Why education system should not simply teach solutions, but the fundamental problem solving skills.

Illustrated using a story of how I taught my kid to make 10 year calendar using less than 10 Pages

Why we should not teach simply the solutions to the problems, instead teach kids the fundamental problem solving skills?

How to create 10 year calendar with less that 10 pages?

Wondering how these two questions are related?

I will share how I taught my 7 year old daughter to make 10 year calendar with less than 10 pages. Then, I will answer why we should not teach simply the solutions to problems by explaining reasoning behind my teaching approach.

I will also discuss few fundamental skills our education system should teach and what can we change in Math education.

How I taught creating 10 year calendar with less than pages?

The conversation I had with my daughter.

I: How long is one year’s calendar ?Daughter : There are 12 months. So It will be 12 pages.

I: How long will be a 10 year’s calendar ?Daughter: Hmm.. that is 10 times 12, so it will be 120 pages.

I: But, we can make 10 year calendar in less than 10 pages.Daughter: (with excitement) How?

I: To do that, we need to do a calendar research to identify patterns.Daughter: How can I do that?

I: I will show you how

I gave 2018 calendar and asked her to check if any of two months look exactly same. After scanning through it, She said — “Yeah, both January and October starts with Monday”. We checked few dates in Jan and Oct to see if they both fall in same weekday and it worked.

I said — Let’s do this. On a piece of paper, write weekdays Monday through Sunday on left hand side, and on the right side, write the month whose first day falls on that week day. This is the pattern she found:

I asked — since months can be combined together, how many pages does a year’s calendar need?

Daughter: Hmm 7 pages, so that means 10 year calendar will need 70 pages, that still a lot.

Yeah, but let’s study 2017 calendar to see if we can find more pattern. She repeated the exercise for 2017 and came up with this table.

I said — Now, we need to figure out a way to combine this together. Let’s say if we write a calendar for a month starting on each weekday, i.e Monday through Sunday on a page with numbers 1 through 7 and I wrote numbers in left hand of side of weekday.

She said — ok then?

Now to create 10 year calendar, we can prepare an index to look up month and year to correct page number based on which day the first of the month falls. This is just like a index end of a book.

She: Oh yeah, I have indexes of words in end of the book. Let me do it.

My Daughter’s calendar research work sheet

I said — So, if we can write index of 10 years in two pages (like 5 years in each page). Then…

She — Oh great idea!, So that means 2 pages for index… plus 7 of pages calendar, so only 9 pages.

Me: Yes!, we can do 10 years calendar in just 9 pages.

Then I made a book with A4 sheet paper and drew grid for her to write indexes and monthly calendar.

Here is the exercise we did to identify more patterns while filling the indexes:

I pulled a 2009 calendar and she looked at first day of each month and wrote corresponding page number at the cell intersecting year and month.

We repeated it for 2010, then asked her to check if there is any pattern between 2009 and 2010. She easily found that each number just got incremented by 1. For eg. index of Jan 2009 is 4 and index of Jan 2010 is 5. And also adding 1 to seven becomes 1 again since there is no eight.

To fill 2011, she didn’t look at the 2011 calendar, instead just followed the pattern by just adding 1 to 2010’s page number. Then we both checked against 2011 calendar and it matched perfectly.

She repeated the same for 2012 calendar, and during verification we found that the pattern changed. For Jan and Feb, pattern remained same just one higher that previous year. But now Mar through Dec was incremented by 2. For e.g index for Mar 2011 is 2 while index for Mar 2012 is 4. I explained her that it’s because Feb has 29 days in leap year which occurs once in every four years. We then corrected all indexes for Mar to Dec.

She filled 2013 (using calendar this time since pattern changed) and we looked for pattern. For Jan and Feb the index got incremented by 2, but Mar to Dec is went back to old pattern of increment by 1.

So the general pattern we found was to increment previous year’s index by 1 except that we need to increment by 2 from Mar of a leap year to Feb of the year following leap year.

Then we applied this pattern to finish up indexes up to year 2018.

Then I showed her how to use to find day of week for any day for last 10 years.

The first step is to find index of page number by looking at the cell intersecting year and month.

Then go that page number to find day of weekday for of date in question.

She enjoyed playing this trick. To make it little fun, when we aren’t sure the answer is correct, we asked voice assistants like siri — “What day is 2010, Sep 5th ?” and it will respond with the weekday.

Why I did not teach her simply the trick ?

We could have skipped the research work and simply asked her to fill the calendar book by looking at the real calendar or using voice assistant.

But, My goal is not to teach her this cool trick of making 10 year calendar with 9 pages, but the process of solving this problem so that she can apply it to other problems.

We should teach kids the fundamental skill of researching subjects and creating theories.

The fundamental skill I wanted to teach is the skill of deeply studying the subject, identifying the patterns and creating generic theory that can be applied boarder problems. Few examples of how this skill is useful.

• Charles Darwin studied numerous plants and animals his during five year voyage which made him propose the theory of evolution.
• Issac newton developed three fundamental laws of motion by closely studying movement of objects and planets.
• The periodic table was developed based on patterns of arrangement of atoms in elements. It’s a important invention that helped us understand all the elements in the universe.
• The music theory was invented by identifying common patterns used by many composers. Without music theory, it is hard to teach music in a standardized way.
• Nobel prize was awarded to Behavioral economist Daniel Kahneman for identifying common cognitive biases of human beings by analyzing human behavioral patterns.

If students learn this essential skill, they can use it in any field.

We should teach kids to make mistakes and correct them.

Another important skill I wanted to teach was to make mistakes and learn to correct them. For e.g in the calendar project I know that the pattern will break for leap year. But I let her do it wrong first time and discover the problem during verification.

Because that’s how we solve problems in real life, but school education teaches only right steps to arrive at the solution.

What can we change in Math education?

The math education should teach the art of problem solving and not simply teach how to work with numbers.

To illustrate with an example. Kids are taught about mathematical constant PI

π = 3.14159...

They are taught π is the ratio of circumference to diameter of the circle. This is a constant pattern no matter the size of the circle. But after that students are given scary formulas to calculate Area of circle, Area cylinder, Volume of Sphere and what not? But what if they are thought.

• Why did we even need that magic number number π?
• How Archimedes found approximate value of π 2000 years before.

Archimedes discovered a brilliant idea to find approximate value of π. Since it’s easy to calculate perimeter of polygons, he drew polygons inside and outside the circle. So,

Perimeter of the Inner Polygon < Circumference of Circle < Perimeter of the outer Polygon

Archimedes realized that by increasing number of sides of polygon, he could computer value of π to better precision like illustrated in this animation.

I wish this creative idea was taught when I was in school. This is a fundamental principle of approximation algorithms used to solve hard problems in computer science and operations research.

Learning Archimedes imaginative skill is more important than to learn formulas using π and calculating areas of various shapes.

Is it not useful to learn facts and how to apply them?

The current education system is created for industrial revolution. It’s goal is to teach facts to create factory workers who can follow the instructions. Unfortunately to get the good grades, many students simply memorize the facts. So most of them doesn’t even know how apply what they have learned.

In Today’s computing and artificial intelligence revolution, intellectual skills are more important than knowing facts. Because everything can be looked up in internet. So students should learn the basic building blocks for solving problems.

To identify what to teach, Ask the question — Is there a fundamental skill they should learn that they can apply to different problem or domain?

Some of the fundamental skills we should teach them are:

• Given a problem, how to generate as many ideas (even wrong ideas).
• How to break down bigger challenges to smaller problems?
• How to figure out the incremental steps that will lead to a solution?
• How to evaluate if they are in right direction and course correct if needed.
• How to iterate and how to make progress?

Education system should not evaluate students for correctness answer, but their ability to think. Measure the quantity of ideas they can come up with rather than perfectness of ideas. Because, the best way to discover great ideas is to generate more ideas.

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