Tuesday, August 11, 2009

Place Value Part 2: Base Ten for Young Students

One of the most fundamental mathematical concepts, yet one of the most poorly understood, is place value. The typical primary school lesson presents only a superficial, nominal understanding of place value. Students learn only to correctly name the place-value columns, or identify the digit in a given column, but they often do not understand the significance of the column names.

In Part 1, The Chocolate Factory, I introduced a middle school activity for rebuilding often weak base ten foundational concepts. The activity extends understanding to place value in other bases. In Part 2, I will introduce activities suitable for much younger children. Young children can construct the meaning of base ten place value through many activities and games.

There is some evidence from Jean Piaget's work as illustrated in the video, that base ten is conceptually out of reach for very young children. If there is demand, I will present some activities that help young children explore “Two Land” and “Three Land.” Years ago I field tested a unit called “The Land of Hand” which of course would be “Five Land” in the terminology of the video.

Today I am going to concentrate on base ten, or “Ten Land.”
1. Morning Circle
Many kindergarten and first grade teachers have a regular morning circle time when they gather the children and go through a structured routine of talking about the calendar, the season, birthdays and other topics using a set of visual materials that are permanently on display.

The teacher prepares a display of three horizontal pockets with transparent envelopes on the front of each pocket. On the side is a cup full of Popsicle sticks and a stack of cards numbered with the digits from 0 to 9. Pocket charts can also be purchased from various vendors. Every morning the teacher takes one Popsicle stick and places it in the far right pocket (as you face the display). Each day the teacher replaces the card in the envelope to reflect the number of sticks in the pocket.

On the tenth day, the teacher places the tenth stick in the pocket and then makes a show of pointing out there are ten sticks. The teacher then bundles up the ten sticks with a rubber band and places the bundle in the middle pocket. The pocket envelopes should now show (empty, 1, 0) representing 1 bundle of ten sticks and 0 single sticks. The teacher goes through the Popsicle stick routine every day.

On the hundredth day, a celebration day in many schools, the teacher gathers the 10 bundles, ties them together with a piece of yarn and places the whole bundle in the far left pocket and changes the display to show (1,0,0) representing 1 packet of 10 bundles, 0 bundles of 10 sticks, and 0 single sticks. The teacher continues the routine until the last day of school at which point the display should show something like (1, 8, 5).

For step-by-step instructions on how to prepare a complete circle time (sometimes called calendar time) display, see this excerpt from the Center for Innovation in Education. Michael Naylor uses the calendar to build number sense.

2. Trading Activities and Games

Playing games is a natural way for children to acquire all sorts of different aspects of number sense. Years ago I checked a book out of the library that was chock full of wonderful tutoring games. The book has long since gone out of print but no matter. I found the author, Peggy Kaye's website. Here is my version of a game she calls "Fifty Wins."

The teacher creates two boards on heavy card stock, one for each player. Each player also has a die. I recommend using extra large die if you can find them. Each player also has a collection of 50+ beans, pennies, or other counters. My own modification involves using the board at first, then doing away with the board and playing with pennies and dimes.

Each child casts their die in turn, and draws the number of counters that matches the number of dots on their die, placing one counter in each of the small squares of which there are nine. Upon accumulating the tenth counter, they transfer ten counters to one of the five squares. The first person to get fifty counters wins. Children learn there can never be more than nine in the one's place, and that the ten's place is precisely groups of ten. If three big squares are filled and none of the little squares, they can see very clearly 3 (groups of ten) 0 or 30.

A modification I have made is to use poker chips for counters. I change the design of the board so that the long rectangle is outlined in one color (say blue) and the squares are outlined in another color (say red). Then as the child accumulates 10 blue chips, the child exchanges the 10 blue chips for one red chip and places it in one of the red squares. The poker chip modification leads quite naturally in the penny-dime modification I mentioned earlier. I have also used the same poker chips with the same color signification for "The Chocolate Factory" activity, blue for leftovers, red for boxes, white for cases.

Another modification of mine which may be considered a weakening of the game is the use of a die to generate numbers. The original game uses a spinner where some of the fields say “Win 10.” At the beginning the child will dutifully count out ten beans and place them one by one in the small squares, only to have to transfer the entire group of ten to a big square. Very soon the child counts out the ten beans and straightway places them in a big square. The opportunity to realize a group of ten in one turn is lost when die are used, but I suppose you could use a set of two dice. I like the die because the child does not have to read words or numerals. With die, the child has only to match, by one-to-one correspondence, the beans to the die spots. There is no need to reference numerals at all, so the game stays squarely focused on number and avoid number/numeral conflation.

“Make Fifty” is just one example of what is known as a “trading activity.” Cuisenaire rods also work well for trading activities. Every ten cubes makes one rod. Any base-ten block set goes one step further where every ten rods makes one flat, and every ten flats makes one cube. Many base ten block worksheets can be adapted to active lessons.

All manipulatives have limitations and some researchers are concerned about the limitations of base ten blocks. Nevertheless, with a good mix of activities, the teacher can address the differing learning styles of each student.

Stuff to Avoid
Generally speaking, worksheets should be avoided. Nevertheless, I like to design special worksheets as data recording instruments for math labs utilizing base-ten blocks and Cuisenaire rods. Students can learn a lot of math without writing numerals. In fact, a foundation of math reasoning skills without reliance on numerals helps children acquire the concept of the difference between numbers and culturally-determined symbols for numbers such as Arabic numerals. Schools “accidentally on purpose” teach children to confuse number and symbol. Cuisenaire has a few such worksheets along this idea, but I have some problems with the worksheet design. Maybe I'll collect my math lab worksheets into some kind of cohesive with comprehensive directions for using them with children and make them available.

Computer-Based Materials

Too many of the computer-based materials, animated mathematics and virtual manipulatives, though so appealing to adults, often has a magical quality to young children. Regrouping happens before their very eyes but they do not understand the mathematical concept and mechanism. They do not get from the computer what I call the psychology of numbers, or how numbers behave. It's just a lot of cool special effects without specific mathematical concept acquisition benefit.


Despite the National Council of Teachers of Mathematics (NCTM) claims to the contrary, calculator studies with the youngest students show no advantage in the development of children's number sense. In 2002, I conducted a major survey of research, research critiques, case studies, and editorials. I periodically asked NCTM to provide me a list of what they characterized as supporting research, but they never did. I found no basis for NCTM's assertion that research backed their recommendation for calculator use in the earliest grades. I found that calculator usage need not hinder the development of math reasoning skills, but it may in fact do so. Teachers report that children become overly dependent on the calculator and have difficulty learning to evaluate the reasonableness of their answers. They trust the calculator more than themselves.

Links and Invitation

The following is a list of links of base ten lessons. They are presented as is. Many exemplify what I believe are the main weaknesses of most base ten teaching.
Lesson plans reviewed by teachers:
Crayola Tally Sticks:
Applet: but better off using concrete manipulatives.
An Unreviewed Collection of various resources:
A favorite resource for getting teaching ideas:

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