LOOKING FORWARD TO Making Math Real: Geometry
“Just the Facts, Ma’am”: In Geometry, Assumptions Are Not Welcome: If It’s Not a Fact, We Say Nothing; We Write Nothing
Imagine a classroom in which the teacher asks a question and a student is jumping out of her seat, frantically waving her raised hand in the unmistakable gesture of “call on me, please!” This affective behavior indicates a student who irrevocably knows that she knows the correct solution. The experience of certainty, “knowing that you know”, is the definition of successful mathematics integration. Guiding students to the awareness of knowing that they know prior to expressing the solutions is the goal and objective of all math education. This “knowing” is deeply integrated in the body – not merely remembering what to do. This depth of body integration is another way to feel and express mastery. When we have learned to mastery, that learning represents the highest level of proficiency, and as such, the learning is fluently and automatically activated when needed, and our ability to sustain that learning is maintained at a high degree of accuracy and consistency.
Math is Surgery
Do you want your surgeon to possess mastery or to guess what to do? Just as in the midst of an operation you would not want your surgeon guessing what to do, so too, in the midst of mathematical operations you do not want to be guessing what to do. Math provides us with specific tools for problem solving applications, and our knowledge of how to use these tools enables us to be precise, direct, and effective with our problem solving. In terms of surgery, I would want my surgeon to be precise, direct, and effective in knowing where to aim those sharp tools… Like surgery, math is never guessing.
Knowing That You Know
Frequently in algebra, the solution to given problems may be a simplified or transformed version of an equation or expression. The solution may not be a conveniently cogent numerical value from which one could easily the check the solution for accuracy. For example, how do we expediently check the following solution for accuracy?
If solutions turn out to be a transformed version of the original equation rather than easily recognizable numerical values, then how do we know with confidence our solution is correct without having to do cumbersome checking of the solutions? Answer: we know our solution is correct because throughout the algebraic problem solving of transforming to equal equations we can sustain the certainty of “knowing that we know” each new transformation is accurate and correct. Knowing that we know before expressing the solution requires the development of a strong executive function directly supporting the ability to sustain a robust working memory picture throughout problem solving. The development of a strong executive function that monitors and regulates a robust working memory picture is how we learn to manage the details in out procedural work (detail analysis), and is also the necessary development all of us need to be successful as independent adults: manage the details – it is how we keep getting a paycheck.
Geometry Deepens What Algebra 1 Starts
As mentioned in the previous installments of this series, from a developmental perspective, the principal cognitive demand of algebra is detail analysis: activating the perceptual working memory picture from the interrelationship of the symbolic details while managing the details throughout the procedural work. In algebra, well-developed detail analysis helps us to sustain knowing that we know so we can effectively manage all of the numerous details comprising extensive algebraic transformation-based problem solving. The developmental outcome is students’ success with simplifying expressions and solving equations because they are able to sustain managing the details and therefore, maintain knowing that they know throughout problem solving.
Geometry introduces a perceptual, structural code that provides factual certainty. The provision of this code of factual certainty deepens the integration of students knowing that they know, because geometry, within all of its problem solving, requires certainty before expressing any statement, solution, or conclusion. Using language and/or physical markings on figures, the codes of geometry tell us explicitly that lines are perpendicular, planes are parallel or two angles are congruent. Applications of geometry also help to string together two or more facts to find new facts, such as two angles are congruent because they are vertical angles; and two lines are perpendicular because the angle they form at their intersection is 90˚. It is never correct to assume lines are perpendicular just because they appear to be in the figure provided; or that angles are right angles, because to the eyes, they look like 90˚ angles. Geometry is not the mathematics of appearances and assumptions, only facts.
For this purpose, it is essential that teachers require students to consistently extend given information, e.g., given language, relationships, markings, etc., to fully “mark up” figures to indicate all known factual information. If there are two intersecting lines forming vertical angles, students need to mark the congruent vertical angles on the figure, not merely hold that congruence in mind. If students are given that the two lines cut by a transversal are parallel, students need to mark the congruent alternate interior angles; and if students are given that on triangles ABC and DEF, sides AB and DE are congruent, students need to mark respective corresponding sides to show that congruence. By fully marking up figures, students are now equipped with multiple facts from which they can begin to string together individual facts to generate new facts. Furthermore, all of the facts marked on the figures hold still on the page, which reduces cognitive load significantly, because the increased cognitive demand of mentally holding multiple visual-spatial facts simultaneously can lead to cognitive overload, which in turn may cause students to lose or weaken their working memory picture resulting in a higher frequency of errors and missed relationship connections.
Developing Deductive Proof
The ability to string together two or more facts is the basis of developing deductive proof. For example, students can prove two triangles are congruent because they can connect and justify a series of individual facts that lead to a specific, certain, and factual conclusion: two triangles are congruent because each of the two triangles has two corresponding sides of equal length and each has an included angle of the same degree measure. When two triangles share congruent corresponding sides and an included angle they make a specific, factual relationship called side-angle-side; and side-angle-side provides sufficient factual basis to be certain that the two triangles are congruent. The progression of connecting and justifying a connected series of individual facts to a culminating factual conclusion is the basis and developmental purpose of geometry. By requiring students to exclusively express, justify, and connect each fact until culminating in a factual conclusion, geometry provides the maximum development of “knowing that you know prior to expressing the solution.”
Geometry Prepares Students for “Court”
Imagine a trial attorney who does not manage the details, assumes facts not in evidence, and asks questions for which the answers are a surprise. Not only does this lead to disasters in the courtroom, but this same lack of preparation also leads to breakdowns for students in math classrooms across the grades, nationwide. Math students are not preparing for a case in court, but they may be preparing for something equally as challenging – the standardized tests. The well-known high school and college entrance exams are more based on measuring executive function development than math content knowledge, which may explain why the test questions mostly do not resemble the content students encounter in their math classrooms, and why these tests contain so many “trick and trap” questions. The ability to successfully navigate the minefield of the math sections on these standardized tests requires a highly developed executive function to sustain a strong math-based working memory picture rather than succumb to the seductive tricks and traps set by the test makers.
One of the major ways the tests attempt to trick the test-taker is to present a perceptual conflict between the mathematical facts and how the eyes perceive the problem. For example, a test problem may present a misshapen quadrilateral in which clearly none of the sides are parallel or of equal length and all of the angles are clearly a combination of acute and obtuse angles – no angles remotely resemble right angles. However, this misshapen quadrilateral has two opposite angles in which the mark indicating right angles has been placed. In the remaining two angles, there are indicators of angle bisectors in each in which each bisected angle has “x˚” and “y˚”, respectively. The question for students to solve is: x˚ + y˚ = ? Lastly, the problem has the ominous statement: “figure not drawn to scale” as an appropriate and necessary inclusion for solving the problem.
The answer to this problem is x˚ + y˚ = 90˚, because, despite the visual impossibility that this misshapen quadrilateral contains right angles, it is nonetheless a rectangle because of the factually marked indicators that two opposite angles are right angles, and half of each right angle is 45˚, and 45˚ + 45˚ = 90˚. The test makers set up a nefarious conflict to determine which processing tool will override the other – the executive processing tool that uses math-based factual knowledge (I know that I know) the figure is in fact a rectangle, despite the powerful visual distractor of the misshapen quadrilateral; or the visual system overrides the executive because there is no way that misshapen quadrilateral could ever be a rectangle, despite the inclusion of “figure not drawn to scale”.
Why do these tests attempt to trick students, because as confusing as math classes have become in our nation’s public and private schools, it is not the specific intent of the curricula or the teachers to present problems to trick students in this way? Perhaps the purpose of these trick questions is to measure students’ executive function development, because admissions teams understand, the higher the executive development, the more likely students will be successful, not only at their schools, but across the board in all of their endeavors for now and on in the future, because a well-developed executive is highly correlated to success in life as an independent adult.
Geometry, the Math of Certainty, Continues the Ongoing K-12 Development of Executive Function
The ongoing developmental outcome of having students engage the executive to “prepare for court” to exclusively express, justify, and connect each fact until culminating in a factual conclusion provides profound development for the integration of “I know that I know prior to expressing the solution”. Therefore, it is imperative in teaching geometry to students across the grades, we structure all geometry lessons to require factual basis, not assumption; and that deductive reasoning requires mathematical knowledge to override the visual distractor of what the eyes may assume to be facts. Any visual information unsupported by facts cannot be determined until factual information is given:
Developing “Knowing That You Know Prior to Expressing the Solution” is Necessary for Success in the Higher-Level Math Classes
As stated in the third installment of this series, “The increased cognitive demands of algebra II, pre-calculus, and calculus require the developmental ability to simultaneously access analysis and synthesis abilities, frequently within the same problem solving tasks.” The simultaneous blending and shifting of analysis and synthesis abilities is a high-level executive function that benefits significantly from the ongoing experiences and integration of knowing that you know. The students’ developmental abilities to hold their working memory picture with certainty makes it possible for them to simultaneously shift back and forth between analysis and synthesis processing without experiencing cognitive overload – an essential developmental foundation for success in higher mathematics.
I look forward to seeing all of you in the first ever Making Math Real: Geometry.