"Buy the ticket, take the ride."

-Dr. Hunter S. Thompson

 

Power Lunch On Pluto

 

 

Algebra in Kindergarten

and

Other Celestial Navigations

 

Algebra (K-5)

 

 

"There are things known and there are things unknown, and in between are the doors."

-Jim Morrison

 

    Algebra is one the cornerstones of mathematics. Many educators are convinced, as I am, that the study of this discipline should not be limited to the middle and high school curricula. In point of fact, the use of algebraic strategies is quite common among elementary school students in China and Japan. However, it is a relatively novel phenomenon in the United States (Stephens, 2004). Notwithstanding, the National Council of Teachers of Mathematics (NCTM) has included algebra as one of the standards for K-12 mathematics (NCTM, 2000). Their intent is not to move the current high school algebra curriculum into the elementary grades. Rather, their goal is to establish standards that foster the development of algebraic thinking in these young learners. These objectives  set the students on an ambitious, but rewarding path. The specific aim is to develop in these youngsters multiple abilities. These include the logical identification  and generation of patterns through the use of concrete materials, the recitation  and recording of their discoveries on T-charts and other conveyances, the understanding of inherent numerical relationships, the use of symbols to stand for unknown quantities, the utilization of these representations to express generalizations, the interpretation of  equal signs as indicating equal values on both sides of the equation, and finally the utilization of  graphs to represent a relationship between two quantities (Laureate Education, 2007). 


    Many educators believe algebra is not only an important component of arithmetic, but is actually a method of interpreting and understanding situations in daily life (Lott, 2000). Pattern identification, which plays an integral role in algebraic thinking, is how the young often make sense of their world (Van de Walle, 2007). As such, they can be recognized, extended, and generalized by children in the early stages of elementary school (Van de Walle, 2007).


Accordingly, open-ended activities, involving pattern identification, will present them with opportunities to use these natural skills in productive ways (Lott, 2000). Thus, teachers should structure authentic situations that include visual, tactile, auditory and verbal experiences to encourage children to identify and utilize patterns in order to understand math (Lott, 2000).


    To introduce patterns at the kindergarten level, the children should begin identifying and discussing these designs using large manipulatives of various sizes, shapes and colors. These can range from using students in the class in alternating succession, (boy/girl/boy), to employing such objects as Cuisenaire rods, cubes, blocks, colored beads, colored macaroni, fruit loops, and pattern blocks (Sauter, 2009). In addition, stamps and stamp pads will allow students to create their own patterns. Finally, so that they can integrate non exemplars as part of their learning experience, have them create motifs that lack patterns. These non examples are often a powerful learning tool. Importantly, as each of these segments concludes, the students should communicate with the instructor, and each other, about what they have observed and discovered (Sauter, 2009).


    At this point the number of items in the patterns can be increased to three or four, or more. Include the environment, indoors and out, as a ripe domain for identifying patterns. Finally, utilize the numbers chart to introduce repeating arrangement of numerals (Sauter, 2009). Have the children count by twos, fives and tens. Have them and identify odd and even numbers, and examine reiterating sequences found in two digit numbers. Lastly they can  study patterns in addition and subtraction,  such as adding "one more" to a number, skip counting, adding ten to a single digit number, as well as adding or subtracting zero (Sauter, 2009).


    In the process of developing algebraic ways of thinking, students learn that the equal sign represents a relation, and means is the same as (Laureate Education, 2007). A fruitful strategy for helping students arrive at this understanding is to explore simple equations, with one or two numbers on either side of the equal sign. The operations can be additive, subtractive, or multiplicative, and the goal is to label them true or false. 


    To further investigate relationships between numbers, open sentences, where a box needs to be filled, or a letter needs to be replaced, can be utilized (Van de Walle, 2007). When the students create and observe these numeric relationships, they begin to practice relational thinking, which is so necessary for algebraic computations (Van de Walle, 2007).


    Students can also learn to generalize and to express their abstractions accurately with the use of  natural language and symbols. Therefore, as the youngsters first become involved in the algebraic process, strict adherence to generic variables like x and y might well be avoided (Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier,, & Wearne, 1997). Alternatively, use significant words from the problems, such as an appropriate initial. Later, when you write or read an equation, emphasize the connection between the math and the story problem by saying the whole word, even if all you write is the initial (Hiebert et al, 1997).


    Another advantageous exercise for inculcating algebraic thinking into arithmetic problems is to adapt single answer math problems into ones that feature various numbers and answers (Blanton & Kaput, 2003). These derivatives will provide the students with ample opportunities to identify patterns, offer conjectures through discussion, justify, and ultimately generalize math facts and relationships.  Concomitantly, as they do this, they can practice computational skills (Blanton & Kaput, 2003). Frequent discussion of questions and results, as well as establishing a risk free algebraically-centered learning environment, are ancillary to the successful application of this evolution (Blanton & Kaput, 2003).


    Specifically, the Handshake problem (“How many handshakes will there be if each person in your group shakes hands of every person once?”) can be adapted into an algebra problem by varying the number of people in the group, or the number of handshakes per person (Blanton & Kaput, 2003).  Use manipulatives and small numbers to begin. Have the children record their results. Discuss the process with the students and support their thinking with timely feedback. Once the children reason that they can’t shake hands with themselves, and reflect upon a sequence of situations involving differing numbers of participants, they are on their way. Have them create number sentences with symbols (letters or pictorials) from the recorded data. Continue to discuss their findings and guide them towards discovering relationships between the numbers. They can then formulate hypotheses that describe what is true about the number of handshakes for a group of any size. Allow them to test these conjectures by plugging in random numbers. This will yield the generalization (they must multiply the total number of people in the group by a number that represents one less than the total). In this manner, the youngsters have begun to use true algebraic thinking in solving derivatively formed algebra problems (Blanton and Kaput, 2003). On caveat is necessary at this juncture. Once the children have been acclimated to the process, be sure that the numbers involved are sufficiently large so that they can’t set down a corresponding model and compute it with case-specific numbers (Blanton and Kaput, 2003).


    In conclusion, I concur that the artificial separation of arithmetic and algebra deprives students of dynamic strategies for thinking about mathematics in the primary grades. This results in predictable difficulties as they confront the discipline in middle school, or later (Lott, 2000).  Learners of all ages can benefit by engaging in the kinds of activities that require them to make generalizations explicit, represent them accurately with natural language and symbols, and demonstrate that they are valid for all numbers (Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996).  Since understanding many of these ideas will take an extended period of time, it is well to begin the process early in the learner’s educational career (Fennema et al, 1996). However, there are challenges inherent to this approach. Many first and second graders,in particular, experience difficulty in coming up with justifications that go beyond examples (Hiebert et al, 1997).  Additionally, students may be understandably tentative about discovering patterns and predicting future values in the primary grades (Hiebert et al, 1997). These neophytes may also experience limited interpretations of the equals-sign (Kieran, 1981). Such obstacles can be overcome if teachers remind themselves that the overarching objective at this age is to develop algebraic thinking slowly and incrementally, rather than precipitating the skilled use of algebra procedures. Consequently, when selecting teaching strategies we must be keep abstractions at a minimum, create authentic contexts for learning, begin with concrete materials and simple patterns, as well as student-friendly representations, or symbols.  We must further introduce and reinforce the concept of the equal sign and relational thinking with true/false and open-ended sentences, and slowly unveil, through discussions of the students’ efforts,  the pathway from the concrete to the recondite (Carpenter, Fennema, & Franke, 1996). “Kid watching” or the vigilant observation of how each individual student structures algebraic understanding, is an imperative correlative that informs each step in this methodology (Lott, 2000). Thus, a child shall lead us, as we guide each student through the propitious avenues of algebraic thinking.

-www.fastoutofthegate.com

References


Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers' "algebra eyes and ears."
Teaching Children Mathematics, 10(2).


Carpenter, T. P. Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: a
knowledge base for reform in primary mathematics instruction. The Elementary
School Journal, 97, 3–20.

Fennema, E., Carpenter, T. P., Franke, M. L. Levi, L. W., Jacobs, V., & Empson, S. B. (1996). A
longitudinal study of learning to use children’s thinking in mathematics instruction.
Journal for Research in Mathematics Education, 27, 403–434

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, p., Murray, H., Olivier, A., &
Wearne, D. (1997). Making sense: Teaching and learning mathematics with
understanding. Portsmouth, NH: Heinemann.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in
Mathematics, 12, 317–326.

Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program One, “Introduction to
Algebra.”  Baltimore: Warrick, P
Lott, J. W. (Ed.). (2000, April). Algebra? A gate! A barrier! A mystery! Mathematics Education  
Dialogues,
3(2).

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


Sauter,S. (2009). Retrieved July 1, 2009 from mathcentral uregina
ca/RR/database/RR.09.97/sauter 1.html

Stephens, M. (2004). Researching  relational  thinking:  Technical  report  of  research.    
: Program  report by Visiting Foreign Research Fellows, May 31, 2004.


Tsukuba, Japan.University of Tsukuba, Centre for Research on International
Cooperation in Educational Development.

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

 

 

"As long as algebra is taught in school, there will be prayer in school."

-Cokie Roberts

 

 

 

Brief Intermission


Leapfrog Addition

 

Here's a nice mathematical magic trick based on properties of the Fibonacci sequence.
Give your friend a card with ten blank lines, numbered 1 through 10. Have your friend think of two numbers between 1 and 20 and write them down on the first 2 lines of the card. Now in each of the successive lines, have your friend write the sum of the previous two lines. For instance, in line 3, write the sum of lines 1 and 2. In line 4, write the sum of lines 2 and 3, etc. until finally in line 10, your friend has written the sum of lines 8 and 9.
Ask your friend to total the numbers. If you've practiced the Multiplication by 11 table, you'll be able to tell your friend the total faster than she can add the numbers (because the total will be just 11 times the number in line 7). Also, you can announce the quotient of line 10 divided by line 9... to 2 decimal places, it will be 1.61!
Let's do an example. Suppose your friend tells gives you the numbers 3 and 7. Her card will then have these numbers:

  1. 7
  2. 3
  3. 10
  4. 13
  5. 23
  6. 36
  7. 59
  8. 95
  9. 154
  10. 249

The total is 649 (which is just 11 times 59, do this in your head with the Multiplication by 11 Fun Fact.
The quotient 249/154 is 1.61 (to 2 decimal places).

 

 

 

Algebraic Variables (K-5)

   

The importance of Algebra cannot be underestimated. Algebra is an entry level skill required in most science, medical, business, and technology-based careers. As this discipline is the first mathematical plan of study to stress number relationships while connecting the concrete to the recondite, Algebra can also be considered  the gateway to higher mathematics.



    One of the most important aspects of Algebra is symbolizing objects and numbers. By being exposed to the material in a sequentially constructed, increasingly sophisticated manner, the learners are able to advance from representing concrete objects and specific numerals before them, to symbolizing varying numbers, and ultimately expressing these numerical relationships in universal terms (Van de Walle, 2007). Learning algebra is a little like learning another language (Van de Walle & Lovin, 2006).  Rather than utilizing words, algebra uses symbols to represent numbers in making statements about things. Variables are an integral part of this depiction. A variable is a letter or other symbol used to accommodate a number value in an expression (number phrases without equal signs), or an equation (number sentences with equal signs) (Van De Walle, 2007).  Consequently, these variables are actually numbers in disguise.


    At the elementary level, there are three different roles for variables (Laureate Education, 2007).The first application is as a specific unknown. This is the most common use of variables in the primary grades (Van de Walle & Lovin, 2006). When students use a box in an equation, such as eight plus box symbol equals twelve, it should not be considered a place to put the answer: rather,  the box represents a specific value that makes the equation true.  In this case the box is, in fact, the variable. 


    Variables are also used to portray quantities that vary. Joint variation occurs when the change in one variable determines the change in another (Van de Walle & Lovin, 2006) For instance, the formula, area equals the base times the height describes a relationship between three variables; as the value of b and h change, the value of a changes accordingly (Van de Walle & Lovin, 2006). The study of joint variation gives rise to the concept of function. (A function is a relationship between variables. If any a value is plugged into the equation, it will yield exactly one h out of the equation. For instance, up to a certain point, a person’s height (h) can be seen as a function of age (a) (Willoughby, 1997). The primary level is not too early to explore this idea (Van de Walle, 2007). Lastly, variables can be used as pattern generators (Laureate Education, 2006). These representations are letters or symbols used in generalized relationships. As such, these symbols convey alliances that provide rules that are true for all real numbers.There is certainly no need for primary students to understand and recognize the distinctions between these three types of variables (Van de Walle & Lovin, 2006).  In point of fact, the following activities are not geared for any specific grade or age group. Alternatively, the students’ academic and developmental levels should dictate the pace and scope of the relevant concept advancement. In general, students at the primary level should be able to accommodate the learning focus of the first two activities
.





Activity One: Variables Representing a Specific Value

Math Focus


Students willuse concrete, written, and numerical representations to develop an understanding of invented and conventional symbolic notations (National Council of Teachers of Mathematics, 2000).


Materials: a balance, Cuisenaire rods, and opaque bags for each pair of students.  

 

    Before beginning this activity, there are a few things to bear in mind. Understanding variables is something that can prove to be a challenge for many students. Children more easily learn to generalize and express their abstractions through the use of natural language and symbols. Therefore, as the youngsters first become involved in the algebraic process, strict adherence to generic variables like x and y might well be avoided (Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier, & Wearne, 1997). It is preferable, therefore, that students derive their symbols from meaningful words in the problems, such as an appropriate initial (Hiebert, Carpenter et al, 1997). Boxes, geometric shapes, smiley faces and even question marks can be used to represent the unknown value (Battista & Brown, 1998).


    Returning to the focus of this activity, children can begin their journey from the concrete to the abstract by becoming familiar with variables that stand for a specific value (Hatfield, Edwards, & Bitter, 1997). A balance and connecting cubes are used in this exercise. The objective is to demonstrate equations that contain an unknown. To begin the exercise, the class is divided into several two-member teams. Each receives a balance and several brown paper bags, each of which is labeled with the letter c. Each one of these bags contains four cubes. The children are not aware of the amount of cubes in the individual bags. Seven cubes are placed in full view on the right side of each balance. The team members take turns placing individual bags on the left side of the balance, and add the appropriate number of cubes in search of one amount that will establish the equality of sets on both sides. When this occurs, the children set aside that bag. When all the students have chosen what they believe to be the correct amount of cubes, the teacher writes the following notation on the board:  3 + c = 7.


He/she then asks, “What number must be added to 3 so that the quantity equals 7?”


Volunteers offer their suggestions and the teacher replaces the c in the equation with the correct response, four. The children then open their bags for verification.  In this fashion, students first solve the equation with manipulatives and then proceed to the numeric realm (Hatfield, Edwards, & Bitter, 1997).


Activity Two: Variables Representing Values That Vary.


Math Focus


Students will model problem situations with numbers and use representations such as tables and equations to draw conclusions (NCTM, 2000)
.


Materials: overhead projector, transparency, paper and pencils.


    Variables used to represent numerical values that vary are best taught in context (Van De Walle, 2007).  Furthermore, by interacting with real life situations, the learners will recognize how using variable expressions provide a convenient shortcut for organizing practical information (Lott, 2000). To begin the activity the following word problem is posed on an overhead.


 “Customers at Wal-Mart receive a coupon that allows them to buy any video for four dollars off the regular price.”


 The teacher asks, “What words describe the price you will pay when you buy any video for three dollars off the regular price?


Answer: the regular price minus four dollars.


At this point prior knowledge is activated as the students discuss variables and equations. One of the students volunteers, “In a variable expression, a letter or symbol is used to represent a number or quantity whose value can change.”


The teacher then inquires, “How do we know this is an equation and not an expression?” (There is an equal sign, it is a sentence and not a phrase, and  the total values on both sides of the equal side are identical).


The teacher continues, “Suppose the variable r stands for the regular price. What expression would represent the sales price?”
Answer: (r-4).


“Could you use a letter other than r?” (Yes).


“Why is r a good choice?”


Answer: (It is the first letter of the word regular).


“If the regular price of the video was ten dollars, how would you determine the sales price? “ (Let r equal ten). Together, the class fills out the following table (Fennell, Bamburger, Rowan, Sammons, & Suarez, 2000).


Regular (r)

Sales price (r-4)

10

(10-4) = 6

11

(11-4) = 7

12

(12-4) = 8

16

(16-4)=12

18

(18-4) = 14

20

(20-4) = 16

                                                     

Reflecting on this exercise, the learners actively constructed their own meaning and understanding regarding variables that vary and the attending concepts. Further, they organized information in a table creating a visual method for finding the solution. Not only did they rely on prior knowledge to negotiate an appropriate meaning for a variable and an equation; they also connected their knowledge of operations to the problem by selecting an operation that represented the given situation (subtraction) (Fennell, Bamburger, et al,, 2000).

                        


Finally, these youngsters established a nexus between the use of algebraic techniques and real world situations (Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996)
.

Activity Three: Variables as Pattern Generators

Math Focus


    Students will use variables to describe, extend and make generalizations about numeric patterns. They will represent and analyze patterns and functions using words and tables.


Materials: Each student will receive copies of papers which depict number machines (with in and out number already included) as well as blank number tables.


    In grades four and especially five, new algebraic ideas emerge and are investigated by the students.  For example, students in these grades are able to make a general statement about how one variable is related to another variable. If a movie ticket costs $7, you can figure out how many dollars any number of tickets cost by multiplying that number by seven. In this case, students have developed a model of a proportional relationship: the value of one variable is always seven times the value of the other, or C = 7 http://standards.nctm.org/document/chapter5/images/mult_sign.gif n (Van de Walle & Lovin, 2006)

                        

    Number machines, in which the student discovers a relationship by studying the pattern established by an example and then applying it to the rest of the numbers, are a proficient way to introduce variables pattern generators. This can also be considered as an introduction to functions (Hatfield, Edwards, & Bitter, 1997).
                        

    The number machines can be presented as stylized boxes with an opening on one side that includes the in number and a doorway on the other side showing the out number (Hatfield, Edwards, & Bitter, 1997).

                        


For instance, the first entry in this particular number machine problem sheet contains five for the in number and sixteen for the out. After deducing that the discovered pattern is to add eleven to the in number to obtain the out numeral, the student can use this rule to determine the value of the other variables (?) and place them in the table (Hatfield, Edwards, & Bitter, 1997).

Number Machine


In                                   Out


5

16

28

39

10

? (21)

0

? (11)

54

? (65)

? (0)

11

After studying the results, the students can devise a pattern generalizer or rule.


The Rule: add 11
                                                                     

As the students advance, the problems involving pattern generators can be posed so that more than one operation is used. For example, if the in number is eight and the out number is fifty, the students must discover a two-step rule: (n plus two) times five (Hatfield, Edwards, & Bitter, 1997).


    Exploring the meaning of symbols through writing and real life scenarios are effective ways for students to learn algebraic representations (Lott, 2000). Therefore, it behooves the teacher to encourage students to next create their own set of numbers, such as the difference in ages among their siblings, list them in a table and discover the generalized rules concerning these relationships. Number machines are optional. Upon completion of these steps, reflecting on their efforts in math journals and discussing these impressions with classmates enriches the lesson, as well as establishing closure.


    Some students encounter difficulties when attempting to understand variables. Variables, in the form of symbolizing quantities that change, are one of the main sources of this confusion. T-charts, tables and graphs can be utilized to clarify the situation. These scaffolds vividly portray the varying combination of numbers as they lend a sequential and orderly quality to the procedure.


    Another common misconception that students fall prey to when working with variables is when they focus on the meaning of “letters.” That is to say, they often think that the variables are simply an abbreviation for a formula (Kieran & Chalouh, 1993). It is sometimes difficult for students to make the transition from thinking of variables as simply letters representing these formularies (l times w) to letters representing numerical values. Therefore, context is vital in introducing variables. When confronting a word problem, the child should be able to discover a natural and relevant nexus between the symbol and the quantity it represents. They can also obtain other problem solving clues from the dynamics of the written piece.


    Accordingly, when presented with the opportunity to develop and make sense of algebraic symbols and notations, abetted by relevant applications and discussions, students will begin to conceptually grasp the meaning behind these variables and mitigate the attending confusion.


    In conclusion, cultivating basic understanding of algebraic variables in the elementary grades is a necessary, developmentally sound prelude to future mastery of many algebraic concepts. The embedded learning objectives can be met through activities that encourage students to gradually progress from symbolizing manipulatives to representing specific numbers. As their sophistication increases, the children can then use variables to represent changing numbers and finally to successfully generate more universal reasoning about these relationships. In this manner we contribute to the transpiring idea of algebra for all students, K-12.

-www.fastoutofthegate.com

 

References


Battista, M. & Brown, C. "Using Spreadsheets to Promote Algebraic Thinking".
Teaching Children Mathematics (January, 1998): 470-478.

Fennell,F., Bamburger, H., Rowan, T., Sammons, K., & Suarez, A.(2000).
Connect to nctm standards 2000.Chicago IL: Creative Publications.

Fennema, E., Carpenter, T. P., Franke, M. L. Levi, L. W., Jacobs, V., & Empson,
S. B. (1996). A longitudinal study of learning to use children’s thinking
in mathematics instruction. Journal for Research in Mathematics
Education, 27, 403–434


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


Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, p., Murray, H.,
Olivier, A., & Wearne, D. (1997). Making sense: Teaching and
learning   mathematics with understanding. Portsmouth, NH:
Heinemann.


Kieran, C. & Chalouh, L. (1993)."Prealgebra: the Transition from Arithmetic to
Algebra". Research ideas for the Classroom: Middle Grades
Mathematics
edited by Douglas T. Owens. Reston, VA: NCTM.


Laureate Education, Inc, (Producer). (2007) [Motion picture]. Program Three,
“Variables and Expressions.” Introduction to Algebra.”  Baltimore:
Warrick, P 


Lott, J. W. (Ed.). (2000, April). Algebra? A gate! A barrier! A mystery!
Mathematics Education   Dialogues, 3(2).

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


Willoughby, S. S. (1997). Functions from kindergarten through sixth
grade. Teaching Children Mathematics, 3(6), 314–318


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


Van de Walle, J.A. & Lovin, L. (2006). Teaching student-centered mathematics
grades k-3.
Boston MA: Pearson.