There are five content standards which represent the mathematics skills that should be taught in each grade from kindergarten to twelfth grade:
number and operations
algebra
geometry
measurement
data analysis and probability
6.
Assessing Math
In addition, there are five process standards which highlight the ways that students acquire and use content knowledge:
problem solving
reasoning and proof
communication
connections
representation
7.
Development of Math Skills
Hierarchies or lists of math concepts and skills that are useful in planning specific interventions are provided by Heddens and Speer (1997) and Stein, Kinder, Silbert, and Carnine (2006). Appendix A includes a scope and sequence skills list that shows a math hierarchy by skill area as well as a hierarchy of what commonly is stressed at each grade level. Although the hierarchy stops at sixth grade, these skills also apply to many adolescents with learning problems because their problems usually involve skills taught in the elementary grades.
8.
Development of Math Skills (cont’d)
Basically, the hierarchy in Appendix A indicates the following skill introduction sequence:
(1) addition and subtraction—first and second grades
(2) multiplication and division—third and fourth grades
(3) fractions—fourth and fifth grades
(4) decimals and percentages—fifth and sixth grades
9.
Readiness for Number Instruction
Several concepts are basic to understanding numbers:
classification
ordering and seriation
one-to-one correspondence
conservation
Mastering these concepts is necessary for learning higher-order math skills.
10.
Readiness for More Advanced Mathematics
Once formal math instruction begins, students must master operations and basic axioms to acquire skills in computation and problem solving.
Operations:
addition
subtraction
multiplication
division
11.
Readiness for More Advanced Mathematics (cont’d)
Some axioms that are especially important for teaching math skills to students with learning problems are:
commutative property of addition
commutative property of multiplication
associative property of addition and multiplication
distributive property of multiplication over addition
inverse operations for addition and multiplication
12.
Assessment Considerations
Examining Math Errors:
Computation
Problem Solving
Determining Level of Understanding:
Concrete
Semiconcrete
Abstract
13.
Assessment Considerations (cont’d)
Many students with learning problems produce accurate answers at a slow rate and use tedious procedures (such as counting on fingers and drawing tallies for large numbers) to compute answers without understanding the math concept or operation. Slow rates of computation are a primary problem of many students with math disabilities.
14.
Formal Math Assessment Achievement Tests
Achievement or survey tests cover a broad range of math skills and are designed to provide an estimate of the student’s general level of achievement. They yield a single score, which is compared with standardized norms and converted into standard scores or a grade- or age-equivalent score. These tests help identify students who need further assessment.
Kaufman Test of Educational Achievement — II (Kaufman & Kaufman, 2004). Assesses concepts, applications, numerical reasoning, and computation.
15.
Formal Math Assessment Achievement Tests (cont’d)
Metropolitan Achievement Tests—Eighth Edition (2000a). Assesses mathematics computation and concepts and problem solving.
Peabody Individual Achievement Test—Revised (Markwardt, 1998). Assesses knowledge and application of math concepts and facts.
Woodcock-Johnson III Normative Update Tests of Achievement (Woodcock, McGrew & Mather, 2007). Assesses calculation, math fluency, and applied problems.
16.
Formal Math Assessment Diagnostic Tests
In contrast to achievement tests, diagnostic tests usually cover a narrower range of content and are designed to assess the student’s performance in math skill areas. Diagnostic tests aim to determine the student’s strengths and weaknesses. No one diagnostic test assesses all mathematical difficulties. The examiner must decide on the purpose of the assessment and select the test that is most suited to the task.
17.
Formal Math Assessment Diagnostic Tests (cont’d)
Comprehensive Mathematical Abilities Test
KeyMath—3: Diagnostic Assessment
Test of Early Mathematics Ability—3
Test of Mathematical Abilities—2
18.
Formal Math Assessment Criterion-Referenced Test
Standardized tests compare one individual’s score with norms, which generally does not help diagnose the student’s math difficulties. However, criterion-referenced tests, which describe the student’s performance in terms of criteria for specific skills, are more suited to assessing specific difficulties.
Brigance Comprehensive Inventory of Basic Skills—II
19.
Informal Math Assessment
Informal assessment involves examining the student’s daily work samples or administering teacher-constructed tests. Informal assessment is essential for the frequent monitoring of student progress and for making relevant teaching decisions regarding individual students.
20.
Informal Math Assessment (cont’d)
Curriculum-Based Measurement
Teacher Constructed Tests
Assessment at the Concrete, Semiconcrete, and Abstract Levels
Diagnostic Math Interviews
21.
Informal Math Assessment: Curriculum-Based Measurement
When progress is assessed within the curriculum to measure achievement, the teacher is assured that what is being assessed is what is being taught. CBM offers the teacher a standardized set of informal assessment procedures for conducting a reliable and valid assessment of a student’s achievement within the math curriculum.
22.
Curriculum-Based Measurement (cont’d)
CBM begins by assessing an entire class with a survey test of a span of appropriate skills.
5 required steps in developing and administering a survey test:
Identify a sequence of successive skills included in the school curriculum.
Select a span of math skills to be assessed.
Construct or select items for each skill within the range selected.
Administer and score the survey test.
Display the results in a box plot, interpret the results, and plan instruction.
23.
Teacher-Constructed Tests
Teacher-constructed tests are essential for individualizing math instruction. The type of test the teacher selects depends, in part, on the purpose of the assessment. To identify specific problem areas, the teacher may construct a survey test with items at several levels of difficulty.
24.
Teacher-Constructed Tests (cont’d)
4 steps to developing and using survey test:
Select a hierarchy that includes the content area to be assessed.
Decide on the span of skills that needs to be evaluated.
Construct items for each skills within the range selected.
Score the test and interpret the student’s performance.
25.
Assessment at the Concrete, Semiconcrete, and Abstract Levels
Most published texts consist of abstract-level items; therefore, they do not yield information on the student’s understanding at the semiconcrete and concrete levels. The student’s level of understanding determines whether manipulative, pictorial, or abstract experiences are appropriate.
26.
Assessment at the Concrete, Semiconcrete, and Abstract Levels (cont’d)
To obtain the type of information required for effective instructional planning, the teacher can construct analytical tests that focus on both identifying difficulties and determining level of understanding.
Items at the concrete level involve real objects.
Items at the semiconcrete level use pictures or tallies.
Items at the abstract level use numerals.
27.
Diagnostic Math Interviews
The diagnostic math interview provides information to determine what math skills to teach the student and how to teach them.
In this technique, the student expresses thought processes while solving math problems.
This technique is often used in administering diagnostic math tests.
28.
Periodic and Continuous Math Assessments
Periodic assessment includes initial testing that generates instructional objectives.
Periodic evaluations include checkups of general progress and in-depth evaluations of students experiencing difficulties.
Continuous assessment focuses on monitoring the student’s progress. It involves daily, weekly, or biweekly assessments.