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 “passing” the placement exam (at least scoring high enough NOT to be placed in developmental (remedial) math). Four suggested courses, which are more useful than “physics or math for poets” courses:
A. A SOAR physics course’s 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 “real world” problems as well as “real world” opportunities for estimation and use of measurement. Better yet, include these topics in middle school science.
B. A SOAR math course’s 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: “Death Valley is −282 feet below sea level.” 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 course’s syllabus might also include:
Problem 5. (An advanced ratio problem). Suppose that 40% (by weight) of a county's trash is paper and 8% is plastic. If approximately 72 tons of the trash consists of paper and plastic, approximately how many tons of the trash consists of plastics?
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 “Pattern Recognition in Math Instruction” 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 students’ eyes would not glaze over, every time a professor mentions percents in a college sociology class.
Students should change 236 centimeters to 2.36 meters and 236 percent 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: " … 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, … '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 … 'understanding' means mastery of a sequence such as A then B then C then D … . It was as though in reading or hearing 'then' the student was understanding 'and'. … [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: " … 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 ‘then’ and ‘because’.”
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 “same“ 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.