Tuesday, April 5, 2011

The Burden of Proof

Vickie Bergman blogs about education and parenting at Demand Euphoria.

It seems like math teachers do not care enough if you know how to get an answer. They care more that you can prove that you know how to get an answer. Here is an example to illustrate the difference:

I was helping an 11-year-old with his math homework. The assignment was to find the greatest common factor (gcf) for eleven pairs of numbers. He is really good at this. In fact, he figured out the answers to all eleven problems completely in his head, in less than one minute. This should be a good thing. This means he "knows" this concept. On to the next concept! Right? No, wait... what does that say on the assignment? Something about "showing your work..." Yes, that's right, even though this child did absolutely no work to get these answers, he has to write down some work. But not just some work. A lot of work.

So for the pair of numbers (60, 18), the gcf is 6. My little friend got this answer immediately upon looking at the numbers, as anyone who knows the concept would. But instead of writing down "6" and moving on, he has to write all this:

     18             60
      ^              ^
   6   3         6     10
  ^              ^      ^
2  3           2  3  2  5

18 = 2*3*3
60 = 2*2*3*5

Both numbers have one 2 and one 3 in common, so the gcf = 2*3 = 6

An assignment that was completed in less than one minute has turned into an hour-long session of unnecessary writing. He has to write this process over and over again, eleven times, to prove that he knows the steps. Even though he did not use these steps to get the answer in the first place! I don't know a better way to make a kid feel like math is boring and hard and requires tedious processes.

I ask: WHY? Why do teachers do this? It would take one minute of a teacher's time, sitting with this child, to realize he knows this concept. Instead, so much of his precious childhood is eaten up filling out paperwork like this, night after night.

This is a child who is brilliant in math. He often arrives at answers to complicated problems in his head, without writing down any work, and without being taught any steps. Sometimes it takes me ten minutes, after he has already told me the right answer, to explain the steps he needs to write down for his teacher's benefit.

On tests, he often loses for not showing work. I have already shared some of my feelings about tests in general (Part I and Part II). But here is another great example of TESTS (Teaching Everyone Some Terrible Stuff). Now this mathematically gifted child is bringing home test scores in the "C" range, even if most of his answers are correct. This is harmful to both his confidence and his interest in math. How can he ever believe he is good at math if he brings home such low grades? And this will only get worse as he gets older.

I can remember being upset about this when I was a student, but fortunately it did not completely turn me off from my favorite subject. And I have both a Bachelor's and a Master's degree in math to prove it.

This practice is ruining math for our children. Can we please make it stop?


  1. I had a really nice response to this, but it got eaten by blogger.


    1. Repetitive practice of a concept that the child knows? Strange. What a waste of time that whole homework assignment was for that child.

    2. Communication of ideas is important. That child should be expected to demonstrate (once) that they can clearly explain their process. Over and over again is ridiculous. Using a prescriptive way of communicating the process? Not necessary. If you don't think communication in math (and related fields) is necessary, see http://en.wikipedia.org/wiki/Mars_Climate_Orbiter#Communications_loss for an example of lack of communication between teams making a huge difference.

    3. The assessment of mathematics should be broken into parts: "Knowledge and application", "Communication of mathematics" and "Evaluation of process." So a child like the one you describe would do well in knowledge and application, and poorly in communication of mathematics.

    4. This fixation on communication likely results from an over-emphasis on scoring part marks on long response standardized assessments.

  2. I agree that students who understand concepts quickly should be required to prove over and over again that they understand it. However, it is important that they explain it at least once.

    One way I use technology for this is to allow students to create some video tutorials explaining how they learned a concept, and teaching someone else how to do it. This take no more than 5-15 minutes of their time and becomes a product they can use for the future (so can the teacher).

  3. Another reason we teachers have students "show their work" is so we can better diagnose any misunderstandings or errors. I agree with your argument that 5 mins spent with the child a teacher would see he can do it. But it was a take-home assignment therefore, very important the child demonstrates his understanding in a written form, not verbal only.

  4. @David, I help this kid with his homework three nights a week, and every math assignment is equally as tedious and time-wasting. It is maddening for him (and me!).

    @Brian, Why does every child need to be able to explain how he did every task? This is an important skill for a teacher to have, but not necessarily for a student. Some people can't explain things well. But that shouldn't take away from the fact that they can "do" the thing...

    @Paige, Why give the take-home assignment in the first place? Why not spend a minute with students like this before assigning the homework? How often does the average person have to write out math solutions in real life (outside school)? Certainly there are some people who never do. Why can't "knowing" the concept be enough?

  5. @Vickie @Paige, this makes me think of something hit on in Freakonomics. Many people who are very gifted in a variety of fields don't know how or why they know something, they just do. It might be interesting or beneficial for others to dissect that (for them!), but it is not helpful to the person who just knows to have them do so.

  6. As a math teacher, it's really important to get students to show work.

    I agree that the teacher shouldn't have had him do it a ton of times AT THIS LEVEL, but at least showing a few. However, as he moves up, it will become more important that he show his work on every question. Two reasons...

    The first is that math builds on itself. x + 5 = 12 is easy - you quickly, without thinking, know it's 7. On the other hand, x + 345 = 457 is a tad harder. In grade 5, when this is introduced, there needs to be a "process" for the question. The reason that we force students to SHOW that process is because it gets more complex still, but the idea stays the same. y + x - 32 = x^2 + 18 is following the exact same idea as x + 5 = 12.

    The second reason is because it allows students to find errors, and reduce problems. In engineering school, the most common mistake was not process, but instead mathematical stupidity. We thought that we could skip 4 steps because we "knew them", messing one up in the process, and then (as our prof said) "killing 120 people because your plane just crashed". An extreme example, but not too far off.

    Therefore, students need to be taught to show their work - I even go so far as to give some tests with all the answers on them, and have students show how to get it. It reinforcese the idea that the process is MUCH more important than the answer, because it tends to be worth more to "using" the math.

    At an early grade, it's annoying because the problems are easy. At a later grade, it's a vital skill because the problems are harder.

  7. @Graeme, As far as your first justification, I just don't think it is necessary for a person to solve x + 5 = 12 and y + x - 32 = x^2 + 18 in exactly the same way. If it's easy, let it be easy! When it gets hard, if steps are needed, then learn the steps.

    And your second justification might hold in engineering school, but most kids are not going to end up in engineering school. It's like making everyone learn how to kick a field goal at 11 years old, just in case they turn into a professional kicker. That would be crazy. If someone is going to be in engineering or any other profession that requires meticulous math detail, they can "learn to show work" when it actually IS necessary. Then they will decide if they are good enough at showing their work, and if they enjoy it enough to keep pursuing that type of work.

    Also, I guess I am arguing this point from a completely different universe, because I don't think all kids should have to be taught complicated, "written-out" math in the first place. So to say that doing in in fifth grade is practice for the later grades doesn't mean much to me either.

    People (including kids) shouldn't have to do things that are annoying. Unless they want to. Then they can choose to go to engineering school. :)

  8. Except that "learn the steps later" is a lot harder than "learn the steps early and build on it". We all learn this way - we don't jump into the hardest stuff without first learning the basics. We start kids off with x + 5 = 12 because we want them to LEARN without being bogged down in the numbers. Then the concepts transfer over to the hard stuff.

    Your second point comes to the argument of "why learn stuff". I'll toss that back at you. What's important to know? Why do we teach students Shakespeare? Who cares why Calcutta is a slum? History is just the old boring past. I'll never need to know what inside of a cell.

    But where do we draw the line? When does it become, essentially, that we "know" nothing?

    Learning for the sake of learning is valuable - it makes us human, it makes us able to converse with each other, and, dare I say, makes us "better people". I didn't care about history in Grade 8, but I sure do love to not sound like a moron now when someone mentions something I learned.

    Giving students the gift of a "love for learning" appeals to me, regardless of what that learning is. Therefore, "forcing" students to learn "annoying" things makes them well rounded people, and opens doors that they may decide they need later.

    Good discussion though

  9. @Graeme, “Learn steps later” is not harder. I would never remember something later that I learned a year or two or ten before. I learn things when I need them. I learned no math in school. NONE! But when I wanted to do a bake sale I figured out money. As an adult when I need to do a budget or calculate percentages and do advanced work in excel I Google or Tweet to figure out what I need on demand. I don’t show my work, nor do I reflect back on what I learned years before. I, btw...never learned those algorithms you mentioned, because I never knew why I needed them and that’s something I need to know in order for me to learn.

    I did have teachers who FORCED me to learn really boring stuff in school about history. I don’t remember a single thing. Spewing information kids don’t care about and forcing them to memorize and regurgitate is NOT learning. Forcing students to do tedious work, read boring textbooks, answer boring questions, and take boring tests amounts to something that gave me the “hate of learning.” School took the fun out of everything. Reading what I was told to read and interpreting it how I was told to interpret it made me hate reading. Poetry meant iambic pentameter. What the hell is that? Poetry was ruined! Math was disconnected algorithms I never learned. History was a bunch of biased facts about people I didn’t know but memorized until I forgot it once I regurgitated it for the test. Science was reading pages in a textbook or looking at manufactured slides and filling in worksheets. I didn’t recover from my hate of writing for many years. Writing a boring report for a teacher has no purpose, meaning or value.

    Like me, many students don’t develop a love of learning by doing boring stuff they don't like for no real purpose and no real audience. Especially today boring kids to death with disconnected, dull, meaningless work like that mentioned in the post turns into a "hate of learning" for students, a feeling by teachers that they have to trick kids into learning, and a budding tutor business because all kids need cheerleaders to make it through the drudgery.

    If students had the opportunity to choose to learn what THEY care about, then maybe they could actually be freed to develop a love of learning, figure out what they love, and we may actually see more than 1 in 3 graduating from high school.

  10. @Graeme, To add to what Lisa said...

    "We all learn this way - we don't jump into the hardest stuff without first learning the basics."

    It's only necessary to learn this way in school. Because this is the way things are taught. But many times in life, we get to jump into something at a higher level and we learn as we go. For example, my husband and I bought a retail store about 8 months ago. We had no clue what we were doing. We opened the day after we paid for it, and we only had a few days to think about it before that. We sure have learned a lot along the way, and we continue to learn every day. If we had hesitated, we would have missed the opportunity. Not to mention that whatever it is I could have learned from a book or a class about how to run a store wouldn't have made much sense until I was actually doing it.

    "Your second point comes to the argument of 'why learn stuff'."

    Not "why learn stuff," but "why force people to learn stuff?" BIG difference. I honestly don't think there are too many facts that every person should know. Babies learn without formal instruction, adults learn without formal instruction, only between the ages of 5 and 17 does our society think we have to force-feed knowledge. I don't understand that.

    "I sure do love to not sound like a moron now when someone mentions something I learned."

    I have a whole blog post coming up on this very topic. I will let you know when it's done. Too much to say here.

    "Giving students the gift of a "love for learning" appeals to me, regardless of what that learning is. Therefore, "forcing" students to learn "annoying" things makes them well rounded people, and opens doors that they may decide they need later."

    I'm more concerned with how schools take away the love for learning. All little kids love to learn. They start to hate it as school makes them think learning is hard, that it's boring, that there are only certain things worth knowing. I'm more concerned with the doors that are closed by school. The doors to art and music and dancing, and other creative pursuits. School even closes the door on math for some kids. I have heard kids say "I'm never doing math again after high school" way too many times. That is a shame. Math is everywhere, it's not just in textbooks. Math could be seamlessly integrated into our lives, but instead it is painfully extracted, made into it's own thing, and forced upon us out of context.

    Enjoying the discussion as well!

  11. What's the Greatest Common Factor for the numbers 18 and 60? The answer is - Who cares?

    My point here is that math should never be about just manipulating naked numbers on page 57 of the textbook, or all the even numbered questions on the worksheet.

    I agree with David - the process should be more important that the product - meaning that if a child were to respond with either the correct or incorrect answer to 5 x 4, my first response is "how did you get that?"

    Now, once I see that the child is reasoning, I don't bog them down with "showing their work" needlessly.

    I work really hard at making sure that the math we do is in a context and for a purpose.

  12. @Joe Bower, first, thanks so much for accepting my invite to weigh in. Next, if the kid understands it, what do your advice for the parent who wants to address the teacher who is demanding that they do so over and over again? And, for the kid who doesn't understand it, how should that be handled?

  13. This is a great discussion. As a math teacher, I also have strong opinions about the topic.

    As for the homework assignment, repetition of the same concept at the same level of difficulty does not serve a useful purpose in my classroom. When I assign homework, it is 4 problems or less and are tiered in difficulty so that I can determine the current skill set of each student when evaluating. It is imperative to see the work to understand the way a student thinks about a topic. While this can be uncovered in a 1 minute conversation with a student, I would have to have 30 1 minute conversations to determine the level of understanding of each student in the class which is not possible.

    As for the greater conversation around knowledge, I separate learning into two areas process and content. I believe that math is primarily process and the purpose of math is not x’s and y’s but to give people a fundamental understanding of organization and logic. I understand that not all students will be getting an engineering degree, but at some point all students will have to use logic. Most people will not have to know how long it will take for two trains to collide, but most will run into a day where they will have to organize a number of tasks in an order so that they will get everything done. Math provides many opportunities to learn new processes and apply processes learned to new situations. These types of skills do drive the success of people in the real world. I think this gets lost in some of the day to day teaching and learning of math sometimes.

    I appreciate the variety of viewpoints and just wanted to add my $0.02.

  14. @Eric, Thanks for chiming in. It's good to hear that some math teachers aren't practicing this kind of "overkill."

    As far as the "math as logic" argument, I have a different take on that as well. I think for us math-types, maybe formal math processes are useful and apply in our real lives. I definitely think about lots of things in mathematical ways. But I have a feeling that is the reason I did well in math in school, and not the result of taking math in school. I know there are people who just don't think in mathy terms. And teaching them math might just be counterproductive. It might make them question their own logic and way of looking at the world. Like somehow any other way isn't good enough because math is better.

    I have tutored many kids in math, and I honestly don't think math helps them think about things differently. The only thing I see them thinking about differently is their own intelligence. I hear "I'm just stupid" a lot. And it breaks my heart. Not everyone is cut out for algebra and beyond, and there is nothing wrong with that.

  15. @Vickie, you hit the nail on the head for me. I never understood math in school. I don't think in those terms, however I can think logically and critically. Math for me did exactly what you said. I felt confident and smart in general, but my jerky math teachers seemed to take pleasure in proving me wrong about my self-perception. If I didn't understand these abstract concepts they told me to blindly believe were important, then I must just not be smart enough.

    I have nightmares to this day about math. Race to Nowhere was dedicated to a teen who committed suicide because of the damage to her self-esteem that was math induced. When I wrote about this I got several letters from individuals and parents confessing that math had resulted in suicidal thoughts and/or attempts for themselves or their children.