Liberal Arts Blog — The Matrix as a Tool for Solving Math Word Problems (and more)

Liberal Arts Blog — Monday is the Joy of Math, Statistics, Shapes, and Numbers Day

Today’s topic: The Matrix as a Tool for Solving Math Word Problems (and more)

Did you ever use a matrix to solve a math word problem on a standardized test? Who taught you the tool? When? Do you remember the day you discovered its amazing power? Wasn’t it a revelation? Well, it was for me. Is there another tool half as useful? If so, please share. The post ends with a footnote about a different kind of matrix — a subspecies that should be the foundation of all of education. The three-by-three matrix is the best metaphor for critical thinking itself. The link below includes 31 matrices covering fields from physics, math, and statistics, to economics, politics. and history to music. art. and athletics. Although of incomparable values these matrices are at very primitive stages of development and eagerly await suggestions for improvement. A final plea is that you do your best to construct a comparable matrix for the field or question that matters most to you and share it. Experts — please chime in. Correct, elaborate, elucidate.


1. Each of four students (boys Jacob and Eli and girls Lauren and Molly) got a different grade on a quiz (A,B,C,D).

2. Both boys scored lower than Lauren.

3. Eli got a B.

NB: What grade did Lauren get? If the girls got both the best grade and the worst grade, what grade did Jacob get? If both boys had scored lower than both girls, which of the above statements would have to be false?


1. Try it without a matrix first.

2. Then try it with a matrix.

3. That is a four by four matrix — eg. the grades and the students as the four columns or rows respectively.


1. It’s easier to solve a spatial problem than an abstract one.

2. Less strain on your short term memory.

3. No bigger problem in life than managing excess information. The key is organizing it in a way that clarifies the options.


1.) This example is taken from Patrick Grim’s “The Philosopher’s Toolkit: How to Be the Most Rational Person in the Room,” a highly recommended Teaching Company course. The photo above is of Grim.

2.) The answers will be provided next week.

3.) The key takeaway is the awesome power of the matrix, the best metaphor for critical thinking. An entire section of the “Liberal Arts Academy” website is devoted to a particular sub-species of matrices.

NB — Please click on the link below and immerse yourself in the miraculous matrices of physics, statistics, music, art, economics. Education properly understood (or in the words of Dumbledore, “to the well organized mind”) should be all about matrix exchange. The website gives you 31 matrices of incomparable value but still very crude and very much work-in-progress. I think of myself as a structurally humble egomaniac. Structurally humble means you know that the depth of your ignorance is infinite. Egomaniac means you know you have something of real value to contribute and are stupid enough to say it. I am all yours — humbly and eagerly awaiting your matrices about anything that matters to you. It is my firm belief that the more important something is to you, the more rewarding the construction of a matrix will be.

Matrices | Liberal Arts Academy

Last three years of posts organized thematically:

PDF with headlines — Google Drive


Please share the coolest thing you learned this week related to math, statistics, or numbers in general. Or, even better, the coolest or most important thing you learned in your life related to math.

This is your chance to make someone else’s day. And to consolidate in your memory something you might otherwise forget. Or to think more deeply than otherwise about something dear to your heart. Continuity is key to depth of thought.