My Math Education Blog

"There is no one way"

Wednesday, May 25, 2016

De-emphasizing Grades

 The Assessment Trap, Part 4. De-emphasizing Grades

(This is a slightly edited version of a post from 2011)
(If you want to start reading at the start of the series, click here.)

When students learn their grade for a given course, what they are learning is how they compare with their peers, which is one indicator of "how they are doing". (See the last post in this series.) Grade or no grade, many students know exactly where they fit in the classroom hierarchy, though some may not admit it to their parents or even to themselves. It is true that some (often boys) overestimate themselves, and others (often girls) underestimate themselves. For those students, knowing the grade may be a helpful corrective. But is it a good idea, educationally, to dwell on comparisons between students?

Like many teachers, I am reluctant to make comparisons between students. Such comparisons are unfair and unproductive. Unfair, because students come from many different family and educational backgrounds. Comparisons between students end up being largely about that. Unproductive, because it is not realistic, in most cases, to expect major changes in the short run. A hard-working C student may need years, not weeks, to become a hard-working B, or even A student. We can point them in the right direction, offer them intellectual tools, help them to improve their work habits, and over the course of their high school career we can see spectacular changes. And we often do -- this is one of the most satisfying parts of working in a strong department.

But paradoxically, the way to get there is not to dwell on the grades. (It's a bit like searching for happiness -- you're more likely to find it if you don't dwell on that as a goal.) At most schools, the conversation is about "what do I need to do to get an A?" (or a B), and of course, that is the subtext of many conversations at any school. The teacher's responsibility is to deflect that conversation towards the specifics of this particular student's needs at this stage. Perhaps the A is already guaranteed, but the student needs to focus on their ability to communicate their ideas better. Perhaps the A is just not going to happen this term, but the student needs to work on developing their symbol manipulation skills, or their ability to write a logical argument. There is always work to do, and a time to stop working, irrespective of where the student stands in the grades distribution at this particular time.

A grades-focused conversation means that in these very common situations (the A is guaranteed, or the A is unattainable at this point) there is little to discuss. It can also lead to grade inflation in a variety of ways: in order to motivate students with the grade, we might make it easier to attain. Or in order to not be hassled, we might make A's more plentiful. Grade inflation is not the end of the world, but if we want to inflate grades, we ought to do it deliberately and not as an unexpected consequence of uncomfortable conversations.

If a student's place on the academic ladder is constantly harped on by the school culture, students can internalize the label and stop striving. This is what is now known as a fixed mindset. Skillful teaching is in part about bringing out students' different strengths to the fore, and building on them, whether or not those lead to a better grade in the short run. For example, a strongly visual student can contribute a lot to a discussion, even if he or she is not yet ready to translate that talent into points-earning write-ups. Over time, such engagement does lead to better grades.

Bottom line: intrinsic motivators (such as interest in the subject matter) are more powerful, longer-lasting, and more meaningful than extrinsic motivators (such as grades.) Our task, as teachers, is to move students from the latter to the former. It is a challenging enterprise, but we must try to keep the focus on the discipline we teach and our own passion for it, rather than on the lines separating our students into A, B, and C. Teaching students to be self-motivated learners, and modeling that relationship to the subject, is a vastly more useful contribution to them as lifelong learners than the Pythagorean Theorem or the quadratic formula.

Next in the series: some of the research on grades.


Monday, May 23, 2016

NCTM Is Its Members

This post's title was also the subject of an e-mail from Matt Larson, the new President of NCTM sent to the membership. You can read it here. Since he asked for suggestions, I replied to his message. Here is what I wrote, slightly expanded.



Dear Matt Larson,

I’ve been a member of NCTM for who knows how long, probably at least 30 years. During that time, I was a speaker at more than 80 conferences of NCTM and its affiliates. I have written and reviewed articles for The Mathematics Teacher. I edited the "Student Activities" department in that journal for a couple of years.

I was excited and impressed that in your recent message “NCTM Is Its Members”, you asked for suggestions. I thought I would take you at your word. I hope you have time to read this message.

For background, here is the rejection letter I received when I submitted a piece to the "SoundOff!" department in The Mathematics Teacher:
We enjoyed reading your SoundOff! submission. We were very interested to read your views and we think our readers would be, too. Your position may of course be somewhat controversial, but that's exactly the type of thing we're looking for in our Sound Off! department; we ask authors for "a short, signed statement, editorial in nature, which forcefully and logically raises a significant issue or advocates a point of view about some aspect of the teaching or learning of mathematics." ( In your piece, your views are vividly presented and your author voice comes through loud and clear - these are two qualities which can contribute to a great Sound Off!However, we must reject your manuscript because it substantially overlaps with a freely available blog post. Even were appropriate citations to be included, the majority of the SoundOff is a direct quotation from previously published material, and so publishing it would be which would be contrary to our policies. We don't accept republication of any material (from websites, books or other journals) in our NCTM school journals. We do, however, thank you for disclosing the original source freely to the MT editor. The Panel would certainly welcome a different submission that expressed your concerns about the CCSSM and referenced the blog post, but was completely rewritten and added substantial new arguments, examples and/or recommendations.
(The MT reviewer was referring to the analysis of the Common Core State Standards for high school math, which I posted on my Web site a couple of years ago. More on this below.)

So here are my two suggestions:


While the rule cited above may make sense in some other situation, it should not apply to pieces that duplicate or overlap with material from a blog or a Web site (with permission, obviously.) In the case of a low-traffic site such as mine,  there is very little difference between the material having appeared on the site, and the material not having appeared anywhere. Given the size of the Web, it is just silly to think that just because people could find it on my site, that they would find it. And in the case of a site with a lot of traffic, such as Dan Meyer’s, republication would help engage the NCTM membership with a vibrant and dynamic math ed community. In either case, NCTM loses nothing, gains possibly useful content, and connects with the online universe. Such pieces would of course be subject to the same review process as any other submissions.

The argument that paying dues should get you something you can’t get for free is specious. Such articles would not constitute a large part of the journals, and they would benefit from the peer review process and the editorial leadership of the journals. In most cases, they would be much improved from the online version. Moreover, they would be selected for publication among the zillions of things available online, so the mere fact that they were chosen for publication by NCTM is a service to the membership, who could not get that combination of curating, editing, and review by randomly surfing the Web.

Changing this rule would require no extra staff or volunteers, and it would increase classroom teachers’ involvement with the journals. I see no downside.


Zooming out from this particular instance: I have done a close reading of the Common Core State Standards for high school, and as I mentioned above, published my analysis online. I received a very positive response to that paper from an author of the CCSSM, an author of Principles to Action, a recipient of an NCTM Lifetime Achievement Award, an ex-editor of The Mathematics Teacher, a board member of COMAP, and many, many math teachers. The most common response from teachers is that they encouraged their colleagues to read my paper. Yet, it is impossible to present these ideas at the conferences of NCTM and its affiliates. I am not just talking about myself: there are very few articles in NCTM journals, and no talks about this most important topic at NCTM meetings. The only sessions about the Common Core are about how to implement it, never about its strengths and weaknesses.

This must change: NCTM’s official stance is that the standards should be regularly revised and updated. Should revisions and updates be the responsibility of a small number of PhD’s who wish to exclude feedback from the rest of the profession, particularly classroom teachers? It is high time that NCTM took leadership in a national and wide-ranging discussion of the CCSSM among its members.

Thank you for reading this far, and good luck in your new position!


Henri Picciotto
"There is no one way."

My Math Education Site:
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I didn't expect a response, let alone a quick one, but Matt Larson replied within 24 hours. He said he forwarded my first suggestion to the appropriate committee. (Great!) For the second, he said there was no policy about avoiding conversation about the merits of the CCSSM, and gave as an example the fact that Zalman Usiskin once gave a talk on precisely that topic. (Moreover, I'm told Steve Leinwand, another superstar, sometimes mentions his concerns about the high school CCSSM in his talks. My own proposal for a session on this topic was rejected.)

In any case, my point stands. No one could credibly claim that NCTM is fulfilling its responsibility to lead a broad discussion of the pros and cons of the CCSSM. Quite the opposite: the Council is very vocal is supporting the positive aspects of the Common Core, and is mostly silent about the negative aspects. There are no panels or debates on this at conferences, and almost all the articles in the journals either do the obligatory genuflection towards the Standards, or ignore them.

Still, I'm encouraged by the fact that NCTM's President is giving more voice to the membership. Let's see where it takes us!


(And here, once again, is the link to my analysis of the CCSSM for high school.)

Sunday, May 22, 2016

The Meaning of Grades

 The Assessment Trap, Part 3: The Meaning of Grades

 (This is a slightly edited version of a post from 2011)

Previous posts [Part 1, Part 2] have focused on the uses of assessment. For many students, parents, teachers, and administrators the key purpose of assessment is to assign grades. Before going any further, we need to think about the meaning of grades.

Grades have no intrinsic, absolute meaning. An A at an elite private school does not mean the same thing as an A at a public school that serves a poor neighborhood. An A at my own school today does not mean the same thing as an A meant 20 years ago. An A in Science does not mean the same thing as an A in History. And on it goes. The one feature of grades that is quite reliable is that an A in a given department at a given school is better than a B, which in turn is better than a C. In other words, the meaning of grades is relative. They are how we compare students to each other.

Almost all teachers will fix how they compute their grades if the outcome does not sort the students correctly. If a student deserves an A, and your calculation yields a B, you will find a way to tweak the percents, or the scores, or the participation points, or the extra credit, or something, to make sure the student does not get cheated by a pseudo-objective algorithm. (Admittedly, if the calculations yield an A, rather than the B we expected for a given student, most of us would let it be.) This makes sense, because teaching is as much an art as a science. Given a small enough class and enough of the right sort of contact with the students, a competent teacher knows better how to sort the students than any formula. (Yes, better assessments yield more accurate grades — that’s what I meant by “the right sort of contact”.)

In the rare case of the teacher who delivers much worse or much better grades than expected by their school, they will be taken aside by an administrator, and told that their practice is out of line. This does not require looking at the students’ work — it is more evidence that grades are strictly a relative measure.

In short, grades compare students to each other. They have no other meaning. This is why colleges are interested in grades. If grades were not about sorting students, they would be useless. Just to be clear: grades do not compare students only to others in the same section of the same class, but with the somewhat broader group of students in the same cohort at the same school. And moreover, it is of course true that the rankings are only meaningful if you accept the teacher's and the school's assumptions and values.

One might argue that grades are a measurement of how well a student meets the standards of a given class. This is true enough, but the standards in question only exist in relation to the specific students currently enrolled at the school. If almost every student met a given set of standards, no matter how valid those are, it could not and would not be used as a way to assess achievement in the class and determine the grade. In fact, such a set of standards would make for a course that is too easy for the given population. Conversely, a set of standards that is met by almost no one makes for too difficult a course. The only standards worth aiming for are precisely the ones that sort students into A, B, C bins.

Some will maintain that this is an argument against a system with no grades at all. Without grades, it would be easier to set your expectations too high or too low, or to have a bimodal distribution, with some students doing very well, others clueless, and little in between. Giving grades can help us calibrate challenge and access in the classes we teach. In other words, giving grades is not per se wrong. In fact, it can be useful.

But that does not negate this fact: grades are about comparing students to each other. Students know it, parents know it, teachers in practice know it. Educators who believe otherwise are deceiving themselves.

In the next post in this series, I argue for de-emphasizing grades.


Wednesday, May 18, 2016

Problematic Uses of Assessment

The Assessment Trap, Part 2: Problematic Uses of Assessment

In my previous post, I listed four legitimate uses of assessment, which make it a key part of instruction. But there are other uses of assessment which I find problematic. Here they are.

1. Assigning grades.

We need to give grades to "let students know where they stand", to enter them into transcripts, and of course, to rank students. Grades are the basic currency of education in most schools, and assessment is how we figure out grades.

I am uncomfortable with comparing students to each other, as this sort of pressure can interfere with learning, and given different backgrounds and experiences it is intrinsically unfair. Some make the claim that grades are about standards, not comparison, but I don't buy it, and neither do students, parents, or college admissions officers. Everyone knows grades are about ranking students, no matter what efforts are made to disguise this obvious fact. (I will return to this in my next post.)

2. Justifying the grades.

Because grades are so important to students' status and opportunities, they are contentious. If a student, parent, or administrator wants to challenge us, or if we ourselves are unsure, we need solid evidence that the grade was assigned fairly. Thus we need some sort of objective-seeming way to justify the grade. From a certain point of view, this is the main purpose of assessment. It provides cover for the teacher and the school if and when grades are questioned, and it attempts to address our concerns about fairness.

3. Preparing students for future assessments.

I am not joking. "We have to do multiple choice tests to prepare you for such-and-such a standardized test." "You have to learn to work under time pressure, because that's what you'll have to do in college." (Or high school, or middle school.) And so on.

There's a bit of truth to this, of course, but only a bit. Giving assessment as the reason for assessment fails to answer any fundamental questions.

4. Manipulating student motivation.

Note that the above three uses of assessment have nothing to do with student learning. Many educators try to balance that by emphasizing assessments as tools for manipulating student motivation. Since not all students are enthusiastic about carrying out teacher directives or pursuing education for its own sake, it is widely believed that grades (along with the points or rubrics that lead to the grades) are the key tool in motivating students to do schoolwork. Unfortunately, this does not work as well as is widely believed. In fact, grades, points, and rubrics shift students' attention away from the subject matter, towards "how they are doing" which in fact undermines their intellectual or emotional engagement with the work. Assessment anxiety can sour a student's entire relationship to the subject matter.

In the world we live in, there is no easy way to escape these problematic uses of assessment, but they should not dominate our thinking. Far from supporting learning, an emphasis on grades, points, and rubrics in fact undermines both motivation and achievement. I will return to this in future posts. Next up: The Meaning of Grades.


Sunday, May 15, 2016

Legitimate Uses Of Assessment

This is the first of eight posts on assessment. Much of the series will focus on some of the traps that are so easy to fall into, and are so damaging to student learning. My experience is primarily in high school math education, but I hope the series will be of use to people in other disciplines and other grade levels.

The Assessment Trap, Part 1: Legitimate Uses Of Assessment

Assessment, of course, plays an important part in instruction. I will start by discussing how it can be used to improve student learning.

1. Fine-tuning the course.

This is by far the most important use, because it affects the most students: not only the ones in the current class, but also the ones who will take future iterations of the class. If a quiz reveals a widespread misunderstanding, it is a sign that something was wrong with the way I taught that topic. Is there another representation of the concept that would help? A different tool that can get the idea across? A different sequencing of topics that would provide a stronger foundation?

2. Diagnosing individual students' understanding and skills
3. Helping students realize what they know and can do

Assessment makes it possible for the teacher and student to celebrate what has been learned, and zero in on the areas that require the most attention. Ideally, some combination of student agency and teacher / school support can help address those challenges.

4. Providing learning opportunities

Many teachers separate assessment from learning. That is neither necessary nor desirable. Assessment can be an integral part of the curriculum: not an endpoint, but a point along the way. This is especially true of at-home assignments and test corrections. (More on those in future posts.)

A quiz or test can contribute to instruction with the inclusion of one or two questions that extend a given topic, or require students to apply what they know in an unexpected context. A challenging question on a quiz can be the launching pad for deeper learning, or even provide an introduction to a new concept. This use of assessments is, to say the least, controversial.

When a student or parent or administrator objects that it is "unfair" to include challenging questions on assessments, they are betraying a certain point of view on education. Unfortunately, it is a widespread point of view: assessments are opportunity to regurgitate ideas that were deposited in their brains by the teacher. If instead you prioritize student understanding over student recall, as I do, you are obligated to provide opportunities for students to engage intellectually with the material, even on an assessment, perhaps especially on an assessment.

Concerns about fairness are of course legitimate. They can be addressed, for example, by appropriate weighting of different parts of the assessment, by labeling questions as "bonus", or in some other way. In my view, removing challenging questions altogether is an unacceptable lowering of expectations. I will return to this in a future post.

Note that all four uses of assessment I have mentioned above can be thought of as formative, in that they focus on student learning, not student ranking.

As you probably surmised from the title of this piece, I believe there are problematic uses of assessment. I will address those in my next post.


Thursday, May 12, 2016

In Defense of Algebra 2

Novelist Nicholson Baker wrote a cover story on "The Case Against Algebra II" for Harper's in 2013. More recently, political science professor Andrew Hacker wrote a book (The Math Myth) challenging Algebra 2 and its sequels. Some day, I may respond to their arguments, but I am addressing this post to the math teachers who don't like Algebra 2.

I already had a conversation on this blog about a related topic in 2013. Rereading those posts, I find I still agree with what I wrote. Today, I want to get more specific and zero in on Algebra 2. I may end up repeating some of the points I made back then, so I ask for your forgiveness if you read those posts and still remember them.

One objection to Algebra 2 is that the course is a hodgepodge of unrelated topics. As I stated in a recent post, I don't consider this to be a matter of principle. There's a place for hodgepodge courses, and a place for themed courses. I only know one proposal to give a theme to Algebra 2, which is to center it on the concept of function. I agree that functions should be a significant concept in the course, but as you will see below, I believe there are other valid topics to include. In other words, I'm OK with a hodgepodge Algebra 2.

Eliminating the Algebra 2 requirement will not affect the children of privilege: no matter what is said in magazines, on social media, or on blogs, they will not be deprived of this course. In practice, the elimination of Algebra 2 would just mean that the less well-off will be kept out of careers in science and technology. But this argument is not what I want to write about. I want to argue that Algebra 2 can and should be a meaningful part of the curriculum. It can help prepare our students to be well-rounded, well-educated adults and citizens. I do not claim most will need it in their careers or for their daily life. Rather, I see math as one of the humanities, a way of thinking that helps us make sense of the world around us, and that is the point of view I take when I think of Algebra 2.

Let me start by agreeing that there are many valid reasons to object to Algebra 2. It is often taught poorly. It can be boring. It can consist of arcane techniques that seem completely divorced from meaning. Some of the topics are obsolete. Some are better postponed to later courses. And so on. I do not intend to defend poorly taught, boring, meaningless memorization of highly technical topics.

In fact, it took my department a long time to sort out how to teach that course. We tried a couple of different textbooks, but no matter what we did, this was the course that students and parents complained about. It was too hard for some students, too easy for others, and boring for all. It was, frankly, the sort of course I described in the previous paragraph.

Things started to improve when we designed our own version of the course:
Bits of curriculum I created to accomplish this can be found on my Web site. Other parts we lifted from books which had not worked for us as textbooks, but nevertheless had some good content. And I will share some specifics below, with a selection of topics which I consider worth teaching in Algebra 2. Some of those help cement previous learning. (If we don't think it's important to do that, why did we teach those topics in the first place?) Some help prepare those students who will pursue math, science, or engineering in their future education. But all are intrinsically interesting, and can be taught well to 10th and 11th graders.

Linear Programming. This is a topic for which it is not too difficult to find worthwhile problems in standard textbooks. It can make a great unit to apply and review many basic algebraic skills and understandings, especially linear inequalities and systems of equations. Better do this in an interesting new context than by rehashing the way these topics were taught in Algebra 1. I wrote an introductory activity for linear programming, and created an interactive GeoGebra applet to accompany it: Letters and Postcards

Exponential Functions and Logarithms. Understanding exponential growth and decay is fundamental to topics that range widely: population growth,  radioactive decay, compound interest, depreciation, and so on. It makes it possible to grasp some issues about the environment and the economy. I live in California. Shouldn't my students understand that one more point on the Richter scale represents an earthquake with ten times the amplitude? Here's an effective intro to logs: Super-Scientific Notation.

Introduction to Dynamical Systems. Iterating linear functions is a great topic to explore in Algebra 2. It yields delightfully unexpected results, introduces limits in a very accessible context, and provides a great environment to introduce subscript notation. Students enjoy some of the applications, such as figuring out the amount of caffeine in one's system if one drinks one cup of coffee every six hours. At the end of the unit, arithmetic and geometric sequences can be introduced as a particular case.

Complex Numbers. We approach the topic visually, as in Chakerian, Stein, and Crabill's Trigonometry: A Guided Inquiry textbook (now out of print.) This rests on an understanding of polar coordinates, and offers a chance to review basic trig. For some students, Algebra 2 is the last math class they'll ever take. All the more reason to end with a bang, completing the journey that started when they learned to count. As they moved up the grades, they became familiar with broader and broader sets of numbers, expanding their understanding of operations and the sorts of equations that had solutions. Complex numbers is a great final stop. (And check out the fun complex numbers games!)

I just picked a few of my favorite units to share here in order to perhaps convince you this can be a great course. You can find more links to Algebra 2 ideas here, or you can sign up for one of my summer workshops: one day on Algebra 2 in Oakland, or two days on Algebra 2 and Precalculus in Saint Louis.


PS: lest you think this course strays far from Algebra 2, here is a list of the topics:  Exponentials, Logarithms, Linear Programming, Parabolas and Quadratics, Unit Circle, Laws of Sine and Cosine, Polar Coordinates, Vectors, Direct & Inverse Variation, Sequences and Series, Functions, Complex Numbers. The one topic I'm sure is non-standard is Iterating Linear Functions, but it's worth it as it helps introduce Sequences and Series.

Tuesday, May 3, 2016


I have a new Fractions mini-home page, with links to three pages on my site. In this post, I'll use it as an excuse to discuss some general ideas about teaching.

Visual Representations

In my Fraction Arithmetic page,  I present a visual strategy for figuring out how to add, subtract, and multiply fractions. (There is also a discussion of this strategy's implications for division, but I suspect that is unlikely to find broad acceptance, as it is somewhat complicated.)

The core idea of the strategy is the use of rectangles on grid paper, and the first step involves choosing the right dimensions for the rectangle that represents 1. What I like about this is that it is directly connected to very basic understandings about fractions, and about operations. For example, if you understand the meaning of 2/3, the meaning of 1/5, and the meaning of multiplication, it is not hard to see how this image leads you to figuring out the product of the two fractions:

On the other hand, if a student doesn't understand the meaning of the fractions, or the area model of multiplication, it is entirely pointless to try to teach them a multiplication trick (multiply the tops, and multiply the bottoms, say). Learning such a trick may help them to get answers in the short run, but it may soon be forgotten, not to mention that it will bring with it some misconceptions, such as "add the tops and add the bottoms" for addition.

Presenting this visual strategy to students who don't understand the basic underlying concepts forces us to shift the conversation to those concepts, which in the end is the only way to make fraction arithmetic meaningful.

Working with this representation also provides an environment to get to the tricks (e.g. using a common denominator) with understanding. In other words, even if you think it is essential to teach those tricks, you'll do it more effectively by investing some time up front with this representation.

One implication of this approach is that it is a good idea to use grid-paper rectangles early on when introducing fractions, as those become a powerful tool further down the line.

Making Connections

This does not mean throwing away the "pies" representation. That way of looking at fractions is not as powerful for the purposes of arithmetic, but it is great for making connections with other ideas: angles, time, money, and of course percents and decimals. 

See my Slices page for several worksheets you could use for this, and some notes on making the connections. While I believe those have a lot of potential, I have not yet really explored it. I hope some of you will come up with some good activities using these, and let me know about them!

Low Threshold, High Ceiling, etc.

Finally, check out the Egyptian Fractions Challenge. I would encourage to work through it yourself, and you'll see that it has the key characteristic of a great activity: it is interesting to the teacher as well as to the student! In fact, it satisfies all the criteria I once listed for a rich curricular activity, and goes further: it is based on the Erdos-Straus conjecture, which number theorists have yet to prove.