All Ashore for Singapore





Math is the Next Great National Export



Singapore Math incorporates the following characteristics:


The principle of teaching mathematical concepts from concrete through pictorial to abstract. Pattern identification is considered integral to the process.

Systematic use of word problems as the way of building the semantics of mathematical operations. Simply put, students learn when to add and when to subtract, relying on the meaning of the situation (rather than "clue-words," as often done in the US schools).

The need for repetitive drill is minimized by clever sequencing of the topics. For instance, the introduction of multiplication facts by 2,3,4 and 5 in the middle of Grade 2 is followed by a seemingly unrelated section on reading statistical data from a graph. In fact, the latter task reinforces the learning of multiplication facts when the scale begins to vary from 2 to 5 objects per graphical unit

The hallmark of the curriculum is the careful guidance of students, done in a child-friendly pictorial language, not only to technical mastery, but to complete understanding of all the "whys". This differs from typical U.S. curricula, which either aim for dogmatic memorization of "rules," or expect students to reconstruct mathematical ideas from hands-on activities without much guidance

Trends in International Mathematics and Science Study in 2003 showed Singapore at the top of the world in 4th and 8th grade mathematics. (This was the third study by the NCES, and the 2007 TIMSS was released in December 2008).Starting in 2002, the Singapore math textbooks have been translated into Hebrew, and implemented in schools

As of November 2007, the state of California allowed state funds to be used for teachers to order the Singapore Math texts.









The methodology I chose to demonstrate pattern identification and exploition was to present a word problem, which includes a pattern, and utilize words and an attending table to identify the repeating arrangement, and to solve the puzzle, as well.  

The posited problem is thus:

“You are invited to a Conga dance. It will take place in a gym. The dance line begins on a white sideline on one side of the gym, and will end at the other sideline.  The Conga dance is easy to learn. In a 4-count pattern (each count includes one movement of the foot, either forward or sideways), you shuffle your feet ahead on counts 1, 2 and 3 and then kick your foot to the side on count 4. If you shuffle I foot (12 inches) forward on each onward count, how many counts will it take you to reach the other line, which is 9 feet away? If the dance was to continue past the line, how far would your next four counts take you?”
After reading and rereading the problem, I noticed that the dance routine included a repeating pattern of 3 steps forward and 1 step sideways. Obviously, the forward movement is going to occur during the first three counts of every four count progression. I have been told that each of these forward counts will move the line one foot closer to the finish. The fourth count in each repeating pattern, a sideways motion, will not advance the line. Therefore, if I create a table, I will more easily be able to track the progress of the line. I will draw a table with, say, 20 columns and two rows. The top row will list the counts and their attending patterns, and the bottom row will track the forward progress for each count, measured in feet. Accordingly, I will label the top row with the word “Counts” and the bottom row with the phrase “Forward Movement in Feet.”  




































Forward Movement in Feet



































After filling in the table I am able to clearly see that for every 4 counts, I move 3 feet forward. This pattern is repeated throughout the dance. Therefore, I will need 4 counts for every 3 feet of progress. To advance 9 feet, I can use the bottom row to count the forward movement in feet. When I reach 9, I will look at the row above (the counts) and note that it is 11. Therefore, it took 11 counts to advance 9 feet. This answers the initial question. Since I understand that the next four counts constitute a total forward movement of 3 feet, I can easily respond to the second query. The solution is 3 feet, which I can verify by itemizing four more counts in the table.



Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Upper Saddle River, NJ: Pearson.





Farmers and Pesky Birds

Alice and Bob are two farmers each wanting to plant a (countably infinite) row of seeds, side by side in a field. Both of them have pesky birds that hinder their efforts in funny ways.
As Alice walks along the row, sequentially dropping her seeds, her bird picks up every fifth seed that she drops. So after Alice "finishes" planting her row of seeds, are there any seeds left? Sure... infinitely many of them.
But Bob's bird behaves differently. Bob walks side by side with Alice, planting seeds in his row. After every fifth seed that Bob drops, his bird picks up the first seed that remains in his row. After Bob has "finished" planting his row of seeds, are there any seeds left?
No! Each of Bob's seeds gets picked up by Bob's bird, eventually! But that is strange: Alice and Bob are working simultaneously and their birds pick up seeds at the same rate... but Alice's row still has seeds left! How can this be?





Proportional reasoning is  a skill that is necessary for completing algebraic tasks  (Hatfield, Edwards & Bitter, 1997).  Algebra consists of finding patterns and developing relationships between variables. People who are able to think proportionally are able to determine these relationships faster and more efficiently than people who lack the ability to think this way. The National Council of Teachers of Mathematics also agrees that students need to solve problems that involve proportions (National Council of Teachers of Mathematics, 2000). Ratios play an integral role in proportionality. Ratios are numbers that compare two quantities in a multiplicative correspondence such as “4 dollars per (one) gallon.” (Van de Walle, 2007).  Proportion results when  two ratios are denoted  as equal to each other such as, "4 dollars per gallon" equals "8 dollars per two gallons" or, “2 teachers for 40 students equal 3 teachers for 60 students.” Consequently, when different values are introduced to make a new ratio they can be seen as in direct proportion to the original correlation (Hatfield, Edwards & Bitter, 1997). Ratios in a ten to one multiplicative correspondence and the resulting proportionality can be unveiled in the following activity that integrates art and proportionality in very concrete terms.
Grade Level 3
Materials: butcher paper, ruler, pencil, crayons, magic markers, favorite objects.

As an initial step in the measurement process, students are encouraged to bring favorite objects from home, ranging from teddy bears to roller blades. They may also choose from an assortment of classroom items such as pencils, toothpicks, erasers, Dixie cups, or books. The dimensions of body parts, such as fingers, arms or heads can also be selected for calculation.

Procedure: The teacher reads “Jim and the Beanstalk” by Raymond Briggs to the class. He/she discusses the implications of being ten times bigger. The instructor then elicits responses from the children such as:
“If I were ten times larger I would join the NBA and make millions.”
“I would work in a supermarket stacking top shelves.”
“I could walk into the deep end of the pool and not drown.”
“I would be able to paint the house without a ladder.”
Procedural directions to the students:
Estimate how large you think ten times your item will be. Get that much butcher paper to start your project.
Measure each item by length and width. Make each dimension 10 times as large and record the measures on paper.
Draw the object’s enlarged size on butcher paper.
Color the item so that it looks like the real one (Hatfield, Edwards & Bitter, 1997).

Algebraic language can be introduced as the lesson progresses. For instance,
“The size of a giant pencil is a function of the real size times 10.”
“The giant size is the same size as the real size times 10.”
 “G = R x 10.”
Graphs can also be created to demonstrate the giant size as a function of the real size times ten. Each object would be placed sequentially at the bottom of the x- axis and the giant-sized dimensions of height would constitute the y- axis. Different colored dots and lines could be used to distinguish the real-sized lines from the giant-sized lines. The straight lines depicting one size can then be traced and superimposed on the other to show that the proportionality is exact. At opportune moments, additional terminology can be advanced, such as “The steepness of the two slopes appears to be the same.” “The two lines are growing at the same rate.”
Tables can be utilized to display, in numeric terms, the ten to one correspondence. An optional essay might be assigned in which the students write about their experiences in a world that has suddenly become ten times larger. These can be posted, along with the art work, tables, and graphs on bulletin boards, to serve as a permanent learning resource center.

As a result of this exercise, the concept of proportionality can be concretely presented with relevant objects in a literary, verbal, visual, written, pictorial,

graphic and tabular modality. This will not only incrementally reinforce the learning material, but address individual learning styles, as well.  In addition, the conceptual bridge connecting the concrete to the abstract (algebraic language and generalized rules) can be discretely integrated at appropriate, teachable moments.



Hatfield, M., Edwards, N., & Bitter, G., (1997). Mathematics methods for
elementary and middle school teachers. Boston MA: Allyn and Bacon.

National Council of Teachers of Mathematics, (2000). Principals and standards for
school mathematics. Reston VA: Author.

Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Upper Saddle River, NJ: Pearson.




Analysis of Change


Analysis of change is one of the learning standards of the National Council of Teachers of Mathematics (National Council of Teachers of Mathematics, 2000). Graphs are one way of utilizing functional relationships to illustrate this change. The typical graph depicts the relationship between two variables, the y or vertical axis, and the x or horizontal axis (Graphs, Algebra and Economic Analysis, 2009). In our case, two disparate subjects (time and distance) are involved. In both graphs, distance (the dependent variable on the y-axis) is a function of time (the independent variable on the x-axis).  
Specifically the directions for this week’s application were as follows. First, we were to analyze the change represented on both a numbered and unnumbered graph. We were then to synthesize these discoveries by concocting a tale in which the storyline represented the movement notated on the graph. Consequently, the math focus for this activity will be interpreting and transposing the algebraic depiction of change from a graphic context to a language-oriented canvas.
         The first graph is featured below followed by the mathematical story that fleshes out the action represented on the graph.
Figure 1 Distance as a function of time.


Bob loves his antique car. The only trouble is that it is not very fast. Last Sunday Bob decided to drive to Tobyhanna. Keeping as close to the right shoulder of the road as possible, Bob soon hit his maximum speed of almost six miles per hour. The road was level and Bob managed to travel four miles during the first forty-five minutes of the trip. However, at that exact point in time, the highway began to climb. The car quickly slowed to a crawl. (The only fast thing it did).  It was now doing three. It took Bob ten minutes to negotiate the next half-mile. At this marker, the road leveled off and Bob drove the next two miles at a more reasonable six miles per hour. Eighty-five minutes for the six and a half mile journey to Tobyhanna was a personal best for Bob, and he showed his appreciation by presenting the mayor with a graph depicting the variations of his journey.


The second graph represents something of a departure in that it is numberless. Consequently, the connected story will be told in more general terms.


Figure 2 Distance as a function of time (numberless version).


Rachel likes to hike. She has also been diagnosed with “graphitis” This rare malady induces her to view outdoor landscapes in algebraic terms. Let us eavesdrop on her thought process as she begins her trip.  
“The first phase of the graph shows a straight linear line with a moderate slope”, she reflects. “This indicates that I am walking in a positive direction at a steady rate.”
Soon the contours of the land change.
Rachel continues, “I am now seeing a steeper positive slope in front of me which means that I am covering more distance in less time. Evidently, since distance is a function of time, I must be going faster, Perhaps this is because of the downgrade in the terrain.”
A few more minutes elapse and Rachel finds that the topography has once again shifted dramatically.  She continues her ruminations.
“Now I sense that the graph’s slope is becoming negative.  This is not good since I am losing ground as I use up time. That means that the association between time and distance has shifted from a direct relationship to an inverse relationship. The reason for this reversal is that I seem to be walking around a huge mound of earth, which is forcing me to temporarily head in the wrong direction.”
Rachel appears chagrined, but things are about to take a turn for the better. Her thoughts reflect this improvement.

“I feel that I am continuing my walk on the identical positive graphic slope with which I began my journey. A steady pace wins the race”, she reminds herself as she strides toward her destination.
  In closing, a graph is a picture of the rate of change of one variable in terms of another. These renderings can be used to illustrate continuous activity. However line graphs portraying atime-distance relationship can sometimes be confusing. Perhaps this is because different aspects of the lines are viewed simultaneously. Therefore, it behooves the scholar-practitioner to study the line formations in sequential order, carefully noting aspects of the slope and whether the time-distance covariance indicates that the rate of increase or decrease is steady, increasing, or decreasing (Hatfield, Edwards, & Bitter, 1997).  In this manner one can ably reconstruct the events behind the shape of the graph and accurately verbalize the changing situations and functional relationships that the delineations represent.





  Graphs, Algebra and Economic Analysis (2009). Retrieved August 10, 2009 from Web site:


  Hatfield, M., Edwards, N., & Bitter, G. (1997). Mathematics methods for elementary and middle
  school teachers. Boston, MA: Allyn and Bacon.  


Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program Seven, Analysis of
Change.” Baltimore.


National Council of Teachers of Mathematics, (2000). Principals and standards for school
mathematics. Reston VA: Author.

Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Upper Saddle River, NJ: Pearson.















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