Note-Making in Math Problem Solving

December 19, 2013

Arguably, a large portion of thinking about math problems is done by people brooding over a sheet of paper, pencil in hand.
So it is perhaps worth asking what a good note-making technique should support, and how it could do that.
(I should point out that what I have in mind is a technique for finding a solution, not for presenting it to other people.)

Here’s an incomplete list of things a note-making technique fit for math should support:

  • documenting chains of thought with enough clarity,
  • using figures, diagrams, tables and equations,
  • adding comments to previous thoughts,
  • storing ideas and checking them later,
  • changing from one chain of thought to another,
  • keeping your mind focused on the problem, making it feel in control and free it from struggling with messy notes,

Here’s a note-making technique that works for me – with some remarks on variations.
The advice is embarrassingly trivial – the later ideas for its usage lead to results that are not, as I hope.

  1. Materials
    I use blank paper in A4 format and a mechanical pencil with an eraser.
    (I write fairly small to get a decent amount of material on one page, so I need a pencil with a fine tip.)
  2. Layout
    I use only the front side of the sheets – I want to see all material that’s on a sheet.
    I take the sheet in landscape format, add a page number and the date and draw three vertical lines to form four columns 1 to 4.
    If the columns are too narrow for your work, reduce their number (but not to one).
    On A3 paper, you can have a larger number of columns, or you can take it in portrait format and use four columns each on the upper and lower half of the sheet.
  3. Writing
    I start in column 1, where I write down the math problem in a box labelled “1” – see the example below.
    In further boxes labelled “2”, “3” etc., I do ordinary math notes, using equations and diagrams as usual.
    To show that one idea is subordinate to another, I indent lines, as in an outliner software.
    When I’m finished with some segment of thought, I draw a horizontal line to form a box, leaving some extra space for later additions.
  4. Using the columns
    If I want to start a new chain of thought, I can start with a new column – this is why having several of them is a good idea, and why the A3 format may be great.
    (Of course, you can start a new page for each new chain of thought, but then it takes more effort to handle the sheets, and it’s more difficult to combine several ideas.)
    Sometimes it’s convenient to use the right neighbour column for comments and reflection.
    If necessary, I use arrows to connect boxes, or I use the referencing described below.
  5. Using several sheets
    If a single sheet is not enough, I insert new ones like this:
    page 1.1 between pages 1 and 2, page 1.0.1 between page 1 and page 1.1 etc.
  6. Using references
    There are two ways of referencing:
    1) I simply reference the box number.
    2) I imagine the 4 columns being separated in four segments a, b, c and d. When I want to reference something starting in the middle of column 3 on page 7, I use the reference 7:3c.
    On the same page, I use just 3c.
  7. Storing and exploiting ideas
    When I come across an idea that I want to check later, I take a note and mark it with a circle “o”. After having examined the idea, I tick off that circle.
    When a sudden idea doesn’t fit in the current line of thought, I note it in column 4 segment d, filling column 4 from bottom to top, if necessary.
    Starting at the bottom leaves the top free for ordinary notes.

Here’s a sandbox example – click to enlarge:


Here are some side remarks on note-making.

  1. Designing good note-making techniques
    The basic idea is simple:
    First, identify useful qualities and activities for solving math problems, and
    second, find ways of supporting them through note-making technique.Here are two examples:
    a) Useful quality: Clarity
    Possible design: If your notes are messy and unreadable, find a layout that gives a clear structure to your notes – I’ve tried this with a four-column layout.
    Find writing material that makes readable handwriting easier – avoid blunt pencils and use checkered paper etc.
    b) Useful activity: Ask questions
    Possible design: Collect questions in one column, number them and find answers while using boxes in other columns.
    (Remember: Trivial ideas are not necessarily worthless.)
  2. Graphic organizers in math
    Some graphic organizers are sheets with prefab structures, where students have to fill in content. They are often used for younger students and simpler problems, and studies show their usefulness (see again here).
    Is it possible to harness their power for more complicated problems?
    Math problem solving is like an expedition, where you cannot plan the path to your destination in advance – you walk the first mile, see what lies in front of you and then decide about your next steps. Sometimes you have to go back, or you have to overcome obstacles you couldn’t see at the outset.
    This is why a single general graphic organizer will probably not work.
    What’s more promising is the flexible use of several specialized graphic organizers – choose from a collection the one most suitable for the current problem situation, and from this sequence of graphic organizers build a solution.
  3. Note-making: An unpopular topic?
    In most of the books about math problem solving I know, there is not much material on note-making, and detailed advice on the topic or reproductions of notes (made-up or real) is very rare.
    There are good reasons for this:
    a) Compared with the math content, most remarks on notes look shallow and arbitrary.
    b) Reproductions from an actual problem solving process need lots of printing space in a book and lots of explanation.

More information on organizing math notes:

Next comes a post on Basic Thinking Tools.