a step in the direction of enhancing mathematical insight

for teachers and the students they teach

Many educators see value in hands-on learning. To me the essential attribute is the ability to manipulate the things one studies, letting the learner explore and tinker, gain experience and familiarity and build intuition.

However, the long-term goal of using *hands-on* is to reach *minds-on —*an understanding of, and appreciation for the abstract. One might say that the point of education is to get learners, in response to objects and events in the world around them, to continually ask of themselves, “What is this a case of?”

Normally, the move from *hands-on* to *minds-on* is difficult because it requires that one move from tangible and manipulable objects to intangible, and thus presumably, non-manipulable abstractions. Many of the mathematical objects and actions that secondary students encounter don’t have easy physical embodiments to manipulate; visual representations of abstractions that can be manipulated offer a means to experiment with ideas, tinker to adjust them, and build conjectures worthy of further investigation and proof. Seeing with the physical eye and manipulating with the physical hand can help in the transition from hands-on objects to minds-on ideas.

It is here that the computer enters. Artfully crafted software environments can present learners with visual representations of the abstractions they study. Moreover, these environments often allow the user to manipulate these representations, thereby mimicking on the computer screen the act of manipulating a tangible object that happens in the context of *hands-on* learning. Computer environments that allow users to display such images and manipulate them are giving the users a *hands-on*[1] experience with an intangible manipulable.

The larger point in all of this is that appropriately crafted software environments can serve to extend the reach of our minds, allowing us to manipulate in a sensory fashion that which we could hitherto only imagine. Further, the ability to manipulate and explore images and their interaction can well led to invention and innovation. It is these interactive images—pictures for the mind’s eye—that give this essay both its title and its impetus.

The teaching and learning of mathematics is intended, at least in part, to help us deal with the complexity of our surround. Doing so requires us as teachers and students to model that complexity and to use our mathematical tools to manipulate those models. Having built models we must also learn to cope with imprecision of these models and exercise good judgment in when and how to use them.

Models of intangible mathematical objects allow us to manipulate elements of the model to understand and explore the relationship(s) among these elements. Such models allow experimenting, interesting problem posing, the generation of ideas and conjectures. However, not everyone is comfortable manipulating symbols that act as surrogates for the objects in our surround. Many people claim to understand better when presented with a visual argument. Indeed we often hear people say “Now I see!” to indicate that they have understood something. This is probably what we mean by developing insight!

Should we consider implementing visual versions of our mathematical models? Mathematical models, *visually* expressed,[2] would consist of *images* that could be manipulated just as mathematical models, *symbolically* expressed, consist of *symbols* that can be manipulated. In many situations our current technology allows us to make such visual mathematical models. Suppose that as a matter of course we were to offer mathematical models in the form of images, screen objects that are reminiscent of, or evocative of the objects of the model in question and allow people to manipulate these screen objects in order to explore the relationships among them?

Consider the potential gains of both allowing exploration of mathematical models, both visual and symbolic, and providing teachers and students with the tools and the encouragement to explore. Students are rarely given the opportunity to control elements of their learning. Allowing students to manipulate and control the images that they use to explore the model of the situation being modeled may produce just the degree of engagement and provocation needed to get them to speculate and make conjectures. This, in turn, may lead them to a better understanding of the issue they are exploring. Further, and perhaps most importantly, it may lead us, their teachers, to a better understanding of what understanding a topic might be.

As teachers we generally agree that assessing how well we have taught and/or how well our students have understood what we have taught is best done by posing a problem that elicits a *performance* of some sort on the part of the students beyond simply parroting what was said to them either orally or in writing. Such *performance* implies change—a situation is presented and the student is asked to transform it in some way that sheds light on the problem. Asking students for performances that involve change implies that the elements of the problem situation should be manipulable in some way by the student. I’ve created a collection of ** Interactive Images** with exactly this purpose in mind. My own use of the site, and therefore the style of many of the questions I pose on it, is for educating teachers and stimulating

In particular, I like to think of three forms of performance – mapping, constructing and deconstructing.

*Mapping* is identifying the correspondence of both mathematical *objects* and mathematical *actions* across at least two different complementary representations; specifically this means interpreting how each aspect of a mathematical *object* in one of the representations is represented in the others and how the *actions*—i.e., the tools for manipulating and transforming *objects*—in each representation are related to the *actions* of the other representations.

Here is an example __[click here to get the live app]__: A function of one variable presented in symbolic form—say *x ^{2}+px+q*—is plotted in the {

Here are some questions that can elicit mapping performance:

• Drag the point around the {*p,**q*} plane by sliding the large YELLOW tick marks on the p and q axes. What happens in the right hand {*x,y*} plane?

• What conditions make the point and the parabola change color? Where are they RED? GREEN?

• What is the shape of the red/green boundary in the {*p,q*} plane?

• In the {*p,q*} plane, the boundary can be thought of as a function *q*(*p*). What is this function?

• How is it related to the discriminant of the quadratic?

• The locations of the real or complex conjugate roots of the quadratic appear in the {*x,y*} plane as large gold dots. Trace the complex roots in the {*x,y*} plane. Can you formulate a conjecture about the path of the roots as you move the point in the {*p,**q*} plane along a horizontal line? Along a vertical line? Can you prove or disprove your conjectures?

And here __[click here]__ is a second example designed to elicit mapping performance.

A rectangle (or any polygon) is drawn in the Cartesian plane and is also depicted as a point in the {perimeter, area} plane. Here are some questions that can elicit mapping performance:

• Every point in the first quadrant of the {width,height} plane corresponds to a rectangle.

• The applet allows you to generate either

•• a family of rectangles by moving the GOLD point along a height = constant/width curve, or

•• a family of rectangles by moving the GOLD point along a height+width = constant curve.

• Can you explain the nature of the curves generated in the {Perimeter,Area} plane as you drag the GOLD point in the {width,height} plane? qualitatively? analytically?

• Can you find the region(s) in the {Perimeter,Area} plane that correspond to all rectangle with a 1:3 aspect ratio? with a 3:1 aspect ratio?

Constructing interactive images involves using the primitive elements of a mathematical topic—e.g. points, circles and lines in the case of geometry—or the constant function and the identity function in the case of algebra – to build more complex mathematical objects. These objects, the relationships among them and the way(s) in which they be manipulated constitute a mathematical model, visually expressed.

Here __[click here]__ is an example with sample questions.

-> Given: A line segment (purple) whose length is fixed and known.

-> Given also a line segment (blue) of fixed length drawn to its midpoint and a third line segment (green) of fixed length perpendicular to it.

• Is it {always, sometimes, never} possible to build a triangle which has one of the line segments as a side and the other line segments as a median and an altitude to that side?

A second example of construction in geometry __[click here]__:

-> The length of one side AB (purple) and the two diagonals AC (green) and BD (blue) of a parallelogram are fixed and known.

• Can you construct the parallelogram ABCD ?

An example of construction in algebra __[click here]__:

• Build a polynomial by multiplying and transforming products of linear functions.

• Enter a target polynomial of order *n* = 1, 2 or 3.

and a second example of construction in algebra __[click here]__:

• Drag the yellow dot in the left panel.

• If the curve in the right panel was a plot of the the function *f*(*x*), what would the algebraic expression of *f*(*x*) be?

• What questions could/would you put to your students based on this applet?

Deconstructing Interactive Images involves decomposing an image into component parts, e.g. hypotenuses of triangles that may be part of a complex geometric diagram in order to uncover relationships among and within the mathematical objects in the image. In cases where the image is a graph, with polynomials or rational functions for example, deconstructing can mean decomposing the functions into the linear functions that were combined to produce them. These more elementary objects, the relationships among them and the way(s) in which they be manipulated constitute a mathematical model, visually expressed.

Here is one example __[click here]__:

• A blue rectangle is inscribed in the green square.

•• What fraction of the area of the green square is occupied by the blue rectangle?

•• What fraction of the perimeter of the green square is the perimeter of the blue rectangle?

•• Drag the GOLD dot. Can you explain the shape of the curves in the right panel?

• Now let a blue square be inscribed in the green square.

•• What fraction of the area of the green square is occupied by the blue square?

•• What fraction of the perimeter of the green square is the perimeter of the blue square?

•• Drag the GOLD dot. Can you explain the shape of the curves in the right panel?

• What questions could/would you put to your students based on this applet?

And a second example of deconstruction in geometry __[click here]__:

• A circle of radius 1 circumscribes a regular polygon of *n* sides. Inside the regular polygon is an inscribed circle. In the limit of a very large number of sides the area and perimeter of both the inner and outer circles approach those of the polygon.

•• Write an expression for *A*(*n*), the area of an *n* sided regular polygon inscribed in a unit circle.

•• Write an expression for *P*(*n*), the perimeter of an *n* sided regular polygon inscribed in a unit circle.

•• Contrast the rates at which *A*(*n*) and *P*(*n*) approach their limits.

• Challenges:

•• The number n of sides grows while the length *S* of each side gets smaller and smaller.

•• How does the product of *n* and *S* behave? How do you know? Can you prove it?

•• The area of a UNIT circle is π and its perimeter is 2*π*.

•• How do you convince a student that the area of a circle is NOT half its perimeter?

• What other questions could/would you ask you students based on this applet?

An example[3] of deconstruction in Algebra __[click here]__:

• Choose factoring to factor a quadratic function *f*(*x*). Then enter your function *g*(*x*) in the form *a*(*x*+*b*)(*x*+*c*).

• What can you learn about possible errors in factoring by examining the difference function *f*(*x*) – *g*(*x*).

• What questions could/would you ask your students based on this applet?

A second example of deconstruction in algebra __[click here]__:

• Enter a function *f*(*x*) in the green box at the top center of the screen.

• Explain how the translation, dilation and reflection transformations of your function are all instances of composing that function with a linear function.

• What questions could/would you put to your students based on this applet?

The central question I have tried to address is How can we use interactive images to enhance and extend the ways learners (both teachers and students) use such interactive activities to scaffold invention and innovation?

Having devoted more than five decades of my professional life to the endeavor, I am remain optimistic about the future of computers and the “*pictures for the mind’s eye*” that can be generated with them in mathematics and science education.

One reason to be hopeful is the amount of attention and concern about the future of mathematics education that is currently being expressed in the media. Given this degree of concern one hopes that society will make the necessary investment of intellectual and fiscal resource necessary to address the issues that it regards as pressing. In an earlier blog[4] I wrote about the one of the reasons a society maintains an educational system that includes mathematics; to provide people with the intellectual tools to model the world they encounter in the practical, economic, policy and social aspects of their lives.

A reason that I’m pleased at the existence of this AMS blog is that public discourse about mathematics education, as well as the consequent question of how well the system we now have helps us attain our goals for educating people in mathematics will increase and become more substantive. I write in the hope that incorporating new visual approaches to mathematics more fully and richly into the educational process may help us move forward in attaining those goals.

ENDNOTES

[1] More properly a* hands-*[mediated by mouse]-*on* experience

[2] Some illustrative examples of what is meant by the notion of manipulable interactive images as well as all of the examples in this essay can be found in interactive form ** HERE**. While these examples were designed to enhance and deepen understanding and insight for

[3] “FOIL” (First, Outer, Inner, Last) is a common school-mnemonic for (but limited to) expanding products of two binomials.

[4] https://blogs.ams.org/matheducation/2018/12/01/a-physicists-lament/

]]>I have been worrying a lot about mathematics education for over a quarter century now. While many university mathematicians who get involved in mathematics education focus on the need for new teaching methods, I have been drawn to examples of failure of the US curriculum to deal properly with basic ideas.

One of the first such ideas I identified was place value, or to be more precise, the base ten place value notational system for whole numbers (and later, decimal fractions). This is the bedrock of school mathematics, and it is used in almost everything that is done day-to-day with mathematics. We ought to try to get this as right as possible, and to have students learn it as well as possible. Yet mathematics education research indicates that we fail rather badly to do so.

Susanna Epp and I wrote a lengthy discussion [Epp, Howe 2006] of the details of the principles and techniques that constitute base ten arithmetic. This was organized around what we called the *five stages of place value*. These are summarized by example as follows.

$$352 \ \ = \hskip 2.4 in$$

$$\ \ \ \ \ \ = \ 300 \hskip .6 in + \ \ \ 50 \ \ \ \ \ \ \ \ \ + \ \ \ 2 \ \ \ \ \ $$

$$\hskip .25 in = \ 3 \times 100 \hskip .38 in + \ \ 5 \times 10 \ \ \ \ + \ \ \ 2 \times 1$$

$$\ \ \ \ \ = \ 3 \times(10\times10) \ + \ \ 5 \times 10 \ \ \ \ + \ \ \ 2 \times 1$$

$$\ \ \ \ \ \ \ \ = \ 3 \times 10^2 \hskip .4 in + \ \ 5 \times 10^1 \ \ + \ \ \ 2 \times 10^0.$$

The first stage, $352$, is just the standard base ten notation for the number. The second stage, $300 + 50 + 2$, indicates that each digit in the number stands for a number of a special kind, and the number itself is a sum of these special numbers. These numbers have a mathematical description – they are digits times powers of 10 (as is made explicit in the fifth stage), but there has been no simple reference term for them in the mathematics education literature.

Recently, the textbook [Beckmann 2017] has used the name *place value parts* for these numbers, and we will adopt this term here. The place value parts are in some sense the atoms of the base ten system, and general whole numbers are like molecules, obtained by combining atoms. The lack of a simple term has impeded focusing on the place value parts as the basic building blocks of the system.

The place value parts have multiplicative structure, and the third and fourth stages make this structure explicit. The third stage displays each place value part as a digit times a *base ten unit *– a place value part with digit 1. The fourth stage exhibits the base ten units as also being products, of a certain number of factors of 10, which is the base of the system. The fifth stage then uses the standard notation of exponents to indicate each base ten unit by simply recording the number of factors of 10 used to make it. It exhibits the number as what might be called a *polynomial in 10*. To indicate the relation between the place value parts and the size of numbers, we also refer to the exponent as the *order of magnitude*.

Together, the five stages reveal the basis for the extraordinary power of base ten notation: it is using all the power of algebra – addition, multiplication, and exponentiation – simply to write numbers, and it employs the clever convention of place value, which allows each place value part of a number each to be indicated simply by its associated digit. This of course is mediated by 0, the symbol for zero, one of the all time great mathematical inventions.

Of course, mathematicians understand this very well, and translate almost unconsciously between the first stage and the last. However, there is evidence in the mathematics education literature that most students do not master this structure. In the 1980s, C. Kamii published several papers [Kamii 1986], [Kamii, Joseph 1988], showing that 3rd and 4th grade students did not understand the significance of the digits in 2- or 3-digit numbers. More recently, E. Thanheiser [Thanheiser 2009, 2010] has shown that many, probably the large majority, of students were arriving in college not understanding the third stage. This is relevant to U.S. mathematics education not only as a measure of current quality, but also for projecting future quality, since as Thanheiser points out, teachers must know the third stage if they are to teach addition and subtraction conceptually. The recent book [Newton 2018] gives examples showing how US K-12 mathematics instruction may fail to develop base ten structure adequately.

It took me several more years to appreciate that, in fact, although mathematicians may take the five stages as common sense, looked at from the educational point of view, each successive stage represents a significant conceptual development, that can take a year or more to get children to understand. The first stage starts in first grade, or even Kindergarten, when students are introduced to 2-digit numbers. In fact, the second stage (aka *expanded form*) is often stated at about this time, and may or may not be used in a conceptual fashion. The book [Newton 2018 ] shows that, even the meaning of the digits in two-digit numbers is often taught inadequately. The third stage has to wait until students can deal with multiplication, so at least until 3rd grade if you follow the Common Core State Standards for Mathematics (CCSSM), although it can be dealt with implicitly with manipulatives somewhat earlier. The fourth stage requires considerable comfort with multiplication, and in particular, should involve appreciation of the Associative Rule/Property for Multiplication, which is arguably the subtlest of the Rules of Arithmetic, so could probably not be readily absorbed before 4th grade, perhaps 5th. The fifth stage requires a grasp of powers and exponents. For powers of 10, CCSSM calls for using whole number exponents in 5th grade, but the general idea of whole number powers comes in 6th grade. Thus, preparing a student to grasp the five stages in a conceptual way requires most of elementary school. And the evidence is, that we largely fail to do that.

The CCSSM pay considerable attention to the issues of place value, and in so doing provide a considerable advance over previous versions of mathematics standards. In particular, one of the summary headlines for Number and Operations in Base 10 in 5th Grade is “Understand the place value system”. However, it does not explicitly formulate the five stages.

One advantage of working with the five stage scheme is that it helps one to focus on the place value parts. A crucial feature of the base ten system is that the operations of addition and multiplication are simply combinations of operations with two place value parts (supplemented with regrouping – replacing 10 of some unit with 1 of the next larger unit). This follows from the Rules of Arithmetic (aka, Properties of the Operations). For addition, the two parts should even have the same order of magnitude.) Moreover, the results of such operations are given by the basic number facts (addition facts or multiplication facts), times an appropriate base ten unit. But it hard to teach or understand these points if there is no name for the place value parts.

The place value parts are also the key to comparison and estimation. Here the salient fact is that any place value part of a given order of magnitude is larger than any part of a smaller magnitude. It is at least 10 times as large as any part of two or more orders of magnitude less. (In fact, although place value parts of only one magnitude difference may be approximately the same size, for example 1,000 and 900, about half the time, the larger part will be 10 or more times as large.) It follows that any number is well approximated by the sum of a few of its largest few place value parts, and for practical purposes, can (and should) often be replaced by this. This is of course the topic of rounding, but provides the perspective that the place rounded to is not so significant at the number of places kept (called* significant digits* in the context of scientific notation).

Perhaps the value of thinking in terms of place value parts is revealed most clearly in long division, the most troublesome topic in whole number arithmetic. Long division in the whole numbers is about the operation of division with remainder. Given a number $n$ and a divisor $d$, we are looking for a quotient $q$ that is the largest whole number such that $qd \leq n$. That is, $n = qd + r,$ with the remainder $r$ being less than $d$. How do we find $q$? If $p = a 10^{\ell} $ is the largest place value part of $q$, then $(a+1)10^{\ell} > q$, so that $(a+1) 10^{\ell} d > n$. That is, $p$ is the largest place value part such that $pd < n$. The converse also holds. So to find the largest place value part of $q$, we should look for the largest place value part $p$ such that $pd \leq n$. Inspection of the long-division algorithm will reveal that this is exactly what it does. The smaller place value parts of $q$ are then found by repeating the process with $n’ = n – pd$, and continuing in this manner. Being able to use the language of place value parts might make the process more transparent.

A common current mantra is that teachers should understand and teach mathematics conceptually. What does this mean for whole number arithmetic? I would contend that it mostly amounts to

- i) understanding the five stages of place value;

along with

- ii) knowing the sums (and differences, when appropriate) of two place value parts of the same order of magnitude; and how this is a consequence of the Rules of Arithmetic together with the structure of the place value pieces;
- iii) knowing the products of two place value pieces, and how this is a consequence of the Rules of Arithmetic together with the structure of the place value pieces;
- iv) understanding how the Rules of Arithmetic combine with ii) and iii) to allow multidigit computation; and
- v) understanding the relative size of place value parts, and how this enables efficient approximation of multidigit numbers with numbers having few non-zero place value parts.

This starts with knowing the five stages, so I would like to see this as a standard part of math preparation for elementary teachers. In seminars I have run with practicing teachers through the Yale Teachers Institute, I have been surprised with the alacrity they show in adapting the five stages to their classroom. One teacher has taken the trouble to write and testify how the five stages seem to have helped her (high school) students.

It seems possible that the five stages can provide an avenue to teaching whole number arithmetic (and beyond this, decimal fractions) in a student-friendly and conceptual manner. Let’s make the five stages of place value a standard part of teacher preparation.

**References**

- [Beckmann 2017]

Beckmann,S. (2017). Mathematics for Elementary Teachers with Activities (5th Edition), Pearson Education, London and New York, 2017. - [Epp, Howe 2006]
- [Kamii 1986]

Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education,. 1, 75-86. - [Kamii, Joseph 1988]

Kamii, C., & Joseph, L. (1988). Teaching place value and double-column addition. Arithmetic Teacher, 35(6), 48-52. - [Newton 2018]

Newton, X, Improving Teacher Knowledge in K-12 Schooling: Perspectives on STEM Learning, by Xiaoxia A. Newton, Palgrave Macmillan 2018 - [Thanheiser 2009]

Thanheiser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40, 251–281. - [Thanheiser 2010]

Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: Refining a framework. Educational Studies in Mathematics, 75(3), 241-251.

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For some writers, the reference may be to standardized testing (SAT, GRE, etc.). Certainly these are gatekeepers. Is this privilege ‘unearned’? I don’t know. That argument is for the College Board and the Educational Testing Service to make. I will argue, however, that the whole practice of judging a person’s fate in life by her or his performance on a single test, even the same test given multiple times, is not a good one (although the question of what such a test does select for is interesting). And this observation holds for any subject matter being tested, not particularly mathematics. So even if this is the ‘gatekeeper’ referred to, it’s not about our subject. And this form of gatekeeping is a matter of practice, of implementation, and not a widespread or deeply-held belief about mathematics. The deeply-held belief is about the nature of testing.

Maybe some writers are talking about textbook mathematics, mathematics as it is taught in a mediocre setting, as a set of rules and procedures. Well, this is not mathematics. This is rules and procedures, more and more imposed on teachers by the requirements of high-stakes state testing. Again, it seems to me that the gatekeeper is the testing, not the subject. And again, this observation is not at all specific to mathematics.

In fact it seems to me that mathematics is less guilty of ’gatekeeping’ than many other academic subjects.

There are many gatekeepers, in any culture. The use of language, both spoken and written, is a much stouter gate than knowledge of mathematics. The reader will shortly see how even one misspelled word can cast doubt on the vallidity [n.b.] of a thought, or even on the intelligence of the writer. We commonly value someone who speaks and writes standard academic English over someone who uses vernacular, or even who has a heavy local accent (and we all have local accents!). Just think of the effect, in a job interview or resume cover letter, of even a single mispronounced word or grammatical solecism. Conversely, we all know well-spoken imposters.

And language is unavoidable. We are constrained to speak in any social situation, and to write in any professional position. This gatekeeper appears unbidden. And often unconscious: we frequently don’t have control over judgments we make on the basis of language.

There are still other gatekeepers. Dress is the most obvious. And some very unfortunate ones: race, class, gender. It is quite human, but at the same time quite de-humanizing, to react unconsciously to people we don’t know by grouping them with others with whom they share external characteristics. Unlike language, these gatekeepers are not routinely addressed by formal education. They are almost always unconscious, hence powerful. And they are clearly ‘unearned’.

Is the privilege of mathematics ‘unearned’? Well, no. I think it is hard earned. Mathematics is the *locus classicus* for addressing logic, the derivation of statements from other statements. And this skill pervades human activity. Further, the better you are at this skill, the more valuable your activity to others. This is why I have argued (in several places) that the teaching of mathematics should be centered on logic, and not on algorithm. The latter, for me, should be a consequence of the former. Even for the 80% of our students (this figure is approximate and variable: see https://nces.ed.gov/programs/raceindicators/indicator_reg.asp) who don’t go into STEM related fields, this legacy of a mathematical education is central.

I have heard arguments about other ‘forms’ of mathematics, sometimes called ‘non-Western’. I would argue that the classification of logic-based mathematics as ’Western’ (or sometimes ‘Greek’) is a misnomer at best, and simplistic at worst.

But let’s not talk about mathematics for a minute. Let’s talk about pizza. Everyone thinks of pizza as Italian, and it is consumed worldwide. But tomatoes originated in America, wheat (probably) in the middle east, basil and pepper in India. The Italians just put it all together.

Similarly, the pieces of what we call “Western” mathematics were lying around for centuries, in different parts of the world. The Greek achievement was a synthesis of these human thoughts. That is, mathematics is ‘Western’ only in the sense that pizza is ‘Italian’. Everyone enjoys pizza. Everyone benefits from logic.

And like pizza, everyone wants mathematics. ‘Western’ mathematics. Algebraic topology. Bessel functions. Lie algebras. Intellectual domains in which a ‘non-Western’ culture had not penetrated, before a cultural influence from the West. I have personally worked on every continent except Antarctica, and everyone wants to learn mathematics the way it has developed ‘in the West’.

But notice that this form of mathematics is now being developed as much in Asia as in Europe. And in fact it always has been. Modern mathematics is ‘Eastern’ as much as it is ‘Western’. And, if Africa and Latin America develop as quickly as we all expect they will, modern mathematics will soon be ‘Southern’ as much as ‘Western’.

What about the other 20%, the students preparing for STEM fields? Let’s leave aside for now the fact that we don’t always know who these students are. Is the status of mathematics ‘unearned’ to this group? Again, no. A knowledge of mathematics is, in fact, an intellectual gatekeeper, or better yet, gateway, into STEM fields. For those who are going to make contributions in these fields, mathematics is vital. And it is growing in importance as the sciences, and even the social sciences, develop.

So yes, it is a gateway for this group. Just as organic chemistry is a gateway for medical school. Do you want to be treated by a doctor who hasn’t studied it?

And do you want to travel over a bridge built by an engineer who hasn’t studied ‘Western’ calculus?

(I thank Paul Goldenberg and Al Cuoco for their help in writing this post.)

]]>2018 was an important year for the Letchford family – for two related reasons. First it was the year that Lois Letchford published her book: *Reversed: A Memoir*.^{[1]} In the book she tells the story of her son, Nicholas, who grew up in Australia. In the first years of school Lois was told that Nicholas was learning disabled, that he had a very low IQ, and that he was the “worst child” teachers had met in 20 years. 2018 was also significant because it was the year that Nicholas graduated from Oxford University with a doctorate in applied mathematics.

Nicholas’s journey, from the boy with special needs to an Oxford doctorate, is inspiring and important but his transformation is far from unique. The world is filled with people who were unsuccessful early learners and who received negative messages from schools but went on to become some of the most significant mathematicians, scientists, and other high achievers, in our society – including Albert Einstein. Some people dismiss the significance of these cases, thinking they are rare exceptions but the neuroscientific evidence that has emerged over recent years gives a different and more important explanation. The knowledge we now have about the working of the brain is so significant it should bring about a shift in the ways we teach, give messages to students, parent our children, and run schools and colleges. This article will summarize three of the most important areas of neuroscience that directly apply to the teaching and learning of mathematics. For more detail on these findings, and others, visit youcubed.org or read Boaler (2016).^{[2]}

The first important area of knowledge, which has been emerging over the last several decades, shows that our brains have enormous capacity to grow and change at any stage of life. Some of the most surprising evidence that highlighted this came from studies of black cab drivers in London. People in London are only allowed to own and drive these iconic cars if they successfully undergo extensive and complex spatial training, over many years, learning all of the roads within a 20-mile radius of Charing Cross, in central London, and every connection between them. At the end of their training they take a test called “The Knowledge” – the average number of times it takes people to pass The Knowledge is twelve. Neuroscientists decided to study the brains of the cab drivers and found that the spatial training caused areas of the hippocampus to significantly increase.^{[3]} They also found that when the drivers retired, and were not using the spatial pathways in their brains, the hippocampus shrank back down again.^{[4]} The black cab studies are significant for many reasons. First, they were conducted with adults of a range of ages and they all showed significant brain growth and change. Second, the area of the brain that grew – the hippocampus – is important for all forms of spatial, and mathematical thinking. The degree of plasticity found by the scientists shocked the scientific world. Brains were growing new connections and pathways as the adults studied and learned, and when the spatial pathways were no longer needed they faded away. Further evidence of significant brain growth, with people of all ages, often in an 8-week intervention, has continued to be produced over the last few decades, calling into question any practices of grouping and messaging to students that communicate that they cannot learn a particular level of mathematics.^{[5]} Nobody knows what any one student is capable of learning, and the schooling practices that place limits on students’ learning need to be radically rethought.

Prior to the emergence of the London data most people had believed either that brains were fixed from birth, or from adolescence. Now studies have even shown extensive brain change in retired adults.^{[6]} Because of the extent of fixed brain thinking that has pervaded our society for generations, particularly in relation to mathematics, there is a compelling need to change the messages we give to students – and their teachers – across the entire education system. The undergraduates I teach at Stanford are some of the highest achieving school students in the nation, but when they struggle in their first math class many decide they are just “not a math person” and give up. For the last several years I have been working to dispel these ideas with students by teaching a class called How to Learn Math, in which I share the evidence of brain growth and change, and other new ideas about learning. My experience of teaching this class has shown me the vulnerability of young people, who too readily come to believe they don’t belong in STEM subjects. Unfortunately, those most likely to believe they do not belong are women and people of color.^{[7]} It is not hard to understand why these groups are more vulnerable than white men. The stereotypes that pervade our society based on gender and color run deep and communicate that women and people of color are not suited to STEM subjects.

The second area of neuroscience that I find to be transformative concerns the positive impact of struggle. Scientists now know that the best times for brain growth and change are when people are working on challenging content, making mistakes, correcting them, moving on, making more mistakes, always working in areas of high challenge.^{[8, 9]} Teachers across the education system have been given the idea that their students should be correct all of the time, and when students struggle teachers often jump in and save them, breaking questions into smaller parts and reducing or removing the cognitive demand. Comparisons of teaching in Japan and the US have shown that students in Japan spend 44% of their time “inventing, thinking and struggling with underlying concepts” but students in the U.S. engage in this behavior only 1% of the time.^{[10]} We need to change our classroom approaches so that we give students more opportunity to struggle; but students will only be comfortable doing so if they have learned the importance and value of struggle, and if they and their teachers have rejected the idea that struggle is a sign of weakness. When classroom environments have been developed in which students feel safe being wrong, and when they have been valued for sharing even incorrect ideas, then students will start to embrace struggle, which will unlock their learning pathways.

The third important area of neuroscience is the new evidence showing that when we work on a mathematics problem, five different pathways in the brain are involved, including two that are visual.^{[11, 12]} When students can make connections between these brain regions, seeing, for example, a mathematical idea in numbers and in a picture, more productive and powerful brain connections develop. Researchers at the Marcus Institute of Integrative Healthhave studied the brains of people they regard to be “trailblazers” in their fields, and compared them to people who have not achieved huge distinction in their work. The difference they find in the brains of the two groups of people is important. The brains of the “trailblazers” show more connections between different brain areas, and more flexibility in their thinking.^{[13]} Working through closed questions, repeating procedures, as we commonly do in math classes, is not an approach that leads to enhanced connection making. In mathematics education we have done our students a disservice by making so much of our teaching one-dimensional. One of the most beautiful aspects of mathematics is the multi-dimensionality of the subject, as ideas can always be represented and encountered in many ways, such as with numbers, algorithms, visuals, tables, models, movement, and more.^{[14, 15]} When we invite people to gesture, draw, visualize, or build with numbers, for example, we create opportunities for important brain connections that are not made when they only encounter numbers in symbolic forms.

One of the implications of this important new science is we should all stop using fixed ability language and celebrating students by saying that they have a “gift” or a “math brain” or that they are “smart.” This is an important change for teachers, professors, parents, administrators – anyone who works with learners. When people hear such praise they feel good, at first, but when they later struggle with something they start to question their ability. If you believe you have a “gift” or a “math brain” or another indication of fixed intelligence, and then you struggle, that struggle is devastating. I was reminded of this while sharing the research on brain growth and the damage of fixed labels with my teacher students at Stanford last summer when Susannah raised her hand and said: “You are describing my life.” Susannah went on to recall her childhood when she was a top student in mathematics classes. She had attended a gifted program and she had been told frequently that she had a “math brain,” and a special talent. She enrolled as a mathematics major at UCLA but in the second year of the program she took a class that was challenging and that caused her to struggle. At that time, she decided she did not have a “math brain” after all, and she dropped out of her math major. What Susannah did not know is that struggle is really important for brain growth and that she could develop the pathways she needed to learn more mathematics. If she had known that, and not been given the fixed message that she had a “math brain,” Susannah would probably have persisted and graduated with a mathematics major. The idea that you have a “math brain” or not is at the root of the math anxiety that pervades the nation, and is often the reason that students give up on learning mathematics at the first experiences of struggle. Susannah was a high achieving student who suffered from the labeling she received; it is hard to estimate the numbers of students who were not as high achieving in school and were given the idea that they could never do well in math. Fixed brain messages have contributed to our nation’s fear and dislike of mathematics.^{[16]}

We are all learning all of the time and our lives are filled with opportunities to connect differently, with content and with people, and to enhance our brains. My aim in communicating neuroscience widely is to help teachers share the important knowledge of brain growth and connectivity, and to teach mathematics as a creative and multi-dimensional subject that engages all learners. For it is only when we combine positive growth messages with a multi-dimensional approach to teaching, learning, and thinking, that we will liberate our students from fixed ideas, and from math anxiety, and set them free to learn and enjoy mathematics.

*This blog contains extracts from Jo’s forthcoming book*: Limitless: Learn, Lead and Live without Barriers, *published by Harper Collins.*

[1] Letchford, L. (2018) *Reversed: A Memoir*. Acorn Publishing.

[2] Boaler, J (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[3] Maguire, E. A., Gadian, D. G., Johnsrude, I. S., Good, C. D., Ashburner, J., Frackowiak, R. S., & Frith, C. D. (2000). Navigation-related structural change in the hippocampi of taxi drivers. *Proceedings of the National Academy of Sciences*, 97(8), 4398-4403.

[4] Woollett, K., & Maguire, E. A. (2011). Acquiring “The Knowledge” of London’s layout drives structural brain changes. *Current **b**iology**:CB*, 21(24), 2109–2114.

[5] Doidge, N. (2007). *The Brain That Changes Itself*. New York: Penguin Books,

[6] Park, D. C., Lodi-Smith, J., Drew, L., Haber, S., Hebrank, A., Bischof, G. N., & Aamodt, W. (2013). The impact of sustained engagement on cognitive function in older adults: the Synapse Project. *Psychological science*, 25(1), 103-12.

[7] Leslie, S.-J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. *Science*, 347, 262-265.

[8] Coyle, D. (2009). *The Talent Code: Greatness Isn’t Born, It’s Grown, Here’s How*. New York: Bantam Books;

[9] Moser, J., Schroder, H. S., Heeter, C., Moran, T. P., & Lee, Y. H. (2011). Mind your errors: Evidence for a neural mechanism linking growth mindset to adaptive post error adjustments. *Psychological science*, 22, 1484–1489.

[10] Stigler, J., & Hiebert, J. (1999). *The teaching gap: Best ideas from the world’s teachers for improving education in the classroom*. New York: Free Press.

[11] Menon, V. (2015) Salience Network. In: Arthur W. Toga, editor. *Brain Mapping: An Encyclopedic Reference*, vol. 2, pp. 597-611. Academic Press: Elsevier;

[12] Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning*. Journal of Applied & Computational Mathematics*, 5(5), DOI: 10.4172/2168-9679.1000325

[13] Kalb, C. (2017). What makes a genius? *National Geographic*, 231(5), 30-55.

[14] https://www.youcubed.org/tasks/

[15] Boaler, J. (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[16] Boaler, J. (2019). *Limitless: **Learn, Lead and Live without Barriers.*

What surprises you mathematically in your classes? When do you witness students’ creative moments? How often does this happen?

When instructors develop an environment where students are willing to put themselves “out there” and take a risk, interesting moments often happen. Those risks can only build one’s creativity, which is the most sought-after skill in industry according to a 2010 IBM Global Study.

How do we get students to be creative? And how does that balance with the content we are required to cover? Below, past and present members of the Creativity Research Group present reasons on why and how we each teach for creativity. We all have different but synergistic teaching practices we engage in to foster creativity in our students. Gulden focuses on having students making connections, while Milos has students take risks through questioning and sharing wrong answers. Emily focuses on tasks that have multiple solutions/approaches; Gail emphasizes the freedom she gives students in exploring these tasks. Mohamed provides time for students to incubate their ideas. Houssein and Paul reflect on their teaching practices and how teaching for creativity has been integrated into theses practices. There is also the thread of opportunities for student self-reflection woven throughout these stories.

One common aspect is that we try our best to saturate our courses with chances for students to be creative from beginning to end. These stories are our attempts at being creative about fostering creativity. Enjoy!

**Gail Tang**

I recently watched an *Ugly Delicious* episode where world-renowned chefs talked about their definitions of pizza and what it meant to them to make pizza. These chefs fell into two types: those who stuck to traditional ways to make pizza and those who departed from these ways. Those who ended up leaving the traditional ways behind did so because they felt stifled to operate under the strict rules of making pizza; they felt their identities were being compromised. With the courage of their convictions, they left to forge their own pizza paths. These chefs rejoiced in their freedom; finally they made pizza in a way that paid tribute to their identities. Chef Christian Puglisi said “If you only look at how it used to be done, or how it’s supposed to be done, you don’t allow yourself to move it forward.” This episode really resonated with me; replace “pizza” with “math” and “make pizza” with “teach math,” – you get the same story of suppressing innovation in the name of tradition. The idea of teaching others in the same way I was taught suffocated me. I was not interested in producing generations of students who could mimic my every mathematical move.

I started with baby steps in Calculus. I found exercises with more than one solution path to the same answer and assigned these without any direction. Students wrote different solutions on the board. Students were not used to seeing each others’ creations let alone creating their own solution paths. The energy in the room was thrilling. Unfortunately, I did not collect this data at the time, but fortunately Houssein El Turkey (see his narrative below) has an example of three students’ work on computing the limit below.

Letting students try problems on their own with little direction has the opportunity to have a profound impact on their mathematical identities. For example, one student started my Calculus 1 as a biology major and ended Calculus 1 as a math major! She wrote in her Calculus 2 weekly reflection:

I think having math ‘done to me’ rather than getting to explore it and have fun with it in high school is the reason I didn’t enjoy math in high school. I love how in your classes we get to try problems our own way and don’t have to use your method. I also think it’s super cool that you encourage using different methods. I would never have considered myself a creative person until I started working with numbers. I’m not anywhere near as creative as I should be, but I feel like math is helping me become more creative. The other night I did a problem with a method that I knew was going to be wrong, but I just wanted to see what happened. It actually helped me understand why that method doesn’t work.

**Emily Cilli-Turner**

A turning point in my thinking about students’ potential for creativity and how to foster it in the classroom happened while I was teaching a Linear Algebra course using the inquiry-oriented linear algebra materials. One task asked for the solution to a system of three equations in two variables; if there was no solution, find the “best” approximate solution. This task purposefully did not define the word “best” so the students would be forced to think about what qualifications the best approximate solution would have. Every group graphed the linear equations (which bounded a triangular region) and most presented “best” solutions as averages of $x$ and $y$ values of intersection points. However, one student was able to find the exact least squares solution by using optimization of functions of two variables techniques from calculus. Once this student presented his solution to the class, the other students were intrigued and could see the drawbacks of their own solutions. This was very unexpected for me. It drove home the point that students can and will be creative when we give them the tools and the freedom to hone their creativity.

In my mind, teaching for creativity has two main components: task design and collaboration. A large part of teaching for creativity is providing students with tasks that involve multiple approaches/solutions. The above episode would have turned out very differently if I had given the definition of the least squares solution and then several problems finding the least squares solutions of a system. The student would have never had a chance to find his original solution because all of the mystery would have been taken away with a provided definition. Yet, if the students had not been working on the task in groups, bouncing ideas off of each other, and using the group whiteboard to do scratch work, I think this vignette would not have happened. Collective creativity can be greater than that of the individual, and the students’ discussions helped that individual student refine his ideas and come up with the idea of using optimization to solve the problem.

**Milos Savic**

I believe teaching for creativity addresses a lot of issues in mathematics education. When a student is trying to be creative, there are many side effects, including more saturation with the content, greater mathematical confidence, and the ability to manipulate mathematics in different or new ways. Also, I believe mathematical creativity allows a student to be more of themselves instead of more like me. Finally, solving problems in STEM fields requires creativity, and I strive to create authentic experiences so that students can engage in being creative. These beliefs either are expressed explicitly (“Class, I want you to play with this idea”) or implicitly through tasks, quizzes, tests, and other requirements.

To generate curiosity, I have students ask two questions for every homework assignment; one is intended to be about the concepts, and the other is about their mathematical processes. I also give many routine problems with little twists. For example, I pose problems that require a student to go backwards instead of forwards (e.g. what non-linear function, when integrated from 1 to 2, is 17?). I also ask them to provide their *own* definitions or theorems using what they know. For example, using what they knew about groups and semigroups, a student created an anti-identity (the additive inverse of the multiplicative identity) and anti-inverses. My actions support these tasks; I celebrate many wrong answers in the class in order to show the process of mathematics and the creative moments within it.

**Houssein El Turkey**

It is interesting how my teaching has evolved to focus on more than just covering the basic learning outcomes in the classes I teach. I have become aware that teaching mathematics can and should include discussions with students on the novelty and flexibility in problem solving (or proving). Now I seek different approaches from my students to show them that there are often multiple ways to solve a problem. I also point out to my students when we build on something we discussed a while back to show that making connections is crucial. Another action I have taken is to explicitly show how certain processes generalize to a bigger picture.

Seeing an AHA moment from a student is one of the best highlights from a class. For example, when I asked my students to factor $x-1$ many of them were baffled but with hints and guidance, some of them came up with: $x-1 = (\sqrt{x}-1)(\sqrt{x}+1)$. The majority were taken by the simplicity and/or originality of the solution and the look on their faces was priceless. These AHA moments have been occurring more than before and they are constant reminders that they don’t have to be ground-breaking in order to have significant impacts on students. To a first-year college student, these simple tricks generating AHA moments can be crucial to show the originality aspect of doing mathematics even though instructors might find these tricks standard.

I also noticed that teaching to foster creativity has lifted my expectations of my students. I now see more potential in them and I work harder to get the best from them as I challenge them with tasks that require incubation and effort. An example of such a task that I used this semester was finding the limit $\lim_{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$ in three (or more) different ways. Students struggled in their first attempt in class to see it as a slope of a tangent line but I left it for them to think about it at home and we picked it up the next class. I was happily surprised that some students had made that connection. Whether it was due to incubation or someone suggested it to them remains unknown to me.

Assigning challenging tasks was accompanied by emotional support through my continuous encouragement and emphasizing that it is OK to fail or not complete a task and it is OK to struggle because that shapes the learning process.

**Mohamed Omar**

Many students come to mathematics with an inherent curiosity, but traditional teaching practices tend to suppress this. It is essential that we tap into the curiosity that students have as a way to make mathematics come alive for them. Infusing creativity in the classroom fosters this in a way unlike any other.

To facilitate creativity, I ask open-ended or open questions on assignments. For example, a central theme in one course was counting the number of regions that a set of hyperplanes partition an $n$-dimensional space into, and determining what data about the set of hyperplanes are sufficient to answer that question. The related open-ended question asked how these results change if we used circles or other geometric objects instead of hyperplanes. I gave students plenty of time to play with these open-ended problems, and supported them along the way. The key to unlocking creativity was to create a structure where students were rewarded for their efforts and diversity of approaches, rather than their final output. To facilitate this, I required a three-page reflection using the Creativity-in-Progress Rubric (To read more about the rubric, please see our short article in MAA Focus Feb/Mar 2016). Students had to submit all their scratch work and all partial results, and subsequently use the rubric to reflect on their problem solving process. For instance, if a student used definitions and theorems from the course in conjunction definitions and theorems from outside resources, they could provide direct evidence for this in their work and comment on how the conjunction occurred. This allowed students reflect on their thinking processes, facilitating the pathway to creativity.

**Gulden Karakok**

Through active learning teaching practices, I plan learning situations that provide opportunities for discovery and making connections. Making connections has been an important component of my teaching, as I believe this process facilitates learning of the new topics and also allows students to transfer learning to other situations. Making connections comes in many forms in my courses — connections between definitions, theorems, various solution approaches, examples, and representations. I often ask my students if they have seen a similar topic, definition or example before. My goal is to discourage compartmentalization of ideas and topics. Unfortunately, our education system seems to train students to see ideas in disparate categories. To address this concern and foster creativity, I have been pushing for the process of making connections. I think asking students to find similarities and differences between ideas from their perspectives and background not only helps students to “own” these ideas but also develop sense of “usefulness” of them. With this ownership, students will be more equipped to be creative.

One example of how making connections promotes creativity comes from my preservice elementary math content course. During a class discussion on definitions of even and odd numbers, one student raised her hand and asked how we can determine “quickly” if a number in base 5 is even or odd (e.g., Is 123 base 5 even or odd?). This particular student was making connections to different bases discussed during the first week. Students worked on this problem and came up with several generalizations. We then discussed connections between those generalizations.

**Paul Regier**

Far too many students are afraid of math. I believe the antidote to their fears concerning math is experiencing mathematics by their own creativity. I suspect that in removing creative exploration from teaching mathematics, we run the risk of damaging our students. Just as the processing of modern food removes most of its nutritional value, removing creativity from math robs students of the most significant benefits they can gain from studying mathematics. Although a few students may appreciate the sugar rush of an already processed solution presented to them, it does not nourish them. What they gain does not last. It does not stick with them.

I teach for creativity by thinking creatively myself! When I lesson plan, after I have some kind of basic connection to the material and to past experience, I try to incubate for at least a day before I think about how I’m going to structure the class. Then I ask myself, “How little time can I spend presenting an idea, so that students are motivated and ready to start thinking about it themselves? What do I subconsciously withhold from students’ experiences (the joy of discovering and creating mathematics for themselves) that I can give conscious attention to?” In my experience, it is much easier to facilitate this kind of awareness by having students work together in groups with limited (but carefully planned) guidance and encouragement from me. It’s may be easy to give students quick answers, but this often takes away the student’s creative drive. Thus, I am learning to acknowledge students’ own thinking to be able to better provide opportunities for their own creative discovery.

**Conclusion**

As this blog ends, we hope that our stories serve as the beginning to your adventures in teaching for creativity! The Creativity Research Group has recently been awarded an NSF IUSE Grant (#1836369/1836371), “Reshaping Mathematical Identity by Valuing Creativity in Calculus”. To learn more about how to participate, or to communicate any of your ideas about fostering creativity, email us at creativityresearchgroup@gmail.com.

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In December 2017, the MAA released the * Instructional Practices Guide* (IP Guide), for which I served on the Steering Committee as a lead writer. The IP Guide is a substantial resource focused on the following five topics:

- Classroom Practices (CP)
- Assessment Practices (AP)
- Course design practices (DP)
- Technology (XT)
- Equity (XE)

The IP Guide was designed with the intention of having independent sections be relatively accessible, so reading it from start to finish is not necessarily the best way to use it — I do recommend that everyone begin by reading the Manifesto and Introduction in the Front Matter of the IP Guide. My goal in this article is to provide three suggested starting points for faculty who are interested in using the IP Guide to inform their teaching, since it can be a bit daunting to identify where to start with this document. I want to emphasize that these suggestions are meant to be inspiration rather than prescription. My hope is that this article might be useful as a roadmap for department leaders incorporating the IP Guide for seminars, workshops, or other professional development activities with their faculty.

My belief is that faculty can be effective teachers using many different teaching techniques — there is no single “best way” to teach. Thus, our goal for faculty should be to gradually expand the teaching techniques they are familiar with, in order to create a “teaching toolbox” full of methods, ideas, and activities. With this in mind, I will frame my suggested starting points for the IP Guide based on the level of previous experience a reader has had with different teaching techniques, assessment structures, and course design frameworks.

**For Faculty With Experience Using Mainly “Traditional” Teaching Techniques, e.g. Instructor Lectures, Problem Sets for Homework, In-Class Exams, etc.**

Many readers of the IP Guide will have had their primary teaching experience be with what I refer to as “traditional” methods — by this I mean that the majority of class time is spent in a lecture format (even if it is somewhat interactive), the students are evaluated using problem sets for homework given once or twice per week and using 3-4 in-class exams, and the goals for student learning listed in the course syllabus are a list of content topics to be covered. For faculty who feel that this generally describes their previous teaching experience, I recommend starting with the following sections.

- CP.1.1. Building a classroom community
- CP.1.2. Wait time
- CP.1.3. Responding to student contributions in the classroom
- CP.1.5. Collaborative learning strategies
- AP.1. Basics about assessment
- AP.3. Summative assessment
- AP.4. Assessments that promote student communication
- DP.1.1. Questions for design
- XE.4. Attending to equity

The selections from the Classroom Practices chapter all focus on short and simple teaching techniques that can be used in any class. For example, CP.1.1. discusses how to handle the first day of class in a student-centered manner, and then the following CP sections build on this by providing examples of techniques such as think-pair-share or paired board work that use 3-10 minutes of class time to engage students. These are natural first steps for instructors who are most familiar with teaching exclusively via lecture.

The selections from the assessment chapter focus on two goals. The first goal is for instructors to be able to increase the quality of exams that are given to students, which is the focus of AP.3. on Summative Assessment. Second, the example assignments in Section AP.4. provide an opportunity for faculty to have students reflect on their experiences in their courses, both from a mathematical and personal perspective.

The final two sections regarding course design and equity are intended to spark reflection on the part of faculty members. For example, section DP.1.1. contains a list of important questions for faculty to ask themselves prior to the start of a course, and section XE.4. provides three concrete examples of ways in which faculty can and should include equity considerations in their teaching.

**For Faculty With Experience Using Some Non-Traditional Teaching Methods, e.g. Think-Pair-Share, In-Class Group Work, Reflective Essays, Lab Reports, Take-Home Exams, etc.**

For those faculty who have some experience with non-traditional teaching methods, I recommend starting with the following sections.

- CP.1.7. Developing persistence in problem solving
- CP.1.8. Inquiry-based teaching and learning strategies
- CP.1.9. Peer instruction and technology
- CP.2.1. Intrinsic appropriateness: what makes a mathematical task appropriate?
- CP.2.2. Extrinsic appropriateness
- CP.2.6. Communication: Reading, writing, presenting, visualizing
- AP.2. Formative assessment creates an assessment cycle
- AP.3.3. Creating and selecting problems for summative assessment
- AP.4. Assessments that promote student communication
- DP.1. Introduction to design practices
- DP.2.2. Designing mathematical activities and interactive discussions
- XE.2. Definitions (in equity section)

For the selections from the Classroom Practices chapter, these sections focus on more “ambitious” teaching techniques that are more approachable once an instructor feels comfortable with smaller-scale methods such as think-pair-share. The three subsections from CP.2. are not discussions of techniques per se, but rather general frameworks through which to consider questions about task design for student activities.

The three sections from the assessment chapter indicated here provide three perspectives on student learning. The first is through the lens of formative assessment, and AP.2. provides both research-based frameworks for defining formative assessment and examples of how to implement formative techniques. Complementing this, AP.3.3. provides a taxonomy for instructors to use when writing exams (summative assessments) to evaluate student learning. Finally, as mentioned in the previous set of starting points, the example assignments in Section AP.4. provide an opportunity for faculty to have students reflect on their experiences in their courses, both from a mathematical and personal perspective.

For instructors with more experience using non-traditional teaching methods, the idea of framing their activities within a more coherent overall course design should be a natural progression. Because it is most natural to begin to engage with deeper consideration of course design from the perspective of activities and discussions, DP.1. an DP.2.2. are the recommended sections to begin with. Supporting an increased awareness of the role of course design, section XE.2. discusses important aspects of equity that should play a prominent role in every course.

**For Faculty With Experience Using Multiple Pedagogical Techniques and Strategies, e.g. Inquiry-Based Learning, Online Courses, Service Learning, Explicitly Designed Formative and Summative Student Assessment, Course Projects, Universal Design for Learning, etc.**

For faculty who have extensive experience with a range of teaching methods, the sections of the IP Guide that will likely be of the most interest are those that dive into some of the theoretical frameworks for teaching and learning, such as the following.

- CP.2. Selecting appropriate mathematical tasks
- AP.2. Formative assessment creates an assessment cycle
- AP.5. Conceptual understanding: What do my students really know?
- DP.2. Student learning outcomes and instructional design
- DP.4. Theories of instructional design
- XE.Equity in practic

The extended Classroom Practices section on selecting appropriate mathematical tasks provides a broad discussion of theoretical frameworks such as Vygotsky’s ZPD theory and cognitive load theory, along with illustrative examples. Further, the discussion of Error Analysis in subsection CP.2.7. provides a refined perspective on how to handle student slips, errors, and misconceptions. The Assessment Practices section on formative assessment provides research-based frameworks for defining formative assessment and examples of how to implement formative techniques. The Assessment section on conceptual understanding is a detailed explanation of concept inventories, including examples and a discussion of how to incorporate items from concept inventories into assessment schemes.

Finally, the discussion of theories of instructional design in DP.4. introduce the reader to the frameworks of backward design, realistic mathematics education, and universal design for learning. Since these three topics are far too extensive to cover within a single section, references are given for further reading. The entire section on equity should be read by every instructor, as the ideas and language from studies of equity in mathematics education are not always widely-known; however, they are especially critical for faculty who have significant experience using ambitious classroom practices but who have not had much experience with explicit equity considerations in their classes.

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Dan Meyer is as close as we can get to a rock star in the world of mathematics education. These days, Dan is known for many things: __3-act tasks__, __101 Q’s__, __Desmos__, __NCTM’s ShadowCon__, to name just a few. But he initially rose to prominence on the basis of a __TEDTalk__ that he gave as a high school teacher in 2010 (I recommend it highly). In his talk, he speaks directly to math teachers, and gives one simple piece of advice.

Be less helpful.

Teachers often understand their job as involving “help,” in some form or another. Most of us would like to think that we help students learn. Dan’s advice can therefore be understood as a challenge to a core aspect of our identity. And yet, the advice has become a __mantra of sorts__ for teachers at all levels.

What does it mean to be less helpful, and why should that be a goal?

In his TED Talk, Dan shows how interesting mathematical questions are often surrounded by scaffolding:

Image credit: __https://www.ted.com/talks/dan_meyer_math_curriculum_makeover__

This scaffolding, in the form of mathematical structure and sequences of steps, works to obscure the interesting question (“which section is the steepest?”, buried in question 4). Students are asked to apply a mathematical structure (a coordinate plane) and accomplish sub goals (e.g., find vertical and horizontal distances) before they even know the goal (find the steepest section). Collectively, the scaffolding works to make math seem like an exercise in rule-following, rather than an opportunity to explore and make sense of interesting questions.

More perniciously, the scaffolding takes away much of the mathematics. Hans Freudenthal believed that structuring was at the heart of mathematics. In this problem, the scaffolding has already structured the problem. There is no structuring—and therefore no mathematics—left for the student to do. By presenting students with a ready-made structure for getting an answer, “mathematics” is reduced to answer-getting. To be less helpful means to remove the scaffolding, and to let students do mathematics. That is hard work for a teacher. In this post, I’ll give an example from my own practice, and discuss how I’m learning to be less helpful.

I have always considered myself a “reform-oriented” teacher—at least in content courses, my classroom practice adheres closely to the “conceptual math” described by Amanda Serenevy in an __earlier post on this blog__. In calculus, I have followed many of the recommendations from the calculus reform movement of the 1990s, including the “rule of four”: concepts and procedures should be explored from graphical, numerical, algebraic, and verbal perspectives. For example, early in the year, before any formal procedures are introduced, students explore derivatives graphically and verbally:

Following the rule of four (actually, I also use a lot of models, so let’s say, the rule of five) we use many strategies to develop differentiation rules and make connections between them. We use numerical analysis (looking at successive differences) to make conjectures about derivatives of polynomials. We couple this with verbal and graphical analyses (such as the examples above), which we also used to make conjectures about the derivatives of sin(x) and cos(x). We use first principles (algebraic) to prove these conjectures about elementary functions. We use models (area and see-saw, respectively) to develop the product rule and chain rule. We use algebraic strategies to write unknown situations in terms of known situations. This allows us to develop new rules, e.g., the quotient rule by rewriting a quotient as a product: and to find derivatives of new functions (e.g., developing the derivative of tan x by writing it as sin x / cos x . Later in the course we develop the logarithm as an accumulator function , and thus we can use the fundamental theorem to find the derivative, .

Finally, we come to a point in the course where it’s time to find the derivative of exponential functions. This is one of the crown jewels in a first-semester calculus course and I want to ensure that students understand it conceptually. As a first-year teacher, my plan was as follows:

- Using the rule of four, students would explore the derivative of y=2
^{x}from graphical, numeric, algebraic, and verbal perspectives.

- Putting these perspectives together, students would conjecture that the derivative of y=2
^{x}is itself an exponential function—in fact, it is a vertical compression of the original function, with a scale factor of about 0.69. That is:

- Do a similar analysis for y=5
^{x}. Here we come to a similar conjecture, only now the vertical scaling is a stretch, rather than a compression:

- Putting these two conjectures together leads to three interesting observations.

**Observation:**The derivative of an exponential function seems to be a vertically-scaled exponential function with the same base.**Observation:**Some scale factors are <1, and they*compress*the derivative, relative to the initial function (e.g., the scale factor for y=2^{x}). Other scale factors are >1, and they*stretch*the initial function (e.g., the scale factor for y=2^{x}).**Observation:**Bigger bases seem to have bigger scale factors. - These observations, in turn, lead to a very interesting question. If the derivative of 2
^{x}is a compression of the initial function (a<1), and the derivative of 5^{x}is a stretch of the initial function (a>1), and if the scale factor generally grows as the base grows, then reasoning by continuity tells us that there must be a base between 2 and 5 where the scale factor is exactly 1. What is this magical base? - Using technology, explore this question. Can we find such a base?

This plan hinges on students developing a good sense of the behavior of the derivative of 2^{x}, that is, steps 1 and 2. Thus, for the rest of this post, I’ll limit my discussion to these steps.

In my first attempt at steps 1 and 2, students competed this worksheet in groups. The worksheet guides students to consider the derivative of 2^{x} from a variety of perspectives.

Throughout, students are asked to reflect on or explain their findings. The worksheet culminates with a prompt to make a conjecture about the derivative of 2^{x}:

Can you guess what happened at this point?

Well, let me tell you what *didn’t *happen. Students didn’t take a holistic perspective on the work that they had done. They didn’t say things like, “from step 1, I see that the graph of the derivative is a vertical compression of 2^{x}. From step 2, I see that the ratio of the derivative to the initial function is about 0.69. From step 3, I see that f ′(x) is some constant times 2^{x}, and I see that the constant is 0.69. Taken together, I conjecture that the .”

Instead, students treated question 8 as if it were independent from the rest of the worksheet. They generally conjectured that the derivative of 2^{x} was x⋅2^{x-1} (i.e., they used the power rule). Even though students had just completed a graphical, numeric, analytic, and verbal analysis of the derivative of 2^{x}, they did not draw on this analysis in answering the big question.

I tried this worksheet for a few years, each time with the same result. What I finally understood was that I needed to be less helpful. Even though I saw the mathematical structure in the worksheet, students didn’t. This is because *I *was the one who was doing the structuring. I created a cookbook-style “recipe for mathematics,” and in doing so, I had done all of the mathematics. The only thing that was left for students to do was to follow steps and get answers.

After a few of years of this worksheet, always with the same good intentions, always with the same disappointing results, I finally took Dan’s advice to heart. I decided to be less helpful.

Dan suggests that we *start* with the big question rather than burying it under a pile of scaffolding. My big question was, what is the derivative of 2^{x}?

To be less helpful, I thought, “why not just ask it?”

So that’s what I did. I put the question on the board, and asked students to explore it. I was really nervous, because I had never been this unstructured before. Structure is comforting to a teacher. Removing it is risky.

Now, can you guess what happened?

Actually, the same thing as when students got to question 8 on the worksheet. They applied the power rule and thought they were finished. Even though we had previously used graphical, numeric, and analytic approaches for finding derivatives of unknown functions, students did not use them here. In fact, they did not even recognize that I had asked a “big question.” Rather, my question seemed to them to be a straightforward application of a known rule.

I learned that I needed to set up the question better.

In math education, we call the setup to a problem the “launch.” Recently researchers have identified the launch as a key phase in a successful problem-solving lesson. When done well, the launch prepares students to engage in a complex problem. When done poorly, the launch either gives away too much, decreasing the cognitive demand of the problem, or doesn’t provide enough grounding, limiting students’ ability to engage in the problem.

A good launch should do three things:

- Help make sure that all students are familiar with the context and content of the problem
- Produce a perplexing question
- Remind students about relevant prior knowledge without “giving away” the solution or solution strategy

Here is how my launch works now:

- Just ask it. Ask students to find the derivative of y=2
^{x}. - When the majority of the class produces x⋅2
^{x-1}as the derivative, have them check this by graphing their conjecture against the numerical derivative produced by their calculators. - When the graphs do not match, this produces a perplexing question: What’s going on here, and what is the
*actual*derivative of y=2^{x }? - Before asking students to explore this question, I give a brief presentation that summarizes our previous strategies for finding derivatives of unknown functions: numerical analysis, graphical analysis, algebraic analysis using first principles, and algebraic analysis by rewriting the unknown function as a combination of known functions.
- Just ask it, redux. What is the derivative of 2
^{x}?

In my experience, this sequence accomplishes the objectives of a good launch. Students now see the question about the derivative of 2^{x} as a “big question,” worthy of exploration, and they have tools that they can use to explore that question. At the same time, I haven’t told them what to do or what the answer is, and so there is still plenty of mathematics to be done by students. The evidence of a successful launch is deep student engagement in the problem, and that has been my experience with this launch.

Groups go at the problem in different ways.

- Some groups use numerical analysis, and explore successive differences. They observe that the n
^{th}difference is always in the form 2^{k}. - Some groups use technology to find numerical derivatives at particular points. They notice patterns in the derivative, for example, that as x changes additively, y changes multiplicatively.
- Some groups use graphical analysis, and notice that the derivative graph seems to be either a horizontal translation of -0.53 of the initial function, or a vertical compression, of the initial function by a factor of 0.69.
- Some groups use algebraic analysis to find the derivative from the limit definition (using graphical or numeric analysis to find ).
- Some groups rewrite the function in terms of known functions. They express y=2
^{x}in logarithmic form and use implicit differentiation to find dy/dx.

I don’t get every approach every year, but each year I get at least a few of these approaches. I have students share their approaches, and we make connections between them as a class. By the end of our class discussion, students are ready to conjecture that , and I can continue with the lesson. All together, this part of the lesson takes about 50 minutes: 10 minutes for the launch, 20 minutes for group exploration, 10 minutes for group presentations, and 10 minutes for me to summarize and connect approaches.

I’m learning that when I’m less helpful, students learn more. This is because rather than going through steps that are initially disconnected from a larger goal, students are consciously engaged in exploring a big question right from the beginning.

I’m learning that *a lot *of mathematics materials are cookbook-style “recipes for mathematics,” much like my initial worksheet was^{1}. I’m learning to use these materials to identify the key question, and then be less helpful by removing much of the scaffolding.

I’m learning that being less helpful is risky. Being less helpful means giving some of the control of the class over to students. Not all students will use every approach in their groups (unlike in my worksheet, where every student did the same thing). Instead, groups have discretion about the strategies that they use. And because the class discussion is dependent on the group work that came before it, the particular strategies that one class sees may differ from those that another class sees. There is less structure and more heterogeneity.

I’m learning that heterogeneity can be a resource. I *depend *on different groups taking different approaches. This allows me to structure our follow-up discussion so that students can see the connections between graphical, numeric, and algebraic approaches.

I’m learning that it actually takes a lot of work to be less helpful:

- Sometimes it takes work to develop the right question—a question that is perplexing, that is challenging yet solvable, and that is likely to lead students to develop the targeted mathematics. In this example, the question was somewhat obvious, but it is not always so easy.
- It takes a lot of work to develop the right launch.
- It takes work to sequence student strategies into a coherent narrative, including analyzing unforeseen strategies on the fly and incorporating them into the emerging narrative.

I’m learning that being less helpful doesn’t mean that I can’t help students. During group work, I often prompt students to develop their strategy further or connect it to a different strategy. Sometimes I’m pretty explicit in this prompting. For example, I’ll suggest that students try to make a table. Sometimes, if no group produces a particular strategy, I’ll introduce it in the class discussion.

I’m learning that being less helpful isn’t a magic bullet—not all of my students are engaged all of the time or learn all I’d want them to learn. But when I have a good question and a good launch, more students are more engaged in the activity and learn more of the mathematics than when I do the structuring for them.

I’m learning that one can learn to be less helpful. These days, it doesn’t feel so risky to ask the big question about 2^{x}, because I have a very good sense of what will happen. And it doesn’t feel quite so risky to ask new and different big questions. I know that I’ll fail sometimes (as in when I just ask the big question with no launch), but I also know that this is part of the process.

1 This includes many so-called “reform-oriented” materials. It’s likely that many reformers share my goal of engaging students in activity, and are trying desperately to shed the urge to explain—the traditional lay-it-all-out-and-then-let-the-student-practice approach, in which none of the puzzling-through is left for the student. However, it seems that often reform-oriented materials replace traditional explanations with a series of leading questions. Much like in my initial approach, these questions do all of the structuring for students and leave little mathematics for students to do.

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In the early part of this millennium, when the math wars were raging, I gave some testimony to the National Academies panel that was working on the report Adding it Up. Somewhat flippantly I said that which side of the math wars you were on was determined by which you were paying lip service to, the mathematics or the students. I was recently invited to give a plenary address at the ICMI Study 24 in Japan on school mathematics curriculum where I decided to expand on this remark, because I think it is worth going beyond the flippancy to map out an important duality of perspectives in mathematics education. What follows is an edited summary of what I said in that address.

In my address I talked about two different stances towards mathematics education: the sense-making stance and the making-sense stance. The first manifests itself in concerns about mathematical processes and practices such as pattern seeking, problem-solving, reasoning, and communication. It is an important stance, but it carries risks. If mathematics is about sense-making, the stuff being made sense of can be viewed as some sort of inert material lying around in the mathematical universe. Even when it is structured into “big ideas” between which connections are made, the whole thing can have the skeleton of a jellyfish.

I propose a complementary stance, the making-sense stance, which carries its own benefits and risks. Where the sense-making stance sees a process of people making sense of mathematics (or not), the making-sense stance sees mathematics making sense to people (or not). These are not mutually exclusive stances; rather they are dual stances jointly observing the same thing. The making-sense stance views content as something to be actively structured in such a way that it makes sense.

That structuring is constrained by the logic of mathematics. But the logic by itself does not tell you how to make mathematics make sense, for various reasons. First, because time is one-dimensional, and sense-making happens over time, structuring mathematics to make sense involves arranging mathematical ideas into a coherent mathematical progression, and that can usually be done in more than one way. Second, there are genuine disagreements about the definition of key ideas in school mathematics (ratios, for example), and so there are different choices of internally consistent systems of definition. Third, attending to logical structure alone can lead to overly formal and elaborate structuring of mathematical ideas. Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.

Student struggle is the nexus of debate between the two stances. It is possible for those who take the sense-making stance to confuse productive struggle with struggle resulting from an underlying illogical or contradictory presentation of ideas, the consequence of inattention to the making-sense stance. And it possible for those who take the making-sense stance to think that struggle can be avoided by ever clearer and ever more elaborate presentations of ideas.

A particularly knotty area in mathematics curriculum is the progression from fractions to ratios to proportional relationships. Part of the problem is the result of a confusion in everyday usage, at least in the English language. In common language, the fraction a , the quotient a ÷ b, and the ratio a : b, seem to be different manifestations of a single fused notion. Here, for example are the mathematical definitions of fraction, quotient, and ratio from Merriam-Webster online:

Fraction: A numerical representation (such as 3/4, 5/8, or 3.234) indicating the quotient of two numbers.

Quotient: (1) the number resulting from the division of one number by another

(2) the numerical ratio usually multiplied by 100 between a test score and a standard value.

Ratio: (1) the indicated quotient of two mathematical expression

(2) the relationship in quantity, amount, or size between two or more things.

The first one says that a fraction is a quotient; the second says that a quotient is a ratio; the third one says that a ratio is a quotient. These definitions are not wrong as descriptions of how people use the words. For example, people say things like “mix the flour and the water in a ratio of 3 .”

From the point of view of the sense-making stance, this fusion of language is out there in the mathematical world, and we must help students make sense of it. From the point of view of the making-sense stance, we might make some choices about separating and defining terms and ordering them in a coherent progression. In writing the Common Core State Standards in Mathematics we made the following choices:

(1) A fraction a as the number on the number line that you get to by dividing the interval from 0 to 1 into b equal parts and putting a of those parts together end-to-end. It is a single number, even though you need a pair of numbers to locate it.

(2) It can be shown using the definition that a/b is the quotient a ÷ b, the number that gives a when multiplied by b. (This is what Sybilla Beckman and Andrew Isz´ak call the Fundamental Theorem of Fractions.)

(3) A ratio is a pair of quantities; equivalent ratios are obtained by multiplying

each quantity by the same scale factor.

(4) A proportional relationship is a set of equivalent ratios. One quantity y is proportional to another quantity x if there is a constant of proportionality k such that y = kx.

Note that there is a clear distinction between fractions (single numbers) and ratios (pairs of numbers). This is not the only way of developing a coherent progression of ideas in this domain. Zalman Usiskin has told me that he prefers to start with (2) and define a/b as the quotient a ÷ b, which assumed to exist. One could then use the Fundamental Theorem of Fractions to show (1). There is no a priori mathematical way of deciding between these approaches. Each depends on certain assumptions and primitive notions. But each approach is an example of the structuring and pruning required to make the mathematical ideas make sense; an example of the making-sense stance. One might take the point of view that the distinction between the sense-making stance and the making-sense stance is artificial or unnecessary. A complete view of mathematics and learning takes both stances at the same time, with a sort of binocular vision that sees the full dimensionality of the domain. However, this coordination of the two stances does not always happen. Rather than provide examples, I invite the reader to think of their own examples where one stance or the other has become dominant. This has been particularly a danger in my own work in the policy domain. I hope that spelling out the two stances will contribute to productive dialog in mathematics education, allowing for conscious recognition of the stance one or one’s interlocutor is taking and for acknowledgement of the value of adding the dual stance.

]]>*From whence this blog*

Nearly twenty years ago Paul Lockhart wrote a brilliant essay, *A Mathematician’s Lament*^{[1]}, on the parlous state of mathematics education. In it, Lockhart laments that mathematics education does not celebrate mathematics as an art and as an important part of human culture. I write this essay in the same spirit, lamenting that mathematics education does not do well in preparing our students to use their mathematical skills to model the world they encounter in the practical, economic, policy and social aspects of their lives.

I have spent many years trying to understand why so many people seem to have difficulty with mathematics. Many people have a distaste for the subject and will go a long way to avoid engaging any use of their mathematical knowledge.

Elementary and secondary schools, the social institution to which we entrust the education of our young, present the subject of mathematics as a “right answer” subject. What other inference is to be drawn from questions like “what is the sum of 34 and 28?” and then “what is the sum of 41 and 24?” Seldom do we see problems like “Make up an addition problem with two whole numbers the answer to which lies between the sum of 34 and 28 and the sum of 41 and 24.” One could follow that problem with “How many such problems are there and how do you know?” And follow that with “Suppose you could use integers rather than whole numbers, how many such problems are there and how do you know?” It is no surprise then that the public at large thinks of mathematics as a body of knowledge in which any question has a unique correct answer and an uncountable number of incorrect ones. On the other hand, I claim that there are important cognitive benefits to be derived from posing problems with multiple demonstrably correct answers and multiple demonstrably incorrect answers.

I have never thought of mathematics as a “right answer” subject. I have always believed that mathematics provides a way of approaching the world and making sense of what one sees, hears and feels there. Indeed this is precisely the reason most often given for the inclusion of mathematics in the curriculum of our K-12 schools. Mathematics offers the promise of allowing people to make reasoned decisions about their daily lives.

Mathematics provides a set of tools that ** can** help to confront the extraordinary complexity of phenomena that surround us. Although the things that interest us and the relationships among those things are inherently complex, mathematics can clarify essential elements in the surround. Mathematical models can often provide great insight into the way our world works. That having been said, our ability to deal with the complexity of our world by using our mathematical models is complicated by the constant need for –

- the exercise of
about the conclusions we draw from our models, and*judgment* - the inherent
of our senses [and their extensions, e.g., telescopes, microscopes, hearing aids, etc.]*imprecision*

*Judgment and Imprecision*

Counting and measuring are essential to all modeling. Even with these simplest acts of modeling, normally considered as an integral part of the K-12 mathematics curriculum, uncertainties about consistency and validity and the need for judgment are present.

Judgment is a serious matter in the act of ** counting**. Does the head count in a ball park include only paying customers? People with complimentary tickets? Attendants, vendors, players, coaches? Clearly, purpose needs to be taken into account.

Imprecision and judgment are serious matters in the act of ** measuring**. Consider the act of making simple measurements, even of such quantities as length or time. No measuring instrument can resolve the attribute it is being used to measure with infinite precision and as a consequence the best we can do is to assign a range of rational values as a magnitude. Additionally, judgment is required in deciding the extent of the attribute to be measured.

Further, judgment is required is assigning magnitude to the distance from Boston to San Francisco. An appropriate choice of unit might be the kilometer. It would hardly make sense to report the distance in millimeters, leaving aside all issues of from where in Boston to where in San Francisco. Reporting that distance in light-years would display a similar lack of judgment^{[4]}.

The need for judgment extends well beyond counting and measuring.

Suppose for example one needs to construct a cube of whose volume is as close as possible to 8 cm^{3}. Deferring the issue of precision for the moment, our mathematical model of a cube tells us that a side length of 2 cm will work. But it also tells us that side lengths of –1+3^{1/2}*i* and –1–3^{1/2}*i* would do as well*. *The decision to discard the extraneous roots of the equation *x*^{1/3} = 1 is a judgment that is made based on the context. The craftsperson fashioning the cube from a block of steel will certainly be able to make the required physical choice and not either of the permissible mathematical choices.

Similarly, our mathematical model of how a ball thrown on a flat Earth moves tells us that there are two times that the ball is on the ground. Newton’s laws of motion have two correct mathematical solutions – the time in the *future* when the ball will hit the ground and the time in the *past* when the ball was launched [neglecting the fact that the ball was almost certainly not launched from ground level]. We are likely to be interested in only one of these times.

*Some postulates about essences*

At least for the sake of this discussion please allow me some postulates about essences –

The essence of

…** learning** is posing for oneself a provocative next question

…** teaching** is posing for students a provocative, engaging next question at the proper moment

…** education** is learning to ask of the world “what is this a case of?” and “what if not?”

…** mathematics** is making, exploring, proving and disproving conjectures

The reader will note that all these essences center on the posing of questions and the making of conjectures. This is particularly true of mathematics. None other than Georg Cantor once said, “In mathematics the art of proposing a question must be held in higher esteem than solving it.” By posing problems, David Hilbert, at the International Congress of Mathematicians in Paris in 1900, set the course of research for much of the mathematics of the twentieth century.

*A physicist’s view of posing problems*

Because as physicists we are engaged in modeling the world, albeit with mathematical tools, we do not always have the luxury of asking questions that have unique correct answers. Our problems, which we pose both to our students and to ourselves, are about models, their predictions and their consequences.

Models are never “correct” – at best they are “adequate.” Unlike an unsolved problem in mathematics, models in physics cannot ever be proven to be “correct.” This observation suggests, at least to this physicist, two guidelines about the posing of problems in mathematics. They are –

- think long and hard before posing a problem that has a single correct answer, and
- pose problems for which answers can be deemed to be “good enough.”

The reason underlying the first guideline is that a unique correct answer can only address the internal consistency of a model but not its validity or utility.^{[6]}

The reason underlying the second guideline speaks to the adequacy of a model—no model can take into account the full complexity of the phenomena being modeled. Judgment must be applied when deciding whether an answer to a problem is “good enough.” Perhaps the most widely known examples of the use of this guideline are Fermi estimation problems.^{[7]}

How might we think about mathematics when posing problems that follow these guidelines?

*Objects and Actions – Measures and Models*

The idea of mathematical ** objects** and mathematical

The power of this idea derives from the fact that essentially all of the languages spoken by people have utterances that are composed of a noun phrase and a verb phrase. A noun phrase tells us about ** objects** and their properties. An associated verb phrase tells us about the

** Measures** are quantifications of attributes of nouns. Examples include the weight of a person or a gallon of milk, the circumference of a person’s waist or the height of a bookcase, the surface area of the body or that of a mountain lake. Other such measures are the volume of a human body or that of a bottle of wine, the length of a person’s lifetime or that of a movie, the number of a person’s red blood cells or the number of people in the US on any given day.

The measures cited here^{[8]} are basic ones. We are equipped by nature with sensory tools to assign some sort of magnitude to these measures or, at least, to order a collection of objects having that attribute.^{[9]}

These fundamental measures can be composed with one another to make composite measures. Composing a distance traveled with a time interval yields velocity as a composite measure. Composing velocity with a time interval yields acceleration as a further composite measure. Composing the number of people in a city with its surface area yields a measure we usually call population density.

** Models** are assertions of relationships among measures. Newton asserted a relationship between the measure of acceleration and the magnitude of the push or pull causing that acceleration. Mendel asserted a relationship between the color of the seeds of one generation of pea plants and the color of the seeds of cross-bred members of the next and subsequent generations.

The reason schools include mathematics in the K-12 curriculum^{[10]} is that the judicious use of mathematical models can help our students make sense of what they see and hear in the world around them. Such use offers the promise of allowing people to make reasoned decisions about their daily lives.

If, indeed, the end goals of our teaching mathematics and our students learning mathematics is to help them use mathematical models in their lives, we must put greater emphasis on modeling in our teaching—and **not** at the expense of reliably and correctly executed mathematical manipulations.

In a future blog, I hope to discuss one approach to doing just that.

*Endnotes*

^{[1]} Reprinted in MAA Online, March 2008, Devlin’s angle

^{[2]} Does one determine a person’s weight before or after a haircut? Does the surface area of a mountain lake include [all, part, none] of the runoff stream?

^{[3]} For example, the π that appears in C = πd is defined as the ratio of a circumference to a diameter, but the magnitude of the number, itself, is not computed from measurements. A measurement of the circumference yield a value, C ± ΔC, and a measurement of the diameter yields a value, d ± Δd. The ratio of these two values (C ± ΔC)/( d ± Δd) is a bounded range of rational numbers. The definition C = πd expresses a ** model** of the relationship between circumference and diameter, perhaps inferred initially from many measurements of circumferences and associated diameters but then determined by logical reasoning.

^{[4]} I recall an incident in which a sixth grade student who fully understood the issue, displayed this understanding in an extraordinary way. I had asked him to formulate an estimation problem. With a huge grin on his face he said, “What is the average size of a postage stamp in square miles?”

^{[5]} Proving and disproving each have a different meaning in mathematics than they do, for example, in law.

^{[6]} A train is normally thought of as a rigid body. Thus, in this model, the caboose begins to moves the instant the locomotive does. Rigid bodies, inextensible strings, point masses are all instances of models that are valid enough and useful enough for many purposes. In the idealized world of such models, problems with single, correct answers can be, and often are, posed.

^{[7]} For example, how long does it take a person to eat his/her own weight in food?

^{[8]} Length, area, weight, volume, time, number.

^{[9]} Measures can be defined within mathematics itself. Consider a collection of rectangles. People of all ages are willing to grant that a 5 x 6 rectangle is ‘squarer’ than a 2 x 15 rectangle. Thus one can pose the problem of defining a measure of ‘square-ness.’ Having done this with both elementary school and college students, I can attest to the fact that many reasonable answers are possible.

^{[10]} I have written elsewhere on the topic of the reasons for the inclusion of mathematics in the school curriculum. See Can Technology Help Us Make the Mathematics Curriculum Intellectually Stimulating and Socially Responsible?, *International Journal of Computers for Mathematics Learning*, 1999, Vol. 4, Nos. 2-3, pp. 99-119.

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How is \(0^0\) defined? On one hand, we say \(x^0 = 1\) for all positive \(x\); on the other hand, we say \(0^y = 0\) for all positive \(y\). The French language has the Académie française to decide its arcane details. There is no equivalent for mathematics, so there is *no one* deciding once and for all what \(0^0\) equals, or if it even equals anything at all. But that doesn’t matter. While some definitions are so well-established (e.g., “polynomial”, “circle”, “prime number”, etc.) that altering them only causes confusion, in many situations we can define terms as we please, as long as we are clear and consistent.

Don’t get me wrong; the notion of mathematics as proceeding in a never-ending sequence of “definition-theorem-proof” is essential to our understanding of it, and to its rigorous foundations. My mathematical experience has trained me to ask, “What are the definitions?” before answering questions in (and sometimes out of) mathematics. Yet, while we tell students that the definition needs to come before the proof of the theorem, what students apparently hear is that the definition needs to come before the idea, as opposed to the definition coming from the idea.

**Why definitions?**

What is a definition anyway? Or rather, what gets defined? We could make a special name for the function that maps \(x\) to \(5x^{17} – 29x^2 + 42\), but we don’t. On the other hand, we give the name “sine function” to \(\sin(x)\), the ratio of the length of the side opposite an angle with measure x to the length of the hypotenuse of a right triangle. We give a name to the sine function, even though it takes much longer to describe than \(5x^{17} – 29x^2 + 42\); in fact, we give it a name in part precisely *because* it takes longer to describe. If we need to refer to \(5x^{17} – 29x^2 + 42\), it’s not that hard, but we do not want to have to write down that definition of sine every time we use it in a statement or problem. We give definitions to ideas for two related reasons:

**Brevity:** It’s clearly easier to write “\(\sin(x)\)” instead of the huge sentence above. Further, packing this idea into a single word helps make it easier to chunk ideas in an even longer statement, such as a trigonometric identity.

**Repetition:** If we have to use the same idea more than once, then giving it a compact name increases the efficiency described above that much more. Sometimes an idea repeats just locally, within a single argument or discussion, and then we might temporarily give it a name; for instance when finding the maximum value \(x e^{-x}\), we would write \(f(x)=x e^{-x}\), so we could then write \(0 =f'(x)\), but we are only using \(f\) this way in this one problem. On the other hand, the ideas that show up over and over again, in many different contexts, such as \(\sin(x)\) or “vector space”, get names that stick.

This begs the question, “Why do certain ideas, or combinations of conditions, repeat?” Consider “vector space”. The idea of \(R^n\) is clear enough, but of all its properties, why focus on the simple rules satisfied by vector addition and scalar multiplication?

First, because several additional examples have been found that satisfy these rules, such as the vector space of continuous functions, the vector space of polynomials, and the vector space of polynomials of degree at most 5. Second, because once the key properties that make up the definition are identified, we may find that the proofs only depend on those key properties: The Fundamental Theorem of Linear Algebra, for instance, is true for arbitrary finite-dimensional vector spaces, so we don’t need a separate proof for \(R^n\), for polynomials of degree at most 5, etc. (Purists may argue that all finite-dimensional vector spaces of the same dimension are isomorphic, but this isomorphism is defined in terms of vector addition and scalar multiplication, just reinforcing the significance of those operations.)

**Choices**

But there are often still choices to be made. Must a vector space include the zero vector, or could it be empty? (Is the empty set a vector space)? For that matter, since vectors are often described as being determined by “a direction and a magnitude” and the zero vector has no direction, is the zero vector even a vector? The answers to these questions are no and yes, respectively, but why? The zero vector is a vector, because it is so helpful for a vector space to be a group under addition, which requires an identity element. (I know — this only takes us back to why are groups defined the way they are. Let’s just take this as a piece of evidence for why groups are an important definition.)

As for the empty vector space, there’s nothing inherently wrong with it, except perhaps for the need for a zero vector as discussed above. (This also takes us back to why groups are not allowed to be empty. Let’s stick to vector spaces for now.) But how would we define the dimension of an empty vector space? How would we define the sum of the empty vector space with another vector space? And then, even if we do make those definitions, how do we reconcile them with this identity?:

\[

\dim (A+B) =\ \dim A\ +\ \dim B\ -\ \dim (A \cap B)

\]

This example shows that, even though we cannot write the proof of a theorem until all the relevant definitions are stated, we do often look ahead at the theorem before settling on the fine points of the definition. At research-level mathematics, we might even modify our definitions substantially to make our theorems stronger, or to deal with potential counterexamples. (For more details on this, read Imre Lakatos’ classic Proofs and Refutations [1].) I will stick to smaller cases where we adjust definitions mostly just to make the theorems easier to state.

**More examples**

Why is 1 considered to be neither prime nor composite? When you first learn this, it may seem silly. The definition of prime is so simple and elegant — an integer \(n\) is prime if its only factors are 1 and \(n\) — and 1 seems to fit that definition just fine. Why make an exception? The answer lies in the Fundamental Theorem of Arithmetic, that every integer has a unique factorization. Well, except of course that we could change the order of the factors around; for instance, it makes sense to consider \(17 \times 23\) to be the same factorization as \(23 \times 17\). And also we need to leave out any factors of 1, otherwise we might consider \(17 \times 23, 1 \times 17 \times 23, 1 \times 1 \times 17 \times 23\), … to all be different factorizations. If we take a little extra effort at the definition, and rule out 1 as a prime number, then the theorem becomes more elegant to state.

Is a square also a rectangle? In other words, should we define rectangle to include the possibility that the rectangle is a square, or exclude that possibility? When children first learn about shapes, it’s easier to simply categorize shapes, so a shape could be either a rectangle or a square, but not both. But when writing a careful definition of rectangle, it takes more work to exclude the case of a square than to simply allow it. Similarly, theorems about rectangles are easier to state if we don’t have to exclude the special cases where the rectangle happens to be a square: “Two different diameters of a circle are the diagonals of a rectangle” is more elegant than “Two different diameters of a circle are the diagonals of a rectangle, unless the diameters are perpendicular, in which case they are the diagonals of a square.”

Is 0 is a natural number? It doesn’t really matter; just pick an answer, be consistent, and move on. It’s even better if we can use non-ambiguous language instead, such as “positive integers” or “non-negative integers.” To be sure, mathematics is picky, but let’s not be picky about the wrong things.

Finally, what about \(0^0\)? If you just look at limits, you’d be ready to declare that this expression is undefined (the limit of \(x^y\) as \(x\) and \(y\) approach 0 is not defined, even just considering \(x \geq 0\) and \(y \geq 0\)). And that’s fine. But in combinatorics, where I work, setting \(0^0 =1\) makes the binomial theorem (\((x+y)^n = \sum \binom{n}{k} x^k y^{n-k}\)) work in more cases (for instance when \(y=0\)). And so we simply *declare* \(0^0=1\), at least in combinatorics, even though it might remain undefined in other settings.

(See here for a list of other “ambiguities” in mathematics definitions.)

In each of these examples, there is a human choice about how to exactly state the definition. This is a great freedom. But, to alter a popular phrase, with great freedom comes great responsibility. If you declare \(0^0\) is a value *other* than 1, now you are limiting, not expanding, the applicability of the binomial theorem. And if you want to declare that \(\frac{1}{0}\) has *any* numerical value, you will have to sacrifice at least some of the field axioms in your new number system.

**In the classroom**

The issues that arise with developing precise mathematical definitions is well-known to mathematicians, but we generally don’t share it with our students enough. If we stop hiding this story from our students, then they will see that mathematics is a human endeavor, and that mathematical subjects are not handed down to us from on high. This can be one factor in convincing students that mathematics, even advanced mathematics, is something they can do, that it is not just reserved for other people. And even students who already “get it” will not be turned off — we should not abandon definition-theorem-proof, we can just pay more attention to sharing why each of our definitions is written the way it is. If students know where a definition comes from, what motivated it, and why we made the choices we did, they may have a better chance of making sense of the idea instead of memorizing the string of words or symbols. (See also my earlier blog post, A Call for More Context.)

An anecdote that Keith Devlin tells, near the end of a blog post about mathematical thinking, illustrates the power of crafting the right definition. To summarize much too briefly, his task was to “look at ways that reasoning and decision making are influenced by the context in which the data arises” in a national security setting. His first step was to “write down as precise a mathematical definition as possible of what a *context* is.” When he presented his work to government bigwigs, they never got past his first slide, with that definition, because the entire room spent the whole time discussing that one definition; later he was told “That one slide justified having you on the project.”

We might not have the luxury of spending an entire hour discussing a single definition, but we can still let students in on the secret that the definitions are up to us, and that writing them well can make all the difference.

**References**

[1] Lakatos, Imre. *Proofs and refutations. The logic of mathematical discovery.* Edited by John Worrall and Elie Zahar. Cambridge University Press, Cambridge-New York-Melbourne*,* 1976.