assessing understanding through reading and writing in mathematics

8/13/2019 Assessing Understanding Through Reading and Writing in Mathematics

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Assessing Mathematics 1

Assessing Understanding Through Reading and Writing in Mathematics

Kwaku Adu-Gyamfi

Department of Mathematics, Science, and Technology Education

College of Education

East Carolina UniversityGreenville, C !"#$#

%!$!& '!#()'*)

adug+amfi-ecu.edu

Michael J. oss!Department of Mathematics, Science, and Technology Education


East Carolina University

Johna "aulconerDepartment of Curriculum and /nstruction


East Carolina University

Assessing Mathematics !

A#TRA$T

The mathematics education community recogni0es the integrality of reading and +riting in

learning and communicating mathematics no+ledge. Unfortunately, many students have yet to

significantly eperience this integrality in their mathematics classrooms despite the po+er these

tools offer teachers for assessing student no+ledge. This paper eplores the integrality of

reading and +riting in mathematics and outlines techni2ues that can 3e utili0ed in mathematics

assessment to create eperiences that promote reading and +riting as tools for articulating

mathematics understanding.


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Assessing Mathematics '

%ntroduction

The ational Council of the Teachers of Mathematics %CTM& ar gues that rather than 3eing

perceived as helpful add(ons to mathematics instruction, reading and +riting should 3e

recogni0ed as an integral part of mathematics learning %CTM, 1)#), !444&. CTM reflects a

3elief that, since reading and +riting necessitates the use of ver3al epressions, num3ers,

sym3olic epressions, and graphical representations, it is a crucial component in the

development of reasoning, communication and connections in mathematics. /n the Principles

and Standards for School Mathematics, CTM further assert that, 5Students +ho have

opportunities, encouragement, and support for speaing, +riting, reading, and listening in

mathematics classes reap dual 3enefits6 they communicate to learn mathematics, and they learn

to communicate mathematically.7 %CTM, !444 p. *4&. 8esearchers in the field of mathematics

education have, over the years, epressed similar sentiments.

9iancarosa and Sno+ %!44*& contend that reading is a central sill in life(long learning.

/n mathematics, no less so, reading is vie+ed as a vehicle through +hich mathematical tet and

contet are negotiated to construct mathematical no+ledge %9orasi : 9ro+n, 1)#$; 9orasi,

Siegel,


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Assessing Mathematics >

and +riting is thus recogni0ed as integral tools in developing and assessing mathematical

understanding %9ishop, 1)##&. Unfortunately, the integrality of reading and +riting in

mathematics have yet to 3e significantly eperienced in mathematics assessment; a crucial

component in the teaching and learning of mathematics.

Assessment

Assessment plays an integral role in mathematics teaching and learning. Some of the

roles identified 3y Croos %1)##& include, 3ut is not restricted to6 reactivating or consolidating

prere2uisite sills and no+ledge; focusing attention on important aspects of no+ledge; giving

students opportunities to practice and consolidate learning; providing no+ledge of results and

corrective feed3ac; influencing students choice of learning strategies and study patterns;

communicating and reinforcing the 3road goals of instruction and desired standards of

performance to students; and guiding the choice of further instructional or learning activities.

Assessment is thus a ey component to 3e addressed in order to ensure purposeful eperiences of

reading and +riting to learn mathematics.

CTM %!444& vie+s assessment as a component of instruction that 3est informs and

guides teachers as they mae instructional decisions and students as they mae @udgment of +hat

is important to learn and their approaches to personal study. The Assessment Standards for

School Mathematics %CTM, 1))$& recommends that classroom assessment should reflect the

mathematics that students should no+ and 3e a3le to do. CTM %!444& posits that assessment

tass can convey to students +hat inds of mathematical no+ledge and performance are valued.

Thus, for reading and +riting to 3e recogni0ed as essential tools in learning and articulating


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Assessing Mathematics $

understanding in mathematics, an assessment regime +here students and teachers have a shared

epectation that learning and articulating understanding through reading and +riting is a

+orth+hile goal. Unfortunately, assessment of mathematical understanding is 3ereft of

purposefully directed reading and +riting eperiences %Mars : Mousley, 1))4&; lacing in

assessments 3ased on reading and +riting

$urrent &ractices of Reading and Writing in Mathematics

Although it may 3e argued that students are continually assessed in mathematics classrooms

through reading and +riting, much of the current occurrences of reading and +riting are not

purposefully directed and as a result serve limited roles %Mars : Mousley, 1))4&.


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Assessing Mathematics *

understanding.

'eed for &ur(osefully )irected Assessment Acti*ities

8eading mathematics is comple and non(linear. /t often re2uires 3ouncing repeatedly

from tet to ta3les to diagrams to sym3olic epressions to graphs and such %


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Assessing Mathematics "

develop assessment materials purposefully geared to+ard +riting to learn mathematics.

Moreover, since standardi0ed test 2uestions are 3ecoming increasingly open(ended,

re2uiring students to read, understand the 2uestion, and then compose responses, the practice of

assessment in mathematics classroom needs to improve in order to 3e compati3le +ith, and to

support and reinforce eperiences +here students read and +riting to articulate understanding.

That is, classroom assessment needs to change to purposefully afford students opportunity to

communicate their understanding via reading and +riting. Similarly, the content of assessment

needs to improve to include items that re2uire students to reflect on ideas, formulate definitions,

read, and epress ideas 3oth orally and in +riting and to communicate their thining %Shield :

Gal3raith, 1))#&. Unfortunately, all too fe+ resources and techni2ues have 3een provided to

teachers to employ in their classes. /n the su3se2uent sections, techni2ues are outlined for

improving classroom assessment through reading and +riting mathematics.

Techni+ues for %m(ro*ing $lassroom Assessment through Reading and Writing in

Mathematics

The premise underlying this article is that in order for reading and +riting to 3e purposeful in a

student s mathematical eperience, a classroom culture +here students read and +rite to

articulate mathematics understanding may 3e needed. Berein, numerous assessment techni2ues

are outlined that teachers can incorporate in assessment to simulate eperiences that promote

reading and +riting as essential tools in learning and articulating mathematics understanding.

These are separated into t+o types6 Aptitude Assessment and Content Assessment.

The former assesses students involvement, interest, and engagement in mathematics.


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Assessing Mathematics #

The latter assesses student s conceptual understanding and content no+ledge. Most of these

assessments have the purpose of delving more deeply into student understanding and revealing

far more in respect to conceptual %mis&understanding +hile promoting reading and +riting as

integral tools in mathematics. Through reading, interpreting, and +riting, each of these

techni2ues forces students to investigate, reason, perform, and or report as they interact +ith

mathematical concepts. Ans+ers to these assessments are never single(step operations

demonstrating one single sill at a time; rather, these assessments al+ays necessitate the student

summoning and revealing more of their understanding in the process of responding to the

2uestionprompt.

Aptitude Assessment Techniques

Aptitude Assessments consider students involvement, interest, and engagement in

mathematics +ithout divorcing such from simultaneously investigating students conceptual

understanding. Some of the ones identified in the literatur e include@ournal +riting %9orasi :

8ose, 1)#); =ay+ood, 1))!; Clare, =ay+ood, : Stephens, 1))'&, epository +riting %enne,

1)#)&, proof +riting, and re+riting lecture notes %Sipa, 1))4&. These can 3e regularly assigned

in the classroom. Eamples of these are provided 3elo+.

Peer Evaluation. Students can 3e ased to mae +ritten @udgment a3out a peer s

performance on a given mathematical tass or 2uestion. This techni2ue assesses a student s

a3ility to understand mathematics produced 3y another and communicate mathematically

regarding such. Thus, as colla3oration is promoted in student +or, so too can assessment 3e a

vehicle for colla3oration. Altogether, the teacher can use the results of this activity to assess the


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Assessing Mathematics )

understanding of 3oth students involved and students gain practice +ith guidance at reading,

2uestioning, and interpreting their o+n ideas and the ideas of their peers.

Two Stage Test. After students receive feed3ac from teachers on tests and assignments,

students can provide +ritten responses to teacher comments. This method of assessment provides

information to 3oth students and teachers %via +riting& on ho+ +ell a particular topic or content

has 3een learned or understood and the area of difficulty and affords students the opportunity to

relearn and rearticulate their understanding 3ased on the feed3ac given.

Portfolios. ?ortfolios have long 3een promoted as an assessment tool to gain insight into

students learning. ?ortfolios afford students the opportunity to longitudinally document their

understanding and progress throughout the school year through purposefully selected artifacts of

their +or and +ritten @ustifications of +hy the artifacts +ere selected to represent their

understanding.

Journal Writing

. Fournaling can 3e utili0ed to prompt students to +rite a3out a specific

concept or rule or to eplain ho+ to perform a mathematical procedur e or to eplain +hy a given

mathematical outcome occurs. /t can also 3e utili0ed as a means for students to record reflections

on materials learned in class, reactions to readings or lectures or responses to open ended

assignments. Fournals can then 3e used to informally assess student understanding of the

included mathematical concepts.

Content Assessment Techniques

Content Assessment techni2ues employ reading, interpreting and +riting to loo deeply into

student understanding. Many of the types of eamples outlined in here can 3e perceived as more

difficult than traditional test 2uestions. This occurs for a num3er of reasons.


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these 2uestions is dissimilar from traditional test items. Second, all too infre2uently students

eperience open(ended 2uestions and even rarer are 2uestions +hich seemingly as the student

to state all he can a3out the mathematical situation 3efore him. Third, some of these 2uestion

types necessitate heuristics +hich employ multiple strategies and the use of multiple

simultaneous representations.


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may 3e incorporated in mathematics assessment.

' $ 4

c!Eample !. "iven that eplain for +hich value%s& of cthis +ill,

have; no real roots; one real root; t+o real roots.

Eample #. Eplain mathematically +hat this diagram demonstrates.

1.!'>!'>!'>!'>1.!'>$!'>$!'>$

1.!'> 1.!'>$ !.>*#"$"*#*$")Eample $.Eplain the follo+ing.


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Assessing Mathematics 1!

Eample %.Tell all you can a3out the polynomial function +hich +ill produce

the accompanying graph off

%&.

As a component of assessment, the investigate and eplain techni2ue furnishes teachers +ith

insight into students understanding +hile promoting reading and +riting as essential tools in the

classroom %Arm3ruster, 1))*&. 9y affording students a chance to unpac these tets and

diagrams, this techni2ue also allo+s students to articulate their understanding in many different

+ays.

&reate 'our Eample

This techni2ue serves the purpose of allo+ing a student to 3oth analy0e a mathematical concept

and to epress the concept in a creative manner +hich demonstrates her understanding of the

concept. The follo+ing eamples serve to illustrate +ays in +hich ideas from this techni2ue can

3e incorporated in assessment.


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Assessing Mathematics 1'

Eample !. Create a real +orld pro3lemscenario +hich +ould use or demonstrate

the concepts in the follo+ing statement.


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Assessing Mathematics 1>

student regarding the underlying concept.

*efinition + or Theorem Altering

This techni2ue focuses on assessing students mastery of 3oth voca3ular y and concepts via their

articulations on purposefully and precisely altered definitions and theorems. The follo+ing

eamples serve to illustrate +ays in +hich ideas from this techni2ue can 3e incorporated in

assessment.

Eample 1. Discuss the validity of this statement6. A function is continuous on the closed

interval a, 3H if it is continuous at each point in the interval.

Eample !. Discuss the validity of this statement6 /f f and hare functions that are

continuous at I c then their 2uotient function is also continuous at c.

The *efinition + Theorem Altering

techni2ue melds suggestions regarding formal epository

mathematical +riting +ith creative mathematics +riting assignments %Alvermann : Moore,

1))1; 9orasi : 8ose, 1)#); Clare, =ay+ood, : Stephens, 1))'; Sipa, 1))4; enne, 1)#);

=ay+ood, 1))!&. Since mathematical understanding is strongly correlated to fluency, through

this techni2ue, teachers can 3egin to assess much regarding student understanding of theorems

they read. % Earp : Tanner, 1)#4; Bel+ig, 8o0e(Tedesco, Tindal, Beath : Almond, 1))); Stahl

:


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Assessing Mathematics 1$

appreciation for the precision 3y +hich mathematics is +ritten.

&onnect epresentations.

This techni2ue is similar to the setch(to(stretch strategy proposed 3y 9orasi et al %1))#&, +here

students develop nonlinguistic representations, and ela3orate on such, of mathematical concepts

discovered in mathematical tets. The follo+ing eamples serve to illustrate +ays in +hich ideas

from this techni2ue can 3e incorporated in assessment.

Eample !. Use tet and pictures to eplain the concept, 5/f f

%& is a polynomial

of degree n, +ith nI 1, then f

%& can 3e epressed as a product of linear factors

in the follo+ing +ay6 f

%& I a%( c &%

( c & %

( c &, +here c , c , , c are1 !

n 1 ! n

comple num3ers and ais the leading coefficient of f

%

&. That is, every comple

polynomial function of degree nI 1 has eactly n%not necessarily distinct&

0eros.7

Eample #. Dra+ a pictorial representation of the theorem6 The points of a line

can 3e placed in a correspondence +ith the real num3ers such that; %1& to every

point of the line there corresponds eactly one real num3er; %!& to every real

num3er there corresponds eactly one point of the line; and %'& The distance

3et+een t+o distinct points is the a3solute value of the difference of the

corresponding real num3ers. ,rEvery line can 3e made into an eact copy of the

real num3er line using a 1(1 correspondence.


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Assessing Mathematics 1*

Eample $. Using 3oth tet and diagrams, eplain the follo+ing theorem6 Jn an

interval, given any nK1 points in the form % ,) & in +hich iI 4, 1, !, ',L, n, and

i i

each is uni2ue, there eists a polynomial function of degree no greater than n

i

that maps through the nK1 points +ithin the interval.

Eample %. =rite the theorem depicted 3y the diagram 3elo+

As +ith other types of assessment techni2ues demonstr ated in this paper, &onnect

epresentationsemploys the process of reading, interpreting and +riting mathematics. Another

version of this techni2ues is provided 3y 8eadence, 9ean, and 9ald+in %!441&, +here students

discuss ver3al concepts through pictures and diagr ams. /t is said that +hen studentsconnect

representations in order to eplain mathematical concepts, their +or and eplanations reveal

much regarding their understanding of the representation at hand %CTM, !444&. Since students

are not told precisely ho+ to mae the connections, 3oth the product and the process are

revelatory of student understanding. Conse2uently as a tool for assessment, &onnecting


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Assessing Mathematics 1"

epresentationsprovide teachers +ith the opportunity to assess student understanding of each

dimension of this process6 during the reading of the representation, the interpretation of the

representation, or the creation of other eplanator y r epresentations.

$onclusion

Teachers can deeply assess student mathematical understanding +hen assessment strategies

employ reading and +riting components. There are multiple assessment techni2ues to promote

successful integration of reading and +riting in mathematics and as mathematics teachers, +e

should eplore these and share our successes. Berein, a fe+ such assessment techni2ues are

outlined 3ased on their emergence in the literature. /t is hoped that these eamples +ill generate

greater interest in assessing mathematics through reading and +riting and +ill encourage others

to develop additional assessment methods.


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Assessing Mathematics 1#

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Assessing Mathematics 1)

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Assessing Mathematics !4

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Assessing Mathematics !1

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assessing understanding through reading and writing in mathematics

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