## On Assessment, IV

Last week, I had ended with an interpretation for an A–D grading scale, shown below (here is a link to last week’s post for reference).

I remind you that this scale is not ideal; the purpose was to come up with some system of assigning grades which wasn’t punitive, but rather which motivated students to learn concepts rather than to avoid losing points on exams.

We continue with a discussion of how to use such a system in practice.

“Now let’s consider this in the context of an exam.  The first part of an exam is a skills portion, with, say, ten short problems of roughly equal length.  Expectations for this part of the exam [meaning a grade of B] are seven problems “essentially” correct, and four problems completely correct.  These expectations are written on the exam for students to see.

Are these expectations too low?  Perhaps.  But then an A [refer to the chart below] means eight problems “essentially” correct, with at least five completely correct. Of course we must ask what it means for a problem to be “essentially” correct — but when in doubt, err on behalf of the student.  (Students rarely suggest that their scores be lowered.)

Then grading is actually somewhat easier, and grades can be assigned as follows, with the abbreviations EC and CC meaning essentially correct and completely correct, respectively (for simplicity, a grade of C is assigned for all other cases not accounted for):

Now this eliminates the need for partial credit — but does require a judgment as to what “essentially correct” means.

This also makes grading much easier.  I would suggest that each problem be marked as “EC,” “CC,” or left blank.  Few comments, if any are necessary.  This is the approach I have taken in BC Fast-Track, and it encourages further learning as it leaves the student in the position of needing to work through their mistakes.

I would have students keep a section of their notebooks for exams and revisions, and there they can keep their reworked problems, should they choose to do so.  Then — as I did in BC Fast-Track — students could visit me periodically with their notebooks and I can take a look at their ongoing progress.  This “additional” work, if sufficiently well done, could boost their grade at the end of the semester.

I think this could have the same effect it did in BC Fast-Track — exams were easier and more enjoyable to grade.  But there were more discussions in my office about reworked exams and sources of error that were initiated by the students themselves, and these discussions were not about points, but about concepts.

Now what about the part of the exam which is intended to be more conceptual?  Let us suppose that there are three problems, roughly comparable in length, and of various difficulties.  Then grades might be assigned as follows:

More details about how this would fit in a classroom environment may be found in a later document [I cannot recall which document is being referred to here].  But this system allows for a more qualitative approach to grading.  Performance expectations are also clearer, but such expectations depend critically upon the nature of the problems given.  Moreover, grades are not assigned punitively, but the emphasis is on doing problems completely and correctly.

For an example, below could be a set of ten skills problems and three conceptual questions for a basic assessment on the rules of differentiation.  This would a 70-minute assessment.  Given expectations for completely correct problems, I think this is reasonable.

Skills questions:

1.  Evaluate

$\lim_{x\to\pi/4}\dfrac{\tan x-1}{x-\pi/4}.$

2.  If $f(x)=e^x\cos x,$ find $f^\prime(x).$

3.  Find

$\dfrac d{dx}\dfrac 7{\sqrt{x^3}}.$

4.  Using the quotient rule, find

$\dfrac d{dx}\dfrac{x^3}{\sin x},$

simplifying as much as possible.

5.  Find the derivative of $f(x)=\left(\sin\sqrt x\right)^{\!2}.$

6.  Find the equation of the tangent line to $h(x)=\sec(2x)$ at $x=\pi/6.$

7.  Using a definition of the derivative, find the derivative of $p(x)=x^2-x.$

8.  Assume that $f$ and $g$ are differentiable functions. Find

$\dfrac d{dx}f(g(x^2)).$

9.  Find the derivative of $q(x)=x^2e^x\cot x.$

10.  Let $f$ be the greatest integer function. Using the definition of the derivative, determine whether or not the derivative exists at $x=0.$

Conceptual questions:

1.  Using the product rule, find

$\dfrac d{dx}f(x)(g(x))^{-1}.$

2.  Suppose that the line $y=6x+a$ is tangent to both $f(x)=x^2+b$ and $g(x)=x^3+3x.$ Find $a$ and $b.$

3.  Suppose that $f$ is a differentiable function. Discuss the following limit:

$\lim_{h\to0}\dfrac{f(x+2h)-f(x-h)}{3h}.$

Stay tuned for next week’s final installment of this series on assessment….

## On Assessment, III

Today, I’ll continue with the discussion of assessment I began a few weeks ago.  Last week, I ended with the observation that “The performance of our students determines our expectations, rather than the other way around.”  Before continuing, I just want to make clear the context for that remark.  When I wrote this essay, IMSA was fully funded by the state of Illinois.

What would it look like to legislators — who approved our funding — if you brought in the top students (relative to their local regions), and then they regularly received low grades, or worse yet, flunked out?  Not good.  And because the admissions process was fairly involved, would this indicate a major flaw in this process?

Yes, some students couldn’t handle the high-pressure academic environment.  But I know that in several of my classes, a student needed to work hard not to earn at least a C.  In other words, showing up, handing in homework on time, and doing a reasonable amount of studying for quizzes and exams would usually guarantee a grade of at least a C.

Now is not the time to dive more deeply into this artificial adjustment of expectations, but I did want to mention that this issue is a significant one, even at presumably elite schools for mathematics and science students.

So to continue with the essay….

“Now there is a natural give-and-take between evaluating student performance and setting expectations.  And, of course, the above remarks are nothing but generalizations.  But they illustrate some of the important issues at hand, and may bear fruitful discussion.

Moving to more concrete issues, I believe that the assignment of letter grades on exams in BC Fast-Track [recall, this was the colloquial name for the Honors Calculus sequence] was, on the whole, successful.  Without going into unnecessary detail, the classroom environment was such that the assigned grades were meaningful to the students.  To give a few examples, I assigned a grade of A+ for truly outstanding work, perhaps only a half-dozen times throughout the entire semester.  The students knew this, and so that accolade truly meant something.

Moreover, an A meant something as well.  It was truly rewarding to see the real pride of a student who, used to earning grades in the B range, began to earn the occasional, or perhaps more frequent, A.  Admittedly, students who made it to the second semester were essentially guaranteed a grade of no lower than a B-.  But this seemed to make an A that much more meaningful.

So student exams had two letter grades on them — one for the skills portion of the exam, and one for the conceptual portion of the exam.  No points were assigned, and few comments were made.  Students were expected to rework problems on which they made errors.

I bring up this point because I think this system of assigning grades really did motivate students to learn calculus rather than accumulate points.  This is the critical issue: I suggest that the way we assign grades does little to disabuse many students that taking a mathematics course is about accumulating sufficiently many — or losing sufficiently few — points.

Let’s take a particular example.  The past few semesters, I stopped assigning half-points on assessments [the usual practice at IMSA].  I might forgive a sign error now and then, but too many on a single assessment would warrant a point or two off.

In the past, I simply considered a sign error as a half-point off.  And so it was.  But consider that without being able to occasionally perform fairly involved calculations, it is not possible to become a successful mathematician.  Attention to detail is as important in mathematics as it is in any number of other disciplines, and we try to develop that skill punitively — you don’t attend to detail, and we will take off points.

Of course one might argue that points are given for work well done — but any of us could, I think, agree that when discussing the grading of an exam, it’s how many points off for a particular type of error that is discussed as much as, or even more than, how many points are given for work correctly done.

And so the idea of “partial credit” is born.  Perhaps now is not the place to begin this discussion, but consider that a student might meet expectations (that is, earn a B) without ever having done an entire complex problem on an assessment completely correctly.  (Some teachers have even gone so far as to give no partial credit.  See On Partial Credit, Letter to the Editor, MAA Focus, February 2002, p. 17.)

Why this system of points and partial credit?  One may speculate as to its origins, and there is controversy even now about its use on standardized exams.  But I cannot help feeling that one function of partial credit is that it allows a teacher to defend the assignment of a particular grade.  “Every sign error is a half-point off.  That’s why you got a B+ instead of an A-.   I have to use the exact same scale for everyone in order to be fair.

But doesn’t this simply shift the responsibility for the grade onto a rubric?  I suggest that many of us would feel competent to take a set of calculus exams — with names removed — and within five or ten minutes, separate out all the A papers.  Of course this is subjective — but no less subjective than saying that this problem is worth six points while another is worth ten, or that sign errors are a half-point off, unless, of course, the derivative of the cosine is taken incorrectly, in which case it’s a whole point.

Thus the assigning of points is no more “objective” than giving a letter grade.  As I’m sure that anyone who has graded a complex word problem based on an assignment of points can attest to.  Consider the student who has the entire procedure correct, but because of a few algebra errors, has no intermediate calculation correct.  The problem is worth ten points.  How many points should the student receive?

Well, of course, you say you’d have to see the problem first.  But I say, no.  The student receives a C.  Having no intermediate calculations correct demonstrates — regardless of what else — that the student has not met expectations.

So is it possible to avoid points altogether?  Perhaps.  Consider the following grading system:

The essay continues in the next installment of On Assessment with a discussion of how to implement this system in practice.

## On Assessment, II

Last week, I had mentioned finding an essay I wrote about assessment in an Honors Calculus sequence I had designed.  I took some time to set the stage for this essay — so now it’s time to dive right in!

A caveat:  This essay was written in 2011.  I will attempt to keep as true to the original essay as I can, though I’ll edit for clarity and updated information/links, and will also [add commentary in square brackets].  So here goes!

“To educate is to illuminate the power of ideas.

This, of course, is an ideal — certainly not the last to be articulated, nor perhaps the highest. But putting practicality aside for the moment, how might we take on such a view of education?

Consider the example of planetary orbits.  Tying together history, technology, politics, physics, and mathematics through a discussion of Kepler’s laws is — yes — illuminating. Of course there is no single idea at play here, but consider this: Kepler was perhaps the first astrologer to become an astronomer.  He, in contrast to so many of his contemporaries, asked not only where a particular planet should be on some future date, but why it should be there.  In his mind, predicting the positions of celestial bodies for the purpose of casting horoscopes for royal personages was not enough — he wanted to know why. That, however, would have to wait until Newton and the application of calculus.  And the power of that idea!

There is no need to present further historical examples.  What is important, however, is to move beyond historical ideas and address the issue of the power of a student’s own ideas.  Two semesters of teaching Advanced Problem Solving [a course I taught at IMSA which emphasized the writing of original mathematics problems] have shown me that having students write original problems motivates them to learn.  They are timid at first — for how could they come up with an original problem?  But after some success, students are excited to create — and as a result, learn about a particular topic in a more profoundly personal way than they might have otherwise.  (Interestingly, the writing of original problems as a teaching tool is common in eighth-grade Japanese mathematics classrooms.  See The Teaching Gap by Stigler and Hiebert, ISBN 0-684-85274-8, pp. 6–41.)

Other assessments are more routine: students are informed that they need to be able to solve linear and quadratic equations, as well as graph linear and quadratic functions.  This is perhaps more manageable — the task is well-defined within narrow limits.  But it is rather different in nature than creating an original piece of work.

Abraham Maslow articulates a similar difference in a discussion of growth theory (see Toward a Psychology of Being by Abraham Maslow, ISBN 0-442-03805-4, p. 47).  Psychologically, why do children grow?  “We grow forward when the delights of growth and anxieties of safety are greater than the anxieties of growth and the delights of safety.”   From an educator’s point of view, this implies making the classroom a safe environment for creativity and exploration.

“The opposite of the subjective experience of delight (trusting himself), so far as the child is concerned, is the opinion of other people (love, respect, approval, admiration, reward from others, trusting others other than himself).”  (Maslow, same source, p. 51.)  Who comes to mind is the student afraid of enrolling in BC Fast-Track [the colloquial name of the Honors Calculus sequence at IMSA] because he or she might not earn an A.  [I did have a very capable student who was so concerned about grades, he opted to take the more traditional calculus sequence.  I actually met with his father, who wanted a guarantee that his son would earn an A in the course.  Given the nature of the course, there was naturally no way I could give such a guarantee.]  Or perhaps the student who shuns a difficult course and takes an easier one instead.  Grades are just that — opinions of other people.

The assignment of letter grades is not necessary for learning, but is merely practical for other reasons.  The assignment of letter grades does nothing to illuminate the power of ideas.

A quick Internet search reveals that the assignment of A–F letter grades is a fairly recent phenomenon, not making its way into high schools until the mid-twentieth century.  In the early twentieth century, grades of E (Excellent), S (Satisfactory), N (Needs improvement), and U (Unsatisfactory) were also used.

The question of whether a letter-grade system of evaluation is the best option for a school like IMSA is perhaps a worthwhile one (some argue against grades at all [this was the link, but it is now a dead end:  http://www.alfiekohn.org/teaching/fdtd-g.htm ]).  That is a question which cannot be answered in the short-term, if the question is considered relevant. So the question is this: given that letter grades need to be assigned this semester, what approach should be taken?

What do letter grades mean?

The only consensus I have heard, so far, is that earning a grade of A, B, or C means that a student exceeds, meets, or does not meet expectations.  Of course, this is reminiscent of E, S, and N.  Moreover, a grade of C is passing, so that a student receives credit for a course even if they do not meet expectations.

But this is not what the grades currently mean.  Essentially, points are assigned to hundreds of problems given throughout semester, whether on assignments, papers, quizzes, or exams — and an arbitrary weighted sum of these point assignments is converted to a letter.  (Of course letters may need to be converted to numbers so that they may be used by an online grade calculator to compute a number which is then converted back to a letter.)

What does this mean about meeting expectations?

Well, nothing really.  I would venture to suggest that “meeting expectations” currently means “enough A’s, but not too many C’s.”  Perhaps this is politically necessary, but expedient.  The performance of our students determines our expectations, rather than the other way around.”

## On Assessment, I

While downsizing for a recent move, I decided to sort through dozens of folders (thousands of pages!) of old problems, notes, exams, essays, and other documents generated in the course of twenty-five years of teaching mathematics.  I found an essay written in 2011 which I though might be of some interest.

This essay might be considered a more thorough follow-up to The Problem with Grades, a satire on assessment practices which I’ve shared with you already.  The essential point in that essay was that how we typically assign grades is punitive:  begin with a given number of points (say, 100), punish students for making mistakes (subtract a certain number of points from 100), and then — magically! — the resulting number reflects a student’s understanding.  (If you think that this is in fact a really great way to assess a student’s learning, you probably should stop reading right here….)

The essay I’d like to share here was written in the context of teaching an Honors Calculus sequence I designed while at the Illinois Mathematics and Science Academy (IMSA).  So before diving in, let me take a few moments to set the stage.

We had quite a broad range of students at IMSA — and so when it came to teaching calculus, some students were bored by the “average” pace of the class.  Of course this happens in a typical classroom; the tendency is to teach to the middle, with the necessary consequence that some students fall behind, and others could benefit from a faster pace or more advanced problems.

Now mathematics educators are looking at ways to design classroom environments where students can work more independently, at their own pace, incorporating methodologies such as differentiated learning and/or competency-based learning.  These ideas could be the subject for wide-ranging (future) discussions; but in my case, I was working in an environment where all students did essentially the same work in the classroom, and assessments were fairly traditional.

In addition to creating a faster-paced, more conceptually oriented course, I incorporated two features into this Honors Sequence.  First, I had students write and solve their own original mathematics problems, as well as reflect on the problem-posing process.  I won’t say more about that here, since I’ve written about this type of assessment in a previous blog post.

Second, I experimented with a different way of assessing student understanding on exams.  Each exam had two components:  a Skills portion and a Conceptual portion.  The Skills portion was what you might expect — a set of problems which simply tested whether students could apply routine procedures.  The Conceptual portion was somewhat more challenging, and included nonroutine problems which were different from problems students had seen before, but which did not require any additional/specialized knowledge to solve beyond what was needed to solve the Skills problems.  (As part of the essay I will share, examples of both types of problems will be included.  I also discuss these ideas in more detail in a previous post, On Grading.)

Moreover, I graded these problems in a nonstandard say:  each problem was either Completely Correct (CC), meaning that perhaps aside from a simple arithmetic error or two, the problem was correctly solved; Essentially Correct (EC), meaning that the student had a viable approach to solving the problem, but was not able to use it to make significant progress on the problem; and Not Correct (NC), meaning that the approach taken by the student would not result in significant progress toward solving the problem.

Letter grades (a necessity where I taught, as they are in most schools) were assigned on the basis of how many problems were CC and EC on the Skills/Conceptual portions.  While I won’t go into great detail here, the salient features of this system are, in my mind:

1.  Earning an A required a fair number of problems CC; in other words, a student couldn’t get an A just by amassing enough partial credit on problems — there needed to be some mastery;
2.  Earning an A required as least some progress on the Conceptual problems; typically, I would include three such problems, which would be assessed more leniently than the Skills problems.  In order to earn an A, a student would need to make some progress on one or two of these.  I felt that an A student should be able to demonstrate some conceptual understanding of main ideas;
3. A student could earn a B+ just by performing well on the Skills problems (or perhaps an A- for performing flawlessly), but an A was out of reach without some progress on the Conceptual problems;
4. The approach was not punitive:  students were assigned CC/EC based on their progress toward the solution to a problem, not how far they fell short of a solution.

Such a system is not new; I have talked with colleagues who used a 0/1/2 system of grading, for example.  Whatever the format, the approach is more holistic.  And given that problems in mathematics typically admit more than one solution, the idea of creating a point-based rubric for all possible solution paths does, in some real sense, border on the insane.

I should also add that our instructional approach at IMSA was essentially inquiry-based.  While such a classroom environment is conducive to having students write original problems and using alternative assessment strategies, it is not strictly necessary.  I have incorporated both the elements described above in a more traditional classroom setting, but with varying degrees of success.  That discussion is necessarily for another time.

So much for a brief introduction!  I would also like to comment that teaching this Honors Sequence was perhaps my most enjoyable and successful teaching experience in the past few decades.  Students who completed the two-semester sequence left thinking about mathematics in a fundamentally different way, and I stay connected with a few of these students many years later.

Next week, I’ll introduce the essay proper, adding commentary as necessary to flesh out details of the course not addressed here, as well as making remarks which reflect my teaching this Honors sequence in the years since the essay was written.  Until then, I’ll leave you with the opening words:

“To educate is to illuminate the power of ideas…..”

## Guest Blogger: Scott Kim, III

Today, I’ll post the third installment of Scott Kim’s blog on transforming mathematics education.  But before jumping into that, I want to share a little about Bridges 2018, which just took place in Stockholm, Sweden.  Because of my move and career shift, I decided not to go — at the time I would’ve needed to make travel arrangements, I didn’t even know whether I’d be living on the West coast or the East coast when I’d need to catch my flight!

In any case, my Twitter feed has been buzzing recently with tweets from Stockholm, and some have featured participants in the Bay Area Mathematics Artists Seminars.  Monica Munoz-Torres tweeted about Frank Farris’s talk on vibrating wallpaper patterns, which you may recall he gave at our March meeting of the BAMAS at Santa Clara University.

And the Bridges Program Committee announced that Roger Antonsen won Best in Show for 2-dimensional Artwork for his piece, “Six Perfect In-Shuffles With 125 Cards and Five Piles.”

Congratulations, Roger!

OK, now we’ll move on to Scott Kim’s commentary on transforming mathematics education.  His next point addresses a prevailing issue in mathematics education:  advances in technology relevant to teaching mathematics are moving along at a rate which outpaces curriculum development.

And it’s not just that.  Even if curriculum could be reimagined at a pace to keep up with technology, teachers would need to be retrained to use the new curriculum with the new technology.  Not just retrained on the job, but while students at university — meaning that institutions of higher education would need to have their faculty keep up as well.  This means resources of time and money, and the willingness and ability of mathematics and education faculty, as well as school districts, to embrace change.  A tall order, to say the least.

I could go on at length about this topic, but let’s give Scott a chance.  Again, if you just can’t wait for the fourth installment, feel free to go to Scott’s blog, where you see the post in its entirety.

## Level 3. The wrong MATH (sailing in the wrong direction)

The mathematics we teach in school is embarrassingly out of date. The geometry we teach is still closely based on Euclid’s Elements, which is over 2000 years old. We continue to teach calculus even though in practice calculus problems are solved by computer programs. Don’t get me wrong: geometry and calculus are wonderful subjects, and it is important to understand the principles of both. But we need to re-evaluate what is important to teach in light of today’s priorities and technologies.

Here are three ways to update what we teach as mathematics.

3a. Re-evaluate topics. The Common Core State Standards take small but important steps toward rebalancing what topics are taught in math. Gone are arcane topics like factoring polynomials. Instead, real world mathematics like data collection and statistics are given more attention. As Arthur Benjamin argues in a brief TED talk, statistics is more important than calculus as a practical skill.

Solution: give kids an overview of mathematical topics and what they are for, long before they have to study them formally.

3b. Teach process. The widely used Writer’s Workshop program teaches the full process of writing to students as young as kindergarten. The process accurately mirrors what real writers do, including searching for a topic, and revising a story based on critique. We need a similar program for the process of doing mathematics. The full process of doing math starts with asking questions. Math teacher Dan Meyer argues passionately in his TED talk that we do students a terrible disservice when we hand them problems with ready-made templates for solution procedures, instead of letting them wrestle with the questions themselves. Here is my diagram for the four steps of doing math. Conrad Wolfram created a similar diagram for his Computer-Based Math initiative.

Solution: give kids an explicit process model for problem solving.

3c. Use computers. In an era where everyone has access 24/7 to digital devices, it is insane to teach math as if those devices didn’t exist. In his TED talk, Conrad Wolfram points out that traditional math teachers spends most of their time teaching calculating by hand — the one thing that computers do really well. By letting students use mathematical power tools like Mathematica and Wolfram Alpha, teachers can spend more time teaching kids how to ask good questions, build mathematical models, verify their answers, and debug their analysis — the real work of doing mathematics. And students can work on interesting real-world problems, like analyzing trends in census data, that are impractical to tackle by hand.

Solution: build and use better computer tools for doing math. Revamp the curriculum to assume the presence of such tools. Emphasize solving interesting problems, de-emphasize or delay learning about the mathematical mechanics for carrying out the computations. In other words, teach mechanics on a need-to-know basis.

Next week will feature the Level 4 of Scott’s remarks.  Until then!

## Guest Blogger: Scott Kim, II

Today, I’ll continue with reblogging Scott Kim’s in-depth post about transforming mathematics education.  You might want to read last week’s post to get caught up.

I will say that the discussion generated quite a bit of interest.  Participants have been actively responding to each other in a very lively email thread.  The comments and discussions are still ongoing — I am having a hard time keeping up with them!  But in a later post, I’ll summarize some key ideas and observations made by members of the group.

But for now, I’d like to turn it over to Scott Kim.  Again, if you’re anxious to read the entire post, please feel free to go to his blog.  Or just be patient….  But you can see by looking at the heading that Scott is addressing a very important issue next.  I can still recall — when teaching gifted high school mathematics and science students — really understanding where the question “When am I going to ever use this?” comes from.

The answer is pretty simple.  Bright students want to know.  When I first started teaching at university, I thought it was the students’ job to find motivation for doing mathematics — after all, they were paying a lot of money for their education.

But I eventually realized that there are only about three months between the end of high school and the beginning of college.  Nothing magical happens to students to transform them into self-motivated human beings, hungering for knowledge for its own sake.

Actually, one of my goals is never to hear the question “When am I ever going to use this?” again.  If I do a good job teaching and motivation concepts, students will already be able to answer that question, and won’t need to ask it any more.

Yes, it’s a more challenging way to teach.  But I can tell you, for me, it has been worth it.

Now I’ll let Scott take over.  Enjoy!  We’ll look at the third level next week.

## Level 2. Lack of MEANING (leaks)

### The most common complaint in math class is “when are we ever going to use this?” And no wonder; the closest most kids get to using math meaningfully is word problems, which are typically dull mechanical problems, dressed up in dull mechanical narratives.

Traditional mathematics education focuses on teaching rote computational procedures — adding, dividing, solving quadratic equations, graphing formulas, and so on — without tying procedures to meaningful situations. Unfortunately most adults, including many teachers and administrators, think this is how it must be. But teaching only the rote procedures of math is like teaching only the grammar and spelling of English, without explaining what words mean, or letting kids read books. Mechanics without meaning is not just deathly boring, it is much harder to learn.

Here are three ways to plug the leaks of meaningless math.

2a. Use math. In our increasingly digital society, kids spend less and less time playing with actual physical stuff. All the more reason to get students out of their desks and into the world, where they can encounter math in its natural habitat, preferably integrated with other subject areas. My friend Warren Robinett told me “a middle-school teacher I knew would, after teaching the Pythagorean Theorem, take the kids out to the gym, and measure the length and width of the basketball  court with a tape measure. Then they would go back to the classroom and predict the length of the diagonal. Then they would go back to the gym, and measure the actual diagonal length. She said some of the kids would look at her, open-mouthed, like she was a sorceress.”

Solution: use problems that kids care about, and excite student interest.

2b. Read about math. Before we learn to speak, we listen to people speak. Before we learn to write, we read books. Before we play sports, we see athletes play sports. The same should apply to math. Before we do math ourselves, we should watch and read about other people doing math, so we can put math in a personal emotional context, and know what the experience of doing math is like. But wouldn’t reading about people doing math be deadly boring? Not if you are a good story teller. After all, mathematics has a mythic power that weaves itself into ancient tales like Theseus and the Minotaur. My favorite recent math movie is a retelling of the classic math fable Flatland, which appeals as much to my 7-year-old daughter as to my adult friends. Here’s a list of good children’s books that involve math.

Solution: read good stories about math in use.

2c. Ask your own questions. In math class (and much of school) we answer questions that someone else made up. In real life questions aren’t handed to us. We often need to spend much time identifying the right question. One way to have students ask their own questions is to have them make up their own test questions for each other. Students invariably invent much harder questions than the teacher would dare pose, and are far more motivated to answer questions invented by classmates than questions written by anonymous textbook committees. Mathfair.com goes further to propose that kids build and present their own physical puzzles in a science-fair-like setting. Kids can apply whatever level of creativity they want. Some focus on art. Some on story. Others add new variations to the puzzles or invent their own.

Solution: Give kids freedom to ask their own mathematical questions, and pursue their natural curiosity.

If we plug the leaks of meaningless math, we will grow a generation of resourceful mathematicians who understand how to solve problems. But are we teaching the right mathematics?  (To be continued….)

## Transforming Mathematics Education: BAMAS, X

This past Saturday marked the tenth meeting of the Bay Area Mathematical Artists Seminars.  You might recall (see the post about Bay Area Mathematical Artists Seminars, VI) that at a recent meeting, we had a very stimulating dinner conversation about the future of mathematics education, with Scott Kim helping to guide the conversation.

Everyone was so engaged, it was unfortunate that the conversation had to come to an end.  So I invited Scott to lead a more formal discussion at a later meeting of the BAMAS.  We met at BAMAS member Stacy Speyer’s place — thanks for hosting, Stacy!

The discussion was quite animated.  Scott prepared a handout based on a lengthy blog post he wrote about various issues revolving around mathematics and mathematics education.  He graciously gave me permission to reblog his ideas.  The post is rather lengthy, so I’ll share it in installments.  You can go to Scott’s blog yourself if you can’t wait to read more.  So without further ado, I’ll let guest blogger Scott Kim take the wheel.  His original post was dated July 6, 2014.

# Navigating Math Education

Imagine that you are a sailor on a leaky boat that is on fire, sailing in the wrong direction, with a quarreling crew. Which problem would you fix first?

Well, that depends. If the leak is slow and the fire is raging, then you would put out the fire first. If the leak is gushing and the fire is small and contained, you would fix the leak first. It makes sense to fix the most urgent problem first.

What you would NOT do is fix one problem and declare victory. If your goal is to get to your destination safely, then you must fix ALL the problems, no matter how difficult. Anything less will not get you where you want to go.

Such is the situation with math education. The problems are so difficult and so numerous that it is tempting to fix one problem, and give up on the rest. And certainly we have to prioritize if we are to make progress. But if we are to get the ship of math education back on course, then we, collectively, must fix ALL the problems of math education. Nothing less will get us where we want to go.

Fixing all of math education may sound impossible or impractical. And indeed it is a formidable challenge. Well-meaning entrepreneurs who have launched successful businesses frequently grind to a halt when they try to start their own innovative schools. Resistance comes from all sides — standardized testing, textbook publishers, parents, administrators, government officials, and the students themselves trying to get into college.

But change is in the wind. America is losing its competitive edge, colleges are becoming impractically expensive, and the internet makes us dream of free education right now for everyone. I say we face the problem with eyes wide open, assess the full range of challenges we face, and look for the smartest moves that get us where we want to go.

With that in mind, here is my survey of the problems plaguing math education, and steps we can take to fix them. I’ve grouped the challenges into four levels that range from the tactical to the strategic: Mechanics, Meaning, Math, and Society.

## Level 1. Faulty MECHANICS (fire)

### The most obvious and urgent problem is that the mechanics of math are taught as a series of blink and you’ll miss it lessons, with little opportunity to catch up.

This one-size-fits-all conveyor belt approach to education guarantees that virtually everyone gradually accumulates holes in their knowledge — what Khan Academy founder Sal Khan calls Swiss cheese knowledge. And little holes in math knowledge cause big problems later on — problems in calculus are often caused by problems in algebra, which in turn are caused by even earlier problems with concepts like fractions and place value.

Here are three ways to fight the fire of poor pacing.

1a. Self-paced learning. The Khan Academy addresses the urgent problem of pacing by providing short video lectures that cover all of K-12 math. While the lectures themselves are fairly traditional, the online delivery mechanism allows students to work at their own pace — to view lectures when and where they want, and to pause and rewatch sections as much as they need. All lectures are freely available at all times, so kids can review earlier concepts, or zoom ahead to more advanced concepts. Short online quizzes make sure that kids understand what they are watching. And with an online dashboard that shows exactly how far each child has progressed, teachers can assign lectures as homework, and use class time to tutor kids one on one on exactly what they need.

Solution: the “flipped classroom.”

1b. Visual learning. I love the Khan Academy. My son hated it, because he, like many students, is a visual learner, and Sal Kahn’s lecture stick largely to traditional symbolic math notation. He would have done better with a visual experiential curriculum. Some kids are primarily audio or kinesthetic learners, some learn best socially. The bottom line is that different kids learn in different ways, and no one way is right for everyone. Education needs to address all learners, not just kids who learn in words.

Solution: teach every lesson three different ways.

1c. Testing for understanding. Nothing can change in education unless testing changes. Traditional standardized tests born of the No Child Left Behind era use multiple choice tests that assess only rote memorization of routine math facts and procedures. The new Common Core State Standards for mathematics, now entering schools across the nation, replaces standardized multiple choice tests with richer tests that include essay questions graded by human beings — a better way to assess mathematical understanding.

Solution: better assessment.

If we douse the fire of poor pacing in math education, we will increase test scores and student confidence. But there is more to mathematics than teaching the mechanics well.

I hope your interested is piqued!  Scott will continue next week….