5:22 AM
Guest :
I agree that using "and" to link numbers together is fine and is
mainly used here in England, in fact I had not hear of the other way until
now. I think it is personal preference to what is used and either way gets
to the same end. As good maths teachers, we shouldn't be prescriptive
in our approach as there are so many ways to teach maths and no one way is
right or wrong. We should be differentiating to include all learners. What this debate does bring out is the fact that the language surrounding
maths can be very difficult to comprehend. For example, in and out are very
different concepts in English, but when looking at fractions 1 in 4, is the
same as 1 out of 4! Bizarre!! This should also be tackled before learners
are able to tackle the actual maths.
Sep 13, 2009 4:47 PM
Guest :
I can't believe there's a debate on this issue. It is mathematically
incorrect to say "one hundred and thirty seven". Not only is it
mathematically incorrect, but it's IGNORANT.
Sep 13, 2009 7:26 PM
Greg Cruey :
Hi September 13th Guest,
I like how EMPHATIC you are. I'm not
sure what's inaccurate about it. How are conjunctions represented
numerically?
Princeton (
http://wordnetweb.princeton.edu/perl/webwn?s=ignorant )gives some basic
definitions for the term "ignorant." a.) uneducated in general;
lacking knowledge or sophistication b.) uneducated in the fundamentals of a
given art or branch of learning, lacking knowledge of a specific field c.)
unaware because of a lack of relevant information or knowledge.
If my ignorance rests in the area of definition "a" it would
have to involve my level of sophistication, since I have three degrees and
11 years of college. Since I'm certified in math in two states and I've
been through a healthy dose of professional development for the Everyday
Math series (University of Chicago), I don't see how I could be classed as
Ignorant under definition "b." And since I'm perfectly well aware
of the "and prohibition" you seem to be advocating as a law of
math, I don't fit under definition "c."
Really, this
discussion reminds me of the sort of academic arrogance that holds that
"ain't" isn't REALLY a word or that there's something wrong with
ending sentences with a preposition...
Sep 21, 2009 7:06 AM
Guest :
I like the explanation for the base 10 blocks and how to utilize these
manipulatives successfully with a class! That's the reason for the
article, and it serves the purpose well. Thank you! (I can't believe the
debate on all else... where do people get time to disucss?!? lol)
Sep 24, 2009 1:13 PM
Guest :
I'm afraid that the larger issue here (bigger than whether or not to say
"and") is actually about the role of manipulatives in teaching
mathematical concepts. Although some students may appear to benefit from
learning about place value through base 10 blocks, sadly the learning often
fails to transfer to symbolic arithmetic (as evident with buggy subtraction
algorithms). I highly suggest reading the literature by Uttal and DeLoache
for a perspective on symbols systems and representation.
Sep 29, 2009 1:57 AM
Guest :
I am a preservice teacher in Australia. In Australia we do use the word
'and' between the hundreds and tens place values and are taught it at
university (college). In our maths (sorry, Americans probably say math but
Australians do not) course we have an American and Australian academic and
it becomes very interesting in the mix up of words. The use of
concrete materials to reinforce mathematics concepts is taught heavily to
preservice teachers. I think it is fantastic for students. Thank you.
Oct 17, 2009 11:55 PM
Guest :
Hi Greg - been reading your articles with interest. Wonder if you can
clarify something? I am a teaching assistant (trained to work
primarily working with dyslexic students) but am doing some 'research'-for
professional development not to publish!- into the difficulties encountered
by dyslexics when doing maths. I think I've got to the point where I can't
see the woods for the trees...I've read Dowker, Cockcroft, Chinn ..the list
goes on....but I can't explain in simple lay man's terms or common English
(oops, let's not start another debate about English!) why dyslexics (and
many others) struggle with place value. I can see why they have
difficulty seeing that 11 is bigger than 5, but why do dyslexic children
struggle when adding/subtracting using the column method-for example? I
used to think it was purely a question of poor sequential memory but am now
totally confused. What is the underlying difficulty? Thanks.
Oct 18, 2009 5:01 PM
Greg Cruey :
A couple of thoughts… Begin by questioning your assumptions. Can you
explain what dyslexia is? I have a graduate degree in linguistics and I’m
certified in reading; but I can’t define it. It’s a conclusion reached most
often through process of elimination. Is it poor eyesight?...no; is it low
IQ?...no; is it a hearing problem?... no; is it lack of access to
education?...no; and when the one runs out of other choices, a reading
problem was determined to be dyslexia. It’s not a technical term. It
doesn’t appear in the DSM.
Why does Johnny struggle with place
value? He probably has his own individual reasons, whether he’s dyslexic
(whatever that means) or not. Neuropsychiatry (using brain imaging) and the
genetic sciences are beginning the process of identifying biological and
physiological causes for learning disabilities. But that’s a ways off yet.
And I’d like to emphasize the plurality of that term – causes.
I don’t believe there is a (one) reason. And since the cause of the
problem is usually beyond our ability to identify, the pedagogy that helps
overcome the problem is more important. That often varies from individual
to individual.
