1.2. Essential, Unit & Entry-level Questions

"The most savage controversies are those about matters as to which there is no good evidence either way."--Bertrand Russell

About Essential, Unit & Entry-Level Questions

Wiggins & McTighe's framework for "understanding by design" suggests using essential, unit and entry-level questions to structure curriculum design. Essential questions have the following characteristics: they "go to the heart of a discipline, recur naturally throughout one's learning and in the history of a field," and "raise other important questions" (Wiggins & McTighe 1998). Unit questions "provide subject- and topic-specific doorways to essential questions, have no one obvious 'right' answer," and "are deliberately framed to provoke and sustain student interest" (Wiggins & McTighe 1998). They are necessary to provide a lens and organizing principles for the essential questions.

Finally, entry-level questions provide "concrete and meaningful experiences, problem, applications, and shifts of perspective to enable an important question to arise" (Wiggins & McTighe 1998). These questions give students an immediate handle by which they can begin grappling with ideas that will eventually lead toward the unit and essential questions. While they may be simple on the surface, they have the potential for expansion into more advanced topics.

Using this framework, questions that suggest themselves include the following:

Essential Question #1

How does mathematics represent and shape the world around us?

In "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Wigner noted that mathematical paradigms bear an uncanny resemblance to the workings of our world. Coordinates and dimension are one area where this comes into play unexpectedly. Not only can we use coordinates to describe a chessboard's battlefield or the layout of a city, we can use them to construct "new" shapes, such as hypersphere, that may turn out to have unexpected applications; after all, non-Euclidean geometry turned out to be important in describing space-time in Einstein's theory of general relativity.

In a broader sense, people use mathematics to construct the world around them. The history of mathematics is embedded in people's concerns and curiosity, whether everyday or exotic, from the use of Pythagorean triples to demarcate right angles in Egyptian architecture to Escher's playful depictions of hyperbolic geometries. This essential question suggests an answer to the ubiquitous question, "Why math?"

Unit Questions

In how many ways can we describe a 2-dimensional world? A 3- or higher-dimensional world? What are the advantages, disadvantages, and ambiguities of different descriptions in particular contexts? How does a (distance) metric reflect or even determine the underlying structure of a space?

These questions bring the topic of mathematical description down to a more manageable introduction, one which many high school students will have seen before in the truncated form of Cartesian coordinates and the distance formula. Despite the questions' restricted scope, they encourage the exploration of meaningful mathematics, focusing on the characteristics and dis/advantages of descriptive systems rather than rote memorization of conversions from Cartesian to polar coordinates or related formulas. In particular, non-Euclidean geometries, transformational geometry and matrix descriptions, and anamorphisms are some "advanced" topics that might emerge from student inquiry.

Entry-level Questions

How can we give directions, specify locations, and describe motion? What ambiguities arise? How do we "translate" or distort (deliberately or otherwise) different kinds of directions and descriptions into each other, and when would we want to? How can you measure the distance between two points? How can you determine the shortest distance between two points?

It is likely that most students have given directions at some point in their lives. Though they probably used words ("You go straight until Vischer's Ferry Road, then take a left and..."), a few may have encountered coordinates in orienteering, chess, Battleship ("Is there anything at A7?") or other contexts. Thus, these questions begin with concrete, familiar and practical aspects of students' lives while segueing into deeper issues in the unit questions.

Essential Question #2

What is a number?

This is a deceptive question in that the answer may seem obvious to people with a wide variety of experience in mathematics, but their understandings of "number" may likewise vary widely. To a child just learning to count, "number" may encompass quantities of marbles or one-to-one correspondence with fingers and toes; to a college math major, it make take on the more abstract forms of fields, groups and rings. Our names for different categories of numbers are indicative of the gradual, sometimes reluctant, extension of (Western) conception: natural, rational, imaginary and transfinite numbers, to name a few. Even with a bachelor's degree in mathematics I remain far from certain that I understand what a number is any better than a child learning to count.

Unit Questions

What properties do vectors have in common with other forms of numbers (natural numbers, integers, etc.)? What can vectors represent?

As a high school student I wondered why I had to learn obscure and seemingly pointless terminology such as "associative" and "identity"; after all, a + (b + c) was going to equal (a + b) + c no matter what, right? Matrices quickly disabused me of this notion, and vectors, which can be considered a special case of matrices, are not as bewildering to the novice as they exhibit inverses, identity, associativity and commutativity under addition. The different forms of vector multiplication pose greater difficulties, however, thus stretching students' conception of what numbers are (scalars, which have only magnitude, vs. vectors, which also have direction) and how they might behave under different operations.

Entry-level Questions

If someone gives you directions as to where to turn and how fast to move, how can you tell where you will end up relative to your starting position? If you have two points, how can you tell someone how to get from one point to the other? Conversely, how can you figure out the most efficient route?

No doubt students are familiar with word problems such as the following: "If Tweedledee leaves home for school at a leisurely 5 mph and Tweedledum leaves from school for home at the same time at a brisker 8 mph, and the two places are 20 miles apart, where will they meet?" Vectors extend such problems into two or more dimensions. Vector addition, as suggested by these questions, allows students to "simplify" a potentially complex "itinerary" and introduces the idea of distance vs. displacement as well as the contexts in which they may be used. In particular, vector addition by graphical "head to tail" means should be accessible to the majority of students.

Realistically, not every question will be addressed, and some will be explored in more depth than others (see 3. Desired Results & Assessments).

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 20 May 2002