Paths To Math: Engaging Students and Teachers

I was in fifth grade when my teacher mentioned the "light year" as a unit of distance while covering an introductory lecture on astronomy. This was the unit used to describe distances of stars from our planet.  The nearest star (excluding the sun), we were told, was about 4 light years away from the earth. After mentioning "light year", the teacher simply gave the definition: A unit of distance equivalent to the length that light travels in one year. The teacher then asked us to figure out how much a "light year" was in more familiar units (such as miles or kilometers).  Before this lesson, we were already made aware of how fast light travels, 186,000 miles per second. We also knew seconds, minutes, hours, days, and year. We had no calculators then. So I had to do the following multiplication with a pen and a piece of paper: 186,000 X 60 X 60 X 24 X 365 1/4. I raised my hand and gave the answer, 5,869,713,600,000 miles. I did not know about significant figures then. Otherwise, I would have just said, approximately 6 trillion miles. Of course, the teacher was quite impressed and she asked how I arrived at my answer so I went on to explain to the class what I did. These were big numbers and I was able to multiply them. But my teacher added that seeing how big these numbers were should help us appreciate how big the universe was.

Mathematics helps us make sense of the world we live in. Numbers are as important as words for these likewise provide us with meaning. To relate and reason with these representations of our world, it is evident that we must first master basic operations. Cecilia  Villabona, a math teacher and assistant principal in the US, was quoted in C.M. Rubin's Huffington Post article, "The Global Search for Education: Finnish Math Lessons":
"We believe that students need to develop self-confidence and trust in their ability to do math, and it is the solving of real-life, simpler problems that gives them the ability to engage in more difficult, abstract tasks. Only the procedural understanding acquired in this manner will empower students to solve the test problems."
Looking back at my fifth grade experience, I could easily relate to Villabona's statement above. C.M. Rubin's article is actually an interview of two teachers, Cecilia Villabona (from the US) and Maarit Rossi (from Finland). These two teachers are the authors of a 3-4 year basic mathematics education program called "Paths to Math":

I registered to see some sample materials of Paths to Math and I would like to share one specific section of a chapter in its pre-algebra module. The section is on the magic square: (The following is taken from

Magic Square

A magic square is a special table: the sum or product of all its columns, rows and diagonals is the same. This number is called the magic sum or magic product. One of the most famous magic squares is in Albrecht Dürer’s engraving from 1514. Notice that he included the year of completion in the numbers of the square.

  1. What is the sum of Dürer’s magic square?
    1. Does the magic square (image 2) have a magic sum or magic product?
    2. Which figures make up this magic square?
    3. What special property does the central figure have?

    1. Complete the magic square of addition (image 3) by using the figures of Dürer’s square.
    2. Compile a new magic square of addition by using the figures of Dürer’s square.
    3. Compare your magic square to the squares of other pupils in class. Are they similar?
  2. The magic square (image 4) is one of the oldest known, dating back to 2800 BC
    1. How do you interpret it?
    2. How does it differ from the magic square of exercise 2.
  3. The figures in the three magic squares do not need to be whole numbers like in the previous magic squares. Complete the magic squares.

    1. Complete magic squares of addition as instructed.
    2. Can you discover a quicker way to calculate the final magic sum?

  4. Create your own magic squares of addition.
  5. This is a magic square of multiplication (image 8). What is its magic product?

  6. Inspect the squares (image 9). Can you find magic in them?
Imagine how a teacher would navigate through the above activity. Anyone who does not see the significance of how a teacher implements this lesson in a classroom is making the wrong assumption that content can easily enter a child's mind. The above are indeed excellent materials. These are, of course, more specific, richer, and deeper in content than what a curriculum provides. Yet, the final outcome still hinges on how the teacher will in fact guide the classroom through this lesson or activity.
Nowadays, there is simply too much emphasis on curriculum and standards. These do not determine quality in education. What still matters is the teacher-student relationship. Yes, I did feel good when I saw that my teacher was impressed with my multiplication skills. But, more importantly, my teacher designed that day in a particular way that we connected. I even had the opportunity to explain what I did to my classmates and at the end, our teacher guided us as well in appreciating something deeper, something beyond numbers, how big the universe was. Good materials combined with a skilled teacher - these are the necessary ingredients for quality basic education. Drawing a curriculum, technology inside a classroom, defining standards - these do not define quality education. Only a skilled teacher does.
In one of Villabona's blog posts, "The Teacher’s Role in Tomorrow’s Classroom", she writes:
In my opinion there is no substitute for an educated skilled teacher. Experience is also important. No computer app, artificial intelligent computer aided instruction program, Internet site or worksheet will educate our students. Some of these will capture their interests and hold their attention for some time but never will replace the teacher.


  1. hi im ronald from zamboanga city i had learn a lot in the information given on this site by Mam Angel C. de Dios, im glad that K to 12 was implemented in the Philippines despite the hardship and many critics who are against it. The Philippines is one of the two the nation around the world that have only have until grade 10, we lack 2 years in our educational system, and many studies confirmed that this one of the factors that decrease our performance in science and math in the findings in TMSS. The lack of preparation to go to college since we lack two years in grade level make it difficult for students to adjust college life and also poverty on which also factors that decrease the student to go to school. I just hope that mam angel would give me more information on the K to 12 for im undergoing my research study on this area. pls emai me at

  2. Thank you for your comments.
    But first, I am not a mam, You probably missed the "About Me" section on the right of this blog. Second, TIMSS cannot be used to justify adding two years to basic education, it is a test given to grade 4 and 2nd year high school students - these exams are for the 4th and 8th year of basic education. I suggest you read "First things First" in

  3. why does it matter if its a test given in the 4th and 8th year? the K+12 proposal is meant to raise average learning at all levels, right?


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