**Stress On Analytical Reasoning (SOAR) **

**by**** Jerome Dancis**

The big
bugaboo for high school and college students is Arithmetic and Algebraic** **word
problems. The big bugaboo for calculus students is calculus
word problems. But, solving
Arithmetic and Algebraic** **word
problems, with a stress on analytical reasoning (SOAR), develops basic life
skills of reasoning and analysis.
It is valuable everywhere, both in quantitative settings and in daily
life. Colleges should require very
basic math and physics courses with a stress on analytical reasoning; the
prerequisite would be ÒpassingÓ the placement exam (at least scoring high
enough NOT to be placed in developmental (remedial) math). Four suggested courses, which are
more useful than Òphysics or math for poetsÓ courses:

**A.** A SOAR physics courseÕs syllabus would
include basic background in energy and power, speed and relative speed, density
and pressure, and measurement; all topics
which appear in newspapers. This would
be an Arithmetic and Algebra I-based science course. This will provide Òreal worldÓ problems as well as Òreal
worldÓ opportunities for estimation and use of measurement. *Better*
yet, include these topics in middle school science.

**B.** A SOAR math courseÕs syllabus might
include topics in

(*) non-trivial, multi-step Algebraic word problems, with Stress On
Analytical Reasoning (SOAR). as well as the
Algebra word problems common to American high school Algebra books a half
century ago.

(*) Arithmetic-level
statistics, including knowledge and understanding of averages,
medians, percentiles, box and whisker diagrams; also being able to read and
draw graphs, charts and tables as well as proficiency with percents and
decimals.

**C.** A SOAR math
course in logical reasoning and deductive logic might include

(*) basic theory
of sets (Venn Diagrams and Boolean Algebra) .

(*) a
deductive-proof-of-a-theorem each day (like the high school Euclidean geometry
course of a half century ago)

(*) mathematical
induction.

**D.** A SOAR Algebraic word problem course for
potential STEM majors might include setting up the Algebraic formula for the maximum-minimum word-problems in
the calculus textbook. (*No* calculus
required. It is likely that many such problems
would be appropriate for an Algebra II class.)

These courses
should increase success in college freshmen
math and science courses as well as an increase in the quantitative reasoning
level of freshmen courses.

Of course, college choice
as to whether these courses are for credit or remedial.

These courses should be
required; they might be given in modular form, so students would *only* take the modules they need. These courses would fade away if and
when the topics become integrated into the Grade 5-11 curriculum.

**All
these courses should provide instruction in reading comprehension and following
directions.**

**Notes **for these SOAR courses:

A math or science SOAR
course might start with:

**Problem 1.**
It is a fact that fat has 9 calories per gram and protein has 4 calories
per gram. If a piece of meat
consists of 100 grams of protein and 10 grams of fat, how many calories does it
have altogether? (Answer:
490 calories; not a trick question)

College students receive
instruction on how to do Problem 1 in an elementary nutrition course on my
campus, (with selective admissions).

**Problem 2 on Speed**.
(**A medium level SAT problem**)
"How many __MINUTES__ are required for a car to go 10 miles at a constant speed of 60 miles per
hour?"

Students who *cannot* do this problem will be *at-risk* in a rigorous high school
physics course.

**Problem
3****.** (**Average Speed**)
We flew from Denver to Boston at an average speed of 500 MPH; we returned from
Boston to Denver at an average speed of 400 MPH. What was our average speed for this round-trip? WRONG answer. 450 MPH

**Problem 4.**
(**Catch-up and Overtake**) As
the clock strikes noon, Jogger J is 2500 yards and Walker W is 4000 yards down
the road (from here). Jogger J jogs at the constant pace of 5 yard/sec. Walker
W walks at the constant pace of 3 yard/sec. How long will it take Jogger J to *catch up* to Walker
B?

Students should understand how to
do Problem 4, *quickly* and *mentally*, in the manner of high school
physics class): First, find the
relative speed: Jogger J is gaining at a rate of 5 yard/sec - 2 yard/sec = 3 yard/sec. Now, Walker W starts out 1500 yards ahead. Jogger J will catch up to Walker W in 1500 yards/3 yard/sec
= 500 sec.

Students should *unlearn*: ÒDeath Valley is −282
feet *below* sea level.Ó This is common in middle school math textbooks. The correct statement in mathematics
and in physics, as well as common English, is that Death Valley is 282 feet below sea level. (It is also
correct in mathematics and physics to write that its altitude is −282
feet, or that it is −282 feet above sea level.) Conflicts between common English usage
and textbook mathematics must be confusing to students.

A SOAR math courseÕs
syllabus might also include:

**Problem 5. ** **(An advanced
ratio problem).**

by observing three
instances. Unfortunately too often
this Pattern Recognition is only valid for some cases. This results in many answers being
marked WRONG. See ÒPattern
Recognition in Math InstructionÓ at www.math.umd.edu/~jnd/Patterns.pdf

**STEM Standard**. Students can solve this (open ended) problem:

**Problem** 6. Find two polynomials, P(n), such that,

P(1) =
1, P(2) =
2, P(3) =
3, P(4) =
4, P(5) =
5,
BUT P(6) NOT = 6.

**Sample solution**: P(n) = n +
(n-1) (n-2) (n-3) (n-4) (n-5) and

P*(n) = n - (n-1) (n-2) (n-3) (n-4) (n-5)

**Percents**. Common Core math includes about two
weeks on percents, easily forgotten by the time students enter high
school. It does *not* even include memorizing that 50% equals a half or being able to do
mentally:

**Problem
7**. Newspaper writes that
14% of elementary school teachers are male. This means that about one in how many are male?

Then studentsÕ eyes would not glaze
over, every time a professor mentions percents in a college sociology class.

Students should change
236 __cent__imeters to 2.36 meters and 236 per__cent__ to 2.36
as fast as they change 236 __cents__ to $2.36.

From
"Reading Instruction for
Arithmetic Word Problems:

If Johnny can't read well and follow
directions, then he can't do math"

(It is at www.math.umd.edu/~jnd/subhome/Reading_Instruction.htm):

A study by the National Center
for Education Statistics noted: " É far fewer [Americans] are leaving higher education
with the skills needed to comprehend routine data, such as reading a table
about the relationship between blood pressure and physical activity, É 'What's
disturbing is that the assessment is [designed] ... to test your ability to read labels,' [commissioner of
education statistics] ... ."

**Converses**. (Understanding the "*one-way*"
significance of implication words.) Ezra Shahn wrote: "In descriptions of
many biological phenomena É 'understanding' means mastery of a sequence such as
A then B then C then D É . It was as though in reading
or hearing 'then' the student was understanding 'and'.
É [But] the sequential relationship is more restrictive, hence more precise and
it is this distinction that many students apparently fail to grasp." Shahn
also wrote: " É it seems that students often misread conjunctions
[including the implication words 'because' and 'then' (as in 'A then B')] so
that they mean 'and'. " Arnold
Arons elaborated: "Crucial to understanding
scientific reasoning and explanation [in beginning physics classes] as opposed
to recall of isolated technical terms, resides in the use of [implication
words] words such as ÔthenÕ and ÔbecauseÕ.Ó

Students should learn the
difference between a statement and its converse and should not expect one to
imply the other. Students should know that a contrapositive
is equivalent to the original statement. This is useful both inside and outside
of math.

A SOAR math course in
logical reasoning and deductive logic might include the basics of geometric
vectors. This includes defining equality of geometric vectors (parallel lines
of equal length, with arrowhead on ÒsameÒ end) and presenting the Distributive
Rule [for scalar products],

a(v + w)
= av
+ aw, when a
is a number and v and w are geometric
vectors. Also, a
proof of this Distributive Rule, preferably, one using similar triangles -- which is an elegant (and simple)
interplay of geometry and algebra.

For
STEM majors, include dot products (which connects
geometric vectors with trigonometry.
This provides the background for the SOAR science course to present the
dot product formula for work (W), due to a constant force (F), namely, W
= F . v).

Also,
include the Distributive Rule for dot products,

u . (v + w) =
u . v
+ u . w, when u, v and w are geometric
vectors. Together with a geometric
proof, which again is an elegant interplay of geometry and algebra. These can be combined to show that the
work going from (point)
A to B
plus the work going from
B to C
equals the work going from
A to C.