MathPASS: A Remedial Mathematics System with Automated Answer Checking Wei Su 1 Paul S.Wang 2 Lian Li 1 Department of Computer Science, Lanzhou University, PRC Department of Computer Science, Kent State University, USA July, 2010 on CICM’10

Wei Su Lanzhou University Outline  MathPASS Overview  Question Prototype and MAML  Answer Inputting and MathEdit  Answer Checking and MACP  User Statistics

Wei Su Lanzhou University 1. Overview  MathPASS is a Web-based intelligent mathematics assessment system  A drill-and-practice system which has been applied in remedial mathematics courses at KSU.

Wei Su Lanzhou University Taking Assignment Page in MathPASS

Wei Su Lanzhou University System Modules of MathPASS User Management User Management core Student Management Account Setting Account Setting Assignment Management Assignment Management Class Management Class Management Question Management Question Management Practice Grade Management Grade Management Questions Generation Questions Generation Answer Checking Answer Checking Answer Inputting

Wei Su Lanzhou University The System Architecture of MathPASS

Wei Su Lanzhou University 2. Question Generation  The questions in MathPASS could be generated automatically by question prototypes on each time when students take the assignment.  A question prototype is a mathematical question template for a specific question pattern containing random parameters satisfying well-defined mathematical conditions.  More than 400 question prototypes have been created in MathPASS by the combined efforts of mathematics teachers and mathematics education experts.

Wei Su Lanzhou University Question Prototypes and Instances

Wei Su Lanzhou University The table also gives the statistic result including taken times, correct times, and the correct ratio of the question. All of the statistic data can be accessed by a MathPASS teacher user who may use these data in preparing his/her course.

Wei Su Lanzhou University MAML-Mathematical Assessment Markup Language  Paper (homework, quiz, examination) elements  Question elements  Question Prototype elements  Description elements  Presentation elements

Wei Su Lanzhou University An example of MAML for Question Prototype A B C B SF SF

Wei Su Lanzhou University 3. Answer Inputting and MathEdit  An intuitive and convenient mathematical expression input tool is very important for a mathematical CAA system to enter answer.  MathEdit, a Web-based visual interactive mathematical expression editor, aims to provide an integrated solution of on-line entering and editing of mathematical expressions.

Wei Su Lanzhou University MathEdit Overview  MathEdit provides convenient and intuitive graphical user interface.  MathML, OpenMath, LaTex, Infix, Picture (Gif, Png), PDF could be generated by MathEdit.  It also provides well-defined API.  It can work with IE and the Mozilla-family browsers  Written in JavaScript and MathML.  or

Wei Su Lanzhou University MathEdit Authoring Environment

Wei Su Lanzhou University The architecture of MathEdit

Wei Su Lanzhou University Some Examples of Math expression  Demo Demo

Wei Su Lanzhou University MathEdit Infix Version

Wei Su Lanzhou University MathEdit Usage Examples

Wei Su Lanzhou University 4. Answer Checking and MACP  Overview of MACP  Data Format of MACP  Answer Grade Rule  Full Simplification Form  Special Syntax Form

Wei Su Lanzhou University 4. 1 Overview of MACP  Mathematical Answer Checking Protocol (MACP) is an access protocol for communication between MACP service and its client.  MACP service is a Web service for checking answers and grading questions to mathematics of middle school and college.  The MACP service aims to grade the user answer through –verifying expressions equivalence –checking the expression form of the answer

Wei Su Lanzhou University Overview of MACP  The standard answer could be provided by the Computer- aided Assessment (CAA) system which is using MACP or automatically generated and computed by the MACP service.  The implementation of MACP is based on Representational State Transfer (REST).  The request data and service response will be encapsulated into JSON code.  In internal, MACP service use computer algebra systems (Maxima) to verify the expression equivalence and compute the correct answer.

Wei Su Lanzhou University A usage example of MACP MathEdit Answer Grade Program CAS Question Generator CAA PHP Proxy MACP Service Web Server MACP Client Browser AJAX POST JSON String MACP Request MACP Response

Wei Su Lanzhou University 4.2 DATA FORMAT OF MACP [{"name":" ", "cmd":" ", "response":" ", "exprs":[{ "expr":" ", "encoding":" ", "role":" ", "value":" ", "form":" ", "op":" ", "var":" " },...]

Wei Su Lanzhou University The Response Data [{ "name":" ", "status":" ", "error":" ", "correctness":" ", "response":[{ "rname":" ", "value":" " },...]

Wei Su Lanzhou University Example 2x 2 + 8x + 7 = 0 [{"name":"c1", "cmd":"(u1=s1&u2=s2)|(u1=s2&u2=s1)", "request":["ranme":"u1.mmlp","rname":"u2.mmlp"], "exprs":[ { "expr":"s1","encoding":"mmlc","role":"stdAnswer", "value":" "}, { "expr":"s2","encoding":"mmlc","role":"stdAnswer", "value":" "}, { "expr":"u1","encoding":"infix","role":"usrAnswer","value":"(-4-sqrt(2))/2"}, { "expr":"u2","encoding":"infix","role":"usrAnswer","value":"sqrt(8)/2-2"} ] }]

Wei Su Lanzhou University Response Data of the Example [{ "name":"c1", "status":"normal", "correctness":"96%", "details":" is not simplest", "response":[ { "rname":"u1.mmlp", "value":" "}, { "rname":"u2.mmlp", "value":" " }]

Wei Su Lanzhou University 4.3 Answer Grade Rule  When can we say an answer is ”right”? –The answer which student entered can be computed with the standard answer by CAS. Obviously the answer is wrong if the computation result is not equivalence in algebra. However we cannot easily say the answer is right even though they are equivalence.

Wei Su Lanzhou University Answer Grade Rule  Hundreds of s from students to report that the answer they entered is correct but the system grade it wrong.  On the other hand, some mathematical teachers also report that some student answers should be treated as ”wrong” or ”partial right” while the system grade them right.

Wei Su Lanzhou University Classification  Most correct answer or best answer be classified into two broad categories: –simplest form –special syntax form  This taxonomy does not separate strictly all the answers into two totally different categories. One part of the answer with special syntax form may also be required as a simplest form.  However this taxonomy has been found to be useful for evaluating the answer of students and solving the issues of partial credit.

Wei Su Lanzhou University 4.4 Full Simplification Form  The question of this category included: –simplification, –calculation, –writing the equation of a function, –writing the domain of expression, –…  Exmaples:

Wei Su Lanzhou University Rule-based Grade Method Judge the Algebraic Equivalence via CAS Convert to Canonical Form Judge whether it is a simplest form Return Result

Wei Su Lanzhou University Canonical Form  Rewrite Rule 1 (Remove Redundant Parentheses)  Rewrite Rule 2 (Commutative Rule of Addition and Multiplication)  Rewrite Rule 3 (Multiplication of Fractions)  Rewrite Rule 4 (Representation of Radical)  The expression will not be simplified during the canonical process and the length and complexity of the expression were not changed during the rewriting.

Wei Su Lanzhou University Simplest forms of a non-symbolic constant expression  Simplest Natural Number (SN)  Simplest Integer (SI)  Simplest Fraction (SF)  Simplest Decimal Number (SD)  Simplest Exponent/Radical (SE)

Wei Su Lanzhou University Checking Rule A  A non-symbolic constant expression is in simplest form when it satisfies one of the following conditions: –It is one of the SN, SI, SD, SF, SE; –If a non-symbolic constant expression is the addition or minus of one SE number and one number of SN, SI, SD, SF, the expression is a simplest form expression. –If a non-symbolic constant expression is the multiplication of one SE number and one number of SN, SI, SD, or SF, where the base of the SE number is different with the other number, the expression is a simplest form expression. –If a non-symbolic constant expression is the dividing of one SE number and one number of SN, SI, or SD, where the base of the SE number is different with the other number, the expression is a simplest form expression.

Wei Su Lanzhou University Simplest Primary

Wei Su Lanzhou University Term  In elementary mathematics, a term is either a single SP, or the product of several SPs separated from another term by a + or - sign in an overall expression. For example, in 3*9+4*x+5*y*z*w, the 3*9, 4*x, and 5*y*z*w are all terms.  A term can be written in form C*S*V where C is a non-symbolic expression, S is a symbolic expression and V is a variable expression.

Wei Su Lanzhou University Checking Rule B  A term of an expression is in simplest form when it satisfies the following conditions: –each variable or symbolic constant only occurs once; –the non-symbolic part is in simplest form.

Wei Su Lanzhou University Checking Rule C  A polynomial expression is in simplest form when it satisfies the following conditions: – each term of the polynomial expression is in simplest form; –the variable parts are different for each two terms of the polynomial.

Wei Su Lanzhou University 4.5 Special Syntax Form  Students need write the answer expression as an appointed syntax form such as multiplication of polynomials, a sum or difference of logarithms.  This kind of answer does not need to be reduced to lowest terms. Contrarily, the question may be lowest terms and obviously the answer should not be same with the question.

Wei Su Lanzhou University Examples  The questions of this category include –factoring, –solving equation, –rewriting the expression as request, –find the function as request, –…  Example: Factor Completely, leaving only positive exponents in your answer

Wei Su Lanzhou University Special Syntax Form  Factoring Polynomial (FP)  Normalized Scientific Notation (NSN)  Sum and Difference of Logarithm (SDL)  Positive Exponent (PE)  …  The user may use logical combination of these special form as a request form. For example a client may send SF|SI to statement the answer could be a simplest fraction or a simplest integer.

Wei Su Lanzhou University Pattern Language for Mathematical Expression  Mathematical Pattern Language provides a concise and flexible means to represent classes of expressions.  A simple example of a mathematical pattern is the expression 2+x^$. This mathematical pattern represents the class of expressions with the form 2+x^[any exponent]. –Regular Expressions--> String –Mathematical Pattern Language--> Mathematical expression

Wei Su Lanzhou University An Example of Mathematical Pattern Language sin$1v+cos$1v sinx+cosx sina+cosa sin2+cos2 sinx+cosy

Wei Su Lanzhou University 5. Statistics for MathPASS NameNumbersNameNumbers Teachers103Students6204 Courses547Assignments3731 Taken Assignments Taken Questions

Wei Su Lanzhou University 6. Future Work  Improve the MACP and answer grade rule.  Add the geometry question in MathPASS.

Wei Su Lanzhou University Thank you ！