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Have you engaged in a “show your work” battle with a gifted math student? Have you heard this line: “If I can do it in my head, why do I have to write it out?”

It might be worth it to quote an interesting study of gifted students and their teachers’ personality types by Dr. Jane Piirto. Piirto found that most teachers prefer Sensing while gifted students prefer iNtuition on the Myers Briggs Type Indicator.

If most educators tend to fall into the S type, how will they meet the needs of the N preferring students that exist in their classrooms…  S type relies on their senses for understanding and learning… If they can not use their senses, learning will be minimized. On the other hand, the N type is quite the opposite. They rely on their hunches or inner sense…

Many gifted students do “just get it” as a result of their intuitive personalities. Many teachers need to see the pieces to understand. This is a major personality conflict that teachers need to be aware of.

Writing Out Lesson Plans

Let’s switch our perspective for a moment. Have you ever been forced to write out your lesson plans? Did you complain to your colleagues that you can teach just fine without typed plans?

Then you understand the frustration a gifted student feels when being forced to write out “the work” to solve for x given 3x = 9. For many of our intuitive gifted students, the solution for x in this case is as obvious as 1 + 1. And you wouldn’t demand proof for that, right?

(Let’s get it straight though, a first year teacher may very well need to write out plans just as a student first wrestling with algebra needs to write out the steps to solve for x given 3x = 9).

The Solution: Increase Complexity

Rather than wrestling with students to “prove” solutions with “work,” simply increase the complexity of the problem so they must do the work out to get it right.

If students can “see” the solution to 3x = 9, first congratulate them on having such an intuitive mathematical mind. Then, differentiate the complexity of the problem so that the child is challenged. Have your student solve for x given 3x – 2 = 7.

This two-step problem may be complex enough to make it useful to a student to show their steps. This is the crux. Once a student believes in the usefulness of writing out the steps, then they have an incentive (beyond avoiding a nagging teacher) to do so. As adults, we know that it’s useful to write out steps because we’ve goofed up enough checkbooks to convince ourselves. But as teachers, we also know that writing out detailed lesson plans is only useful in certain situations.

Are your primary students refusing to write out the steps to solve 21+ 30? Increase the complexity to 35 + 21 + 30. Can they do that in their head? Congratulate them (because it is impressive) and then push the complexity another step.

I wrote more about increasing complexity here.

We must be careful not to admonish our intuitive learners for being intuitive. As teachers of the gifted, we must set up learning environments that our best for our students. And if they’re doing it all in their heads (and getting it right!), then the environment needs to change.

Read more about showing work in math here.


11 Ideas

  1. ljconrad 19 September 2010 at 11:06 pm #

    I agree with you on not always showing work, but what about State Assessments? Our state (PA) requires that steps be shown, or points are lost. Perhaps this accounts for lower test scores of some gifted student? I really appreciated the POV that teachers and gifted students often have personality conflicts. My children’s teachers seemed well aware of the situation, but simply did not care. Hopefully, this will change for future generations.

  2. David Radcliffe 19 September 2010 at 11:24 pm #

    Most students think that math is about “getting the right answer”. I had much the same attitude until my junior year of college. It’s certainly a good idea to ask gifted students to solve more challenging problems, but it’s also important to address the root misunderstanding. Showing work is a part of communicating the solution to a problem. What good is it to solve a problem if you can’t convince others that your solution is correct? Also, showing work prepares students for more theoretical (proof-based) mathematics.

  3. Ian Byrd 20 September 2010 at 12:41 am #

    Interesting, California’s state test is multiple choice (at least in elementary). Before excusing anyone from any work (including showing steps), they should sufficiently prove their ability to do it correctly. In your case, I would make my own assessments mirror the state test and demand work on tests.

  4. Ian Byrd 20 September 2010 at 1:24 am #

    I think we’re both trying to address the root problem, which is to convince students that there is a useful purpose to showing steps in math. For these kids (I’m specifically thinking of some 6th grade boys I’ve had), showing them that the work can benefit them today is more effective than trying to convince them that someday they’ll take a math class that will require proofs (which, believe me, I have attempted). I want to move them towards that math class today, rather than dangling it like a threat. For some kids, that may mean asking them to give me the square root of negative one (and watch them squirm as long as they can before writing something down and then realizing they still can’t figure it out).

    I want to be careful not to turn these kids off from math in elementary school because they see it as a chore. Again, I’m talking about a certain group of natural, intuitive math students. If that happens, they run the risk of never making it to their junior year of college (and discovering that their teachers were right all along).

  5. kathee 20 September 2010 at 5:00 am #

    I’ve had this “discussion” with my son many times. I think it’s basically a good argument (that and noting that the teacher isn’t psychic and that while a correct answer is lovely, partial credit cannot be expected to given for anything not on the page). This WILL become more important in college. However, the problem (at least in our experience) many teachers are uncomfortable either with math in general (and so cannot offer increasingly difficult problems to the degree some particularly intuitive students can’t “hold it all in their head”) or are uncomfortable with a variety of mathematical proofs (my son had to drop a Algebra II class due to frustration with a teacher who would only allow things to be solved “the Saxon Way”, although my son was having greater success at explaining concepts to fellow students than she was).

    Fwiw, in my son’s case there is a history of serious dyslexia and dysgraphia, which I think don’t adds to the appeal of writing things out. While this would probably surfaced in other arenas than math, I think it is a consideration if there’s a student that just HATES writing out a lot of math. Or “refuses”.

  6. D Davidbate 25 September 2010 at 10:30 am #

    What everyone misses is … speed IS an integral part of the process … writers learn this early on. If you write with pen and paper, your thought process , your words, your voice limits itself, adjusts to the speed at which you can copy down what it is you have to say. Similar to “observing a thing changes a thing” … by recording a thing, a thought, writing down your work and the steps …. you change that entire process. Example, Try to dictate a story or an essay rather than use a keyboard. At first it is uncomfortable and challenging but you will adapt. As you get better you will note differences in your style, the very words you choose in fact. Because your vocabulary is not limited by your spelling or rate of typing etc. The thoughts flow faster and faster like an improvisational actors mind as they practice their craft. Wait until you are good at this. Do this for weeks or months … And then …. take up pen paper and try to write a new story or essay. It is painful. Your thoughts get backed up like cars in rush hour traffic jam. The flow is disrupted. Thats my 6 year old son when I ask him to do grade 7 math on paper instead of in his head. And your examples of making it harder so as to demonstrate the need for paper …. been there, done that and what happened was the horns started honking and the cars slammed on their brakes …. Jazz musicians, improv comedians, they know the beauty the joy that my son knows. The high! Yes he is just like an actor stepping off the stage after killing the audience …. I know that feeling …. he knows that feeling … and he gets it every time he does math in his head. Now how do we mess with that and not kill it??? Because a lot of Jazz musicians, and other talented people will tell you you can’t.

  7. Ian Byrd 25 September 2010 at 6:14 pm #

    Yes, speed is essential. Speed makes the next step in math possible. Fractions are much easier once you can divide in your head. But developing speed (or flow) requires work. Remember that a jazz musician also practices hours and hours and develops skills that push him to be able to improvise and play “in his head.” If he doesn’t, he’ll stagnate and end up repeating similar licks. Comedians put in hard work for their craft, writing and rewriting joke; appearing at open-mics and collaborating with other comedians. This makes the real show go smoothly. Dictating a story is possible, but writing a great story requires sitting down, editing, rereading, and rewriting. Math requires work in order for speed to develop. The traffic jam is okay because, with work, it will became another Autobahn for your son to speed through. Be careful not to push too far, but remember that if your son has done math in his head up to this point, he’s facing a challenge. And challenges are ok. I tell my students that our muscles only grow when we push them past their usual limits. Then they rebuild stronger. The only time we grow intellectually is when we face challenges.

  8. kathee 4 October 2010 at 5:15 pm #

    Perhaps a problem is that so often the “showing” of the work is only allowed in a traditional arithmetic/algebraic format, not allowing for the fact that some students (many gifted kids!) are more intuitive or visual thinkers. For example, the Pythagorean theorem can be proven with triangles alone. Gaspard Monge didn’t start with arithmetic calculations either (although his visual method was evidently a military secret for a long time). I think it is good for students to consider (and reveal) how they came to their conclusions but there can be more than one way to get there.

  9. Frustrated 16 December 2010 at 11:32 pm #

    I’m 16 years old and I’m a gifted grade 11 student. Last year, my school’s gifted program was cut and I was placed in the regular program. I noticed the change immediately. I found that there was a lot less time spent doing actual thinking and more time spent trying to explain how I got my answer on paper.

    Just yesterday, I received my physics test back and found that I had lost a total of 15% from the total mark because I didn’t properly explain my work in the format expected, although the steps that I showed were perfectly understandable and easy to follow. I had already figured out the correct answers to the word problems in my head within minutes but was forced to spend 5 – 10 wasted minutes drawing diagrams and writing out each and every step. Even then, I had more than half the total marks for the questions deducted because I didn’t rewrite all the given information from the questions.

    Every one of my tests are the same. The answer is placed at the very top of the page and after several minutes, the step by step explanation follows underneath it. It is extremely frustrating, especially when I get partial marks due to my work because both, my teacher and I know that I know how to do the question.



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