Transcript Slide 1

The language of mathematics in the classroom.
Some typical weaknesses of the students in using
this language and ways which promote them to
master it.
Dr. Rebeka Pali
Department of Mathematics
Polytechnic University of Tirana, Albania
1
"The universe cannot be read until we have learnt the
language and become familiar with the characters in
which it is written. It is written in mathematical
language, and the letters are triangles, circles and other
geometrical figures, without which means it is humanly
impossible to comprehend a single word.”
Galileo
2
Introduction


As other languages the language of
mathematics has two main components: the
semantic and the syntax.
semantic of the language of mathematics the symbols of this language, its terms and
words, and
syntax - the structures and its inner
construction.
3
Introduction
Mathematics uses different signs. The entirety of all these specific
signs form the mathematical symbolic.
The finite system of letters, clearly distinguished between them forms
the alphabet of the mathematical language.
The alphabet of the mathematical language is the basic "material",
and by means of it, based on some defined mathematical rules are
structured the words and sentences of this language.
In this alphabet are included: all letters and punctuations signs of the
everyday language, and the mathematical symbolic.
4
Everyday language plays a very important
role in mathematics






It is a flexible tool, and mastery of it helps students in
simultaneous usage of specific languages.
It serves as an intermediate link between mental
processes, specific symbolic expression and their
logical framework in certain mathematical activities.
It serves as an intermediate link between student’s
experience and specific needs of their mathematical
thinking, especially the need for interpreting and using
reasoning that are different from the everyday ones.
It has an important role in creating mathematical
concepts through argumentation.
It makes possible the interpretation of mathematical
results.
It interlaces the basic mathematical concepts.
5
Specifics of the language of mathematics

One of the particularities of the language of mathematics which is
not present in other languages is its symbolism: theorems
expressed by variable “x”, can be applied when “x” is replaced by
“b” or “2x+7”.

Everyday language starts to be internalized by children as a
spoken language, through voice sounds, related between them, in
certain ways. While the language of mathematics starts to be
internalized by children alongside spoken language as a written
language.

The language of mathematics using terms of everyday language
gives them a rigorously determined meaning. In the everyday
language synonyms are considered as a wealth of the language.
From the other side many terms have double or even multiple
meanings. Differently, the language of mathematics is
distinguished for the precise meaning of its terms. Each term of it
represent a fully determined meaning or object.
6
Specifics of the language of mathematics

Using or interpreting of connectives might cause that the meaning
of a compound sentence in mathematical language to be different
from its meaning in everyday language.

The language of mathematics and the everyday language differ
regarding to the intention, suitability or implications of a sentence.

Small changes in verbal formulation in the mathematical language
might influence the interpretation of the solution from a way to
another.

Another difference between the math and everyday languages
comes from the fact that the everyday language use many
expressions with indexes, which are understandable from the
context but can not be written in the symbolic language.
7
What does it mean to master the language
of mathematics ?

As a first step, to master the language of mathematics is necessary
to master the native language. The language of mathematics includes
it and can not exists without the arsenal of the native language.

Alongside, mastery of the mathematical language presumes the
clarifying of the meaning of terms of the everyday language, which
in mathematical language take a rigorously determined meaning.

A special importance in mastering the vocabulary of mathematical
language that is used in the school, have “the cue words“.

Also, it is important for students to recognize and use patterns.
Pattern recognition is the key to much of mathematics.

Mastery of mathematical language require that student to be
familiarized with statements or sentences that have widely usage
in mathematics.
8
What does it mean to master the language
of mathematics ?

Of a great value is the effort to make students aware about the
fact that the major part of mathematical sentences or
statements have two truth values: true or false.

Mastery of mathematical language is strongly relied in the
knowing and precise using of the symbolic.

Another important demand in mastery the language of mathematics
is the ability of students to do the two-way translations.

One more aspect to be mentioned is finding of a balance between
the spoken language and written language.
9
Study

The data are based on the observations.

8 schools - 612 students - 20 teachers

Methodology:
Observation and recording of the verbal work of
students and teachers;
written tests;
interviews with teachers;
study of the math textbooks.




10
What is observed?







The knowledge and usage of symbolic of proper level;
The knowledge and usage of the terminology of the
proper level;
The scale of respect toward the conventions of the math
language;
The way teachers use the math language;
The ability to pass from symbolic language in
terminology or in everyday language and vice versa;
The ability to use the mathematical language in giving
explanations;
The ability of students to use a precise language in the
formulation and solving problems.
11
What is observed?







The ability of students to make questions and to answer
to questions;
The ability of students to argue and to solve problems;
Their ability to express their ideas and to participate in
the discussions;
The possibilities that the students really have to work in
groups;
The questioning techniques that teachers use;
The evaluation techniques in mathematics;
The interaction that is going on in the classrooms.
12
Findings
Analysis of:

Some typical weaknesses of the students regarding
the usage of the mathematical language.

Classroom learning environment and teaching
techniques
13
Some of the students mix up the terms that stands for the
operations with terms that stands for the results of the
operations.
A - students express precisely
B – have difficulties with “production” and “quotient”
C – manipulate easily only with “sum” and “difference”
D- are not able to interpret results
60%
50%
40%
A
B
C
D
30%
20%
10%
0%
Results
14
Some of student’s responses



the double of its difference with 3 – is
translated: 2x-3 instead of 2(x-3);
the half of a number is 7 less than the
number itself – is translated:
½ e x = 7 < x, or ½ e x = 7 – x, or
½ x < x 7 < x instead of x/2<x-7.
triple of its addition with 7- is translated: 3x+7
instead of 3(x+7).
15

A great number of students have difficulty to
do double-sided translations.
Write with mathematical symbols
•
The half of a number is 7 less
than the number itself;
•
A number squared is 6 less
than the number itself.
•
The quintuple of a number is
equal to the double of its
difference with 3.
•
The quotient of a number with
4 is greater or equal to triple of
its addition with 7.
A – do precise translations
B – can partially do the
translations
C – are not able to translate
60%
50%
A
40%
B
30%
C
20%
10%
0%
Results
16
Inversely,
Write by words the
following equalities or
inequalities:
 2 x = x + 8
 x: 2 = 3( x-1)
 4·x>2+x
 x:4=x-3
60%
50%
40%
A
B
C
30%
A – make precise translation
B – make partially
C – can’t express in words
20%
10%
0%
Results
17
Some of student’s responses

“x: 2 = 3( x-1)” – is translated: “ a number x divided by 2
is equal to its difference with 1, multiplied by 3”

“2x = x +8” – is translated: “ double of x is equal to x
enlarged 8 times”

“4x > 2 + x” – is translated: “ quadruple of a number is
greater than that number added with 2”, ( don’t use the
term ”sum” but use the words “added with”).
18
Many students don’t know the properties of
arithmetical operations and don’t name them
correctly.

Say in words the
equalities: a) ∆· 0 = 0;
b) ∆ · 1 = ∆ ;
c)
(∆·○) □ = ∆ ( ○·□ )
60%
50%
40%
A
B
C
30%
A – Know the properties and
generalize the answer
B – can apply the property but
can’t generalize the answer
C – can’t say by words the
properties and don’t know them
20%
10%
0%
Results
19
Some student’s responses

“ when multiply a number by zero the sum is zero”; ( the
result of multiplication is named “sum”)

“when multiply a number by 1 the production is equal to
addend”, ( mix the term addend with the factor)

“for all numbers that are divided by 1 the production
doesn’t change”, (mix the quotient with the production”

“ a + 0 = a is translated :” the sum is added with that
number itself” etc.
20
A considerable percentage of the students don’t
know and respect the conventions of the language
of mathematics regarding the order of doing
operations and using brackets.
It is easier to decide the order of the operations of a given
expression that to create an expression when the order of the
operation is given.

Determine the order of
doing operations: (20+13) ·
4 + 76.
51 % of the students gave
correct answer.

Write a numerical
expression which has this
order of doing operations:
multiplication, addition,
addition.
40% of the students created
correct expressions.
21
Unclarities in naming of the variable
• Students mix up the concept about the variable with the
number concept.
They solve the equation x + 5 = 8 and give the answer:
“the solution of the equation is x = 3”. In fact, x = 3 is a
new equation equivalent to the first one and the answer
should be: ”the value of x equal to three is the solution
of the equation, number 3 is the solution”.
• This is reflected in written work as well.
They have to find values of “x” that satisfy the inequality
x  12, in general they write: x=1,2,3,4,5,6,7,8,9,10,11
instead of x1,2,3,4,5,6,7,8,9,10,11.
22
Unclarities in naming of the variable
Students show difficulties to

transform different expressions;

to evaluate the values of an expression for given
values of its letters;

to evaluate the values of the function for given
values of the variable;
to make reduction of similar terms.
23
Unclarities in naming of the variable

(I)
(II)
(III)
(IV)
(V)
In the textbook:
“Choose the correct answer for “x ≥ 7”:
smaller than or equal to 7;
greater than or equal to 7;
not smaller than 7;
not greater than 7;
at least 7;
Such alternatives in fact correspond to “  7 ”.
24
Students presents difficulty and inaccuracy
in using the terminology and symbolic.

The symbolic notation “f(x) = 4x + 3” sometimes is
red by words as: “multiply x by 4 and add 3” instead
of “multiply by 4 and add 3” .

Students are not clear about the fact that:
The function f is the rule,
f(x) is the value of the function for a given value of its
argument, and
f(x) = x -8 is the symbolic notation that is used to present
the function in a analytic form
o
o
o
25
Other difficulties and inaccuracies in using
the terminology and symbolic.

Some students are not able to use the algebraic
symbolism as a mean to express and to think about
numerical relations.

They have incorrect imagination and usage of the
symbol of equality . ( they write:“2x = 30 = x = 15”)

Can not distinguish an equation from an equality with
a letter which is true for all values of the letter.
 The symbolic they use for sets is incorrect.
26
There are some weaknesses of the
students regarding solving equations.

they don’t recognize the pattern - a pattern of
operation and order represented by symbols.

feel a kind of uncertainty regarding the number of
the solutions of an equation. They speak about “the
solution” of the equation and not about “the set of
the solutions”
27
A considerable number of the students
hesitate or have difficulties to give
explanations.

“Explain why each positive
number is greater then each
negative number.”
35%
34%
33%
32%
A
B
C
31%
A – give exact explanation
B – try to give explanation but they are
not clear
C – can’t give any explanation
30%
29%
28%
27%
Results
28
Student’s responses
• “ because numbers are getting bigger when we go from the
left to the right of the numerical axis”,
• “because the positive numbers are on the right of zero and
the negative numbers are on the left of zero”, or
• “ because the positive numbers are greater than zero while
the negative numbers are smaller than zero”.
• “ negative numbers are lined up under the positive numbers
”;
• “ the positive number has no sign, while the negative
number has the sign”,
• “because are far from the zero”
• “ the negative numbers are behind of zero, positive numbers
are ahead of zero”
• “because the number which has “+” sign shows addition,
while the number which has “–“ sign shows subtraction”
29
The capability of students to formulate a
problem.
Two cases:
 to formulate a problem based on the math knowledge
that students already have;
and
 to formulate a problem that respond to a given
numerical expression.
30
First case
“Formulate a problem
based on your
mathematical
knowledge ”
A – can formulate problems which
are conform to their knowledge
B – formulate problems which are
not conform to their knowledge
C – can’t formulate a problem
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
A
B
C
Results
31
What students do?







give more or less data than is needed;
do not keep relation between the data. Example: “ Three workers
must prepare 100 models. The first one prepared 30 pieces. The
second one prepared the double of the first one, while the third
on e prepared the triple of the second worker”;
try to remember the formulation of problems from the textbook;
do not formulate the main question of the problem, sometimes
they presume it and do not write.
during the solution are not focused to the question that problem
propose but try to find all results that can be generated with given
data,
do not use in a proper way the “cue words”.
some of them formulate two problems at the same time,
32
Second case
“Formulate a problem which respond to
the numerical expression 150+ 4·30
+ 2 ·20 “.
What students do?

conceptualize the given numerical
expression:
150+ 4·30 + 2 ·20 as (( 150 + 4) · 30
+ 2) · 20

use as data the productions “4·30”
and “2 ·20” but the formulation
doesn’t respond to the given
scheme
80%
70%
60%
50%
A
B
C
40%
30%
20%
A – can formulate a correct problem
B – try to formulate but don’t respond to the
given expression
C – can not formulate the problem
10%
0%
Results
33
Inaccurate usage of logical connectives

Write the set of odd numbers
between 23 and 30. Find some
numbers that don’t belong to
this set.
50%
45%
40%
A great number of students are
able to find the elements of a set
when a common property of them
is given. But, if we give two
common properties they are
concentrated to the last one.
35%
A- can find the elements correctly.
B – can find some of the elements
and don’t take in consideration
both of characteristics.
C- do not fulfill any demand
10%
30%
A
B
C
25%
20%
15%
5%
0%
Results
34
Misunderstanding in using of the words
like each, any, all, at least, some, not all,
etc.
Is given the chain 6, 12, 18,
24, 30, 36.... Is it true
that:
 All numbers of this
chain are multiples of 2;
 This chain include all
multiples of 2;
 Not all multiples of 2
belong to this chain;
 This chain include
multiple of 4;




In general, students respond exactly
to the first question but get confused
to the second one because the word
“all” has changed its place in the
sentence.
They understand the sentence " Not
all multiples of 2 belong to this chain”
as “none of the multiples of 2 is
included in this chain”,
They are not sure if in the sentence
“This chain include multiple of 4” is
talk about “all” or “some”.
It seems that it is difficult for students
to show and to write a empty set.
Usually they write it: “A = 0”, and
say: “empty set include only zero”(?).
35
Classroom learning environment &
Teaching techniques

Students have limited possibilities to speak and discuss with
each other.

Has no efforts for an active inclusion of all students.

Students work individually and not in groups.

Generally, teachers don’t enter deeply in questioning of
the students to find reasons of their weaknesses.

Teachers tend to make closed questions.
36
Classroom learning environment &
Teaching techniques
• Poor evaluating techniques.
• It happens that teachers interrupt the natural curiosity of
students and their prospector spirit because they don’t
hear to the students.
• Teachers don’t take in consideration the age
development characteristics of the students.
• There are a few efforts to enable students to read
mathematics.
37
How can teachers do to promote students
to master the language of mathematics.

Compile exercises and problems which
develop the language of mathematics at
students

Design learning activities which make
students to speak “mathematics”.
38
Samples which have to do with the way we name the
mathematical objects.
•
•
•
•
•
Give the proper term for the following symbolic
notations:
a) 3x+5; b) x+5=12; c) 2 < 6; d) 2 + 4; e) x=2x-7;
f) x<4; g) 2 + 3 = 5; h)x(a+b)=xa+xb.
(ex.
3x+5
expression with letters)
Write an equation of the variable “x”, and an inequation of
the variable “x”.
Write an numerical expression and a numerical equality.
What is the difference between “ 3∙2+4” and “3∙x+4”?
Give the proper term for the following numbers: 2; 13;
5/9; 4,5; 2/3; 4 1/3 ; 5/15; 0,6; 22; -4 ;
39
•Game
problem: Write the expression for each step in the
number game.
Steps
Expression
Your Number
a.Pick
any whole number.
b.Add 12.
c.Double that number.
d.Subtract 8.
e.Divide by 2.
f.Subtract your original number.
g.The result is always _________.
•Word
problem: Sixty-three more than four-fifths of a number
equals 111. What is the number?
40
The following are examples where is required to
generalize concrete examples in symbolic
representation of math facts:
•
•
•
•
•
•
As you know 1 · 5 = 5, 1 · 97 = 97 . Write in symbols this fact about
multiplicand 1.
56 + 49 = 49 + 56. Name the property that generalizes this math
fact.
As you know 12/1=12,321/1=321. Write with math symbols this fact
about division by 1.
As you know 88+0=88,112+0=112. Write with math symbols this fact
about addition of zero.
As you know 2(3+5)=23+25. Write with math symbols this fact
about addition and multiplication.
Write with symbols the sentence: “ If you associate in different ways
three addends of a sum the result doesn’t change” .
41
The concentrating of the attention to the method of
solving helps students to feel secure in solving all
equations of that type.
•
•
•
•
•
•
•
What would you do to solve the equation : x-32=358?
Write your answer as an imperative sentence.
Write an abstract fact which correspond to your method.
What would you do to solve the equation : 42x=252?
Write your answer as an imperative sentence.
Write an abstract fact which correspond to your method.
Which is the rule that allows to pass from (a) to (b):
(a) x -7 = 12
(a) 5 x = 25
(b) x = 19
(b)
x=5
42
Some examples which would promote students in giving
explanations:
•
•
•
•
How do we add a negative number to a negative number? Explain
your answer.
22. How do we multiply a negative number to a negative number.
Explain your answer.
23. How can we write a decimal number as a fraction? Explain your
answer.
How we do multiply fractions?
Express your answer in everyday language;
Express your answer with math symbols.
•
•
Why positive numbers are greater than negative numbers?
Using numeric axis explain the following inequalities:
-2 < 0 ; 7 > 0 ; 5 > -2 ; -5 < -1 ;
43
•
•
•
•
•
•
•
•
What is the sign of the number (-5) · x?
What does it mean to solve an equation?
What does it means to prove an equality?
What does it mean to simplify an expression with letters?
Are equivalent the following pairs of equations?
2x+4=20
and
2x=16;
x²=9
and
x=3;
2x=10
and
2y=10
When a set is called subset of a given set? Draw a Venn’s
diagram to show this relation.
How we can find the elements that belong to the intersection of
two given sets?
How do you find the area of a square. Give your answer by
words and with math symbols.
44
The two-way translations can be stimulated by the
following examples:
•
Write with math symbols the given sentence: “The quotient of 32 with x
is not smaller than double of x”.
•
Say by words the following equality: 4(x+2)=5x.
•
Write in words, as you understand the following symbolic notations:
x – 2 = 12; x /3 = 21; x² = 16; x + 7 = 19 ; 2(x + 4) > 3.
( ex.: the difference of number x with number 2 is equal to 12)
Write with math symbols the sentence:
“One number is twentyeight more than three times another number.”
•
•
Turn into an equation the given problem: “Twelve pencils and one
rubber costs 14 leks. The rubber cost 4 leks. How much does it cost one
pencil?”
•
Find and formulate the characteristic properties of the elements in the
given sets: A = { 1,2,3,4,5,…} ; B = { 2,4,6,8, ....} ; C = { 5,10,15,20,...}
45
Rules and conventions of the mathematical
languages:
•
•
Write an expression without brackets in which the first
operation should be done the multiplication and the
second should be done the addition.
Write an expression which have this order of
operations:
addition, multiplication, division
multiplication, addition, multiplication.
•
•
Decide the order of operation in the given expression:
5(3∙2+5)+7
Are equivalent the expressions: “x + 5x” and “ (x +5)x “
46
The correct use of the symbols:
Which of the following symbolic notations is written wrong:
•
•
•
•
•
•
•
•
•
2 < x > 10;
4 > x < 5;
-2< x < 5
What represent the symbolic notations: ( 2 , 5 ) , { 2 , 5 }
Which of the following symbolic notations is written right:
{1 , 6 } = { 6 , 1 }( 1 , 6 ) = { 1 , 6 }
5  { 3, 4 ,5 } 5  { 3, 4 ,5 }
x<2x>5
{ 0, 1 }  {0,1,2, 3, 4}
Which is the difference between 7 and { 7 } ?
Which is the difference between “ ”and “” ?
Write with symbols “ the set of numbers smaller than 10”.
Write the set that has the only one element, 2.
What represent the symbol [AB] ?
Which of the following symbolic notations in written right:
a + b = B + a ; a + b = b + a ; A + B = a + b;
The sentence “1 < x < 5” is the short form of an sentence that
has two parts. Could you rewrite the sentence in the long way?
47
To improve the language of mathematics that they use
for function:
•
•
•
Give the mathematical expression in terms of “x” which
express the rule: Add 20 and then divide by 3.
Create the function : “divide by 4” in the set: A= { 4,8,12,16,20}.
Let be y = 3x. When x = 4, y = 12. When x = 5, y = 15. When x
change, y change as well. Can you tell what remains the same?
•
What represent the following symbols:
a) f ;
•
•
•
b) x;
c) f(2);
Let be f(x) = 2x. Is it any difference between f(x) and
2· f(x)?
What is the difference between “f” and “f(x)”?
From a given plot, find the relation between x and y.
48
Students can be practiced in formulating problems:





Formulate a problem when is given a number sentence.
Example: How would you create a problem situation
represented by 1∙4=4?
Formulate a problem when a picture is given.
Create a problem situation by yourself and formulate the
problem.
Formulate a problem including in the data the multiple
of a number.
Formulate the inverse problem of a given problem.
49
To enrich the math vocabulary :
•
•
•
•
•
I’m a number. I’m greater than the production of 8 with 4. I’m
smaller than the difference of 550 with 510. I’m even number. Can
you find me?
I’m a production. I’ve two factors. One of my factors is 7 less than
30. the other factor is 5 times smaller than 20. Can you find me?
Which is the great number of the sweets:
seven portions of eight sweets each;
eight portions of seven sweets each
fifty portions of one sweet each
one portion of twelve dozen sweets.
I’m a six-digit number. To write me you need five zero. You don’t
need to use the decimal comma. One of my digit is 2. can you find
me?
I’m a fraction. My denominator is 14. I’m not part of a whole but I’m
the whole. Which fraction am I?
50

To practice students in using the “ cue words” :


Complete the text of the given problem. (Is given a text where
the cue words are missing. Students are asked to complete the
blank places.)
To interpret graphical information:





Use diagrams or pictures to solve problems.
Use tables to solve problems.
Interpret given drawings and graphs.
Find and discuss on the location of given points in the coordinate grid.
Make and interpret graphs that show a certain information, etc.
51
Learning activities
o
o
o
o
o
o
o
o
“Math vocabulary” lists.
Word grouping activity
Brainstorming activity
Labeling activity
Process description activity
The chain of questions activity.
Picture dictation activity
Cooperative problem solving
52