My Reflective Journal post

I want to share an event that I witnessed today the 18th January 2017, as I drove to work along Thika Rd.

Event: A passengers' minibus abruptly without warning cut in front of me and then after a few metres he/she changed lanes.

what was interesting? His manoeurves though dangerous, creates the feeling of him moving faster and arriving earlier than others. There might be some truth to this. Also he will endup driving a longer distance than the one who keep to one lane. It may look fun overtaking many cars but the driver is just raising his stress levels in order to meet his commercial target. The interesting bit is that the driver might not be aware of this; that most of the things we do are in futility.

 What were my feelings? I always use this road and I also use the lane that is mostly preferred by matatus i.e. the outer lane. I find it moving faster and I always have this idea of researching on this phenomenon. So I am always on the lookout of such behavior from matatus but this one caught me off guard.

I also wondered why was the driver bulling others? I wondered if the law can solve this behaviour or is it a problem of our upbringing right from when we were young? Was he taught the value of respecting others? was he taught the value of being patient? where was he rushing to? for your information I think we arrived in the CBD basically at the same time. could the passengers politely tell the driver to just drive along one lane? are passengers to blame for reckless driving?

what did I learn? I learnt that maybe laws are not enough. we need to teach a certain culture to our children. Culture of respect, patient, no-love-for-money, and generally good values.

implications? it will be better if authorities focus on passengers. I suspect that passengers can easily change the behavior of drivers. the customer is always right. As a teacher I am in a better position to educate my students on the good of standing up against wrong things even if you wont be popular.

School fires and dysfunctional learning approaches and philosophy

Ask any school going child or their parent, what they want to be in future. The common response is that they want to be great lawyers, doctors, pilots, university lecturers, media personalities and any other career that is deemed to make major returns in terms of materials accumulation and prestige. Try to pose to the same people the question, how they want their eulogy to read. You will be amazed by the responses. Most of them would wish people to know that they were such great parents, humble, sincere, caring, a doting parent, a loving spouse, a person of impeccable character and the likes, qualities to do with character. These are the words of one experienced lecturer in a local private university.

If all teachers could stop teaching and facilitate learning in our schools, and focus on transformative education with a lot of self-reflection, then there is likelihood that schools fires and other ills may be a thing of the past. We may even see the death of the mentality that, until one of us in power, we cannot progress economically and socially.  What most people lacks but ideally wish for, is a transformative education, one that transforms an individual holistically, and as a consequence transform the society, and everybody appreciate life and humanity in general. So far we have only managed to acquire an education that explains our society ills by apportioning blame to others. That’s why highly educated politicians and the clergy are blaming a third force behind the schools fires. None is taking time to self-reflect on how by either commission or omission, we could have contributed directly or indirectly to the current anarchy. How often do we blame the police for our reckless driving that result to road carnages?

Our society lacks respect to those holding divergent opinion and those of different profession, in particular the teaching profession, transport service, meteorology, sanitary and cleanliness and the police service.  The society should recognize, respect and appreciate the role of teachers in mentoring the youth and by default shaping the society. The good news is that at least some schools are positive in this. A case in point is what was reported in the media about how a school in ol kalou avoided destruction by summoning parents and together with the students addressed the causes for their children grievances.

Something unique came out of the ol kalou school meeting. The students demanded for more play than study.  It may have surprised most parents but a keen observer will agree that we do not let our kids be kids. Play is essential. Figure the situation that is currently live in most private primary schools where each day, children carry homework (read schoolwork) that they sleeply tackles until late in the night and supervision of a house-help or a tired parent. A few hours of sleep and he wakes up by 4am. The routine is repeated seven days in a week (Saturday tuition and on Sunday to finish weekend homework). This scenario is repeated and maintained until the child finish class 8 in the wild goose chase for 400 points and no learning. Note that, our kids start schooling very early in life, at age 3. Hopefully, to spend the next 19 years of their life in an exam oriented education that promises nothing much but a bleak and hopeless life after school. This is pure torture and very unhealthy, both physically and mentally. When do we expect this child to play and be a child? We should learn from the Finnish education system and relieve our children the burdens of a purist education system.  Any educationist worth his salt is aware that children learn a lot through play, the child develop physically and intellectually through play.

There are a lot of injustices and inequalities in our school system and the society in general. The students are very aware of these injustices and inequalities. Thanks to the learning they get through the much accommodative technology. They are also aware that it is not the right way that gets its way to solving societal issues but the most dramatic, selfish and militant kind of way that gets its way. They have perfected what our politicians and clergy are good at. We advocate for stiff cut throat competitions in national examinations where the entry behavior are diverse while at the same time schools are endowed differently resource wise. No wonder it is not strange that the politicians now want to reintroduce the ranking system. Our students are conscious of the fact that even if they were to score highly in national examinations there is no guarantee that they will pursue courses that may lead to the career of their choice. They are living witness to academically challenged classsmates who join the parallel programs and pursue the course of their choice. They have seen how the government shuns brilliant students who end up in the private sector contradicting the immense investment in their education. Students who struggled academically are the current role models of our children. Since they have ‘exceled ‘in politics, church business, tenderpreneurs name them. I am not undermining the genuine efforts and hard work of academically challenged students but social justice, prudence and integrity should take over. Since this is a contradiction of what tutors preach in our school that, academic excellence for a bright future. 

The other probable cause to the anarchy in schools is our culture, how we have been brought up.  Our current culture determines how we behave and how we approach life’s issues.  Road carnages (these are not accidents), political thuggery disguised as fighting for rights (read self-rights), arsons in schools (read teenage rebelling against unrealistic societal expectations through the education system that rarely address their needs and wants), impunity in the private and public sectors, among others are a manifestation of a culturally confused society.  I will give a simple analogy, a keen observation of a washroom in a public and a private university. It is a clear indication of the differences and inequalities in how children are brought up. The presence of toilet tissues and clean toilets seats in a private universities and the direct opposite in public universities tells all.  There is an atmosphere of respect of properties and persons no matter the ranks. The lack of dialogue and informal interactions between students and tutors is a recipe for disaster. Such informal interactions create a sense of belongings and acceptance by members of that community. There is a sense of ownership and the members are inclined to be each other’s brother’s keeper.  That’s why upon the slightest provocation the students cause much damage to release the pent up anger and frustrations. In public institutions we have a stratified society that propagates segregation and your-problems- are- yours attitude. The best facilitates are for senior administrations going down to students.

The remedy to current fires was to give them a break, and let the raging hormones cool down. Unfortunately this was not done and as a consequence more dorms were lost to arson.  Remember teenage loathe laws. True, these are criminal activities but applying the law and imprisoning the child may not be the best option. They just need guidance and probation.  To prevent future occurrence of such cases plus others that may not capture the media attention is to transform the learning approaches and philosophy in our school system. It is a high time school tutors establish a body or association that focus on professional enrichment, similar to an initiative by local private universities. This can be facilitated by school boards.  Tutors should make it a mandatory habit to regularly update their teaching-learning skills, knowledge and attitudes. We should respect and appreciate our children and let them be. Recognize that a child is a unique being and is capable of making independent and rational thoughts. Reassess the boarding and transport (for day scholars) facilities and all the concepts behind it. A lot of resources are channeled in these facilities and especially transport of children in primary school (but is is pathetic and a disaster waiting to happen).  The boarding facilities should be cognizance of the learners’ primary school background. The feeling that a boarding school is a prison should be no longer the case. The Spartan thinking should cease. Again it is high time schools privatize boarding facilities.

power of personal reflection

what are your experiences with personal reflections? what are the gains from thinking from your feelings? what are the benefits of writing from emotions? If an incident happens in class how do you reflect on it?

why do students hate maths?

Most students do not like maths. They find themselves in a maths class since it is a compulsory subject or a common unit that the institution requires you to pursue for you to graduate. The fact that majority of students, especially those pursuing arts and humanities based courses, do not see the relevance of maths in the future career should be a great concern to facilitators of learning. Is the knowledge, skills and attitudes that can be acquired through learning mathematics relevant to the future life of a person? Is learning new knowledge, acquiring new skills or attitudes really important to a lawyer, a psychologist, a theologian etc? That is one of the questions the students of such programs plus their lecturers, always ask. The negative attitude towards maths from early ages in the learning institutions may be a reason for such arguments, not forgetting the level of ignorance.

The lack of shortcut to solutions of maths problems may be a major discouragement to such students and lecturers. If say it is a linear system, it either have a unique solution, many solutions or no solution, but you have to prove beyond doubt that it is either of the three. This is unlike in some scenarios in the humanities or arts where a problem may have many solutions, depending on whoever is talking. A person may put forward a case and always assume to be right as per his own justification. Alternatively differing camps may arrive at a consensus even if the problem is still unsolved. This can be attested by the way politicians agree on certain political issues. The problem is never solved, but if they were to think mathematically or scientifically then there is a high likelihood that the problem wont reoccur. Unfortunately most of these politicians lack a strong mathematics (scientific) reasoning.

Hence a student will sit in a maths class so as to get the right grades and finish his degree and thereafter supposedly go out in the world and make money. That thinking is normally captured by newspaper articles on how a billionaire  makes a lot of money and yet he dropped out of the education system. They wrongly assume that the only ultimate goal of school learning is the ability to make money. Hence if a learning experience is seen to be an obstacle to finalize my education journey to that goal, then it is not important. Maths is often a victim since you cannot break its laws and appreciating the working of these laws require patience. This thinking kills the essence and the beauty of learning and moreso learning maths: the logics in maths.

Some fundamental problems that afflict our society can easily be sorted out if people understand simple laws of (maths) nature. For example, water follows its course, it always find its own level. Even in traditional African societies we had sayings that echos this simple but complicated law. Most of our roads are in pathetic condition simply because the 'engineers' did not really comprehend the implication of gravitational forces, laws  governing fluid flows, vector laws among others while constructing roads. A murram road can withstand harsh wet conditions for a long time if it is engineered well. An engineer who can forego shortcuts and stick to the laid rules can leap big in the long run than the shortcuts of corruption. What joy will you get if you leave in luxury but not in peace since you fear your poor neighbour will mug you? Good well constructed and maintained earth/murram roads that are far cheaper than tarmac roads can change the lives of many poor people as they wait for tarmac roads.

These are laws that directly or indirectly can be captured by those who like maths, and by a keen facilitator of maths learning. A system that requires a unique solution has no shortcuts. For a system to be balance, it needs the right inputs/variables. To avoid conflict of whatever nature, all the systems must be in tandem with each other. Learning of whatever kind must be seen to add value to the learner. There is no learning that is useless, otherwise it is not learning. Such values should be capture very early in the cycle of learning. The mentality of waiting to 'use' knowledge after graduating, almost 16 - 20 years of education is disservice to oneself and humanity in general. It is paramount to teach history such that an individual can appreciate the impact of knowledge, skills and attitudes throughout man's history of growth and development. As a student, one should strive to diligently inquire the relevance of such knowledge and skills without bias in-spite of his weakness of the same. Anything that is, is important, the fact that you do not see its importance not withstanding.

Individualized Education Programme IEP

Learning is not a hard science. There is no single formula that can be applied to all learners to achieve the desired learning outcome.Furthermore, each learner is unique and so is the facilitator of the learning process and as a consequence the learning process should be unique to an individual learner. That complicates the learning process in a school setup where the availability of qualified and passionate facilitators of learning are scarce, to allow a 1:1 ratio. Hence an individualized education programme will then be tailored made for the so called special cases only. Unfortunately this contradicts the assumption that we are all unique and we all need individualized education. In most cases we target the marginalized minority on the negative side i.e. those who have poor handwritings, get all sums wrong, the easily agitated, those who have many spelling errors etc. What about those who are excellent as per our standards? How do we ensure they exploit their potential to the fullest? What about the normal student? are they just normal or conformist, or have they mastered  what the society want and follow what the society advocates to be the right standards? Despite the fact we all complain of a poor education system that cannot address the societal challenges, not forgetting we are part of the system.

Reflection on what kind of learners i have and what kind of teacher i am, can have great benefits. Analysing critical incidences in and out of our classrooms can help discover the uniqueness in each one of us. Of course reflection per se can be hard and bias. The assumption is, if you are highly exposed to the right knowledge, skills and attitudes, you can be in a better position to utilize these abilities - reflecting and CIA, to their fullest.

This post welcomes replies on what potential you discovered, what insights you had, when you reflected on your IEP and took whatever steps you undertook. Sharing of experiences can be beneficial to any reflective individual.

Rewriting an Exam Question

When answering exam questions some weak students normally rewrite the question in the Answers booklet before providing the solution.why do they rewrite it?it does not add any value. Moreover it is a waste of precious time. No matter how many times you tell them they always repeat.Some explain this behaviour by saying the tutor may confuse the solution and its respective item.

Basic techniques of differentiation

Techniques of Differentiating

The following are basic techniques of differentiating. 

(a) Power rule

So far we have seen that given; y=f(x)=kx^{n}

Then the gradient function is; dy/dx=knx^{n-1}

Example 1: Differentiate the function y=3x^{5} with respect to x

solution: dy/dx=15x^{4}

Example 2: Find the derivative of the function y=2x^{3}-6x+3

solution : dy/dx=6x^{2}-6

Example 3: Differentiate the function y= [6x^{3}-7x]/[x^{2}]

solution: we need to first simplify the function by rewriting it as;

y=6x^{3}/x^{2}-7x/x^{2}=6x-7x^{-1}

Hence we have;

          dy/dx =6+7x^{-2}

(b) Product rule

Let y=f(x)=uv where u and v are both differentiable functions of x, then the gradient function is given as;

dy/dx= u dv/dx+v du/dx

Alternatively;

 (uv)^/=uv^/ +vu^/

This rule is appropriate when it comes to integrating product functions.

Example 1: Given y=(3x^{4}+7x)(5x^{7}-3x^{2}+9x) determine the derivative.

Solution: by definition dy/dx=u dv/dx+ v du/dx

we let,

u=3x^{4}+7x then du/dx=12x^{3}+7  and we let v=5x^{7}-3x^{2}+9x  then dv/dx =35x^{6}-6x+9

Therefore; 

dy/dx=u dv/dx+ v du/dx=(3x^{4}+7x)(35x^{6}-6x+9)+(5x^{7}-3x^{2}+9x)(12x^{3}+7)

Example 2: Given y=3x^{5}sinx , determine the gradient function.

Solution: by definition dy/dx =u dv/dx+v du/dx

We let, u=3x^{5} then du/dx =15x^{4}

Also let, v = sinx    then  dv/dx =cosx

Therefore; 

dy/dx=u dv/dx+v du/dx =3x^{5}cos x+15x^{4}sin x=3x^{4}(xcos x+5sin x)

Example 3: Let y=3x^{2}e^{x}, find dy/dx

Solution: We let u=3x^{2} then du/dx=6x

Also v=e^{x} then dv/dx=e^{x}

Hence;

dy/dx=(3x^{2})(e^{x})+(6x)(e^{x})=3x^{2}e^{x}+6xe^{x}

(c)  Quotient rule

Let y=f(x)= u/v  where u and  v are both differentiable functions of  x, then the gradient function is given as;

dy/dx=[v du/dv –u dv/dx]/v^{2}

Example 1:  Given y= 4x^{3}/[3x^{2}-4x]  find dy/dx;

Solution: Let u=4x^{3} then du/dx =12x^{2} and v =3x^{2}-4x then dv/dx = 6x-4

dy/dx=[12x^{2}(3x^{2}-4x)-4x^{3}(6x-4)}]/[(3x^{2}-4x)^{2}]

=[36x^{4}-48x^{3}-24x^{4}+16x^{3}]/[{(3x^{2}-4x)^{2}}]

=[12x^{4}-32x^{3}]/[(3x^{2}-4x)^{2}]

=[4x^{3}(3x-8)]/[(3x^{2}-4x)^{2}]

Example 2: Find the derivative of the function;

y= 2x/sin x

Solution: Let;

u = 2x then du/dx  =2

Let; v= sin x then dv/dx=cos x

Hence;

dy/dx=[2sin x -2xcos x]/sin^2x=2[sin x – x cos x]/ sin^{2}x

(d) Chain rule

Let y be a differentiable function of  u  i.e.  y=f(u)  and that u  is a differentiable function of x i.e. u=g(x), then y is a differentiable function of x i.e. y=f(g(x)), and;

dy/dx=(dy/du )(du/dx)

Example 1: Suppose y=(3x^{5}+7x^{4})^{4}, find dy/dx

Solution:

Let u=3x^{5}+7x^{4} then  du/dx=15x^{4}+28x^3

Also y=u^{4} then  dy/du=4u^{3}

By definition

dy/dx=(dy/du)(du/dx)=(4u^{3})(15x^{4}+28x^3)

=4(15x^{4}+28x^3)(3x^{5}+7x^{4})^{3}

=4x^{3}(15x+28)(3x^{5}+7x^{4})^{3}

Introduction to Differentiation

Definition 1:
To understand the concept of basic differentiation, we need to consider the gradient or slope or steepness of a straight line of the form y=mx+c where y and x are variables, m the gradient of the line and c is the y-intercept. Now the gradient of a straight line is the change in the vertical distance over the change in the horizontal distance. That is;
Gradient, m=(change in vertical distance)/(change in the horizontal distance)

Note that the gradient of a straight line is constant, but the gradient of a curve keeps on varying.

Definition 2:
To determine the gradient of a curve at point we need to use a gradient function. Given a function;
y=f(x)=kx^{n}
Then the gradient function;
dy/dx=knx^{n-1}

Example 1: Let, y=3x^{5},then dy/dx=15x^{4}

Example 2: Let; y=2 , then dy/dx =0

Remark 1: Note that, y=2 can also be written as y=2x^{0}. Recall that x^{0}=1 where x is not 0

Example 3: Differentiate the following function with respect to x; y=(2x-3)(4x+5)

Solution: Note that another term that is used to mean, 'finding the gradient function' is 'differentiate' with respect to a certain variable. One can also refer differentiation as 'determining the derived function'.
In this example, we need to expand the RHS to get;
y=8x^{2}-2x-15 then dy/dx =16x-2

Remark 1: The rule so far discussed is the most basic rule of differentiating. Some functions may not be differentiated at all over certain intervals or may not be differentiated using the above rule. Some differentiating techniques are discussed in the next section while others are beyond the scope of this blog.

P.S.
1) Next post will be on techniques of differentiating
2) This is an excerpt from the textbook, Basic Mathematics; by Kahenya NP

Incident 4 The Missing Girls

It was on Thursday mid morning at around 10.25am. This was immediately after a short break and learners were expected to attend science lesson. Due to the harshness of their class teacher who was also their science teacher twenty pupils did not attend science lesson and therefore only twenty three learners were in the classroom ready to be taught. The other twenty hid behind the classroom and others in the latrine.

This was the day I was expecting to be assessed for the second time because we had already communicated by the assessor that morning and even gave her direction to my school. I was expecting her to be in the classroom for assessment in the same class. When the teacher realized that some of the learners were not in the classroom she decided to report the matter to the deputy headteacher who also reported the same matter to the headteacher. The headteacher did not take his time to know what could be the root cause of the misconduct instead ordered boys from standard seven and eight to go and look for the standard three pupils who hid within the school compound. They were assumed to be within the school compound because the school is fenced and has lockable gate and a gate keeper, later on the learners were found in the latrines except two girls who were missing for the whole day.

On Friday very early in the morning the parents of the two girls came in tears (sic) asking for their beloved daughters, None of the teachers could explain the where-about of the girls. The Headteacher asked the teachers on duty to assemble learners in the assembly ground so that he could make an announcement concerning the two girls. During the announcement one of the boys raised up his hand and informed the headteacher that the previous evening he saw the two girls in one of the herdmen house and he expected that they slept there.

In addition the boy declared (sic) to the Headteacher infront of the two parents that the herdman had been luring school girls using money and therefore he had a chain of girl friends from our school. Above all, the herdman had been sickly and some weeks ago he was diagnosed H.I.V. positive. When the two parents heard that, one of them fainted and the other jumped on the headteacher in front of his office (sic). The situation was unbearable teachers tried to bring calm but it was in vain. The parent who fainted later recovered and went with the other parent and the deputy headteacher and two other teachers to where the girls were. Then girls were found still sleeping. To them it was not a big deal; one of the them declared to the deputy headteacher that even her parents knew she was befriending the guy (herdman). And many are the occasions she has ever taken shopping to the mother from the guy.

The long story ended in the police station whereby the herdsman was jailed because of child abuse. The girls were  later transferred to another school.
P.S.
(1) This is unedited version of the incident.
(2) Give reflective comments on this incident. Reflect on so what... and now what situation of the incident in respect to the teaching/learning process.

Incident 3: Std 3 CRE lesson

The following is unedited version of the incident as reported by Mary (not her real name):

It was a standard 3 CRE (christian religious education) lesson. The topic was 'Jesus Resurrection'. We learnt the purpose of Jesus being born, dying and resurrection. I explained that when we die our soul goes to heaven (sic) and our body remains here on earth (sic). Therefore in the case of Jesus, it seems God put the soul back into Jesus' body after three days.

After the lesson, it was questions time to check whether i had achieved my objectives. Question 3: which part of our body goes to heaven when we die? Answers: Child 1 gave: brain. Child 2 gave: mind. Child 3 gave: Heart. Child 4 gave: legs. I got curious and i further inquired why he said legs. The boy replied,'Teacher! it is because the other day my dad was struggling (sic) my mom while lying on her. Her legs were raised up and she was screaming in pain while saying, ooh my god, am coming'.I was extremely shocked, breathless...a moment of silent followed. I hoped other learners had not heard this. I didn't know what to say or do.

I recollected myself very fast and advised the learners that it was not the legs but the soul that goes to heaven. Afterwards i called the boy and inquired further. He disclosed that they live in single room. I was a bit mixed up because i felt i needed to talk to his mother but i did not know how she would respond or react. I prayed about the issue, gathered courage and called her. On hearing the story, she was honest enough and apologized. she also promised to check on the situation. Being a very private matter i left it at that.

How would you interpret the above incident?
Now what... next?

Incident 2: Class 2 pupil’s Traditional Brew Present

It was during the second week of my teaching practice when my assessor Mr Achi* came for assessment during teaching (sic). Earlier I had prepared my learner’s psychologically (sic) that the external visitor (sic) would come to class. The learners are always excited about visitors.

Prior to the day (sic), I informed them t0 clean their uniform and to come to class early. When the assessor came, learners were quite cooperative. Learning took place well as intended but during the lesson one of the learners had brought a present to the visitor. The boy stood and asks for permission to present the present to the visitor but I told him to wait.

At the end of the lesson, the boy stood and walked steadily to the visitor and presented a bottle of traditional brew (traditional brewed alcohol) to my assessor. The rest of the class laughed and tried to run out of the class but I controlled them.

Immediately, I allowed the class to go for the long break and called the boy and asked why he did so. The boy told me that it is a normal drink at their home. I advised him not be using the brew and later I called the parent and talked with her on the dangers of brew to young kids and the health effect in general.

NOTE:

·         No editing has been done.

·         Names have been changed to protect privacy.

What was most interesting or bad aspect about the incident?

What new lessons did the teacher learn?

What is the way forward as far as the teaching/learning is concerned, in regard to this incident?

Incident 1: Class Two Incident

On the morning of 14th March 2015, i woke up early than usual since i was the teacher on duty and i was also expecting external supervisor from the university. I was able to arrive at school at about 7:00 a.m. Pupils began tickling in one by one and soon learning started.

At about 10:00 a.m our school chairman came to school to inform us that the county governor would be visiting our school in the afternoon. We hurriedly held a brief staff-meeting and shared duties amongst ourselves. I was charged with responsibility of ensuring that the environment (sic) becomes clean and tidy. i asked pupils to collect litter and some to sweep their classrooms.

It all begun when it started raining in the afternoon. It was a heavy downpour and word came around that the governor was not going to make it. Kamau* (not his real name), one of our trusted class three prefect came running to the office and reported that something weird was happening in class two. Being the teacher on duty  i hurried to investigate. As i neared their class i could hear the whole class singing loudly ....'kila mtu na demu wake.. '(everybody hold your girl). I could clearly recognized the voice of the most notorious boy nicknamed 'Jangili' (Thug) leading the singing. When i entered the class, i was shocked to find some of the kids were half naked.

We were able to guide and counsel the kids who led others into this ugly incident. Jangili and his close associates (sic) confessed  that they have been doing thus on their way home, the reason being that, they have been seeing their parents doing 'jig jig' at night. The kids vowed not to repeat it again.

It came to our understanding that class two pupils had not being doing their assignments and homework since their class teacher has been reluctant to mark the classwork. We also learnt that Jangili and his close associates (sic) had joined a group (sic) that had been abusing drugs.

I learnt that each child has a unique characteristics/traits and should be handled differently. Learners have individual differences.
NOTE:
The name of the university, school, author, teacher, pupil, governor and county has been omitted to protect privacy.

Critical incidences in our classrooms

In the next posts I will share some of the incidences that my students experienced when they were undertaking their teaching practice. They analyzed the incidents from a reflective point of view. They were guided by reflective models. The goals of these posts is to hear your comments on theories/opinions on why such incidents happen in our classrooms. Theories to explain the why. I would also like to hear what feelings you could have experienced if you were in such a situation. And finally the lessons we have learnt or can learn from such incidences and the way forward.

sequences and series

A sequence is a set of terms which are written in a definite order obeying certain rules e.g. 2,4,6,8,10,... is an infinite (since it goes on and on forever) sequence obeying a certain rule which is; for you to get the next term in the sequence you add 2 to the preceding one.In particular this sequence is made of multiples of two or even better the even numbers.

A sequence with definite terms is said to be a finite sequence e.g. 3,6,9,12. Note that they are no three dots after 12 i.e. the three dots normally denote that the sequence goes on and on forever. Hence this sequence has 4 terms. Hence  it is a finite sequence.

A series is the sum of the terms of a sequence e.g. 2+4+6+8+10+... is a series. In general a finite series is the sum of terms of a finite sequence i.e. a1+a2+a3+...+an. While an infinite series is the sum of terms of an infinite sequence i.e. a1+a2+a3+...+an+....

The most common sequences are the arithmetic sequences and the geometric sequences. An arithmetic sequence is a sequence that proceeds with a common difference which is normally denoted by d.In general an arithmetic sequence is a sequence of the form; a, [a+d], [a+2d],[a+3d],...,[a+(n-1)d] where a is the first term, n  is the number of terms in the sequence and [a+(n-1)d] is the last term which is either denoted as l or Tn (i.e. the nth term). For example; 2,4,6,8,10,... is an arithmetic sequence with first term a=2 and common difference d=2.

An Arithmetic series also referred to as an Arithmetic progression is the sum of terms of an arithmetic sequence i.e. a+[a+d]+[a+2d]+[a+3d]+...+[a+(n-1)d] e.g. 4+7+10+13+...is an infinite arithmetic series with first term a=4 and common difference d=3


On the other hand a geometric sequence is a sequence that proceeds with a common ratio normally denoted by r .In general a geometric sequence is a sequence of the form a,ar,ar^2,ar^3,...,ar^(n-1),where a is the first term, r is the common ratio,n is the number of terms in the sequence and ar^(n-1) is the last term or the nth term of the sequence. For example; 2,4,8,16,32,... is a geometric sequence with first term a=2, and common ratio r=2

A geometric series or the geometric progression is the sum of terms of a geometric sequence i.e. a+ar+ar^2+ar^3+...+ar^(n-1) e.g. 3+9+27+81+...is an infinite geometric series with first term a=3 and common ratio r=3.

Solving a quadratic equation by completing the square method

I will demonstrate by two examples. Example 1 has the coefficient of x^2=1 and Example 2 has the coefficient of x^2 greater than 1
Example 1
solve by completing the square method; x^2-7x+10=0 (note also the coefficient of x =-7)
solution
The first step is to take the constant 10 to the RHS to get;
x^2-7x=-10
Next we add a constant k to both sides to get;
x^2-7x+k=-10+k ....(*)
Next we find the value of k to 'complete the square' on the LHS by making it a perfect square i.e.;
k=(1/2*the coefficient of x)^2
k=(1/2*-7)^2=(-7/2)^2
Now we write equation (*) as follows;
x^2-7x+(-7/2)^2=-10+(-7/2)^2
The LHS can be easily factorized as below;
(x-7/2)^2=-10+49/4
(x-7/2)^2=9/4
We next find the square root of both sides to get;
x-7/2=±3/2
x=7/2±3/2
x=3.5±1.5
x=5 or 2
Example 2
Solve by completing the square method; 2x^2+x-3=0 (Note that the coefficient of x^2=2 unlike example 1 above)
Our first step is to make the coefficient of x^2=1 by dividing every term by 2 to get;
x^2+x/2-3/2=0
Next we take the constant to the RHS to get;
x^2+x/2=3/2
Next we add a k to both sides to get;
x^2+x/2+k=3/2+k.....(**)
Next we find the value of k to 'complete the square' on the LHS by making it a perfect square i.e.;
k=(1/2*coefficient of x)^2=(1/2*1/2)^2=(1/4)^2
Equation (**) can now be written as;
x^2+x/2+(1/4)^2=3/2+(1/4)^2
the LHS can be factorized to get;
(x+1/4)^2=3/2+1/16
(x+1/4)^2=9/16
Next find the square of both sides to get;
x+1/4=±3/4
x=/1/4±3/4= 4/4 or -2/4 
x=1 or -1/2

Solving by a quadratic equation using the Quadratic Formula

Given a quadratic equation ax^2+bx+c=0 then x=(-b±√(b^2-4ac))/2a. This is the quadratic formula. Example 1 Solve the following equation using the quadratic formula; 3x^2+7x+2=0 solution. In our case a=3, b=7, and c=2. We next need to replace a, b,and c with 3, 7, and 2 respectively in the formula to get; x=(-7±√(7^2-4*3*2))/2*3. x=(-7±√(49-24))/6 = (-7±5)/6 = -2/6 or -12/6 =-1/3 or -2.

Solving a quadratic equation by factorization

A quadratic equation is an equation of the form ax^2+bx+c=0, where a, b, and c are known constants and a  i not zero e.g. 2x^2+3x+7=0; 3x^2-4x=0; 7x^2-9=0 etc.

To solve by factorization, first factorize the LHS i.e. ax^2+bx+c, by first finding two factors which i will call T1 and T2 such that; (i) their sum is b i.e. T1+T2=b and (ii) their product is ac i.e. T1*T2=ac. Next replace bx with T1x and T2x to obtain ax^2+T1x+T2x+c=0. Then carry out group factorization of the LHS.

I will next demonstrate using two examples;
Example 1
Solve by factorization; x^2+7x+12=0
Solution
In this example our a=1, b=7 and c=12. Hence we need two factors T1 and T2 such that;
(i) T1+T2=7 and (ii) T1*T2=12. The two factors are T1=3 and T2= 4.
Next we replace 7x with 3x and 4x to get;
x^2+3x+4x+12=0 ......(***)
The next step is to factorize the LHS by group factorization i.e. we take the first two terms x^2 and 3x and factorize [i.e. x^2+3x=x(x+3)] and then factorize the next two terms i.e. [4x+12=4(x+3)]
Such that our equation (***) above becomes;
x^2+3x+4x+12=x(x+3)+4(x+3)=0
Note that (x+3) is common thus we have;
(x+3)(x+4)=0
(x+3) and (x+4) are the two factors of x^2+7x+12. Since their product is 0 it implies that either;
x+3=0 or x+4=0
Hence x=-3 or x=-4.
-3 and -4 are the solutions or the roots of the quadratic equation x^2+7x+12=0
Example 2.
solve 3x^2+10x+8=0
Solution
In this example our a=3, b=10 and c=8
Hence we need two factors T1 and T2 such that (i) T1+T2=10 and (ii) T1*T2=24
The two factors are 4 and 6.

Next we replace 10x with 6x and 4x to get;

3x^2+6x+4x+8=0 ......(******)

The next step is to factorize the LHS by group factorization i.e. we take the first two terms 3x^2 and 6x and factorize [i.e.3x^2+6x=3x(x+2)] and then factorize the next two terms i.e. [4x+8=4(x+2)] 

Such that our equation (******) above becomes;

3x^2+6x+4x+8=3x(x+2)+4(x+2)=0

Note that (x+2) is common thus we have;

(x+2)(3x+4)=0

(x+2) and (3x+4) are the two factors of 3x^2+8x+10. Since their product is 0 it implies that either;

x+2=0 or 3x+4=0

Hence x=-2 or x=-4/3. 

-2 and -4/3 are the solutions or the roots of the quadratic equation 3x^2+8x+10=0

self-drive

When it comes to maths,your own initiative plays a key role in learning new concepts.A self-driven person who is curious and adventurous and more so an inquisitive person can gain a lot from the world wide web.Though skill of research is crucial.The world will be at the tips of your fingers.

Difference between permutation and combination

The difference is that permutation is an arrangement where order is important while combination is a selection and order is not important. Think of a sandwich made from a slice of bread, an omelette and salad. You can arrange the three ingredients in the following order (assume you are laying the sandwich on the plate); a) bread, omelette,salad b) bread, salad, omelette c) salad,bread, omelette d) salad, omelette, bread e) omelette, bread, salad f) omelette,salad, bread The above is 6 arrangements (permutations). Now, assume 6 people ordered (a) to (f) sandwiches for breakfast, would they be right to say they ate 6 different types of sandwiches? of course NOT!It is just one COMBINATION (selection) of sandwich.

Solving system of linear equations

One can solve a system of linear equations using substitution method, elimination method, graphical method, crammer's rule among others. Some websites offer applications/solver that show you how to work out the sum.  One can get a variety of resourceful sites if you are good in research skills. As long as you know exactly what you are searching for,and you are patient, inquisitive and curious enough,and also you do not have a phobia for computers, then you have no problem as long as you can make use of the google search engine.For the time being, it basically has enough for you.