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Pre-algebra

If the student is taking this branch of mathematics, they are expected to show their work because they're expected to solve specific problems in a certain way. Ex, when they're solving for a variable they're supposed to manually find the value of x by isolating it rather than entering the left and right hand of the equal sign in the calculator (in slope-intersect form) and finding the intersection point. If they only get the correct answer without showing work, we give them 2 points. 8 more points if work is shown.

Algebra

Students in algebra are expected to know how to get the correct answer. We don't care how they do it, as long as it's the correct answer. Full credit is given for the correct answer, basically.


Are we being too "harsh" on our pre-algebra students? Should we let them get full credit for the correct answer or should we make sure they're ready for algebra with our current system?

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Full credit should be awarded to correct results achieved by correct means. –  Mark Fantini Jul 8 at 3:27
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Seconding Fantini's sentiment. With the additional remark that "correct" = "logically valid" may mean something other than the method spelled out in textbook examples. –  Jyrki Lahtonen Jul 8 at 7:50
    
While I agree with Fantini's sentiment, I don't really like comments that are aimed at trivializing the question. I think it's a good question. –  Chris Cunningham Jul 8 at 12:36
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Real mathematicians write proofs. An answer in mathematics is not the result alone; it's the logic of the proof. If a student knows this, it should be more motivating than a simple command that work be shown. –  Cory Jul 8 at 16:02
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I was told in my courses here that "magical answers" (i.e., just plop down a number/formula) gets the "magical grade" 0... –  vonbrand Jul 8 at 18:14

6 Answers 6

up vote 14 down vote accepted

Personally, I'd say "no."

The students who don't write out the steps in their algebra classes appear in calculus thinking that they should be able to write down all the answers without any intermediate steps. I even have some students in Calculus 2 who think that there is some kind of value in not writing down the steps. None of these students can complete any calculus problems without errors.

I'd prefer that in all math classes, student submitted solutions should be solutions: I should be able to read them and be convinced that the final answer is correct. "x = 7" out of nowhere isn't convincing anyone, and I don't see it as a useful submission.

In my opinion, the earlier this standard becomes expected, the better.

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I agree, and in fact I think if anything one might relax the issue in pre-algebra, but certainly not in algebra. I thought the main algebraic ideas in pre-algebra were what it means to solve an equation and maybe some very basic English (or other appropriate language) translations to algebraic expressions (i.e. subtract four from three times a certain number can be expressed as $3x-4).$ On the other hand, a main topic in a (purely) algebra course is methods for solving equations, and you want to see students exhibit the methods to judge whether they appear to have learned them. –  Dave L Renfro Jul 8 at 14:41
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The word 'convince' is the absolute key for promoting cooperation from students who are inclined to produce an answer alone. Give a hypothetical wherein another student provides a one-line answer that's different from theirs, have them defend their own answer against the alternative, and explain that this defense needs to be a part of the answers that they provide. –  NiloCK Jul 8 at 17:07
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To expand on that, make sure that students understand (and make sure that this is your understanding as well) that they approach all problems as if the asker doesn't already have the answer. –  NiloCK Jul 9 at 12:35
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I'll add this: one objective of assigning problems to students to see if they can do them, and how they approach them, because this makes it easier to support the student. For a student who shows no work, whether or not they get the final answer correct, it becomes extremely difficult for a teacher to decide how to support this students continued growth. –  David Wees Jul 22 at 10:50

I'll agree with what's been said, and say no.

Showing the work means showing that you've grasped the concept itself. Being able to produce the answer is more often than not the easy part. Often times, the answer is immediately apparent when reading the question, and one might be tempted to "show off" and simply state that I can tell you the answer without doing any work, and hope that the reader will be impressed by this.

It is impressive. It means that you've gained a very intuitive understanding of the concept. But with that, you should be able to tell the reader exactly why it is so intuitive and obvious. That's the really impressive part, and therefore should be the thing that awards full credit.

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It depends on the mark allocation but in my opinion I would say no. I've seen before that a learner somehow arrives at the correct answer in homework but none of the work is correct. At the end of the day we are assessing a number of things, not just the answer. A learner must demonstrate the appropriate skills and knowledge when answering questions.

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Great Question.

In school, and in math especially the goal is to model systems. Wether the system be large, or complex. Super applicable or abstract. Our goal as educators is to teach students how to think deeply about the concepts, and to apply the skill sets.

For me, mandating showing work allows me a new dimension of teaching. It demonstrates an understanding of the conceptual, and of being able to follow a modeling process. For you as a teacher, it will allow you to differentiate your teaching to meet the needs of students who are lacking understanding or follow through in a particular area.

Showing work is equally helpful for the instructor and student. For the student, it forces them to compartmentalize their work in a framework that they can communicate to others. It also allows them to track mistakes, organize methods for problem solving, and build a framework that is applicable in all areas of life. For the teacher is allows you insight into the needs of the student, and a map for how your students are thinking differently.

And on the practical note, habit building is crucial. And in any field in industry these students will need to be able to document their work in a way that successfully communicates how they've solved the problem.

I'm an Undergraduate researcher, and this is something one of my favorite advisors told me, when I didn't think I needed to be thorough in the way I documented my findings. "If you can tell someone the answer, but you can't explain the process, or teach anyone to get there. More often than not your solution falls on deaf ears" Essentially. Communication is key.

The best Mathematician I've ever met, Dave Prince, always told me, "What works, is Work!". So if you don't show it then it doesn't work. We would give 20% for correct solution, and 80% for process.

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Professors worldwide should use the procedure as the basis of grading. Three reasons.

  1. It allows the examiner to catch potential misunderstandings/mistakes, and it gives him/her the opportunity to signal them to the student.
  2. It rewards students in a way such that the work is more valuable than simple fraud (it can be easy to copy a classmate's answer, but it degrades the whole methodology, turning it into something similar to a multiple selection exam).
  3. As mathematics builds up, topics from first courses start getting trivial, and can be forgotten by students. Sometimes, a small error generated by this can ruin the answer, and deprive a student from a good grade, even if the actual course's theory is correct in the solution. Grading by procedure allows the student to receive a good (if not the whole) grade, because he has demonstrated that he/she understands and applies the actual course theory.
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My last undergraduate exam (many years ago...) was for the course "Microeconomic Theory II" - not a 4th-year course, but we had freedom to schedule. I was aiming to get an overall "Excellent" grade in my BA (= above 8.5/10 as grades are measured in my country). Moreover, this was my last exam, so I wanted it to be a triumph (vanity is never far away...). We knew that the exam would require drawing diagrams, but also performing basic algebraic calculations (finding the extremum of a function -cost minimization, utility maximization, etc). So I went in with pencil, ruler, eraser, sharpener, millimetre paper to draw the diagrams, and glue-tape to glue them on the exam papers. But no calculator - we were allowed calculators only in Statistics, to perform linear regression.

And indeed, there were the diagrams, and I was excited to draw them in such a high-quality (and flashy) manner, and of course, there was the "find the extremum" part. I did -and the end result was not a nice looking, round number. I got suspicious: we knew the professors were usually giving numerical exercises with nice round solutions - as a gift to the students, and perhaps a little easier (to the eye) to grade afterwards. So I re-checked the whole calculations twice (so in all, I did them three times). I could not find anything wrong, so I thought, "hey, this time, no nice and round solution". Apart from this little worry, I was pretty sure I had answered everything perfectly (i.e. completely and correctly).
After the exam ended I realized what I had done, three times in a row: I had "divided a multiplication": there was in front of my eyes something like "$3 \times 4$" ($=12$) and I have repeatedly calculated it as $3/4$ (=$0.75$). Down goes your triumphant final exam...

I got a 10/10. I am not saying that this was a fair grade (because there were maybe other students that performed at the same level as me without making the silly mistake), but obviously the professor saw that it was a silly mistake, and decided to ignore it, (impressed, perhaps, by the unexpectedly executed diagrams). But, my point is, that the only reason he could see it as a silly mistake, was because I had written down all the steps leading to the solution.

So while the correctness of the end-result is important, it does not really convey anything about what the student knows, if it is presented alone (and leaving also aside issues of cheating, etc). It is the arrival method that gives the instructor something to evaluate (the journey and not the destination, as it is said in other contexts...). So I would agree that "answers out of nowhere" should get a "nowhere" grade.

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