[Numeracy 581] Re: Cognitive levels on the GED math section and its implications for instruction

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ROBERT G STEINKE steinkedb at q.com
Thu Oct 14 11:15:31 EDT 2010

Mark and all:

There have been other posts back to you, Mark, on your comments. I'm
starting from where you left off hoping others will be able to follow this

Your point about Procedural versus Conceptual questions on the GED gets to
the heart of the matter: What CONCEPTS are our adult students missing? There
are two fundamental concepts of number relationship that textbooks assume
everyone grasps by 3rd grade. If our adult students missed one or both of
those, they can do procedures until kingdom come and not be able to
understand the relationship of the numbers in word problems. I am talking
about normal, intelligent adults. What they lack is a sense of the physical
relationship of numbers. This seems to have gotten blocked at an early age
and they have have gotten by in math on rote skills ever since.

I say all this based on a model of number sense development in young
children that came out of the University of Georgia in the 1980s (Les Steffe
and his colleagues, including Paul Cobb). As a teacher of GED math and first
level developmental math at a community college, I find the Steffe-Cobb
model of the 3 Stages of number sense in children applies to adults as well.
The behaviors I mention below were observed by Steffe and Cobb in young
children. I find these same behaviors in the adults I teach.

What adults may be missing is: 1) the physical sense that each whole number
is exactly one more than the number before it and one less than the number
after it. These are the adults who, when subtracting, are always off by 1
because they count the digits rather than the distance between the digits.
For example, a person missing this concept will subtract 11 - 4 by counting
down and saying "11 10 9 8" and give the answer as 8 because he/she has
counted four numbers. These people may have a lot of computational skills,
but they have no sense of WHY they are doing what they are doing. They just
do it.

The second missing concept (and the more frequently missing one) is: 2) the
sense that a number (for example, 9) and all its parts (3 and 6 or 2 + 2+ 2
+ 2 + 1) exist within and at the same time as the larger number. For
students lacking this concept, when I add 4 + 7 it turns into 11 and the 4
and 7 disappear. I only have the 11. If I don't know that the parts and the
whole exist at the same time, how can I understand the relationship of
numerator (parts) to denominator (whole) in fractions?

I wrote an article for Focus on Basics about this a couple years ago. In the
article I also tell how I get started with students to make them aware that
they have to change their perception of number relationships from this
"either - or" thinking about parts and whole to "both - and" thinking about
number relationships. Here's the link.


I invite you to read the article and ask questions either here or at my
school e-mail: dorothea.steinke at frontrange.edu

Dorothea Steinke
adjunct math faculty
Front Range Community College, Westminster, CO

-----Original Message-----
From: numeracy-bounces at lincs.ed.gov
[mailto:numeracy-bounces at lincs.ed.gov]On Behalf Of
Mark.Trushkowsky at mail.cuny.edu
Sent: Wednesday, October 13, 2010 2:41 PM
To: The Math and Numeracy Discussion List
Subject: [Numeracy 575] Cognitive levels on the GED math section and its
implications for instruction

I am interested in hearing what people are thinking about the kinds of
questions our students face on the GED.

Often when people talk about what is on the GED, and what students need to
learn, and what teachers need to learn, the conversation remains limited to
content areas - things like, "students struggle with the pythagorean theorem

I am looking for ways to work with teachers in exploring the model of the
three cognitive levels that the GED uses to categorize the math questions.
According to the GED Testing Service,the test "assesses different ways of
applying math skills through the use of different question types". The
three types are based on Bloom's taxaonomy and divided up as follows - 20%
of the questions are Procedural questions, 30% are Conceptual questions and
50% are Application/Modeling/Problem Solving questions.

To read a more detailed description of each cognitive level, check out:

To go back to my earlier example, I don't think it is a procedural issue,
when students get questions dealing with the relationship between the legs
and hypotenuse in a right triangle. Students who are drilled with examples
of "A squared plus B squared equals C squared" start to make mistakes when
the GED asks a question which tests students' conceptual or problem-solving
levels of understanding. So for example, students will often do "A squared
plus B squared equals C squared", even when given the hypotenuse and one of
the legs and asked to find the other leg. They do this because their
understanding it only at a procedural level. Unfortunately 80% of the GED
math is trying to assess if they understand math concepts at a deeper level.

Many teachers who focus on content only, also focus (often without
realizing) on procedural type questions. So for example, for students
working on area, they will work on problems where they are given the sides
of a rectangle and asked to find the area. To me, a question like this is
procedural because it only requires students to "select and apply the
correct operation or procedure to solve a problem".

But 25 out of 50 math questions are going to be
Application/Modeling/Problem-Solving - add 15 more questions at a Conceptual
level and that is 40 (out of 50 questions that require a deeper than
procedural level of understanding. Students are far more likely to face a
question like: "One of the legs of a right triangle is two feet longer than
the other leg. If the area of the triangle is 24 square inches, what is the
length of the hypotenuse?" No one can prepare for that question on
procedure alone. On a procedural level, students need to know how to find
the area of a triangle (and knowing how to find the area of a rectangle
would probably help as well), as well as how to caluclate the hypotenuse of
a right triangle when given the legs. But on a deeper, conceptual &
problem-solving level, students need to understand why they are given the
area, and how to use it. They also need to understand the relationship
between the area of a rectangle and the area of a triangle - if they don't,
they might think the legs are "4 feet" and "2 feet". And they'll need to
know how to develop a strategy to go about this problem, to implement that
strategy and to gauge whether their strategy is working.

To not help our students learn how to do math on a level deeper than
procedure is to leave many of them unequipped to do better than guess for
80% of the test.

How are other people working with teachers to help them feel more
comfortable and capable developing these deeper cognitive skills in their


Mark Trushkowsky
Mathematics Staff Developer
CUNY Adult Literacy and GED Program
101 W. 31st Street, 7th Floor
New York, NY 10001
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