Recently, I received an email from a concerned parent about how her daughter’s math assessment was graded. With the email, I received a copy of the graded assessment. The student is in the 8th grade taking an Algebra class.

I was mixed about whether to write about someone else’s grading methods / policies / standards. If you are a teacher, you will likely know what I mean. Often times there are in-class, verbal instructions that are given, that do not find their way onto paper. Part of the reason for this is that sometimes there are meaningful clarifications that have to be made because some students may have misinterpreted or misunderstood (or not understood at all) the instructions given. Another reason is that rather than write an \(n\)-page, legally-worded document explaining the instructions in an airtight manner, it is simpler to give “common sense” verbal instructions. Additionally, even if there are no changes to the instructions given on the exam, there is still an understood context. For example, textbooks often have the instruction $$\mbox{Simplify the expression}$$

But what does “simplify” mean? In one sense, we can argue that “simplify” is vague. In another sense, we can recognize that there is an implied context.

For example, on the last Intermediate Algebra exam I gave, I had the instruction “Find all real solutions for \(x\) to the equations below. If no solutions exist, state why.” And one of the problems under this set of instructions was \(y^{4} – 2y^{2} + 1 = 0\).

Oops.

Some students said, “There is no solution because there is no \(x\).”, while other students “understood” the question to mean “Solve for the only variable …” (or maybe they didn’t even read the instructions). Regardless, during the exam, I announced that that particular question was null and void as it was badly worded with respect to the instructions given. But if a third party, not privy to the in-class nuance, were to see my exam, the reaction could be, “What the heck kind of question is this?”.

Next, on the matter of the actual task of grading, different teachers have different styles. On one extreme, some provide feedback on the assessment, while on the other extreme some just give points (or minus points). And again, the in-class nuances do not show up on paper.

For example, when I grade an assessment, I will typically scribble something out, squiggly underline an erroneous or confusing part, and then just write how many points the student received on that question. If a third party were to look at the manner of grading, then an easy criticism would be, “How is a student supposed to learn what they did wrong if this is the type of feedback they receive?”. Unfortunately, what happens in class is not accounted for in the criticism. I, for example, go over the assessment with the class when I hand back the assessment. As I do the assessment review, I explain where points were lost, what mistakes students made, what the correct answer(s) is (are), etc. I tend to not write corrections on the assessment since, ultimately, I want students to be able to correct the assessment for themselves and I typically offer such opportunities. Also, by working out the assessment in front of the class, it gives me and the students opportunities to discuss (as a class) some recurring / class-wide mistakes. It’s a way of letting students know that more than one person made the same mistake.

Anyway, the larger point here is that what you are about read is based solely on the assessment that was given to me in paper form. I do not know what has transpired in class, let alone the expectations that the teacher has set forth. Please keep that in mind before going forth with pitchforks (at me or the unnamed teacher).

I have blacked out some identifiable information like the student’s name, the date, and the type of assessment. I provide my commentary after these two images. The first is page 1, the second is page 2. Page 2 was on the back so I blacked out the parts of page 1 that seeped through.

## The Assessment

## Some Thoughts

Generally speaking, I thought the student did a fantastic job on this assessment. The work was clean, the penmanship legible, and the mathematics sound. I have to say that I was a bit stunned by how the work was graded. There are some good parts about the grading and some not-so-good parts.

### The Good Stuff (Broadly)

I like that the teacher checked every step and didn’t just gloss over the work. I also like the detailed feedback that was given — though I don’t agree with much of the feedback. Regardless, there is a consistency in the feedback and an acute awareness of what the expectations were.

### The Grading (Mostly Not-So-Good)

I thought that the lost points here were unnecessary.

Problem #1: the student loses two points because they didn’t write \((4)\frac{x}{4}\) as \(\frac{4}{4}x = x\) and that the “multiplying by four on both sides” was done as left-multiplication on the left side and right-multiplication on the right side? \(6\cdot 4 = 4\cdot 6\) — there is no need / requirement that the multiplication has to be done as left- or right-multiplication. It’s a different story if we were discussing, say, multiplication of matrices.

I have similar commentary on problem #2.

I do like that the teacher filled in the “\(=\)” for when the student handled the additive constant on the left-hand side.

Problem #3: What?! Lost a point for left- vs right-multiplication?! Or something else?

Problem #4: What??! Lost points because \(-5\) wasn’t rewritten as \(+ (-5)\)? And an apparent order of operations error! Erm, \(3(-7 + 9) – 5\) does indeed equal \(-21 + 27 – 5\). It also happens to equal \(3(2) – 5\) via “order of operations”. However, distributing the \(3\) does not break rules!

Problem #5: Again, the same nitpicking of the student’s work. Although, I do *like* the fact that the teacher called out the missing parentheses around \(-7x\). I *may* have nicked the student for a point on that, but in all likelihood, I probably would have just called it out and shown the visual difference between \(\frac{-1}{7}-7x\) and \(\frac{-1}{7}(-7x)\).

Problem #6: Five points lost!! For what?! For not first writing \(\frac{3x -7}{5} = \frac{1}{5}(3x – 7)\)?? That’s excessive and entirely too prescriptive.

Problem #7: Holy smokes! Eleven points lost!! For what?! For not writing what axiom allows that \(-6 = +(-6)\)? For not further explaining why \(57 + 6 = 63\)? Again with the left-, right-multiplication and the “lack of” regrouping? For not writing \(1x\) instead of just \(x\)? (Also, it looks like that the correct follow-up should say \(0x\), which is false.)

A defense I can think of in favor of the grading is that maybe all the axioms were written on the board or a reference sheet was available explaining each axiom; maybe the teacher had written on the board that all multiplication has to be done in the same way (which is still not a correct way of teaching this).

But I’m really puzzled as to what is going on here! This type of grading (and hence possibly teaching?) of math is entirely too rigid. The teacher’s perspective here is unknown to me, but it seems that the student’s perspective is lost on the teacher. This is an 8th grader! Taking Algebra! She’s using semicolons to denote end of line! That’s a programmer in the making! (Kudos to the teacher for teaching them about programmatic line breaks, but shame if that were part of the prescription for working through the math.) We don’t have to treat students with kid gloves forever, but this just seems like militant grading.

There’s only one other thing I can think of that would make someone grade like this: it is having to explicitly ensure that students know explicit standards. This is where we get into tricky territory. One of my fears is that “grading / assessing via standards” will eventually lead to “checklist-based grading” like this. I understand that at the grade school level there are a myriad of standards that students have to be tested towards. But this seems excessive either on the side of the standards requirements or the teacher (or both). How did other students do on this?

This is the stuff of math misery. If I were asked to grade this assessment, it would land in the neighborhood of 99-100. I *may* have docked a point for the lack of parentheses on problem #5. This student clearly knows what is going on and there is no need for this level of inflexibility in the grading.

This is a student that is showing an aptitude for Algebra (and possibly Mathematics, in general) now. Motivate, nurture, and encourage her now! (And of course, for students who don’t show aptitude for the material now, motivate, nurture, and encourage them now, as well!)

On a related, but unrelated note, look at how all the answers cleanly worked out to integers. At least there were fractions in the original equation! And to reprimand myself immediately for that, this is one assessment of probably several (to come), so it is entirely likely that there will exist problems with fractions as answers. I just hope those assessments won’t be graded in the way this one was.

Readers, what are your thoughts? Is this excessive? Par for the course? Somewhere in between? Please leave comments below. Other educators chiming in will be greatly beneficial! Remember, all we have is this one assessment.

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Josh ZManan,

I am attempting to comment prior to reading your or others’ responses. As a quick background I have taught middle school math for ten years. My levels have ranged from Math 7 to high school college prep geometry (8th graders). Each year I have taught both inclusion and honors levels classes. Back in the 90s when I was in middle school I considered myself an average math student. I often went in to school early in the morning or after school for extra practice (thank you Ms. Oates).

After looking at Page 1 of the test I have to say I do not agree with two components of the grading style. As Manan stated in his original post, I am not in the walls of the class, but I still struggle to make sense of some of the thinking by the grader. Specifically, why did the students lose points in question 1? The grader wrote, “change to x.” Again, I am assuming they literally wanted the student to show the 4 being divided by a 4, thus becoming 1. I would argue the student did show this, in their next step they wrote “x=24,” displaying evidence that the 4s became a 1.

Secondly, in questions 5 and 6 I perceive the student lost points because their pen did not write the correct marks to display multiplication and subtraction:

#5- The teacher wrote, “multiply.” I believe the student did, did they lose points because they did not put () around the -7x? Again, when I analyze the row by row evidence the student did multiply. Also, on the right side of the equation why did the reciprocal of -7 have to go first? Doesn’t the commutative property allow us to show the multiplication that way?

#6 – The teacher wrote, “change to +.” I am assuming the grader wanted the student to “add the opposite,” and change the problem to …+(-7)…. Still the student showed the knowledge of applying inverse operations a received the correct answer.

Ultimately, I would not have marked points off on either problem because the student showed the ability to apply inverse operations to isolate a variable. Although the student may not have mirrored my methods they displayed accurate evidence of application. A “C” grade does not accurately assess the knowledge of the student. I firmly believe we must leave our own mathematical comfort zones. We must allow our students’ minds to go and reach our endpoint or goal in unique paths. It is our job to continue to question and defeat our own agendas to allow the students to struggle with material and find their best fit/method on mathematical concepts. As always, this is easier said then done and I still struggle with this on an annual basis.

To end on a positive, I liked how the grader analyzed the student’s work on a row by row basis. I believe it is important to identify errors or weaknesses within the process. At times, one reoccurring error can take a grade to a D or a C or possibly even an F. Does a D accurately display the student’s knowledge if only one reoccurring error plagues an assessment?

Thank you for sharing and have a great week.

Josh Z

David RadcliffeProblem #1 has a minor error. The original equation had -9 on the right side, but it was changed to 9. I am surprised that the teacher overlooked this error.

Problems #2, #3, #4, and #6 are entirely correct, and no points should have been deducted. The teacher’s objections strike me as more stylistic than substantive.

Problem #5 has a minor error (missing parentheses) and it’s fair to dock a point for this.

Problem #7 is very difficult if we insist on following the axioms strictly. Julia’s solution is nearly correct, but there are at least two missing steps. In order to rewrite (1/9)(9x) as x, we need the associative and identity axioms of multiplication, not just the inverse axiom.

In my judgement, the lowest score that could be given for this assessment is 93 (deducting 2 points for #1, 1 point for #5, and 4 points for #7), but I would probably give it a score of 97. This is outstanding work.

Manan ShahPost authorYou’re right about #1! I actually missed that and just followed the student’s work. ðŸ™‚ But even there I’d probably still be forgiving if I just graded the exam without my own solutions first. Nice catch and thanks for the detailed and thoughtful comment!

mrdardyI have to say Julia that you are being exceedingly kind in spirit and understanding here. I agree with the rather harsh assessment of your teacher’s grading policies here. Never mind the fact that this is the work of an 8th grader. I’d be pleased with my high school students producing work like this. Strikingly harsh penalties on this grading.

mrdardyTo be clearer than I was on the first visit – I agree with Manan’s critique of the grading here. Overly harsh penalties for pretty small technical problems in the work presented.

Julia KI do concur with primarily everything said here. What I’d like to address is that there were no axioms for the students to use as a reference guide, although they did take notes on all of them prior to this quiz, and the teacher expected everyone to study & know them (I am the student that took this quiz). I understood this, but I was absent for about a week, thus, I studied to my best ability with the time given and took it later. However, I have no recollection of her telling the class nor me these rules she followed while grading this quiz (i.e.- adding the opposite, not putting the equal sign when using the additive axiom of equality/adding on both sides, etc.,) When I asked a classmate if she told them about any of these exceedingly strict rules she wanted the class to follow, he told me that she hadn’t. As a student who had no distinct power over the authority of a teacher, I wasn’t surprised with the fact that he seemed overall okay with this, but I found it unfair. On the other hand, I would say that the teacher portrayed these steps while practicing problems with us, but never actually explicitly told us each of them in a way 8th graders would understand.

Overall, I really do enjoy her teaching—I feel that she provides students with a good understanding of the concept, even when it’s not in the curriculum. Despite her ways of grading, she essentially helps students immensely with their comprehension, which I like.