The Growth Mindset thread pointed out that often we see the right answer as the goal of math. This issue came up in several responses. I think context is everything. Math, on the job and in application during the day is absolutely 100% about getting the right answer. That is the ending goal. Learning math is a different process. I think we need students to understand the difference between the two, but no classroom should lose focus that the end result is people who get the right answer, not people who understand the concepts but fail many times on getting the right answer. How can we ensure that students learn the difference between the process of learning and the end goal of doing?

I like your question. Maybe the answer is to celebrate the process of problem solving more than the answer. For example, after having several teams of students work on a problem, have them give presentations to the class about their thinking and strategies. Then have the class discuss the merits of each team's techniques. Following this could be a test where each student must solve a similar problem by themselves, and the grade is more about the technique than the answer.

If my doctor misdiagnoses me I don't care how good his process was. If the bridge I'm driving on collapses I don't care how good the process was. Results matter. Even in social sciences I stress to students that there may not be one right answer all the time but there are absolutely wrong answers. I think it is a misunderstanding of growth mindset to assume that end results don't matter. Growth Mindset instead says one should accept that making mistakes is the way to ultimately achieve a right answer.

I agree that the answer itself is important; Rockguykev makes great quasi-analogies to show that. I think this is essentially what you're saying (learning vs. end goal), but the key difference in those analogies are that those are people that are in their fields having gone through all sorts of learning and are now expected to implement that learning, whereas we're working with kids, going through the initial stages of the learning process. A closer analogy might be the use of the bathroom (stick with me here...ha). We fully expect that there will be some "accidents" along the way as they figure out the use of a toilet when they're young, but once they have that down, the expectation is that they are accurate with that (we won't get into boys and accuracy ). I fully expect and hope that there'll be some struggles in their initial learning of concepts, but expect that they learn from their mistakes and are accurate eventually. I always tell my kids that they will always make mistakes in their lives, but the key is that they celebrate what went well, they determine what didn't go so well, and then they figure out a way to ensure the "not go so well" doesn't happen again. Isn't that what we hope for from everyone? With that mantra, it steps a bit away from "the answer does matter" or "the answer doesn't matter".

One step further.... If grading with partial credit, which part should carry more weight - concept or accuracy? Is it possible to pass a test but not get any of the problems correct? Should we at any point in the learning process just grade on the final result? For example, homework and quizzes can be graded such that the concept holds more weight but the final test requires both with students being well aware of this fact from the start.

Tough question, as this seems to play at least a bit differently in standards-based grading versus letter/percentage/point grading. In addition, there seem to be different beliefs about grades: some feel as though it's supposed to directly represent one's knowledge/conceptual base, whereas others feel as though it's more a combination of work ethic, effort, conceptual understandings, and correct answers. To complicate it further: in some areas, you literally either get a right or wrong answer dependent on your understanding of the concept, whereas in others, there are numerous elements/concepts/calculations at play. As an example, a student determining the angle measurement of an angle is straight forward, but proving a theorem in a long-winded proof requires a large variety of steps along the way. I remember a college mathematics course where honestly, the professor expected very few to get most of them "right"; the process there and all the steps/ingenuity in between was more important to him. Thus, at least to me, it highly depends on the situation.

It absolutely should NOT be possible to pass a test and get every answer incorrect. It wasn’t at any of the colleges I went to and it would not be acceptable in the workplace either. I guess this is now an American thing where we accept wrong answers so long as the student(s) “tried.” Ridiculous. As Master Yoda once said, “Do. Or do not. There is no try.” And this is the quote I have posted at the front of my classroom. At my school students get an A, B, C, or F. We don’t give D’s because D’s are not good enough to meet society’s standards. If you get anything below a 70%, you get an automatic F.

I was reading Jo Boaler's book Mathematical Mindsets and I agreed with most of it. However, somewhere in the book, she said that we shouldn't take points off for mistakes or give kids who made mistakes a higher grade to kids who made mistakes. This went a little bit too far for me. I agree that the end goal of math is to get the correct answer with strong understanding and with the ability to explain your thinking. I do believe that it is important to normalize mistakes so kids know how to learn from their mistakes. A lot of times, it seems like kids stop when they make a mistake and immediately ask for help or shut down. They should learn that mistakes happen and that they can inspect their own mistakes and help their classmates inspect their mistakes to correct their answer with understanding. This can't happen if kids are afraid to make mistakes. In addition, it is also important for students to learn how to solve problems that they haven't been explicitly taught how to solve. This requires students to be able to try a variety of problem solving strategies -- and likely make mistakes before arriving at a correct answer. I'm sure that this is an important skill for future careers as well!

I agree with this. It depends on what your purpose for grading is. My school does standards-based grading, so I've stopped giving "points" on a test. I simply mark answers as right or wrong, and then I write sentences of feedback on the process. I note errors in calculation, as well as missteps in a procedure or process. No one "passes" or "fails" the test. We don't have letter grades or percents. It's all about learning from mistakes and making growth over time. Perhaps it all depends on the grade-level, too. I teach elementary and think that this method is developmentally appropriate. You high school teachers may argue that it's not as appropriate at your level - which may or may not be true. I don't have the experience to know. From my perspective, a student who makes a simple calculation error (therefore, getting the answer wrong) but demonstrates understanding of the concept and process should receive a higher "grade" or "score" than a student who cannot show solid understanding of a concept or process, regardless of having the correct answer.

Explain how a student can have the right answer on a complex problem without demonstrating understanding unless they just put the answer and nothing else. Also, how does someone demonstrate understanding unless they describe it, especially in math where procedural knowledge in problem solving can sometimes mask understanding.

Well, that's just it. I often have students who just make a guess and, occasionally, they end up with the correct answer simply by chance. I also have had students who would solve a problem incorrectly, but they would somehow happen to get the correct answer. I can't remember the specific example, but I know that this happened on a test over area and perimeter last year. Students can demonstrate understanding by showing all of their thinking with equations and visual models. They don't always have to describe their thinking in words, but they can do that too. I'm not sure if that's what you're looking for or not. Maybe I'm misunderstanding your question...

No you are not misunderstanding. Good answer. Students can write an answer down and understand everything even without writing a step or as you said they can guess. You probably use other knowledge about them to determine what they were doing, but if at some point in the learning of the content they figure it out, you won't know the difference from the correct answer. That ah-ha moment could have come between the last time you saw them work a problem and the test. Solving incorrectly but getting the right answer. Sometimes it is solved incorrectly, but you have to make sure that they just aren't able to think differently. For example, sometime physics problems can be solved different ways. I was in a discussion about a student who solved a physics problem differently than the teacher had ever seen before. Was it right or wrong? Well after a lot of thinking, it turned out that the way it was solved work work in every case. It was just a novel and different way of looking at it. I don't believe students always need words to describe the understanding either, but we can't always depend on a series of equations, a right answer, or a set of correct steps. As you said, visual models and equations can demonstrate understanding. I think we have to be very careful when evaluating students that we really know what we are asking and seeing.

I don't disagree with you at all. I almost always require my students to show their thinking on tests and assignments, though. So, while it's true that they might understand something even if all they write down is a correct answer, I don't accept that work. I hand it back and make them show how they arrived at that answer. I've also had students solve problems in novel ways, as you've described. I won't argue with their work as long as I can follow it, even if it is solved differently than I would have solved it. What I was referring to in my previous post, though, wasn't this type of situation. If I remember correctly, it was something along the lines of a student determining a missing side length of a polygon incorrectly, which I knew because they labeled it wrong and showed their steps for determining that length. However, despite that incorrect side length, they somehow were still able to get the correct final answer through miscalculations or other errors in a later step (again, I can't remember the specifics at this point). To me, this doesn't demonstrate solid understanding. Sure, they knew some steps, but they didn't have a firm grasp on the concept as a whole. Even though they wrote down the correct final answer, their steps along the way showed that they lacked full understanding. So, in that case, I would say that their correct final answer doesn't deserve full credit (if you are giving credit - again, I don't do points or anything like that in my class). Finally, I do agree with you that I use other knowledge of my students beyond their performance on a test. All of my observations in class factor into their final grade on the report card. With some students, it's obvious that they copied someone else's correct answer on their test. With other students, it's clear that all they did was rush and make a calculation error. I try to base final report card grades on more than just a single test.

In my experience teaching students Physics, I have seen correct algebraic work (using the equations only, no values), and get incorrect answers because they failed to resolve units (not realizing they needed to convert), or converted incorrectly (1mm = 1000m). I have seen students incorrectly convert more than once in a problem and just so happen to cancel out their incorrect work, thereby getting the correct answer. They get the point for the correct answer, points for correct algebraic work, but lose points for the conversions and "calculator plan" since the values they intended to use were incorrect from their conversions. They also drop significant zeros and then get the wrong rounded answer because of it.

I'm teaching summer school algebra 1. Last week we had a test on quadratics. There was one page where they were given 4 quadratics to solve, using any method they wanted. The catch was they had to use a different method for each quadratic (i.e. quadratic formula, CTS, factoring, square roots).so they used each method once. There was one problem where the answer was "no real solution". I had a student write that for one of the problems, yet I gave him 0 out of 5. Why? He wrote that he used the factoring method to figure this out since the quadratic did not factor. Obviously factoring can only tell you if you have rational solutions, not real ones, and so, while he happened to write the correct answer, his reasoning and process was completely erroneous so I didn't award him a single point. On the other hand, I had a different student who knew to apply the quadratic formula, but made a sign error (they messed up with the b^2-4ac), and so, they wrote that it had irrational solutions based on the quadratic formula. This student received 2 or 3 out of 5 (I forget which). Although their final answer was not correct, they did at least proceed down the correct path.

An example I've mentioned before, I once bought a stack of books at a library sale, 25 cents each. I set them on the table in groups of four to make it easier for the librarian to count. She asked me why I did that. So I counted them up for her, and she said, "No. We must do it this way." She then wrote out a multiplication algorithm on paper showing every detail including putting in the zeroes. This was fine, she still calculated the correct amount, and if that was an easier method for her, that's fine too, I'd think; the problem was, she didn't have a clue that 4 books equaled a dollar. I've seen this in third grade. Students, when they're taught to put this here and put that there, tend to develop confusions about the purpose of algorithms. Algorithms are shortcuts towards obtaining "the answer". Along with this, "the answer" tends to be thought of as getting an answer "right" on a handed in paper rather than obtaining information. The standard algorithms in use today, which I believe should be taught, are what most people view to be the most efficient and easiest method of obtaining accurate information. "Showing one's work" is the easiest way to demonstrate on paper to the teacher that the student is applying the algorithm correctly, but we must be careful that the students do not begin to view the penciled notations as a must for solving the problem. In reality, outside of the classroom, such notations are not required and can even be cumbersome. For example, when I calculate in my head, I don't always go from right to left; sometimes thinking left to right is easier so that I can remember what I'd already calculated. I know beforehand how much to "borrow or carry" and just throw that in as needed. Or perhaps I won't even use a standard algorithm, such as organizing 25 cents into groups of four. Or a sale that's 20% off. It's often easier to find 10% off and then multiply by 2. I'm concerned that students view mathematics as obtaining grades rather than applying what they've learned. Real math is application. Even in theoretical math, there is a purpose beyond just supplying an answer in a way that pleases a teacher. I'm concerned how in elementary school, especially, most arithmetic is presented as a list of algorithms in a text or on the board to solve, not that this is bad, but this is not all there is to math. Usually in a text, at the end of the page are a couple of token "story problems" where the students must take the numbers and arrange them according to that lesson's algorithm and solve. I'd much rather emphasize realistic situations to then use the algorithms to resolve that situation. In learning the algorithms, more complete understanding occurs when students apply the algorithm according to the mathematical reasoning for how the algorithm works. Just only developing the habit of "carrying" or "borrowing" (to use older terminology) when the top or bottom number is too big or too small or the answer space can only hold one digit does lead to correct calculation and correct annotations but so does entering the numbers into a hand held calculator. The answer is right, but the understanding might not be, and the student is unprepared for future mathematics.

You hit on some very excellent points here, well said. The thing I want to add to this conversation, just something for people to stew on is that this is "teaching for the test" at work. Sometimes we're so focused on teaching what to think in order to pass a test that we forget to teach how to think. And in fact, the system is stacked that way against students. The whole system, not just maths. We break the cycle by not teaching for the test, but rather teaching for the experience. The analogy I like to use is that maths are poetry. Spoken or written, rhyme or not, but a language. Science, on the other hand, is a stage performance. Both of them, however, are beautiful in their own right.

Uh.... Did you have an actual point to make other than just cutting my post up into chunks that make no sense? Or did you not get what I meant?

I agree with a lot of your post, but I disagree when it comes to your comment on elementary math being presented as a list of algorithms. That's certainly how it was when I was a child many years ago, but it's not how math has been taught in past 5+ years. With the Common Core standards (and, in some schools, prior to that), elementary math has had a greater emphasis on problem-solving and number sense as the overall goal. Algorithms have been de-emphasized significantly in recent years. Sure, they are still taught, but students have many strategy options for solving problems beyond the algorithms.

I'd almost simplify elementary math as "solve these problems, and here I'll teach you some suggestions on how you might do that."

Yesterday, I stumbled upon an interview on YouTube that addresses the above discussed issues, titled India Questions Math Genius Professor Manjul Bhargava from NDTV.