Using Base Five as a Context for

Introducing Research Concerning

Children’s Mathematical Thinking

Steve Blair

Grand Valley State University

Abstract:Mathematics courses for preservice elementary teachers often include

experiences involving number systems with non-standard bases in order

deepen students’ subject-matter content knowledge. We will discuss and

share examples concerning how we broaden this approach in ways that

introduce students to the wider body of research involving children’s

mathematical thinking. Our students do not convert between bases;

rather we have them solve word problems within the base five number

system in order to experience problems as children would. Research

concerning children’s mathematical thinking is then introduced as

students reflect upon their own thinking. Parallels can then be drawn

between how our students solve problems within base five and how

children learn within base ten. Our goal is that such experiences will

deepen our students’ pedagogical content knowledge regarding whole

number concepts and operations.

Discussion Question:

Do you include activities involving non-standard bases (e.g. base five) in your courses for pre-service elementary teachers?

If so, What types of activities do you use? Why do you use them?

My first answer (circa 1999):

I used base five activities to help my students “unpack” and “deepen” their understanding of place value(unitizing, consistent regrouping, and positional notation).

Much of my students’ understanding of the base ten number system tended to be procedural and routine. For example, when asked how many tens are in 234, many of these students will say 3 because “3 is in the tens place”, i.e. 234 is 2 hundreds, 3 tens, and 4 ones. They were not able to flexibly view 234 as 23 tens and 4 ones.

By using “Xmania” activities, my students could not rely upon their memorized procedures and therefore had to “make sense” of the problems using tools such as concrete models. Once they had constructed a meaningful understanding within Xmania, it was relatively easy for them to connect back their understanding within the base ten system.

Example Activity:Xmania Riddles (Place Value Problems)[note in Xmania O,A,B,C, and D are used as numerals]

1 I have a collection of BC small cubes and D sticks. What number represents the total amount in my collection?

2 The amount of my total collection is ABC. If I have exactly AA sticks, how many small cubes do I have?

One thing that always struck me about using

Xmania activities with pre-services elementary

teachers was how the unfamiliar context put

students into the “children’s shoes”. My students’

developing understanding within Xmania seemed

to parallel how children develop an understanding

of base ten. This led me to consider broadening

my use of Xmania activities from a purely subject

matter content knowledge (SCK) perspective to

one that included pedagogical content knowledge

for teaching (PCK).

There is one important caveat here:

my students’ development seemed to parallel

that of children so long as they were not allowed to solve Xmania problems by converting between bases.

When allowed to convert between bases, the

activities degraded from a “making sense”

perspective because my students tended to find

procedures to convert and then use their base

ten procedural knowledge uncritically.

My current (evolving) answer:

I still use Xmania activities to address my students’

SCK concerning place value, just as before.

However, I have also broadened this approach to

more directly address students’ PCK with regard to

teaching whole number operations. Specifically, I

use Xmania as a vehicle for introducing research

concerning children’s mathematical thinking.

The body of research involving Cognitively Guided Instruction has been used to help in-service elementary teachers better understand how children’s understanding of whole number operations develop when they are encouraged to solve word problems

in meaningful ways.

One important result of the CGI research is

that students’ strategies for solving

meaningful problems develop in a natural

progression of three general stages:

Another important result of the CGI research is that simple word problem types can be classified according to how they are interpreted by children. Knowledge of these types can then be used by teachers both to assess children’s understanding of the operations and also to plan for appropriate

instruction.

OK, so why use Xmania?

Professional development materials based upon CGI research has generally been used with in-service teachers. They are able to make sense of it’s importance/relevance based upon their experience teaching

children whole number operations.

My pre-service teachers do not generally have this kind of experience. They often find it hard to believe that children won’t “just solve it the right way”. For example, my students generally see a Join-Change-Unknown problem as inherently a subtraction problem because “that’s the correct way to do it.” They focus on how they would solve it symbolically, not on how the children can

solve it by making sense of the context.

My students didn’t have the experience they need to realize the usefulness of the CGI research, they needed a context that would challenge their own understanding – my idea is to use Xmania as that context.

Xmania Word Problems (handout)Consider the following Xmania problems. Explain

how an Xmanian child might solve each problem

using a direct modeling strategy. Then write a

number sentence to represent the solution. What

are some likenesses and differences in the

problems’ structure?

1 George has D banana trees and on each tree

are AC bananas. How many bananas does he

have on all his trees?

2 Fred the giraffe is C times as tall as Henrietta

the stork. If Henrietta is D feet tall, how tall is Fred?

3 Joe has a total of AOC postcards in his collection. His scrapbook has AB pages. If he puts all of his postcards

in his scrapbook so that each page contains the same number of postcards, how many will be on a page?

4 The school is CA yards tall. Each story of the building

is D yards tall. How many stories does the school have?

5 Mia also has a total of AOA postcards. She decides

to put AC postcards on each page of her scrapbook. How many pages will she need to use for all of her postcards?

6 The python is BD feet long. It is AB times longer than the rattlesnake. How long is the rattlesnake?

Xmania Word Problem (notes)All six problems involve operating with equal-sized

groups.

Number sentences:

1) D x AC = ?

2) C x D = ?

3) AOC ÷ AB = ?

4) CA ÷ D = ?

5) AOA ÷ AC = ? or AOA ÷ ? = AC

6) BD ÷ AB = ? or BD ÷ ? = AB

Problems 1 and 2 are both multiplication problems (total

number or amount unknown),

Problems 3-6 are all division, but 3 & 4 are partitive division

(number in each group or part unknown) while 5 and 6 are quotative division (number of groups or parts unknown),

Problems 1, 3 and 5 represent discrete grouping/partitioning

contexts while problems 2, 4 and 6 represent multiplicative

comparison contexts,

Children’s direct modeling strategies will generally differ depending upon the context of the problem.

Xmania / CGI Reference List

Carpenter T., Franke M., and Levi L. (2003). Thinking Mathematically,

Integrating Arithmetic & Algebra in Elementary School. Portmouth, NH: Heinemannn.

Carpenter T., Fennema E., Franke M., Levi L. and Empson S. (1999).

Children’s Mathematics, Cognitively Guided Instruction. Portmouth, NH: Heinemannn.

Fuson K. (2003). Developing Mathematical Power in Whole Number

Operations. In J. Kilpatrick, W. G. Martin and D. Schifter (Eds.), A Research Companion

to Principles and Standards for School Mathematics (pp. 68-94). Reston, VA: National

Council of Teachers of Mathematics.

Philipp R. and Cabral C. (2005). IMAP, Integrating Mathematics and Pedagogy

to Illustrate Children’s Reasoning. CD published by Pearson, Merrill Prentice Hall.

Schifter D. and Fosnot C. (1993). Reconstructing Mathematics Education,

Stories of Teachers Meeting the Challenge of Reform. New York: Teachers College

Press.

An abbreviated account of CGI can be found in several “methods” textbooks, such as:

Van De Walle, J. (2004). Elementary and Middle School Mathematics, Teaching

Developmentally, fifth ed.. Boston: Pearson.