Assessing Mathematics Pre-requisite Skills
By Lisa Zack-Swasey, M.Ed., SAIF
When a student can’t read, we test for reading readiness and acquired skills by
administering tests of phonemic awareness, sound-symbol correlation, decoding, reading
comprehension, reading fluency, listening comprehension, auditory discrimination, and a
host of other tests which will help pinpoint the reasons that the student has not learned to
read. But when testing for problems with mathematics, we often rely on the
standardized mathematics subtests alone to determine why a student has difficulty
learning mathematical concepts. We analyze the mistakes made on these subtests and
determine that a student can’t add with regrouping or has trouble computing fractions.
But why can’t the student perform the math computations? What is causing the reversal
problems, difficulty with place value, or weak fact mastery? What about math pre-skills,
cognitive readiness and number sense? What part do these skills play in the acquisition
of mathematics?
There are many reasons for math difficulties. Sometimes learning disabilities,
which manifest in other academic areas, also affect math ability, such as reading
difficulty or memory deficits. Mathematics has a language of its own, with its own
syntax; a student has to “translate” symbols into English and then back again. Math
anxiety may hinder math acquisition; some students have a true math disability. A student
may not have acquired the prerequisite skills needed for math understanding or might
lack cognitive readiness or number sense—s/he might not have the foundation skills
needed to learn arithmetic and mathematics.
Can a student who has difficulty sequencing, follow order of operations? Can a
student with visual-spatial problems tell the difference in size and location of a number
and its exponent or discern geometric shapes which are similar? Can a child with no
estimation skills “catch” his/her calculation errors? Can a student who has weak spatial
orientation/space organization tell the difference between 23, 32, or 2/3? Without the
ability to use deduction or induction a student cannot determine or apply mathematical
processes or formulae. Although these skills are called mathematical “pre-requisite
skills”, they are skills needed to compute math at all levels of complexity. Pre-requisite
skills become support skills as math skills develop.
There are at least 8 mathematics pre-requisite skills which are worth exploring in any
assessment of mathematics skills:
1. Sequencing—this skill is necessary for working with numberlines, math charts, place
value, multi-step math processing, counting forward and backward, following order
of operations.
2. Spatial orientation/space organization- the ability to know relative and absolute
positions of objects; parts-to-whole and whole-to-parts (the value of a number.) Also
needed for geometry and graphing.
3. Pattern recognition and its extension—this is the ability to see and recognize
patterns, a skill that is necessary for memorizing fact families, times tables,
multiplication and division facts, exponential growth, developing math formulae,
understanding numerical relationships.
4. Visualization—the ability to hold and manipulate data in the mind’s eye; this helps
with long term memory and ability to work on one step of the problem while
remembering the steps done before, or picture an easier problem in order to compute
a more complicated one, including signed numbers, algebraic expressions, equations,
algebraic word problems involving geometry and trigonometry.
5. Estimation—the child should estimate before a mathematical procedure to know what
kind of answer to expect (Is this a reasonable answer?); needed for metacognition.
6. Inductive Reasoning—taking lots of examples and arriving at a general rule or
principle. The student examines particular cases, identifies relationships among these
cases and generalizes these relationships. (What is next in a pattern is an example of
inductive reasoning.)
7. Deductive Reasoning—going from a general principal and applying it to specific
cases. Students solve problems containing clues in which they have to draw
conclusions from facts presented in words, function machines, diagrams graphs,
charts or tables; also needed for understanding of rudiments of symbolic logic.
8. Classification—the ability to sort and group by attribute/s. This is necessary for place
value, data collection, graphing, classfifying geometric shapes, completing Venn
Diagrams.
Mahesh C. Sharma, Factors Responsible for Mathematics Learning. Center for
Teaching/Learning of Mathematics. Cambridge, MA, 2003.
Assessing Math Pre-requisite Skills
A traditional assessment battery, which includes a cognitive abilities test and
achievement test, can assess some of these prerequisite skills. An evaluator can assess
sequencing ability by analyzing any of the subtests which test for it. Tests, which assess
fluid reasoning, might give insight into a student’s ability to reason deductively or
inductively. Visual memory tests might offer insight into a student’s ability to visualize.
If the evaluator is aware of the prerequisite skills needed for mathematics, s/he can use
findings from these standardized subtests to help identify weak math areas.
Listed below are some informal “tests” which an evaluator can also use to assess
prerequisite skills needed for mathematics.
Patterning, Sequencing, Math Fact Mastery, Fluency
An easy way to assess pattern recognition and sequencing, as well as math fact
mastery and fluency, is to have your student count forward and backward starting at an
arbitrary number. For example, younger students might start counting by 2s starting at
17—17, 19, 21, 23……….39. Then have the student count backwards by the same
number starting at the number on which s/he ended—39, 37, 35………17. If the student
has difficulty starting the pattern, the evaluator can provide every other number until the
counting is more fluid. Younger students can count forward and backward by any whole
number. If preschoolers or kindergartners cannot count, then assess pattern recognition
by placing a series of objects and asking the child to add to that series, e.g., dime, nickel,
dime, nickel, dime, nickel…or use a series of colored beads, e.g., red, green, blue, blue,
red, green, blue, blue, red, green, blue, blue… If the student has difficulty with the pattern
presented, or if the pattern is too easy, then adjust the level of difficulty so that useful
information regarding skills can be gained from the task.
Counting with older students—start by having the student count forward and back by a
whole number. You can start with an arbitrary number or
start at 0 to assess whether the student knows the multiplication facts, e.g., 0, 9, 18, 27,
36……… or 0, 4, 8, 12, 16, 20…
Assess knowledge of addition and subtraction facts by starting at an arbitrary
number, e.g., “Count by 5s starting at 22” –22, 27, 32, 37, 42….. (Can the student add by
5s?) Have the student count backward by the same number beginning at the last number
reached when counting forward. Counting backward helps to assess a student’s
subtraction facts, as well as, ability to reverse thinking.
To assess pattern recognition, ask, “Do you notice a pattern?” “Yes, every
other number in the one’s place repeats.” 22, 27, 32, 37…..
Intermediate students can count backward into the negative numbers, e.g., 17, 14,
11, 8, 5, 2, -1, -4………
Middle and high school students—test students with a set of whole numbers, e.g.,
“Count up by 4s starting at 17”--17, 21, 25, 29, 33………”Now count backwards by 4s
starting at____ (the number ended when counting up).”
If the student is able to easily count forward and backward with whole numbers,
test with fractions and/or decimals, e.g., “Count by fourths starting with 15—15, 15 1/4,
15 2/4, 15 3/4, 16, 16 1/4…… or “Count forward by two-tenths starting at 17—17,
17.2, 17.4, 17.6, 17.8, 18……… Notice whether the student is able to use equivalent
fractions, e.g., Does the student say 5 2/4 or 5 1/2? Does the student easily count the
whole number or does s/he have to say the fraction (or decimal) and then rename as a
whole number, e.g., 15 4/4 or 16?
A child who has difficulty counting forward might have trouble visualizing a
numberline or number chart, might have weak number sense, or might have had difficulty
memorizing math facts. Inability to count backward might indicate weak number sense,
weak subtraction facts or immature cognitive development. Children who are in the pre-
operational stage of cognitive development have difficulty reversing their thinking.
Difficulty counting might also substantiate weak math fluency.
See if the student notices a pattern as s/he counts—Do numbers repeat? Do the
numbers end in even numbers? Odd numbers? Do they skip a number? Skip 2 numbers?
If a student is a visual learner, then the number pattern can be written as the student
counts and the child may be able to recognize the visual pattern presented.
If you have the student start counting at 0, you can assess pattern recognition of
multiplication facts. Can the student count by 3s? 5s? 10s? 9s? 7s? Does the student
count up on fingers, or have the facts memorized? Does this task give you information
regarding fluency?
If an older student counts by fractions or decimals, do they convert the whole,
e.g., 14 1/4, 14 2/4, 14 3/4. 14 4/4 or 15? What insight gain you can in regard to the
student’s number sense?
Inductive and Deductive Reasoning, Sequencing
The Quantitative Reasoning subtest of the Differential Abilities Scales (DAS)
provide excellent insight into a student’s ability to sequence, as well as, inductive
reasoning. The easier level of this subtest requires the student to complete a series of
numbers following a specified sequence. The more difficult items provide a series of
number pairs, and the student must provide the missing numbers after determining the
relationship, or rule, between the pairs provided. If the DAS is not used to assess
cognitive ability, an informal assessment can be used. Present the student with sequences
such as: 5, 10, 15, 20, 25 (add 5 each time)….. or 100, 90, 80, 70, 60 (subtract 10 each
time)….. or 1, 0, 2, 0, 3, 0, 4, 0 (add one, then zero in between)….and ask the student to
extend the sequence and tell you the rule (how they did it.) If these are too difficult,
present sequences which increase or decrease by one or two. You might present older
students with sequences, such as, 1, 4, 9, 16, 25 (series of square numbers)…. or 3, 5, 8,
13, 21 (add the two previous numbers)… or 3, 6, 18, 72, 360 (multiply two previous
numbers)…..or 2, 4, 8, 16, 32 (double the previous number).
Test for deductive and inductive reasoning by giving students a “function
machine” or providing a rule and a number and asking the student to complete the set.
For example, Rule: add 5, input is 11, output is ? (16); input is 22, output is ? (27). Rule:
subtract 3, input is 30, output is ? (27); input is 14, output is ? (11). Rule: multiply by
10, input is 3, output is ? (30); input is 15, output is ? (150). To test for inductive
reasoning, give a series of numbers and ask for the rule or the relationship, e.g., 11, 22 -8, 16 -- 10, 20 Rule: ? (multiply by 2 or double the number.) Adjust the level of
difficulty for the age and grade of the student. Use fractions, decimals, and negative
numbers, or exponential numbers for older students.
Estimation
Estimation is a skill which is important for metacognition. However, while
mathematically-able students use estimation while completing math computations,
students who have difficulty with math probably do not use estimation as a math tool.
Although an evaluator cannot ask a student to estimate while completing the math
calculations subtests during the administration of standardized achievement tests, s/he
might ask a student to estimate the answer to similar problems after the standardized test
is administered. “Test the limits” by asking the student to estimate the answer to
problems which were answered incorrectly. Evaluators should not assume that all
students use estimation as a tool on a regular basis, but rather consider it a discrete lesson
to be learned in the math curriculum. Students who cannot estimate might also have
difficulty reading numbers, with place value and rounding numbers. These skills might
be worth assessing if the student cannot estimate.
Visualization
Although tests of visual memory can be good indicators of a student’s ability to
remember what they see, they might not be a good indication of ability to visualize a
mathematical concept. As an evaluator assess mental math computations, ask students,
“How did you do that? What did you see? Did you see a mental whiteboard or piece of
paper with numbers? Did you draw a picture in your head?” Notice whether the student
uses paper and pencil for math applications or word problems presented during the math
achievement tests. After the standardized testing takes place, while working on informal
tasks, ask a student to draw a picture to represent a word problem. Observation during the
math computations subtests of the achievement battery can also reveal whether or not a
student has the ability to visualize.
Spatial Orientation/Space Organization
The Rey-Osterreith Complex Figure Test (RCFT) can give good information in
regard to a student’s visual-spatial skills. Students who have difficulty with this test
might have difficulty perceiving and/or drawing geometric figure, perceiving place value
of digits, understanding that computation problems laid out horizontally are the same as
those placed in vertical arrangements, or counting the number of sides of an illustrated 3D figure. If the RCFT is not used, ask a student to draw some simple geometric shapes,
tell which of two numbers is larger, both number containing similar digits, e.g., Compare
326,174 to 362,147 or choose numbers with different place values and see if the student
can tell the difference, e.g., 146,274,098 and 16,498,247. Can a student read a number
in the millions (or billions), but cannot read it when given a number with a decimal? Can
they tell you which digit is in the hundreds place? ten thousands place?
Visual Clustering (Visual Imagery)
For younger students, assess visual clustering by quickly showing a student a
cluster of dots or small objects which are arranged in configurations similar to dots
represented on the sides of a die or a domino. If the student can tell the number in the
cluster without counting the dots or objects, the evaluator will know that s/he has
“numberness” of the single-digit numbers shown. This quick assessment confirms
whether the student has a feeling for the numerosity of the group, and therefore a good
understanding of number concepts which will be needed for arithmetic operations.
Classification
For younger students, I use Attribute Blocks and ask the child to sort by one
attribute (color, thickness, size, shape). Then I ask them to sort by two attributes or more.
Older students can sort numbers: even and odd, prime and composite, perfect squares,
number of digits, greater or less than a given number. The number 49, for example, is
odd, a perfect square, composite, less than 50/100, is more than 10/25 has a 4 in the tens
place, is a multiple of 7. Children need to understand that numbers and shapes can have
more than one name. For those students who are working in the Everyday Math
program, give the child a Name Collection Box to see how many names they can give for
a given number. (This “routine” uses numbers, addition/subtraction/multiplication/
division combinations, tally marks, coin combinations, words, expanded notation, and a
variety of different names for numbers.)
Summary: Evaluators can help determine the cause of math difficulties by assessing
math prerequisite and support skills (sequencing, spatial orientation/space organization,
pattern recognition, estimation, visualization, deductive and inductive reasoning.
Assessing the student’s ability to count forward and backward will give the evaluator
insight into number sense, math fluency, facility with addition, subtraction, and
multiplication, visualization, pattern recognition and sequencing, and deductive/inductive
reasoning. Informal observations and questions during or after achievement testing can
also help to reveal the development of math prerequisite skills. These pre-requisite skills
become support skills as math concepts develop, so if a student has difficulty with
mathematics, these skills are worth assessing for students, no matter what the age or
grade level. Recommendations for addressing weaknesses in these skill areas will be
valuable input into the evaluation team meeting process and will ensure mathematical
progress for the student who needs math support.
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