[Numeracy 585] Re: Cognitive levels on the GED math section and its implications for instruction

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Susan Jones SUJones at parkland.edu
Mon Oct 18 14:18:35 EDT 2010

I see a lot of useful practices that involve students discussion problems, and I think it can work if the students manage to make that connection between the concepts, what we call them, and the problems and procedures.

I'm thinking, after reading the article, that a more guided and structured way to make sure they get the concept they're missing is quite possible. I agree that things seem to ahve been "blocked" from developing as any of a number of compensatory rituals have replaced mathematical thinking.

I'm going to be pondering ways of melding the "part whole" discussion with the progression I learned about at a workshop this summer. It was being applied to basic place value, but essentially students worked with concrete stuff to figure out the math idea, then learned what it was called and practiced calling it that. Then they learned the symbols for that idea, *and* what they were called -- instead of being happy with the correct scribblings. THen they were walked back through the process of using verbal language and the concrete *and* the math symbols and practicing being able to move back and forth between all three domains.

Only slightly tangentially, though, was that I also appreciated the storytelling element of good teaching. I remember my seventh grade math teacher teaching us the transitive property in terms of love... it was a giggle but never forgotten. Just this morning, an algebra student here told us that when he's explaining to little guys why fractions need common denominators, he tells 'em "to add fractions, they have to have the same kinds of shoes. High Heels can't be added to work boots." No, it's not very conceptual, but it *does* bring in a critical visual-kinesthetic element and does deliver the concept that you should look at things you might not normally notice if you want to get it right, just as if you were Sherlock Holmes, you'd notice things like shoes, but otherwise you might forget to look.

That connects to the idea taht "make it look right" and "make it feel right" can be used as an asset instead of a liability if we teach the *right* important stuff to look for and feel. (Changing a mixed number to an improper fraction is often a motor memory action.)

> 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

> thread.


> 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.


> http://www.ncsall.net/fileadmin/resources/fob/2008/fob_9a.pdf



> 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

> questions".


> 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:

> http://www.acenet.edu/Content/NavigationMenu/ged/etp/math_test_descriptio.htm


> 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

> students?


> Mark


> Mark Trushkowsky

> Mathematics Staff Developer

> CUNY Adult Literacy and GED Program

> 101 W. 31st Street, 7th Floor

> New York, NY 10001

> 646-344-7301



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Susan Jones
Academic Development Specialist
Center for Academic Success
Parkland College
Champaign, IL 61821
sujones at parkland.edu