[Numeracy 366] Re: Missing Concepts

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Michael Gyori tesolmichael at yahoo.com
Wed May 12 12:15:48 EDT 2010

Susan raises a very important point.  Why the need to diagnose? This ties in with my severe reservations especially about high-stakes multiple-choice assessment tools. The wonder we can discover in the act of responding - beholding what goes on in the minds and hearts of our learners - is reduced to 4 or 5 distractors. 

I was delighted that my graduate school didn't require a thesis. Little did I know how stretched I would need to become to produce the required exit portfolio, a testimony of my learning.  Of course, there is much I would now change - but the fact that my work is alive in my mind far outweighs the ability to answer a few questions and then forgetting 90% of them.  In hindsight, a thesis would have been far easier to write and evaluate.

Michael A. Gyori
Maui International Language School

From: Susan Jones <SUJones at parkland.edu>
To: numeracy at nifl.gov
Sent: Wed, May 12, 2010 5:33:37 AM
Subject: [Numeracy 365] Re: Missing Concepts

I don't "use anything" for diagnostics, but it's one of the things I do
best.  The thing I look for is a response that shows the student is
making connections and comprehending the problem as if it were something
in words, or, if the student is prone to verbal ruts, then making sure
s/he can also visualize it. 
  Evaluating can be tricky, though many students experience delight
when that integration happens; it's the "it clicked!" phenomenon. 
I'm hoping to get a framework of these building blocks at the "making
math real" overview.  AFter that basic skill of being able to "see"
seven when you see the number seven (and I'm absolutely NOT a
visualizer, but something like it happens)  , which I do think is
missing more often than we realize, then place value is a big deal.
      While riding home last night I remembered www.learner.org, where
Annenberg Media has some pretty good math lessons and idea.  I'm going
to look at them again, because I think that these, too, incorporate
making a transition from concrete experience, through *student
language,* then math language, *then* symbolic language.  How much of
our instruction starts with a cursory whisper of math language, then
plunges into the symbols?

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

>>> "Andrew Isom" <isom at centerforliteracy.org> 5/12/2010 8:59 AM >>>

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

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,

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" con
cept. 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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