December 21, 2013
Following the posts on note-making in math and on basic thinking tools, here come first ideas on thinking tools for math problem solving.
Finding a solution to a math problem can be broken down into parts in several ways – for example by asking questions.
A first useful catalogue of questions could look like this, where each main question is followed by more specific ones.
(The catalogue is massively inspired by the famous list of questions from George Polya’s classic book “How to Solve It”. – Polya’s list is reproduced here.)
The next posts shall give some answers to these questions.
How can I get started?
- How can I approach a problem? Are there established methods that work for a large variety of cases? How can I use them?
- How can I represent a given problem, using equations, diagrams and other means that allow a math treatment?
How can I construct a solution?
- How can I generate seminal ideas? What methods can I use to generate ideas?
- How can I exploit these ideas?
- How can I start from what is given and work forward?
How can I start from the end and work backward?
- How can I decide whether to proceed with an idea or whether to try a new one?
- How can I deal with multiple ideas?
Should I produce several of them and pick the most promising one, or should I start with one and remodel it until it works?
How to deal with obstacles?
(This part is closely connected to the previous and the next one.)
- How can I make sure I realize there are obstacles that need attention?
- How can I use established ways of dealing with obstacles?
- How can I analyze an obstacle?
- How can I generate ideas to overcome an obstacle?
How to deal with other troubles?
- How can I deal with frustration, with the impression that I’m no good at math, with a lack of interest?
How to look back?
(This part is not exactly necessary, but a great opportunity to improve problem solving skills.)
Remember: Looking back should happen not only at the end, but also on the way!
- What’s bothering me? What doesn’t feel right?
- How can I clarify these intuitions?
- How can I check if my solution is correct?
- What can I learn from what I’ve done so far?
Finding answers to these questions may lead to a collection of math problem solving tools. Read more in the next post.
December 19, 2013
Following the post on a simple note-making technique, here are some ideas on basic tools for thinking on paper:
- a simple symbol and
- a key question it stands for
– like this:
stands for “What’s wrong here?”
stands for “What could I do?”
stands for “What would be logical?”
- Most important:
I see no need to use the symbols all the time – when my work flows without them, everything is fine.
But when I get stuck, using = “What’s wrong here?” very often helps.
- The symbols are designed for quick writing.
I gave the question “What would be logical?” the most simple symbol since I use it most often.
- The questions are designed to be “fail-proof” in the sense that they should lead to some progress in practically every situation.
- The exact phrasing of a question seems to me a matter of personal liking.
Instead of “What would be logical?” you might try “What would be natural?” or a simple urging “So?!”.
- There are countless other possible symbols and questions. The above three questions, with their focus on obstacles, options and next steps, seem essential to me.
- As described earlier:
If the question “What could I do?” leads me to several options that are worth trying, I use for each of them a circle “o” as a reminder for examination. Later I can tick off the options I have tried.
Here’s the next sandbox example – click to enlarge:
Next comes a post on Math Problem Solving Tools.
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.
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.)
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.
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.
- 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.
- 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.
- 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.
- 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.
- 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.)
- 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.
- 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.
December 19, 2013
Which practical methods could help me solving math problems?
I will approach this question by splitting it into two parts:
Q1: How can I make notes that support thinking about math problems in the best possible way?
Q2: How can I use thinking tools that are helpful in solving math problems?
In Q1, I find the following aspects important:
- How can notes help me to develop ideas and to think straightforward?
- How can I add reflections on my thinking, to see what works well and what doesn’t?
- How can I deal with multiple approaches and chains of thought without getting lost?
- How can I store sudden ideas for later examination?
In Q2, I’m puzzled by the following things:
- How can I find useful representations of the problem?
- How can I generate and exploit seminal ideas that eventually lead to a complete solution?
- How can I deal with obstacles, mathematical and otherwise?
Answers to Q1 and Q2 are obviously closely linked, so I’m aiming at an “integrated method of math problem solving” – it’s not just about note-making and not just about thinking tools, but about combining both for good results.
This is the program for the following posts.
The next post is about note-making technique.
I’m focused on these topics for several years now, and I have written previous posts about them, see here or here. What I present now is a thoroughly revised version.