Developing Mathematical Communication Competence For High School Students In Teaching Algebra

DEVELOPING MATHEMATICAL COMMUNICATION COMPETENCE FOR HIGH SCHOOL STUDENTS IN TEACHING ALGEBRA Luong Phuong Anh Buon Don High school Le Tuan Anh (Ha Noi National University of Education) Nguyen Thanh Hung (The University of Da Nang University of Science and Education) Abstract Mathematical communication competence is one of the five core components of mathematical competence that high school math needs to form and develop for Vietnamese students In traditional teaching, teachers have also more or[.]

FOR HIGH SCHOOL STUDENTS IN TEACHING ALGEBRA

Luong Phuong Anh Buon Don High school

Le Tuan Anh

(Ha Noi National University of Education)

Nguyen Thanh Hung

(The University of Da Nang - University of Science and Education)

Abstract: Mathematical communication competence is one of the five core components of mathematical

competence that high school math needs to form and develop for Vietnamese students In traditional teaching, teachers have also more or less paid attention to developing their students’competence However, due to the content-based approach, they still have not yet had a complete and right view on the teaching methods to developing mathematical communication competence General constructivism, especially the social constructivism of Vygotsky with studies on the knowledge formation process of children, stresses the role of social interaction as the foundation of the mathematical communication category within the math classes In this article, some pedagogical measures to develop mathematical communication for students in teaching algebra in high schools are presented The theoretical basis of these measures is the constructivist teaching process In this process, teachers take the responsibility for designing lessons to increase both the quantity and quality of mathematical communication situations in the classroom because it is the foundation to form mathematical knowledge for students.

Keywords: mathematical communication, constructivism, constructivist teaching, teacher, student.

1 INTRODUCTION

Studies on mathematical communication competence have been carried out by many researchers in some aspects as follows: reasoning; its role in the 4.0 technology era; its relation to other mathematical competencies; some teaching models to develop mathematical communication skills

Regarding the concept of mathematical communication and mathematical communication competence, mathematical communication is the process that shows the mathematical ideas and understanding verbally, visually in writing, numbers, symbols, pictures, graphs, diagrams, and words (Ontario Ministry of Education, 2005; Merriam-Webster, 2017) Alternatively, mathematical communication is a necessary skill in mathematics, mathematical communication competence is the ability to understand mathematical problems through the written, verbal, and graphical communication

of others and the ability to express one’s mathematical ideas in many different ways (for example, Ansari, 2003; Sumaro, 2009; Pisa, 2009) Mathematical communication competence is divided into components besides the expressions of those components (NCTM, 2000, 2003; Greenes and Schulman, 1996; Ansari, 2003; Kennedy &Tip, 1994, Sumaro, 2003) For instance, NTCM (2003) divides mathematical communication competence into the following components: (1) communicate their mathematical thinking coherently and clear to peers, teachers and others; (2) use the language

of mathematics to express mathematical ideas precisely; (3) organize and consolidate mathematical thinking through communication; (4) analyze and evaluate the mathematical thinking and the strategies

of others Thus, mathematical communication requires various cognitive skills and exchanges of ideas, including listening comprehension, reading comprehension, and speaking and writing (expression) Individually for mathematics, expression can include the representation of mathematical ideas in non-verbal ways

In parallel with the extensive analysis of the content, what the authors mentioned above also affirm that mathematical communication is a vital process for learning mathematics because through it students reflect, clarify, expand ideas and understanding about mathematical relationships and mathematical reasonings; Moreover mathematics itself is an exact, powerful and clear communication tool Sumaro (2003), NTCM (1989), NTCM (2000), Wichelt (2009), Polya (1973), Lim & Chew (2007), Roland G Pourdavood & Patrick Wachira (2016), Laney Samson (2019) According to Ezrailson et al (2006), students keep only 20% of what they hear, 30% of what they see, and 50% of what they hear and see However, when teachers focus on interaction and communication in the classroom, students will keep 90% of what they say and do when they participate in discussions In this study, communication is clearly an important factor in improving the learning quality of students

Following those research results, some teaching models to develop mathematical communication skills are given: Problem-based teaching in geometry for junior high school students; contextual teaching for primary school students; research-based teaching to improve 4C skills for students (Titin Masfingatin

et al., 2019; Ina Riyati & Suparman, 2019; Kembara et al., 2018) Besides, Lexi Wichelt (2009) guides teachers to create an environment for students to practice mathematical communication skills or as Abd-Qohar (2011) affirms that the learning model to develop mathematical communication skills for students is also a learning model in which learning is where discovery, interaction, collaboration, and construction activities take place to acquire new knowledge

2 CONTENTS

2.1 Mathematical communication capacity

According to the Mathematical Education Program (2018) issued by the Ministry of Education and Training of Vietnam, mathematical communication capacity includes four criteria: (1) listening comprehension, reading comprehension and recording the necessary mathematical information presented

in writing or spoken or written by others; (2) present and express (spoken or written) mathematical contents, ideas, mathematical methods in interaction with others; (3) effectively use mathematical language (digits, symbols, charts, graphs, logical connections, ) in combination with common language

or physical movements when presenting, explaining and evaluate mathematical ideas in interaction (discussion, debate) with others; (4) reveal confidence when presenting, expressing, asking questions, discussing and debating content and ideas related to mathematics Accordingly, teaching mathematics

to develop mathematical communication competence means teaching students to express mathematical thinking effectively verbally, using representations and in writing Some activities teachers to develop students’ mathematical communication skills: (1) Listen attentively and see students’ ideas; (2) Ask students to respond to and assess their ideas orally and in writing; (3) Assess the depth of understanding

or ideas expressed in student’ discussion; (4) Decide when and how to present mathematical notation in the language of mathematics in students; (5) Monitor student participation in discussions, deciding when and how to motivate each student to participate (Paridjo & Budi Waluya, 2017)

2.2 Vygotsky’s Social Constructivism and Constructivist Teaching Methods

The study of Lev Vygotsky (1934) – The theory of social development which focuses on the fundamental role of social interaction in cognitive development (Vygotsky, 1978), emphasizes that community plays a key role in knowledge formation, social learning precedes development, learning is oriented in the zone of proximal development and most of the important things children learn are through social interaction He defends his views through two categories: the More Knowledgeable Other (MKO) and the Zone of Proximal Development (ZPD) MKO refers to a person with a higher understanding or competence than the learner for a new task, process, or concept ZPD is a higher development compared

to the current zone of development where the children can only solve problems with the support of MKO

He encourages teachers to use collaborative exercises to help children develop weak competencies with help from their peers who are more competent and the teachers must recognize the ZPD zone of students

to choose appropriate teaching methods

Constructivist teaching is a teaching style in which teachers design situations and students participate in building, creating, and transforming their knowledge and skills to be suitable to new situations and gain new cognition and skills Knowledge is actively constructed by the perceiver Clements and Battista (1990) argued that learning is a social process in which children gradually get involved in the intellectual activities of people around them In particular, in addition to participating in discovering and finding, students also participate in the social process, including explanation, exchange, negotiation, and evaluation

Some models of constructivist teaching: Bruner’s discovery teaching, collaborative teaching, reciprocal teaching, teaching by the case method, discovery, and problem-solving teaching The steps

to design and conduct a constructivist teaching phase can be implemented as follows: select teaching content; design constructive case; design questions and activities; organize and guide students to participate in constructing; consolidate new knowledge and skills

2.3 Measures to develop mathematical communication competence for high school students when teaching algebra in the direction of constructivist teaching

According to the Mathematical genneral curriculum (2018), the algebra in high school is inherited from the algebra and arithmetic background in elementary and junior high school and is allocated

as follows:

Sequence, arithmetic progression, geometric progression x

The biggest feature of mathematical language in high school algebra program is high abstraction Some definitions and concepts are developed from the definitions and concepts that students have learned, but are brought to a “level” that requires a higher level of thinking when receiving the concept and applying it For example, the concepts of equations and inequalities are presented in the form of propositions containing variables, and the concepts of logical propositions are given to prepare for the presentation of formal logical reasoning in mathematics In addition, the algebraic language in high schools with symbols and numbers is relatively close to students, so the confidence level of students in communicating when learning this content is somewhat higher than in other subjects of mathematics,

so the teaching process of algebra will have more potential and favorable opportunities to develop mathematical communication competence for students

2.3.1 Organize for students to form knowledge of the mathematical language and improve skills to use mathematical language effectively

a Purpose of the measure: Help students understand and grasp the vocabulary semantics

of mathematical language used in algebra; grasp and use proficiently the syntaxes of mathematical language, mathematical symbols in the process of presenting their mathematical ideas and solutions

b Scientific basis of the measure: Mathematical language undertakes two functions as the means

of mathematical communication and the tool of mathematical thinking In the process of learning mathematics, students often use two functions of mathematical language to perform mathematical communication and form mathematical knowledge As Vygotsky (1962) concludes: Language influences the cognitive development of children To perform mathematical communication, students must know the mathematical language, and if students want to have good mathematical communication skills, they must have the skills to use mathematical language effectively Also according to Kim & Pilcher (2016), one of the factors that affect listening comprehension skills is a person’s vocabulary

c Content and the methods to take measures:

To develop mathematical communication skills, teachers are first required to equip students with mathematical language in parallel with the process of teaching mathematical knowledge in schools These two categories have an extremely close relationship and cannot be separated The knowledge of mathematical language shows mathematical knowledge and vice versa, mathematical knowledge is the entrust place to send the semantics and syntax of the mathematical language On the other hand, the formation process of the mathematical language knowledge of students must also strictly follow the nature of the cognitive process according to the view of constructivism with the characteristic of the cognitive stage of high school students This is the stage of formal manipulation and logical thinking Therefore, the methods to equip mathematical language for students according to the process of 3 activities with the characteristics of constructivist teaching are proposed as follows:

- Activity 1 Approach vocabulary and mathematical rules: The activity of forming a new mathematical language needs to be constructed on the knowledge of the mathematical language that students have already known According to the content of mathematical language knowledge that needs

to be formed, the teachers decide to select the context for students to approach and form knowledge in one of two following ways: From practical situations, internal mathematical situations, using analogy, generalization, and concretization to form new concepts and terms; generalize the prior known concepts

- Activity 2 Practice mathematical vocabulary, rules: This is an important step in transforming from the state of approaching new knowledge to the state of applying it to solve mathematical tasks

and create active reflexes in using this new knowledge in specific suitable situations It is necessary

to be noted that in the mathematical language, mathematical terms (words, phrases) can be seen as mathematical concepts while the main rules are relational concepts Mathematical language knowledge should be taught to some extent as teaching concepts Some math tasks teachers can assign to students are specifically suggested as follows: implementing some language activities to both reinforce and develop mathematical terms for students (restate the definition in their own words, change the restate expression, express the definition in different language types); analyze, highlight important ideas that are included

in the definition); Identify terms, mathematical symbols, rules; expression exercises (exercises to fill

in the blanks with appropriate words, phrases, and symbols to make correct statements); Language conversion exercises (natural language to mathematical language and vice versa, propositional language

to aggregation language and vice versa, propositional language, aggregation language to graph language, etc.); Exercises to apply the characteristics of concepts and terms to problem-solving activities and practical applications

- Activity 3 Consolidation (reconstruct knowledge of the mathematical language): This task is designed with a spreadsheet in which students need to recount the words, phrases, symbols, mathematical rules they have learned, explained, or illustrated Students can perform this activity immediately after approaching knowledge about mathematical language or after completing a lesson in class

Spreadsheet:

Title of the lesson………(1)

No New terms (words, phrases) (2) Symbols, notations (3) Rules (4) Interpretation (5)

(1): Students give the general name for the main math content in that lesson by themselves (2); (3); (4): Students list again the mathematical language they have just learned in the lesson (5) Students interpret concepts, symbols, and rules according to their own understanding and give illustrations by themselves (if any)

Example 2.1 Teaching to form the terms “Set, element, subset”

- Activity 1 Startup - Students are activated with the knowledge learned about sets, subsets, and

elements Students are divided into groups of 3 students to perform the task

Question 1: Set is a concept that had already been introduced in middle school Let’s give an example

of a set in math or in everyday life

Or: State a sentence that includes the word “set” (This sentence is for students who do not remember the concept of sets)

Question 2: Let’s specify the element of each set in each of the examples given?

Question 3: Among the group’s work results, are there any subsets of another set?

Question 4: When is a set A a subset of a set B?

Knowledge formation:

+ Set is a fundamental concept of mathematics

+ In mathematics, capital letters A; B; C, etc are often used to denote sets Lowercase letters a; b; c; etc are used as the symbol for the element

+ The element contained in set A is denoted by a A ∈

+ Set A is a subset of set B if every element of set A contained in set B It is denoted by A B ⊂ .

- Activity 2 Practice vocabularies and notations

Objectives: Receive the semantics of the term “set, a subset of a set”; be proficient in the rules of the form

used to illustrate sets; Use propositional language to express relationships between objects of the sets

Exercise 1 a) Let’s state the concept of a subset of a set b) Is the above statement a proposition?

c) If the above statement is a proposition, use mathematical notations to write the statement as an equivalent proposition

Knowledge formation:

A B ⊂ ⇔ ∀ x x A x B ∈ ⇒ ∈

Exercise 2 For the following sets, list the elements of each set

a) The set of natural numbers that are less than 10

b) The set of vertices of triangle ABC

c) The set of results when rolling a coin once

Exercise 3 Fill in the blank “…” to get the correct proposition.

a) “Set C is…of set D if set D contains every element of set C”

b) “Every natural number is a … of the set of natural numbers”

c) The set of vertices of a quadrilateral includes …element(s)

Exercise 4. A is the set of rectangles; B is the set of rhombuses What do you comment about the

relationship between set A and set B?

- Activity 3. Consolidate - Complete the spreadsheet

The regular implementation of this process by teachers in the teaching process is to create opportunities for students to form mathematical language knowledge, practice using mathematical language exactly and effectively However, rigor should be avoided especially in activity 1 because there are some symbols of the term appearing mainly in the internals of mathematics It is very difficult

to give situations and problems in real life

2.3.2 Develop skills to receive and reflect information (reading comprehension, listening comprehension)

a Objectives of the measure: Students develop skills of refining and receiving information when

approaching a certain math content, both when reading or listening Students proactively and actively receive, reflect and transform information accurately and completely by mathematical language

b Scientific basis of the measure: The formation of new knowledge must go through two stages of

assimilation and accommodation according to constructivism Without approaching and understanding

at least some of the “meaning” of the information provided, students cannot form new knowledge The communication process cannot continue and be effective if students do not know what they have been reading or listening to Reading comprehension and listening comprehension are the first elements of mathematical communication competence

c Content and methods to conduct the measure: Students perform reading comprehension and

listening comprehension activities when reading materials such as textbooks, school exercise books, reference materials when approaching a math task and listening carefully to the teachers, other students

or MKO talk about certain math content These are activities that students must perform in any school environment, whether in a traditional or modern teaching environment However, for modern teaching, this activity has the following differences: Firstly, MKO is not only limited to teachers in the class but also other students in the class as well as indirect MKOs through video recordings Secondly, due

to new requirements in terms of competence development, especially mathematical communication competence, modern teaching pays much attention to the content understood by students when reading and listening This means that students have to reconstruct spoken or written language products about what they have received

To foster students’ reading comprehension and listening comprehension skills, teachers can guide students to practice according to the following activities:

- Activity 1 Write words, terms, mathematical content, graphs analysis, drawings (within their

ability) immediately when students read or listen (keywords): For text reading, the method of underlining the words, phrases, symbols, and mathematical terms can be used

- Activity 2 Connect those discrete mathematical symbols and terms into initial mathematical

messages

- Activity 3 Set up the texts, language products/Solve tasks: Find the connection between messages

that have been newly discovered to solve the requirements given or find new messages In this step, students will check, supplement, learn extensively about the given data, find the connections between them, and write down that correlation into new complete data

According to the purpose before reading or listening to be divided into two types: Reading or listening to approach, receive pure information; read or listen to information to solve specific tasks For the first type, students perform reading and listening actions when approaching a concept, a theorem and a solution to a problem Then, with their own purposes, students identify the core information approach and transform it into their own personal texts and language products For the second type, to solve specific tasks, students will not receive information in a general way, but find out the information

to explain or find solutions for a problem given

Example 2.2 Teaching functions with absolute values

if

if

y x

= − = 

− − <



With the graph of the function as the figure beside.

Remark From the above example, we can find out the way to

- Draw the graph of the function y = f(x).

- Keep the graph of the function y=f(x) on the right of the Oy

axis and remove the part of the graph on the left of the Oy axis.

- Take symmetrically the part of the graph on the right of Oy

to Oy.

Conclusion: The combination of the two parts of the graph

Let’s read the math text above and do the exercises

Exercise 1. Let’s present the way to draw the graph of the function y = 2 x − 4 presented in the above mathematical text

Exercise 2. Let’s comment on the characteristics of the graph in the figure.

Exercise 3 Can you do the same drawing again? Is there a faster way to draw? Compare the result

of your assertion with the “Remark” given above

Instructions for teaching the above exercises according to the process of 3 activities proposed:

Teacher’s activities Student’s activities

Exercise 1

Activity 1 Let’s write the basic math information you read in the first paragraph.

The information you can read on the

figure

(Pay attention to the special points)

The function y=2x−4 is the combination

of 2 functions:

2 4 if 0

y= x− x≥ ; y= −2x−4 if x<0 (1) The graph of the function y=2x−4 is a V shape The two rays on the figure correspond

to the graphs of the two functions above

y= x − includes the graph of the function

2 4

y= x− on the right of Oy and the graph of the function y= − − 2x 4 on the left of Oy

Can you draw the graphs of these functions? Linear function, I can draw its graph as

straight lines

find out the way they used to draw the

graph of the function y=2x−4

Draw the straight liney=2x−4 on the right

of Oy and the straight line y= − −2x 4 on the left of Oy The combination of the two half-lines above is the graph of the function

Exercise 2 Comment on the characteristics of

the graph The graph includes two branches that are symmetric over Oy (2)

Exercise 3

Activity 2 - If you know one of the two branches, can

you draw the other? Absolutely, I just need to take the symmetry over Oy

So for the above case, do we need to

draw a graph of the function y= − − 2x 4?

Absolutely not, I just draw a graph of the functiony=2x−4 then take the part of the graph on the right of Oy that is symmetric over Oy and get the part on the left of the graph Oy That means I have a V-shaped shape like in the picture

Can you present again the way

you draw the graph of the function

2 4

y= x− ?

To draw graph of the function y= 2x− 4, we draw graph of the function y= 2x− 4on the right of Oy, then take symmetry over Oy The combination of the two graphs above is the graph of the function y= 2x− 4

Activity 3 That way is faster Let’s compare what

you have just drawn with the comment

mentioned in the text

Comments in the text are the general case of the graph of the function y f x= ( )

Three aspects of reading and listening comprehension skills: Collect information; analyze and interpret the text; Response and evaluate To successfully solve a problem, students must first be able to read and understand well, so that they can understand correctly the content of the problem and connect the ideas expressed in it to fully grasp and find out the way to solve the problem successfully It can be said that teaching reading comprehension skills, listening comprehension skills, and teaching problem-solving skills go hand in hand in math education

2.3.3 Developing speaking skills through mathematical discourse activities

a Objectives of the measure: Help students participate in a conversation about a mathematic

topic Help students clearly and coherently present mathematical solutions and ideas; be able to argue, criticize the mathematical solutions or ideas of other people, or defend and explain their own solutions

or mathematical ideas

b Scientific basis of the measure: According to Stein (2007), mathematical discourse is that students

conjecture, talk about math, agree or disagree on a certain problem The difference in manner between talking about mathematics generally and mathematical discourse stems from a constructivist view of learning in which knowledge is created through interaction with the environment It is also consistent with the view that mathematics is not about memorizing and applying a range of processes but about developing an understanding and explaining the processes used to come to the solutions

c Contents and implementation methods of the measure: Students must be allowed to discuss regularly

A lesson should begin with many talks and discussions in which students do the following activities:

Activity 1 Teachers provide mathematical problems within the ZPD zone for students to study and

finding their individual ideas and solutions

Activity 2 Present ideas and solutions to the activity group or the class.

Activity 3 Interpret, justify the answer.

When asking students to explain and justify their answers, the focus should be on explaining exactly what steps and the strategies they performed to get to the results It means that the question focuses on the process, not the end product

Example 2.3. When teaching functions and values of functions, teachers guide students to gain

mathematical knowledge “determining parameter values of a function when knowing domain of the function” The problem is given: Let’s determine m so that the equation x4−4mx m2+ 2− =1 0 has a unique solution ZPD zone of students in general when learning about this content:

Students know how to solve biquadratic equations, understand how the unique solution is required + Students do not know that if the biquadratic equation has a unique solution, then that solution is 0 The task of students is to find solutions to solve the problem, clearly explain their strategies to find out the answer, give new mathematical statements, and work together with other students in the class to perform the mathematical discourse Teachers give suggestions, follow, and observe to guide students

to connect the knowledge formed in the previous lesson with new requirements, and finally give their own “statements” These statements may be true or false, complete or incomplete, but all of them are significant because they represent student participation and the exchange of mathematical ideas

The mathematical discourse activities of students when performing the above problems are illustrated by activities 1, 2, 3 as follow: (Teacher: T; Student: S)

Activity T/S Math speaking/writing content

1 T (1) This is a biquadratic equation and you can completely solve it by using its features

- If x0 is the solution of the equation (1), then −x0 is also a solution of the equation (1) Therefore, if the equation has a unique solution, then x0 = −x0 ⇒ x0 =0

- Replace x0 =0 to (1), we have m2 = ⇒ = ± 1 m 1

- With m =1: (1) ⇔ − x4 4 x2 = ⇔ = 0 x 0 hay x= ± ⇒ =2 m 1 Removed

- With m= -1: ( )1 ⇔ +x4 4x2 = ⇔ = ⇒ = − 0 x 0 m 1 (Accepted)

Ask the teacher: “why x0 = −x0?”

T (2) Don’t ask me, ask your friend

T Does anyone has another way?

S 3 Can we use the method of solving biquadratic equations?

T (3) You can ask other students in the class

S 4 I think it is possible to use quadratic equations after we change the variable

2=

x t

S 5 Condition t≥0.

T It is great if one of you present the method

S 5 Can you (S3) present and explain the method on the board?

S 3

Set x t t2 = ( 0) ≥ (1)⇔ −t2 4mt m+ 2− =1 0 For each value t>0, we have two values x that satisfy this equation Therefore, only t =0 satisfy Then:(2)⇔m2= ⇒ = ±1 m 1

S 6 It is similar to the method of S1