[Numeracy 364] Re: Missing Concepts

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Andrew Isom isom at centerforliteracy.org
Wed May 12 09:59:50 EDT 2010

This is what I have been struggling with, and what motivated my request for a diagnostic that I posted a few weeks ago. I was trying to find something akin to what my colleague uses in literacy, where it breaks the skills down to the smallest constituent elements. I don't know of anything like this in numeracy. The most basic diagnostics that I know of just test basic computation.

I did, though, recently come across this site http://makingmathreal.org/

One of the "articles" (really, in actuality, a sales pitch,) states:

"One of the principle sensory-cognitive developments necessary for being automatic with the math facts is symbol imaging, the visuo-perceptual ability to perceive, hold, store, and retrieve sequences of numbers and/or mathematical symbols. Sensory-cognitive tools, such as symbol imaging, enable us to express what we know – they provide a direct conduit in both directions connecting processing to intelligence. Sensory-cognitive development for math refers to the specific ability of using the visual, auditory, and kinesthetic-motoric senses to engage and support the successful central processing of numerical and/or mathematical symbols. Students with under-developed sensory-cognitive abilities often have limited access to memory and are characteristically challenged by learning, retaining, and applying the math facts, recalling formulas and definitions, remembering the sequences and structure of multi-step problem solving, integrating concepts with their respective procedures, and managing all the details in their procedural work."

I'm inclined to believe this, because I feel that the root of the disjuncture in a student's understanding lies very deep. In the many years that I have been teaching math, I have repeatedly butted my head up against the tip of this cognitive iceberg. I explain the steps involved in whatever the problem requires in various ways, I use visual and kinesthetic activities, I make games involving the skill required so students can practice the skill, and yet I still get blank stares when they're confronted with the same type of problem that I had just felt they had acquired a few days prior. So, perhaps the "symbol imaging" is something that I'm wont to believe in just because I don't have any other answers. I'm also inclined toward cognitive explanations, but it's hard to crack the "black box" of the brain.

So my question is, is there a way to probe where the process breaks down at the most elementary level? So the "equal distance" concept could very well be a part of it. Is "symbol imaging" prior to that? It would seem so, right? What about viso-spatial perception? If this complex of skills is underdeveloped, then students cannot encode the information in the first place. Then there's the process of categorizing that information in order for it to be accessible. Just as chess experts can glance at a configuration of pieces on a board and have it memorized without little to no conscious effort because there's a whole cognitive system of concepts that are inter-related that can hold this information. I feel that a good analogy is that of a clean/organized room versus a messy/disorganized room. It seems that for some learners, when we hand them some piece of information or a skill, they toss it in the room on top of a heaping pile of undifferentiated stuff. So when they go back in to access it in the future, it has by then blended into the undiscernable heap.

Can we access this level, and even if we can, is there a way to improve these processes? Or is there something akin to Wernicke's and Broca's area of the cerebral cortex which has a "critical period" after which its development slows immensely? Just as people can't learn a language near as easily after age 8 or so.

Sorry to go on and on, but if anyone has hung on this far, has anyone out there used anything to diagnose learners other than a test with standard math problems on it, i.e., basic computation, words problems, etc.?

Andrew J. Isom
Math Specialist
Center For Literacy
North Philadelphia Community High School
isom at centerforliteracy.org
(215)744-6000 ext. 210


Message: 2
Date: Mon, 10 May 2010 10:56:40 -0500
From: awayman1 at kirksville.k12.mo.us
Subject: [Numeracy 358] Re: Missing concepts
To: numeracy at nifl.gov
Message-ID: <380-220105110155640552 at kirksville.k12.mo.us>
Content-Type: text/plain; charset=iso-8859-1

This idea that lack of math understanding comes from missing concepts
makes sense to me. If these are concepts "math people" have, and
even textbook writers assume that all adults have them, how can we
know what we need to teach for increased understanding? My questions
therefore are:

1. Where can I find an ordered list of the necessary math concepts
an adult needs to have to understand math?

2. Is there a test already developed to check for these necessary

In other words, how do we identify the problem so we can begin to
"fix" it?
Thank you for any help you can give.

---- Original Message ----
From: steinkedb at q.com
To: chip.burkitt at orderingchaos.com, numeracy at nifl.gov
Subject: [Numeracy 356] Re: Guest Presenters
Date: Sat, 8 May 2010 09:19:24 -0600

Thank you, Chip, for asking about "understanding" math.
If you want to know about understanding math you have to look at how
people think about number relationships. That means going back to -
does number sense develop in childhood? Why do some people "get it"
some don't? (More on that shortly.)

As adults, these people may understand part/whole relationships in
areas of life but may not apply them in math. These are the people we
see in our classes. They don't like math because it has never made
any sense
to them.

Can you teach adults those missing concepts [the ones textbooks
assume all
adults have]? Of course. However, first you have to identify which
concepts the adults are missing.

I am finishing up a student learning research project on the campus
where I
teach looking for the first of those concepts, which is the sense
that each
counting number is exactly the same-sized "1" more than the number
it. I call that concept the "equal distance" concept. It appears that
10% of the adults in the first two levels of our developmental math
[class 1: whole numbers, fractions and decimals; class 2:
pre-algebra] may
lack that "equal distance" concept. This is in a group of over 300
adult students.

As teachers we try to identify the concepts students are missing. Are
looking deep enough? Once we know what is missing, then it can be
Then people can develop "understanding" of number relationships.

Dorothea Steinke
Adjunct faculty, Front Range Community College, Westminster, CO
GED math/developmental math

Annell Wayman, Director
Kirksville Program
Adult Education and Literacy
1103 S. Cottage Grove
Kirksville, MO 63501
(660) 665-2865-phone
(660) 626-1477-fax
awayman1 at kirksville.k12.mo.us

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