Teaching Algebra in Secondary Schools: Challenges and Strategies

This article discusses the challenges faced in teaching algebra in secondary schools and suggests effective strategies to improve students' understanding of the subject. It also includes an overview of the topic on National Curriculum, common misconceptions, and assessment on the topic.

PGCE MATHEMATICS
[Author Name(s), First M. Last, Omit Titles and Degrees]
[Institutional Affiliation(s)]

Abstract
Algebra is found in all the secondary school mathematics. One of the objectives of National
Curriculum is to ensure that there is provision of mathematical literacy to all the willing students.
As a result, the student should be capable of providing solutions to the problems presented to
them algebraically right from their first year in secondary schools. The competence in the
applications of algebra allows for the use of the knowledge in other areas of mathematics.
Understanding of the ways to solve the word problems in algebra will definitely assist in making
of the connections between different concepts that have been learnt in classes. The profanely in
solutions of the word problems that are gained from algebra can be used in generation of the
solution in other mathematical problems. This level of stability and flexibility normally
strengthen the ability of the student to move between various representations that assist in the
selection of the appropriate method to use. This capability constitutes one of the main cognitive
steps in the mathematics required for abstractions.

Contents
Abstract.......................................................................................................................................................2
Introduction.................................................................................................................................................4
An overview of the topic on National Curriculum.......................................................................................5
Effective Mathematics teaching and learning..............................................................................................6
Common Misconceptions............................................................................................................................8
Variable...................................................................................................................................................8
Expressions..............................................................................................................................................9
Equations.................................................................................................................................................9
Construction of equations.......................................................................................................................9
Assessment on the topic.............................................................................................................................9
Word problems....................................................................................................................................9
Conclusion.................................................................................................................................................15
References.................................................................................................................................................16
Appendix...................................................................................................................................................18

Introduction
I spent over two months in a secondary school, planning, observing and teaching lessons and
making work related to algebra. This particular exercise has given me a proper insight into a
number of issues that are connected with the teaching of algebra. For example the question of the
usefulness of the curriculum for the secondary students. My general feeling has been that we
need to teach mathematics that will be of use to our students. Just as Einstein puts it” Teaching
should be such that what is offered is perceived as a valuable gift and not as a hard duty “The
lesson for mathematics should aim at equipping the students with the skills that will later help
them in their future life(Southall 2017).
Having illustrated the same, there is an area for more learning of advanced mathematics. The
students that have aptitude for this subject should not be discouraged just by the mere fact that
some fractions have no interest. In the process of making observations or audit, some weaknesses
were identified in the learning of algebra. The difficulties in the beginning of Algebra may rise
from the methods of generalization that is normally involved. Also the use of letters in problem
solving of algebra differs from every day. In the previous audit, the student were asked in one of
the task to solve a problem in which they to relate heights. They were told that Jumna is 10cm
dollar than David and David is h cm.The students were asked what they could write as Jumna’s
height. The answers that were received included: h10cm, 10h, 11, 12 and many more that were
not correct.
This study clearly indicated the misconceptions by students in the use of the letters as labels for
objects, use of products as sum thereby giving answers that did not reflect any knowledge of the
applications of the equality. Understanding of the ways to solve the word problems in algebra
will definitely assist in making of the connections between different concepts that have been

learnt in classes. The profanely in solutions of the word problems that are gained from algebra
can be used in generation of the solution in other mathematical problems. This level of stability
and flexibility normally strengthen the ability of the student to move between various
representations that assist in the selection of the appropriate method to use. This capability
constitutes one of the main cognitive steps in the mathematics required for abstractions.
Algebra is found in all the secondary school mathematics. One of the objectives of National
Curriculum is to ensure that there is provision of mathematical literacy to all the willing students.
As a result, the student should be capable of providing solutions to the problems presented to
them algebraically right from their first year in secondary schools. The competence in the
applications of algebra allows for the use of the knowledge in other areas of mathematics. This
definately ensure there is firm foundation for solving other relevant orioles in the future. If the
student’s poor performance in algebra is not checked, the county’s ability to develop will be
suppressed in technological and scientific lines(Solanki 2018).
An overview of the topic on National Curriculum
One of the fundamental qualities of a good teacher is his or her ability to explain a particular
topic in various ways. Each student has a specific preferred method for grasping concepts of
mathematics although sometimes they may not be aware of tithe theories of learning points out
two types of understanding-instrumental and rational(Stewart, Redlin & Watson 2016).
Instrumental comprehension is more algorithmic; the understudy will take in a succession of
ventures for taking care of a specific issue, and will just apply this to any comparable issue,
winding up with the correct answer, yet with next to zero comprehension of how or why the
technique works. Social comprehension, then again, alludes to a substantially more extensive
level of perception; the understudy can perceive any reason why a strategy works has some grip

of the mechanics behind the means and can apply their more extensive information to related
issues which a simply instrumental learning would be insufficient for. At GCSE level and
beneath, there is little preferred standpoint to the understudy in picking up a social seeing instead
of an instrumental one as far as exam results, thus an educator ought to know about these two
methodologies and ensure that the manner in which they instruct takes into consideration both
instrumental learning what's more, social comprehension. Practically speaking this typically
shows itself as fathoming a progression of precedent issues by a given grouping of steps, yet
additionally clarifying the thinking behind each at each stage(Simsek,Xenidou, Karadeniz &
Jones 2018).
Effective Mathematics teaching and learning
I am now going to put into the consideration, a lesson that I planned and taught to students on the
multiplication of the brackets. In this particular case, I decided to use an approach of geometry.
The introduction of the topic was through consideration of a garden whose dimensions were
unknown. There was gradual extension and eventual incorporation of double brackets(Lichtblau
2017).
In my opinion, this method of teaching has several advantages. The method appeals to both
rational and instrumental learner. Most of the pupils have been taught bracket multiplication by
use of Smiley method. This method involves the use of a series of lines that indicate that each

term in the first bracket must be multiplied by the second term. Although this was equally
considered and results into the correct answer, there is little assistance to the student on
understanding of how it works and why. The use of the analogy between the multiplication and
calculation of area may bring an idea of square. This will possibly explain why a number
multiplied by it is called a square. The concept also explains the mixed terms(Pimm & Johnston
2016).
Majority of the students found this approach very useful to them despite the fact that it took them
longer to solve the first few problems. Considering that the topic was introduced in a similar
manner, it will hopefully mean that the topic will be understood and remembered better in the
future. This very concept can be extended to give illustration on why a negative-negative
multiplication results into the positive answer(Leahy & Wiliam 2015).
When there is removal of strip from the edge of the garden, it is simple to see that we are
actually subtracting to get the final area. Removal of the strip from the two sides gives the final
results
The students can follow with very little thought that removal of an entire strip from each side is
the same as removing the final part two times and therefore it needs to be added back at least
once. This type of explanation should help in the understanding of the students on the idea of

bracket multiplication. Also this will appeal to their sense of geometry(Gaster,Renney &
Mitchell 2018).
I am increasingly becoming aware that that whatever is taught in school is normally done with
different aim towards improvements on the understanding of the mathematical concepts by
students. The focus of the lessons has actually shifted from understanding and exploring
mathematics to fulfilling the criteria of exam preparation. This is because the students are being
subjected to numerous national exams(Kaput, Blanton & Moreno 2017).
Although this has been helpful to students who have been performing well in mathematics as
compared to the previous generation, the technique of exam is detrimental to the proper
applications of the concepts of core mathematics (Searle & Khuri 2017). Tutors feel compelled
to teach particular syllabus some of which do not allow time to develop fully the ideas that they
may see very important. They too force students to learn what appear to be disconnected from
the facts for the sake of exam. It appears that teachers feel unduly restrained in their methods as a
result of the pressure from the departmental heads in the cases where the heads of education have
opposite idea in regard to the same matter:
"We want all schools to ... take ownership of the curriculum, shaping it and making it their own.
Teachers have much more freedom than they often realize to design the timetable and decide
what and how they teach" (3) (Finch, Johnston, Shaida, and Winterbottom & Wiseman 2014).
Common Misconceptions
There are several misconceptions in regard to teaching of mathematics.

Variable
Most of the students had misconceived the letter for the algebra could be possibly ignored and it
was one particular value. From the responses by the student, there was no identification of the
misconception that different letters cannot have the same value.
Expressions
There was misconception of the inappropriate conjoining of the terms just from the beginning of
the algebra. This was indicated with the student confusion on the product and sum applications.
This possibly suggested that students were not familiar with some of concepts (Foster 2015)
Equations
Majority of the students were not able to calculate effectively the multiplicative inverse of the
equation. The equation that was challenging to the student was, Solve for x: 24x=6.This was the
same as dividing both sides of the equation by the coefficient of x which is 24 thus the answer
becomes ¼.The inverse operation that used division and multiplication were found to be more
difficult than mere subtraction and addition of the few concepts (Kanatani 2015).
Construction of equations
There was requirement of combination of the knowledge of the letter as a quantity to assist in the
solution of the word problems .This too became a challenge (Cuthbert 2016).
Assessment on the topic
Word problems
In my assessment test, word problems were used to test the understanding of the concepts by the
students. The sample question included the following;
i. Mary has y bananas and Koja has x bananas.Otieno counts the number of bananas each of
the individuals have and finds that they are equal. Mary then said that this could be

written as y=x but Otieno said that x and Y are basically different letters and cannot be
the same. Who do you think is right in this case?
ii. Tom has m pens and Basil has three more pens than Tom. How many pens does Basil
have?
iii. A ball costs ten shillings while a shirt costs y shillings more than the ball. What is the
cost of the shirt?
iv. Which number is four less than x?
v. In a packet there are x number of pencils. A lady has three packets of pencils and due to
her generosity; she gives her friend five pencils. What is the number of the pencils she is
remaining with?
vi. Given that c is the number of the cups while p is the number of the plates, write in terms
of algebra that there are four more plates than cups.
vii. There are five more goats than dogs. Write in algebra
viii. Taking b to be the number of the books and p to be the number of the pens and that there
are twice as many books as pens, represent this information using algebra.
ix. In a class there are b boys and g girls. Given that there are 3 boys for every girl, represent
this information using algebra.
In question 1, the student processed the letter as an alphabet and not as a quantity. The response
given by the student was “X and Yare different letters”
In question 2 and 3 there were processes sing and transformation errors. This was due to the fact
that the student were transforming more as division or product instead of sum (Chuang, King &
Leinster 2016).

S1: We can find it by dividing by Y
S2: Three times X will give us 3x
S1: It will be equivalent to 3 toys
S4: Y plus 10 equals to 10Y
The inability to have transformation was the core reason why there were several errors. The
student had proper understanding of the question and also acknowledges the relation of the two
quantities.
S1: 4*C*P
S3: P+b+2=2bp
S2: b=3+g
In this particular equation student wrongly processed using improper operations and also
algorithm that was faulty.

Personal Practice
The one double and two single algebra lesson that I gave were observed by a researcher before
professional learning programs. The general pattern was that as I explained the concepts, the
students listened and I wrote notes and worked an example on the blackboard. In the process of
the lessons students were asked several questions. The student then copied the worked examples
and notes in their exercise books. I gave more examples while one or two students were given
the opportunity to work out the solution on the blackboard. The individual class work was given
and I went round checking and giving individual student feedback on their generated solution. I
then wrote the right solution on the board accompanied with comprehensive explanations. The
student copied the explained examples and corrections into their notebooks before I gave the
homework and closed the lesson. The most frequently occurred activities included the following:
 Teacher explaining the concept
 Students keenly listening and writing notes
 Teacher asking question.
The knowledge of the students of vocabularies of mathematics was considered very important.
Whenever I explained a new word like coefficient, I related its importance to the literacy of
mathematics.” There are some words which are normally common and we use them in
mathematics, such as coefficient. After this lesson now, when you will be doing other topics in
mathematics, we will be referring to coefficients. I expect you to know it now and you will now
be transferring this knowledge to other topics when it comes to that time. Therefore ensure that
you get it now.”

Although I identified and corrected some of the misconceptions by students about the conjoining
terms, I myself did not seem to be aware of some of these misconceptions about the letters as
labels or as objects (Riccardi, Tardioli & Vasile 2015). In this particular lesson I explained that,
“the letters stand for something in the algebra. You will properly understand when I finally
attach a number to the equation.”
In order to build on this explanation, I introduced a simple equation of 7t-3t
Me:What should we have to attach to?
Student A: Table
Me: Seven tables minus three tables. Actually we don’t want to use tables now but instead, what
will be the possible answer?
Student B: 4t
This particular misconception was addressed in another lesson. During this lesson, I used the
student’s names.
Me: You are all familiar with oranges. How many of you have eaten oranges?
All students carried up their hands up.
Me:Owili Jared Ochieng is having 4 tomatoes and Ruth Agwena is having three tomatoes. How
many tomatoes do they have altogether?
Joint response from student: Seven tomatoes

I later explained this concept for the student as follows: The above problem can be written in
algebraic form using t to denote tomatoes. Assuming that t represents tomatoes, Jared has 4
tomatoes while Ruth has 3 tomatoes. In case we want to use t we say Jared has 4t instead of
saying Jared has 4tomatoees.Simliar case applies to Ruth 3t.Now altogether how many tomatoes
are they having?
Joint response from students: Seven tomatoes (Langrall 2016)
Similarly we can have other examples like 4p +3p where p can represent pawpaw.
Figure 1: Blackboard writing (Southall 2017).
I found that one of the most difficult things in the teaching of mathematics was not as many as
have thought to be trouble of understanding the subject matter, but rather the clarity of
articulation. I as a mathematician have learnt quickly at the secondary school level and as from
then I have used the ideas and the concepts to handle nature. In many occasion, being familiar
with the materials to be taught is very important. It allows the tutor to teach while applying the
knowledge in the wider context and the relationship with other branches of mathematics. This

feature may be a handicap on its own since it becomes harder for the tutor to be in students shoes
and have the concept explained in the initial stages.
In order to provide the remedy to the above challenges, the Bell-Lancaster method is used. The
fundamental idea behind this technique is to have students not just as mere recipients of
knowledge but also as co-teachers. In future I would put into practice the use of this method of
Bell-Lancaster. Most of the researches have indicated that explaining a concept of mathematics
to a third person is one of the effective methods of learning it yourself. The other advantage of
this concept is that the students will find it easier to have relevant issues discussed amongst
themselves than with their teacher. The teacher with the concepts will quickly move to correct
the misconceptions and steer the students in the right directions. At the end of the period of
teaching, I realized that language is very important in the teaching and learning of mathematics.
This should be seen to relate to the real-life contexts. Language is actually not isolated but rather
transferred to other real life activities.
Conclusion
Although it is common that the student feel algebra is difficult to be understood. Applied and
solved. I think there will be some solutions that tutors can effectively do including giving both
realistic and unrealistic problems, watching applications of algebra on videos and also use of
software on the algebra. In addition this ways will assist in the motivation and general
encouragement of the learners to focus on the concepts of algebra. These ways will sure
stimulate students that will try and apply it in their real life. The teachers must consider giving
practical problems continuously to students so as to assist in the identifications of the
misconceptions or the mistakes (Chambers & Timlin 2015).

References
Chambers, P., & Timlin, R. (2015). Ensinando matemática para adolescentes. Penso Editora.
Chuang, J., King, A., & Leinster, T. (2016). ON THE MAGNITUDE OF A FINITE
DIMENSIONAL ALGEBRA. Theory & Applications of Categories, 31(3).
Cuthbert, J. (2016). Adjusting the Scottish block grant abatement: the algebra of CM and
IPC. Fraser of Allander Economic Commentary, 40(3), 73-78.
Finch, W., Johnston, R., Shaida, N., Winterbottom, A., & Wiseman, O. (2014). Measuring stone
volume–three‐dimensional software reconstruction or an ellipsoid algebra formula?. BJU
international, 113(4), 610-614.
Foster, C. (2015). The convergent–divergent model: An opportunity for teacher–learner
development through principled task design. Educational Designer, 2(8), 1-25.
Gaster, B. R., Renney, N., & Mitchell, T. (2018). OUTSIDE THE BLOCK SYNDICATE:
TRANSLATING FAUST’S ALGEBRA OF BLOCKS TO THE ARROWS
FRAMEWORK.
Kanatani, K. (2015). Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for
Computer Vision and Graphics. AK Peters/CRC Press.

Kaput, J. J., Blanton, M. L., & Moreno, L. (2017). 2 Algebra From a Symbolization Point of
View. In Algebra in the early grades (pp. 19-56). Routledge.
Langrall, C. W. (2016). The rise and fall of probability in the k–8 mathematics curriculum in the
United States. In Proceedings of the 13 th International Congress on Mathematics
Education.
Leahy, S., & Wiliam, D. (2015). Embedding formative assessment.
Lichtblau, D. (2017). East coast copmuter algebra day 2017. ACM Communications in Computer
Algebra, 51(2), 66-69.
Pimm, D., & Johnston-Wilder, S. (2016). 6 Teaching for mathematics learning. Learning to
Teach Mathematics in the Secondary School: A Companion to School Experience, 92.
Riccardi, A., Tardioli, C., & Vasile, M. (2015). An intrusive approach to uncertainty propagation
in orbital mechanics based on Tchebycheff polynomial algebra.
Searle, S. R., & Khuri, A. I. (2017). Matrix algebra useful for statistics. John Wiley & Sons.
Simsek, E., Xenidou-Dervou, I., Karadeniz, I., & Jones, I. (2018). The conception of substitution
of the equals sign plays a unique role in students' algebra performance.
Solanki, V. (2018). Whitehead group of the Iwasawa algebra of GL2 (Zp) (Doctoral dissertation,
King's College London).
Southall, E. (2017). Yes, but why? Teaching for understanding in mathematics. SAGE.
Stewart, J., Redlin, L., & Watson, S. (2016). Precalculus. Nelson Education.

Appendix

UK National Curriculum

PGCE/SD Mathematics: Lesson Plan1
Check secure entry routines: Calm entry, uniform, equipment, prompt start
Starter Activity
Time Activity Differentiation AfL
13:25
Introducing concepts of
algebra
Stretch it activity:
Multiplication of letters
with digits Blackboard in up on the wall
Walking round to class room
checking how students are

solving problems.
Activities to secure progress
Time Activity Differentiation AfL
13:35 Main lesson’s activities start Student give opportunities
to choose letters
Blackboard in use
13:40
13:42
Student answer 4 algebra
related questions
Students ask to copy into
exercise books sample
questions on algebra.
Different latters used
13:48 Students write down two
things that they know about
algebra.
Use mini blackboards
13:51 Sample of algebra
representation slide
Student ask to identify Some of the pupils will
Asked students to write down
in the books name of sample
problems related to algebra
Student asked why they
prefer specific letters for use.

13:54
14:00
14:02
14:06
14:14
sample of Algebra equation
from the chart.
Allowed to discuss
surroundings and share their
ideas
Pupils share they findings
with the teacher and the
rest of the class
Area explain and unites
Recap -plenary
Student asked to stand
behind their chairs
Class dismissed
look for more not obvious
Pupils work on number of
questions set by teacher
Different set of questions
allow establish level of
topic understanding
Teacher walks around the
classroom to see progress
Mini blackboards help to
assess progress
Mini blackboard in constant
use

14:15
Homework Task (if applicable)
Orderly closure of lesson and departure
Reflection
In what ways did lesson match my expectations?
All students were engaged. Actively participated in the lesson.
Comment on pupil progress in this lesson
Continuous assessment throughout the lesson indicates good results.
What could I change/adapt for future lessons?
Reduce explanations time; ask more questions.
PGCE/SD Mathematics: Lesson Plan2
Date
17.10.2018
Period
3
Group
7
Teacher
Trainee Assignment data

Topic Solution of algebraic word
problems
Observer Assignment data
Inclusion:
SEND
EAL
PP
PP, FSM, K, N Support
Staff
Key
Objectives
(Big Idea)
Mental evaluation methods
Learning
Objectives
(these can
be
adapted to
your
school
context)
Secure
Understand simple means of forming equations
Develop
Applications of different approaches
Challenge
Some students vised the knowledge gathered before in problem
solution.
Link to prior learning Simple multiplication concepts

Link to future learning Variation of variables
Key vocabulary Conditional word “If”
Teachers’ Standards 1
2
3
Resources Presentation using power point
Anticipated
misconceptions
Confusion of selected letters for labels
Check secure entry routines: Calm entry, uniform, equipment, prompt start
Starter Activity
Time Activity Differentiation AfL
11:10
end12:0
5
Tag names on pupils' desks,
do ‘Do It Now’
Extension
multiplication, addition
and subtraction combine
Walking round the
classroom when students
solve problems

Activities to secure progress
Time Activity Differentiation AfL
11:22 Explanation of ways of
forming equations
Provision of considerate
equations
11:30
Student to do new
questions related to
Algebra
Set of questions
accommodate more weak
students
Moving around the
classroom to see work
progress
11:40
Explanation of equation
formation
11:45 Questions given to class
Explain about formation of
Multi-level ability
questions to be solved
Moving around the class and
check note books

11:51
11:54
11:59
algebraic equations
Questions given to the
class
Plenary
Set questions that
accommodate all level of
students Check students' progress in
their note books and mini
black board
Plenary: Check outcomes referred back to LOs
Student work out given problems by applying methods learnt; set of final questions with the use of
blackboards to help assessing instant improvements.
Homework Task (if applicable)
N/A
End of lesson and departure
Reflection
In what ways did lesson match my expectations?
The lesson was covered within the anticipated time.
Comment on pupil progress in this lesson
Student indicated positive responses in understanding.
What could I change/adapt for future lessons?
Improvement on time management