ERIC ED495307: Algebra Rules Object Boxes as an Authentic Assessment Task of Preservice Elementary Teacher Learning in a Mathematics Methods Course

2007·0.58 MB·English

by  ERIC

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Algebra Rules Object Boxes as an Authentic Assessment Task of Preservice Elementary
Teacher Learning in a Mathematics Methods Course

A Research Study Presented at the

Annual Conference of the New York State Association of Teacher Educators (NYSATE) in
Saratoga Springs, NY, April 28 [Conference presentation title: Preservice elementary candidate
assessment of algebraic understanding for accreditation]

and at the

Annual Conference of the Association of Mathematics Teachers of New York State (AMTNYS),
October 26-28, Saratoga Springs, NY [Conference presentation title: Using hands-on materials to
write algebraic generalizations (Grades 5-8)].

Authors
Audrey C. Rule and Jean E. Hallagan

Contributing Preservice Teacher Authors
Carolyn Agil, Pam Belcher, Ricci Benitez, Erica Bisbo, Tiffany Brown, Laurie Cleveland, Bruce
Dannan, Michelle Derby, Amy Freelove, Megan Green, Hannah Griffin, John Grosso, Mike Hanley,
Kim Hierholzer, Megan Hohman, Amanda Howe, David Laing, Todd LaLone, Anne Licup, Tiffany
Lynch, Melissa Mahlmann, Nicole McHale, Ashley Millerd, Samantha Morgan, Aaron Pascale
Ashley Petreszyn, Joanne Poppelton, Danielle Rizzo, Shannon Shaw, Tori Sivers, Eileen Smith,
Suzy Stahl, Andrea Szymanski, Andrea Tucker, Joe Wasiluk, Katie Waugh, Tina Welch, Meghan
Wheeler, YoAnna Yerden, and Patrick Young.

Abstract

The purpose of this study was to describe elementary preservice teachers’ difficulties with
understanding algebraic generalizations that were set in an authentic context. Fifty-eight preservice
teachers enrolled in an elementary mathematics methods course participated in the study. These
students explored and practiced with authentic, hands-on materials called “object boxes,” then
created sets of their own object box materials. Each algebra rules object box contained materials to
illustrate and describe four different algebraic generalizations, or “rules.” The variables “n” and “z”
were used in each of the generalizations. For each generalization, there was a set of objects
attached to a piece of mat board that showed three cases of the generalization for different values
of “n.” Two sets of cards accompanied these objects, giving word problems, defining variables,
stating equations, and explaining the algebraic generalizations. Students matched word problems
to the object sets, defined variables and checked their work, then wrote algebraic generalizations
for the object sets and used the reverse sides of the equation cards to check their work. Projects
were graded with a rubric. Students were then surveyed about their difficulties. Results of the
analysis showed that students were able to make an assortment of authentic materials in a variety
of contexts and enjoyed the creative aspects of the project, but found the algebraic content
challenging. The most common mathematical difficulties were being able to define the variable,
and identify the pattern. Examples of effective student materials are provided.
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Algebra Rules Object Boxes as an algebraic generalizations through this
Authentic Assessment Task of Preservice authentic assessment.
Elementary Teacher Learning in a
Mathematics Methods Course Preservice Teachers’ Conceptions of
Algebra
Introduction and Literature Review In her seminal review, on the
Because it is critical for elementary learning and teaching of algebra, Kieran
students to be taught algebraic concepts (1992) noted an “enormous gap in the
(National Council of Teachers of existing literature on teaching regarding how
Mathematics [NCTM], 2000; Rand algebra teachers interpret and deliver that
Mathematics Study Panel [RAND], 2003), it content” (p. 356) and also “the scarcity of
is equally critical that pre-service methods research emphasizing the role of the
courses include instruction on how to teach classroom teacher in algebra instruction” (p.
algebra effectively. The NCTM (2000) 395). To date, there is scant research that
advocates that algebraic concepts be reports on the practices of algebra teachers
presented in a meaningful and authentic (Doerr, 2004; RAND, 2003), and in particular
context, and be relevant to students lives. preservice teachers. Current studies
While recent initiatives call for further study demonstrate that elementary level students
on developing and promoting effective are capable of algebraic reasoning (Kaput &
algebraic teaching practices for elementary Blanton, 2001a), yet many elementary
teachers (RAND, 2003), there is an preservice teachers (Zizkas and Liljedahl,
insufficient body of research on teachers’ 2002) may not appreciate the role of
knowledge in the area of algebraic instruction algebraic generalizations in the elementary
(Doerr, 2004, Kieran, 2006) including the use curriculum nor do they understand ways to
of authentic assessment and manipulatives connect generalizations to an authentic
(NCTM, 2000). Therefore, three bodies of context. Bishop and Stump (2000) examined
literature informed the conceptualization of preservice elementary and middle school
this study, research on effective teaching and teachers’ conceptions of algebra. In a
learning of algebra generalizations, authentic semester course, the preservice teachers
assessment practices, and use of concrete engaged in college-level algebraic
manipulatives. experiences involving generalization,
The purpose of this study was to problem solving, modeling, and functions.
inform and improve the teaching and learning They also explored algebraic activities for
of algebraic generalizations with preservice children involving variables, functions and
teachers. A previous study (Hallagan, Rule, pattern generalization. Bishop and Stump
& Carlson, in review) investigated the effect found that many preservice teachers did not
of making algebra rules object boxes on understand what distinguishes arithmetic
preservice teacher knowledge of algebra, from algebra, and of those that did make the
finding a significant improvement. Our distinction, a majority held a procedural
current study focuses on the types of perspective even at the end of the semester
difficulties preservice teachers encountered course. The few preservice teachers that
while engaged in this project, the errors they held a conceptual view of algebra in the
made in the materials they created, and the beginning of the semester valued algebraic
successful sets of materials they devised. generalizations at the end. A similar finding
These objectives enabled us to understand was reported by Goulding, Suggate, and
preservice teachers’ development of Crann (2000) who examined differences
between preservice teachers. The preservice
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teachers were assigned weekly research to carry out a broad based authentic
read on elementary mathematics involving assessment task with their future students
algebraic proof. Then, they reported on the unless they have this experience
problem in class. During the preservice themselves. Many elementary education
teachers’ presentations, they asked the other majors have weak, fragmented knowledge of
preservice teachers to try the activities teaching mathematics (Ma, 1999; Hill,
themselves before discussing the solutions. Schilling, & Ball, 2004), yet when preservice
The weaker presentations reflected the pre- elementary teachers engage in reform-based
service teachers’ inability to think deeply curriculum materials and related
about the actual responses. assessments, Lloyd and Frykholm (2000)
found that those who struggled the most,
Authentic Assessment in Mathematics learned the most and also were able to
Assessment practices are central to identify ways in which they could help future
effective teaching. The current reform elementary students. Hence, it is significant
movement in mathematics education calls for that preservice teachers have experiences in
teachers to simultaneously improve the reform-based mathematical tasks and
quality of classroom assessment practices assessments.
while increasing instructional emphasis on
problem solving. Leaders in educational Using Object Boxes in Teaching
reform argue the form and content of Mathematics
assessments must change to better Manipulatives have been shown to
represent thinking and problem solving skills; be useful in motivating students, focusing
additionally, the way assessment is used in their attention, and helping them to
the classroom needs to change in a conceptualize abstract mathematical
corresponding manner (NCTM, 2000). concepts (NCTM, 2000). The use of
Despite the reformed vision of assessment in manipulatives can help students make
mathematics classrooms, vast differences connections between abstract mathematics
exist in how teachers use and view and a concrete representation.
classroom assessment (Stiggins, 1999). Manipulatives can increase student interest
Traditional assessment practice is pervasive and understanding, serving as a bridge to
as evidenced by a comparative study of successful mathematical learning.
TIMSS data, where most lessons in the Our study focuses on a specific type
United States and Great Britain emphasized of manipulative material called an “object
procedures (Stigler & Hiebert, 1997) as well box.” An “object box” is a set of objects and
as others (Battista, 1999; Manouchehri, corresponding cards that are housed in a box
1997; NCTM, 2000). In contrast, reformed and used for instruction. Montessori (1964)
models of assessment are interactive devised the first object boxes for teaching
between teachers and students, and reading and writing of words. Rule (2001)
between teaching and learning. then expanded this concept to include many
A reformed perspective of phonological awareness exercises. Object
assessment includes a process of constant boxes have been used very successfully in
development and structured, purposeful science to increase descriptive vocabulary
experiences where teachers conceive of (Rule, 1999; Rule, Barrera & Stewart, 2004),
assessment as a way to understand and to teach form and function concepts (Rule &
enhance students’ learning rather than just Barrera, 1999; Rule & Furletti, 2004; Rule &
checking for mastery (NCTM, 2000). Rust, 2001), and science words with multiple
Preservice teachers cannot be expected to
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meanings (Rule & Barrera, 2003; Rule, of cards was in the box with the objects and
Graham, Kowalski, & Harris, 2006). the second set was in an envelope for later
Recently, object boxes have been use. The front of each card in the first card
created for reviewing/ teaching mathematical set showed a word problem referring to one
concepts with preservice teachers and their of the sets of objects. The student read each
elementary students. Rule, Grueniger, word problem and matched the card to the
Hingre, McKenna, and Williams (2006) corresponding object set. Then the student
reported that preservice teachers defined what the variables “n” and “z”
significantly improved their knowledge of represented. These definitions were listed on
numeration, algebra, geometry, and the reverse side of each card for self-
measurement through making “mathematical checking.
mystery object boxes” for elementary Students then attempted to write an
students. These materials included a set of algebraic generalization using “n” and “z” for
items and corresponding clue cards that each of the four object sets. After writing
described numeration, algebraic, geometric these algebra rules, they removed the
or measured aspects of an object. The remaining second card set from the envelope
student read the clues and attempted to and tried to match the algebraic equations
locate the object that satisfied them. written on the card fronts to the object sets.
Sometimes the equations provided did not
Algebra Rules Object Boxes match those the students devised and
In the investigation by Rule, allowed the students to revise their thinking.
Grueniger, Hingre, McKenna, and Williams After all cards had been paired with object
(2006), preservice teachers reviewed sets, students examined the reverse sides of
mathematical concepts through planning and the cards to determine the correct
creation of an object box for use with corresponding object set and to read a brief
elementary students, thereby improving their explanation of the algebraic generalization.
knowledge of mathematics. Our current In this manner, the cards and object sets
study drew upon this idea by asking guided students through the process of
preservice elementary teachers to devise a devising an algebra rule for the sets of
new type of mathematical object box called objects.
an “algebra rules object box” for use with Example sets of cards for algebra
elementary students. rules object boxes created by the authors are
Each algebra rules object box included in Appendix1.
contained materials to illustrate and describe
four different algebraic generalizations, or Authentic Learning and Assessment
“rules.” The variables “n” and “z” were used This project involved students in an
in each of the generalizations. For instance, authentic learning experience. Authentic
four generalizations from one of the sets learning has four major components (Rule,
created by the authors and used by 2006): real-world problems that engage
preservice teachers as an example were: z = learners in the work of professionals; inquiry
n2, z = 5n2, z = 2n, and z = 6n + 2. For each activities that practice thinking skills and
generalization, there was a set of objects metacognition; discourse among a
attached to a piece of mat board that showed community of learners; and student
three cases of the generalization for different empowerment through choice.
values of “n.” Students were involved in making a
There were two sets of cards that useful curriculum material that many were
accompanied these object sets. The first set able to immediately use during their
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practicum experiences and others will use in mathematics methods course at a mid-sized
the future. Orion and Kali (2005) identified college in central New York State
three teacher factors that positively participated in the study. Preservice teachers
influenced student conceptual worked in groups of four students in general,
understanding: “a) openness towards but two groups had only three students, for a
innovative teaching methods, b) scientific total of fifteen groups.
background and c) enthusiasm, and
willingness to invest time and effort in Procedure
teaching” (p. 392). Therefore, enthusiastic Students explored and practiced with
investment of time in making innovative sample algebra rule object boxes provided
curriculum materials is an important and real- by the course instructor, then worked in small
world part of becoming an effective teaching groups of four students each to create their
professional. own. Each preservice teacher supplied a set
Inquiry, problem solving, and critical of objects representing three cases of an
thinking occurred as students devised algebraic generalization and the two
algebraic equations for objects and as they corresponding cards. Preservice teachers
found/ created objects for new problems. were required to check the work of other
Metacognition occurred as students reflected members of their group, thereby increasing
on aspects of algebra and of the project with the amount of discussion among members
which they encountered difficulties. Group and providing opportunities for informal peer
work allowed students to discuss their ideas evaluation of products. Additionally, when the
among a community of learners and projects were completed, groups examined
conference presentations along with the work of other groups and used the given
publication of results of this work allowed rubric to score them, although the instructor
preservice teachers and their professors to scored all projects and her scores were used
share ideas with other mathematics as the final grades.
educators. Finally, students chose the The project integrated several
objects and algebraic generalizations they important aspects of the course: knowledge
wished to illustrate, thereby feeling of mathematics, mathematics pedagogy,
ownership in the project. The assignment group work, and technology integration.
given to preservice teachers to work in small Technology integration into the project had
groups to construct their own algebra rules the following components:
object boxes constituted an authentic
assessment of their knowledge consistent • Students were given a PowerPoint
with Newmann (1993) in that the students template to write over and create
were active in their own construction of their own fronts and backs of clue
knowledge, and the problems extended cards for the objects.
value beyond the school environment. • Students used digital cameras to
photograph the objects and learned
to insert images into PowerPoint.
Method • Students learned how to crop and
improve lightness and contrast of
Participants images.
Fifty-eight preservice elementary • Students learned how to use
teachers (48 females, 10 males; 58 Euro- superscripts for exponents.
American) who were college juniors or
seniors and who were enrolled in a
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Assessments Table 2. Preservice teacher reactions to the
Students were provided with the algebra rules object boxes during the second
rubric with which their projects would be day of use of example sets of materials.
scored. The rubric is shown in Table 1. There were 15 groups responding in total.

Table 1. Rubric for scoring preservice Reaction Frequency
teacher algebra rules object box projects. Materials are interesting 15
Materials are colorful/ attractive 11

Can touch/ manipulate the materials 6
Criteria Yes Border No
The materials help me figure it out 4
Labeled box with 4 objects 1 ½ 0
The materials keep my attention 3
and 8 cards?
A pattern can be seen in the materials 3
Cards neat and durable? 1 ½ 0
Fear – what is all this stuff? 3
Set contains several types of 2 1 0

algebraic expressions?
Are the algebra rules 4 2 0 Preservice teacher reactions to the
mathematically correct? materials indicated that the hands-on
3 examples of different cases 1 ½ 0 materials increased their interest in the work
included in each object?
(unanimous response of fifteen groups) and
Printout of PowerPoint card 1 ½ 0
that they found the sets of materials colorful
file included?
and attractive. This observation is important
Total Points out of 10
because Rule, Sobierajski, and Schell (2005)
As students worked with the showed that preservice teachers who viewed
example algebra rules object boxes, they hands-on materials for mathematics as
completed a questionnaire that asked about “beautiful” or “attractive” performed better
their reactions to the materials and their mathematically with the exercise as well as
difficulties during this initial exploration. The noting that they felt more motivated. Several
results of this survey were compiled and groups remarked that the materials helped
analyzed. Additionally, the final projects were them understand and figure out the concepts
examined for errors and trends. through manipulation, seeing patterns, and
focusing attention. That was the initial intent
Results and Discussion of the authors in creating these algebra rules
object boxes. Focus of attention on the
Questionnaire Responses activity through touching and manipulating
Groups were asked to complete a the objects is important because research
short survey the second day of working with shows that attention is more important than
the object boxes, which asked preservice time on task (Wittrock, 1986).
teachers to tell their initial reactions to the A few groups had an initial reaction
algebra rules object boxes and to describe of concern as to what to do with the materials
their difficulties in solving the algebraic because this approach to algebra was very
generalizations using the object boxes. new to them. Related to this fear are the
These were open-ended questions and the difficulties preservice teachers indicated they
resulting responses were tallied according to encountered while working with the example
general category of response. Table 2 shows object boxes. The group responses to this
reactions to the materials. open-ended question are shown in Table 3.
Although the students had done
some initial work with identifying and creating
patterns with color tiles and sets of printed
symbols before using the algebra rules
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object boxes, they were somewhat less related to squaring. Early on in the lessons
familiar with defining variables from story leading to this project, several preservice
problems. Therefore, they noted this difficulty teachers expressed their confusion in
in using the new materials. Some also had understanding what squaring meant. Many
trouble seeing the pattern in some sets of were helped by using square color tiles to
materials and interpreting the word problems. make squares of different sizes and therefore
Some preservice teachers were unfamiliar “see” square numbers as square shapes.
with or confused terms such as area, Two of the story problems with
volume, perimeter and diameter. The accompanying sets of objects preservice
instructor took time to review these terms for teachers devised for the equation “z = n2” did
them and rulers were provided so that not show a repeating square arrangement of
preservice teachers could measure the items (as other more successful sets made
objects rather than rely on estimation. by other groups did). Instead, preservice
These difficulties reveal the teachers attempted to fabricate a story of
incomplete knowledge base many preservice items being added that just happened to
elementary teachers have regarding work for the equation, but the scenario had
mathematics. Concrete activities, such as the no repetitive basis to allow other values for
object box described here, help preservice “n” to be substituted. This indicates that the
teachers build a stronger foundation that preservice teachers in these groups did not
supports more abstract reasoning. truly grasp the idea of squaring.
In two other projects, the equation “z
Table 3. Difficulties preservice teachers = 2n2” caused confusion with some
encountered in working with the example preservice teachers interpreting it a (2n)2
algebra rules object boxes. rather than 2 x n2.
Finally, the last mathematical error
Mathematical Difficulty Frequency occurred when preservice teachers
Difficult to define variables 13 misinterpreted the equation “z = 4n +3” as “z
Difficult to identify the pattern 7
=4n+n” with n = 3. The fact that they only
Interpreting the wording of the problem 5
supplied an object for n=3 and not a set of
Determining what is wanted 4
objects for n of different values probably
Understanding the terms: area, volume, 4
perimeter, diameter allowed them to overlook their mistake.
All of algebra is difficult 4
Table 4. Mean preservice teacher scores on
Estimating the inches without a ruler 3
different aspects of the project. Standard
Algebra Rules Projects Created by deviations are shown in parentheses.
Preservice Teachers
Criteria Mean Score
Table 4 shows mean scores on
Labeled box with 4 objects and 8
different aspects of the projects. In general, 1.0 (0.0)
cards?
most groups of preservice teachers produced Cards neat and durable? 0.95 (0.1)
quality algebra rules object boxes containing Set contains several types of
2.0 (0.0)
the required components. However, there algebraic expressions?
Are the algebra rules mathematically
were some mathematical errors made that 3.74 (0.4)
correct?
bear discussion because they shed light on
3 examples of different cases
the most difficult aspects of the project for included in each object? 0.92 (0.2)
preservice elementary teachers. Printout of PowerPoint card file
0.93 (0.3)
Four of the five group projects that included?
Total Points on Project out of 10 9.55 (0.5)
contained mathematical errors had errors
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Examples of Effective Materials objects that accompanied this story is shown
Preservice teachers created many in Figure 2.
sets of clever, creative, and effective
materials to illustrate algebraic Figure 2. Sets of basketballs in cages to
generalizations. Examples of their work are illustrate Z = n2.
shown in the following sections with
comments.

Z = n2. Although, as noted previously,
several groups had difficulty with squared
variables, many preservice teachers were
able to produce effective examples for z = n2.
Here are four examples.
Bowling pins. Boy Scouts are
bowling and are allowed to set up as many
Marbles in boxes. Bobby is storing
pins as they like as long as they are
marbles in compartmentalized boxes. Write a
arranged in a square. Write a rule for the
rule for the number of marbles each box can
pins in their games. The set of objects that
hold if each marble needs one square inch.
accompanied this story is shown in Figure 1.
The set of objects that accompanied this

story is shown in Figure 3.
Figure 1. Sets of bowling pins to illustrate Z =

n2.
Figure 3. Sets of marbles to illustrate Z = n2.

Basketballs in cages. Mr.

McNamara asks his gym students to place
Donut boxes. Daylight Donuts sells
basketballs into three different cages. The
their yummy product in square boxes. Write
number of basketballs that fit in a cage
a rule for the size of the boxes if different
depends on the size of the cage. Write a rule
numbers of donuts are placed flat inside. The
for the basketballs and cages. The set of
set of objects that accompanied this story is
shown in Figure 4.
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tissue. Write a rule for this. The set of objects
Figure 4. Sets of donuts in boxes to illustrate that accompanied this story is shown in
Z=n2. Figure 6.

Figure 6. Sets of toilet tissue rolls used to
illustrate Z = 2n.

Carnations. A florist uses three
purple carnations for every rose in flower

Bears in cages. A zookeeper is arrangements for a wedding. Write a rule for
cleaning bears’ habitat areas and needs to the number of carnations in arrangements of
put the bears in cages. If the number of different sizes. The set of objects that
bears that can fit in a temporary cage accompanied this story is shown in Figure 7.
depends on the area of the cage bottom,
Figure 7. Sets of flowers used to illustrate
write a rule for the number of bears a cage
Z=3n.
can hold. The set of objects that
accompanied this story is shown in Figure 5.

Figure 5. Bears in cages used to illustrate
Z=n2.

Watermelon seeds. Joe eats
watermelons all day but picks out the seeds.
If each slice of watermelon has 9 seeds,
write a rule for the number of seeds in
different numbers of slices. The set of
objects that accompanied this story is shown
in Figure 8.

Figure 8. Watermelon slices with seeds

illustrating Z = 9n.

Z = a n. All projects showing a generalization
of this form were done correctly, indicating
the familiarity of preservice teachers with
mathematical expressions for multiplication.
Some interesting examples follow.
Toilet tissue. A fancy hotel has a
strict policy for the number of toilet tissue
rolls provided for each guest. For every
guest, the maid supplies two rolls of toilet
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Propeller blades. Planes at an Z = an + b. Preservice teachers thought of
airplane show were arranged in groups of clever ways to illustrate equations of this
different sizes. Write a rule for the number of type. These are shown in the following
propeller blades in a group if all planes have sections.
four propeller blades. The set of objects that
accompanied this story is shown in Figure 9. Racing tires. A special race has
teams composed of cars with one
Figure 9. Propeller sets with blades to motorcycle. Write an equation to determine
illustrate Z = 4n. the total number of tires in each race. The
set of objects that accompanied this story is
shown in Figure 11.

Figure 11. Sets of vehicles used to illustrate
the “racing tires” problem with the equation
Z= 4n + 2.

Rollercoaster hills. Claire likes to
raise her hands when she rides over
rollercoaster hills. She wonders how many
times she would do this on different groups
of rollercoasters. Write a rule for this. The set

of objects that accompanied this story is
Farm animals. In the spring, each
shown in Figure 10.
animal mother at the farm gave birth to twins.

If one mother always watches over one or
Figure 10. Sets of rollercoasters, cleverly
more groups of twin babies, write a rule for
made from drinking straws, used to illustrate
different-sized groups. The set of objects that
Z=3n.
accompanied this story is shown in Figure

12.

Figure 12. Sets of animal groups to illustrate
Z=2n+1.