Tall, Dark, and Mysterious


Data retention

File under: Those Who Can't. Posted by Moebius Stripper at 4:56 pm.

One of these days, I should post about how much I like my students. I really do. I have a handful - as in, I can count them on the fingers of one hand - that I wish weren’t in my classes, but the overwhelming majority of the ones I’ve actually spoken more than a few words to are mature and eminently reasonable people; of those, a substantial portion are also putting in the effort required to master the material I teach. Swear to God; I don’t know what I did to deserve this.

Anyway, my precalculus students wrote a test the other day. I tend to make my tests “semicumulative” - 90% of the test is drawn from new material, and the remaining 10% is culled from topics they were exposed to before the previous test. On this test, their second, I put two questions that could have been on Test #1; of these, my students were to choose one, and I’d count the better one. One of my top students came to me after the test and mentioned that she’d done almost perfectly on the new material (she had), but couldn’t get more than half marks on either of the “old” questions. She wasn’t complaining; she was commenting that she’s clearly able to learn and understand the material, but that she has trouble retaining it. She asked me if I had any advice.

I didn’t offhand, but I told her that hers was a valid and worthwhile question, and that I’d think about it over the weekend and see if I could come up with any useful advice. I know that there are a number of experienced high school and university math instructors reading this blog, and I’d love to hear if you have any insights into how math students (and students in general) could better retain the material they learn - as well as how math teachers could teach for better retention. (Please ping this post if you think your readers might have any ideas!) Every year in grade school and high school, math teachers spend several weeks reviewing the previous year’s material, so this is clearly a pretty widespread problem.


  1. Talk about what you will teach, teach it, and then talk about what you taught.

    We tend to naturally focus only on the second item, maybe the first, but we rarely address the last one. I’m as guilty as anyone.

    In some settings, it is possible for the student to cover the last item himself. For example, if you give an significant assignment *after* the students supposedly learned the material, then all three phases have been covered. Alas, in a fast-paced course, the last step often gets forgotten.

    - Daniel Lemire — 10/23/2004 @ 6:56 am

  2. I’m a cognitive psychologist which means I study how people’s memory works. I would say that your student’s problem is she is not studying for only short time use. In order to learn material that you want to keep with you, you should space your studying, overstudy the material and make the memory really strong, refresh the memory by studying again. The stronger the memory, the less it will “fade” as time passes.
    There are many ways to make memories stronger, but you still have to maintain the memory or all is for naught.

    - DrD — 10/24/2004 @ 12:50 am

  3. As a student, I tend to forget anything I haven’t used for more than a couple of weeks - probably because of my bad study habits - though if I’ve learnt something once, I’ll very quickly pick it up a second time. Could you get your students to keep on using older material - work it into the newer stuff, or have mini-quizes on it, or weekly exercise sets or something (or encourage them to do this for themselves)? I like your semi-cumulative tests idea. I might suggest it to friends of mine who teach.

    - Zoe — 10/24/2004 @ 2:51 am

  4. (1) distributive practice. don’t study the material all at once, the night before the exam. instead, space out your studying over several weeks. the more often you study the material, the more it gets integrated into long-term memory. studying it just once or only a few times is a recipe for forgetting it. the more often the brain is called upon to utilize information and skills, the less likely it is they’ll be forgot. in this respect, the brain is very much like a muscle — the more it gets used in a particular exercise, the stronger it is.
    (2) study the material at night, just before going to bed. one of sleep’s functions is to integrate short-term memories into the fabric of long-term memory. the fresher the material is in your mind right before you go to sleep the better.
    (3) take a piece of paper and fold it in half lengthwise. in the left column, write a list of key questions or problems from the older material, and in the right column write their solutions or answers. then cover up the right column and quiz yourself on the material. this forces the brain not only to solve problems and demonstrate comprehension, but to remember the material without any references or aids just as one must do on a test. take this quiz every couple nights — especially for older material. (it only takes 5 or 10 minutes.) this powerfully reinforces the old material and makes forgetting it virtually impossible.

    - wes — 10/24/2004 @ 8:12 pm

  5. While retention is clearly a problem, I think that the real problem is that people expect (near-)perfect retention rather learning to deal with imperfect retention. What I do (and tell my students to do) is to review more or less continually. I look back at things and if I have doubts about how well I understand something, I review.

    This isn’t just a matter of constantly reinforcing memory, however. It’s much more important than that. When I go back over something, I often understand things differently — in a different context, with more knowledge — than I did when I originally learned it. That is, I learn it better. I’d guess that your student would discover the same thing — that the second (or third, or fourth) time around, her understanding becomes more complete.

    Eventually, we all forget the things we don’t use regularly. But the more and better we’ve learned something, the easier it is to bring ourselves back up to speed (e.g., paging through a book may suffice). We shouldn’t expect learning to be a linear process. You don’t just learn something — you learn it, and keep on relearning it.

    While there are some things you could do to help your students retain material, I think that it’s most important to tell your students that (i) forgetting is inevitable, and (ii) they shouldn’t be afraid to go back over old material (besides which, it’s easier and more pleasant the second time around). For this reason, I often prefer to inflict material on students at a relatively high speeds, knowing that having time to revisit things is often more valuable than discussing things carefully the first time around.

    [MS: I will get around to writing you a proper email one of these days … though I’ve been thinking this for more than a year.]

    - Trung — 10/24/2004 @ 11:02 pm

  6. I agree with all the commentary, especially Wes’ comment #3 about revisiting key questions. I have binders full of those sheets from every course I have ever taken.
    I have one disagreement, also with Wes: there is no evidence that learning something immediately before going to bed improves long term retention.
    There is abundant evidence that working up until bedtime interferes with sleep, and about 10% of a general group of people will have sleep onset insomnia. I wouldn’t encourage habits that further disturb sleep.
    Study until a set time, and then (paper deadline crises aside)put the work down and do something to relax until you are sleepy; only go to bed when you are sleepy; never stay in bed trying to fall asleep or back asleep; maintain a consistent wakeup time;…and all the rest of that good sleep behaviour advice so few of our students follow.

    And I am glad you like your students Ms. TDM, Continuing to like the students is a challenge that many older teachers falter or founder over, and you can’t teach well if you are indifferent to them, much less contemptuous of them.

    - littoral zone — 10/26/2004 @ 12:49 pm

  7. Thanks for all of the excellent comments and suggestions.

    Daniel Lemire - it’s funny, I’ve recently made a point of going through that third step - talking about what I’ve just taught - but for an entirely different reason. My reason was that I finding that so many of my students were completely bewildered when I gave them a word problem that differed even slightly in form from the ones I’d done in class or given in the homework. They seemed to truly believe that the reason I was giving them that question about boats was so that they’d learn how to solve a problem involving a boat going 10 km in 2 hours with the current and 10 km in 3 hours against the current. (On the test, I had the current die down to half its original speed on the way back. Three quarters of my students were baffled.) So I’ve taken to going over what we just did, along with a “here’s what you should be thinking about; here are the key ideas.”

    Trung - (BTW, I keep meaning to send email to all the people I haven’t talked to in too damned long, too) Your comment about forgetting, and learning to deal with imperfect retention reminded me of how often I have students mention that “there are so many formulas to memorize” in my class. Lately I’ve taken to asking “which formulas?” (since I really don’t know; this is not a formula-heavy class!) and then telling the students that the formula they’re thinking of comes from somewhere very specific and that instead of memorizing it, try to understand where it comes from. And upon reflection, I think that this is much of the problem with students not “remembering” basic algebra. Now, it’s not like I have retained all (or even a majority) of the math we learned at our alma mater - but there are some aspects of the subject such that “forgetting” them strikes me as being analogous to forgetting how to conjugate a verb used in everyday conversation, and less like forgetting the date on which the Treaty of Versailles was signed.

    Wes and littoral zone - I’ve given my students advice about timing their studies, but I have never heard anything one way or the other about studying before bed. (I’m sure there’s a big difference between studying before bed at a reasonable hour, and studying before bed when you’re ready to die of exhaustion.) I do tell my students that if their exam is at 8:30 am, say, they should get in the habit of doing math in the morning, even if they’d rather be sleeping.

    - Moebius Stripper — 10/26/2004 @ 1:50 pm

  8. About morning tests, and studying before bed: I’ve never heard any data on time-of-day stuff, but as to sleep v.s. studying, if the goal is problem-solving, sleep is extremely important (take note, MIT Mystery Hunters), whereas for short-term recall, studying is better. So, cram the night before for your ecology test, but sleep the night before for math.

    Your post, and the first comment to it in particular, reminded me of the main comment that I’m going to write on my math prof’s teaching evals at the end of the term… I don’t know how directly relevent it is for your classes, since I word it in terms of proofs, but perhaps you do this already for explaining methods….

    Point is, one of my very best profs had a habbit of giving the proof in class, and saving enough time at the end of the class to give the whole proof again, in an out-lined “what did we do” form. In that way, complicated arguments with many steps can be reviewed quickly at the structure level. I’d expect that for precalculus, too, this would work to increase understanding: after explaining all the steps to solving the problem, review the structure of the solution and the reasons for the steps.

    Many profs sketch arguments before giving them. This can be helpful, but I find I’d rather see the sketch at the end, because I can better relate it to the details. When I’m listening to details, I dutifully take notes, but don’t have time to think back to the overall structure. When I hear the structure, I can refer back to the details.

    Point is, and it’s a point you’ve made many times on this blog, math needs to be understood as a collection of methods, not a collection of facts. Your students have probably learned math as a collection of facts (formulas and such), but in fact much better is to learn the procedures and structures.

    - Theo — 10/29/2004 @ 10:48 pm

  9. Hi Moebius, I graduated from Occidental College two years ago in mathematics and I feel your pain when it comes to teching math to children who don`t even have the basics…my advice to you is to relate the math and its topics to what the students like the best..For instance, a few months ago, I spoke about the mathematician by the name of Euclid and the parallel postulate. The example that I gave involved parallel parking and the difference between two lines crossing at one point and the correct way of parallel parking..I posed the question, if I park halfway on a curve, am I parallel parked. Well, the answer is no because I have crossed the line at some point. I have done two things, proved that two lines can cross at no more than point and given my audience an example that if they do not understand in the classroom, they can apply this to real life………

    - gerald — 4/13/2005 @ 10:11 am

  10. Hi

    I tell my students that they need to spend about a third of their studdytime on old chapters. When i talk about what we will be doing the comming classes i mention what chapters are extra important to what we will be doing so they can spend some time brushing up.

    Also tell your student to teach eachother, nothing helps you understand a subject as teaching it, and when you really understand it its harder to forget.

    I also try to teach so my student can see where the new material fit with the old so they dont have to remmber as much and that they can think for themself to fill in the parts they forgot that day.


    I know this entry is ancient but maybe someone beside me decidied to read this from the start…

    - Per Sjunnesson — 4/25/2005 @ 6:31 am

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