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1. Introduction
When
visiting Bethel Home and School in Himo, Tanzania January/February 2015 I had
discussions with math teachers Joseph, Martin and Jumanne. We agreed that I
should write a memo giving them more examples where the pupils must use math to
answer practical problems from real life. Most of the examples in the memo have
connection with the surroundings, especially at the Bethel home.The picture shows Joseph teaching algebra late evening January 31th 2014.
After Youth School
the pupils can apply for Upper Secondary, more than 90% do. Then they can
choose between three study programs preparing for college/university level and nine
study programs preparing for different vocational education, ending with a certificate in a practical occupation.
In Norway education at all levels are free, but at university/college the
students have to pay the text-books and school material. There is no uniforms
or food served at any level.
It seems
like the math Syllabus and assessment in Tanzania today is much the same as the
Norwegian version when I myself was in Primary School from 1951 to 1958. The
math subject in Norway was at that time “pure” in the sense that there was very
little text, but a lot of numbers, fractions, parenthesis etc. – algebra.
In Tanzania
the teacher seems to have an extremely important role. The students shall be
able to remember and express what she or he has told them. That goes for all
the subjects.
The
Norwegian teacher still has a very important role, but it has changed quite a
lot. Norwegian teachers shall give lectures as they always have done. In
addition to that they shall also give content to the Curriculum frame when
needed, they shall organize the students in groups working with some issue part
of the time, they shall have individual discussions with the students and with
their parents about what is most necessary to improve the next weeks, etc. Continuous
assessment shall be a part of the learning process and has a function that is
very different from final assessment - exam.
Also the
role of the students is quite different now compared to 1958. The students
shall still be able to repeat what the teacher has explained and what is said
in the textbook, but the students shall also be able to find out for
themselves, individually or in groups – this is a way of preparing for later
education and life itself. Later in their lives there will be no teacher around
who knows the correct answer to everything. A part of this is, among other
things, also to stimulate creativity, ability to cooperate and ability to be
critical to what they see, hear or read. The role of the students has also
changed in other ways, for instance in the continuous assessment.
In Norway
the Curriculum and the teacher role started to change in the 1974 curriculum.
This curriculum also stated that the teachers should base much of the teaching
upon what they found in local nature, culture, production etc. Norway is not a
very big country measured in km2 (Tanzania: 945 000 km2,
Norway 320 000 km2), but the distance from south to north is
long, which give very different climatic conditions in different parts. The
landscape varies – mountains in some part, flat in other parts. The conditions
at the coast where there is a lot of fish to catch or cultivate - is very
different from the inland etc.
So the
schools were told to make local curriculums. More practical math was part of
this. (Later on Norway have had new curriculums in 1987, 1997 and 2006 with
changing goals and principles. The curriculums from 1974, 1987 and 1997
emphasised what the pupils should work with, become familiar with etc. The 2006
Curriculum stated much more clearly what the pupils should actually learn.
Basic skills are much more emphasized in this version)
The math exam at the end of class 10 is
divided into two parts. The first part is algebra mostly with little text. The
second part has different problems where there is a lot of text. Quite a number
of the pupils may be good solving the algebra-problems in Part 1, but find it
difficult to extract the math part hidden in the text in Part 2. I will send
you a translated version of National exam Class 10, May 2014.
As I understand assessment as part of National
exam in Tanzania, like in many other countries around the world, is based on
the multiple choice-principle. So far in Norway there is no or very little assessment
or exam of this kind.
Through hard and enthusiastic work from students as well as teachers, the
results at exam are excellent in all subjects at Bethel School.
If we once again shall compare math in Tanzania (Bethel School) and average
Norwegian schools, I have the impression that your students are more clever in
algebra. The main reason for that is probably that you practice a lot, even late
in the evening the board is full of math tasks.
In Norway, we emphasize that students
should understand, we don’t practice and drill as much
as you do, numeracy skills are not automated. (Personally I think it’s valuable and often necessary to
automate – in sports, music, work situations, driving a car, behaving, etc.).
In different international tests, like TIMSS, the conclusion is that Norwegian
students are not very good in algebra compared to other countries.
The way exam is organized and the content and choice of assessment will
give direction back to the teaching. This is a fact that is well known around
the world. Your students do very well at exam, so my proposals are not meant
for big changes. But I would like to give some reasons why it could be a good
idea to spend a small part of the time trying to connect math with real life.
Good for the brain and getting a deeper understanding of math
Practical problems will represent a challenge both for the cognitive and
the creative parts of the brain. The students will meet examples how to include
math to solve problems.
Can be funny and also show the relevance, importance and usefulness of math
This can be valuable for the motivation
of the students, even very young students if you find examples the young ones
have a possibility to understand. Practicing like this will be very well taken
care of in their memories, also the math part of it.
The students will meet this kind of math later in their education
To be honest I don’t know how much this is true when it comes to lower and
upper secondary education, but at University level math applied on practical
problems is some kind of a core in many studies and professions, like engineer,
economist, physician, statistician (like myself), auditor etc
Math is important in many practical occupations
Bricklayers, carpenters, tailors and many others will have benefit planning
and performing the job, if they have a certain math skill.
Daily life in society and family
Reading newspapers, listening to discussions and speeches you often will
meet explanations and allegations that demands some math understanding to catch
and evaluate. A certain level of math in the population is important for
everyone individually and for the society and democracy as a whole.
In your future family it’s valuable if you can plan your economy - what
your income permits you to do when it comes to food, house to live in/build,
education for your children, helping your parents etc. Is it necessary and possible
to increase the income? Should I try to save some money? Other questions like
how much maze or beans/other vegetables or fruits you are able to grow, can be
part of such planning.
Everyone will meet questions like this later in life. It will be a big
advantage if you have had some practice in connecting math to real life.
2. Examples of using math on practical
problems
In some of the problems it’s necessary that the students work in
groups. It’s possible in all the types of problems. Let one of the
students from one of the groups present the way they solved the problem and the
conclusion.
In many of the problems you can let the students make a guess what will be the
answer. It’s a good habit to have to make a consideration if the conclusion you
have reached is reasonable. Many times this has saved me from giving very wrong
answers in math and statistics.
1 Constructing 90º
Give the students a thin rope that is at least 12 meters long and give them
a measure. In this task it’s not relevant to ask them to make a guess.
Tell them to start making a plan for
building a house. They shall mark one of the corners. Ask them to construct 90º with the rope, using
Pythagoras.
They can operate in the playground at school or at home or perhaps in a
place where a house is going to be built. They have to choose the length of the
two legs and calculate the length of the hypotenuse. In a right triangle where
the two legs are 3 and 4 m, the hypotenuse is 5 m. The rope is long enough for
that. (As an alternative you could give them a much shorter rope, long enough
for a right triangle with legs 60 and 80 cm and hypotenuse 100 cm. Or you could
do both)
2 The cistern at home
The shape of the cistern at home is approximately 3 m long, 2 m broad and 2
m deep. That gives a volume of 12 m3.
1 m3 is 1000 liters, so the cistern can hold about 12 000 litres.
The cistern is filled with water that is pumped up from the river. Godbless
told me that this takes about two hours.
Start asking the students to
guess how many litres the cistern can hold and write the answers down.
There are many ways of
estimating how many litres the cistern can hold. If we assume that: the cistern
is perfect in the sense that all the walls are vertical, the bottom is horizontal
and the angel between all the walls and also the bottom is 90
we can do as follow if we use a stick that is
more than 2 m long:
·
Remove 200
litres (or another amount of water) from the cistern. It doesn’t have to be
full. Set marks on the stick and measure how many cm the water level has
dropped. A drop of 1cm represents how many litres? The volume of water
is 200cmx300cmx1cm = 60 000 cm3. 10cmx10cmx10cm = 1 000 cm3
= 1 litre. So 1 cm drop means that 60 litres is removed. If we remove 200
litres the drop will be 3,3 cm.
·
Add 200
litres and measure the increase of the water
If you don’t assume the
walls and the bottom to be perfect, but assume that the pump delivers the same
amount of water all the time, you can:
·
Take the
time filling up a certain amount of water for example 10 buckets each holding 20
litres with water from the pump. Then take the time filling the whole cistern. (Another
possibility is to add the 200 litres and measure the increase with the stick)
If we assume that the
cistern holds 12000 litres and the pump needs 2 h:
2 h is 2x60x60 = 7200 sec.
That means 0,6 sec per litre. To fill
200 litres: 0,6x200 = 120 sec. If 120 sec is the result of the experiment and
it takes exactly 2 h to fill the whole cistern the conclusion is that it holds
12000 litres. (Because 12000/200 = 60. 120x60 = 7200. Probably it will take
more or less than 120 sec to fill the 200 litres and probably it will take more
or less than 2 h to fill the cistern.)
3 The tree in the yard at home
The task can be to calculate
how many litres of tree there is from the ground and up, let us say 1 m. If the
branches are lower than that you can reduce to half a meter. The students will
learn and forewer remember that 10cmx10cmx10cm equals 1 litre.
Again – start with letting
them make a guess and write this down. Perhaps the teacher should make marks for
instance with two thin ropes at the beginning and at the end of what they shall
measure.
The task is to estimate how many litres between the ropes.
I can see at least two
different ways.
1. Use a rope no three in the middle between the other two and
measure the circumference. When you know the circumference you can calculate
the diameter and the radius and therefore also the flat contents of the
cylinder.
2. A more problematic but also useful way is to measure the
diameter directly. Perhaps two or three students must cooperate. One holding
two sticks horizontally, one stick at each side of the three (the same middle
spot). When the sticks are parallel the students can measure the diameter.
4 Running
You need a stopwatch or two.
And the students must have the possibility to run let us say 50 m as fast as
they are able to. Also before calculating this task ask them to guess the
answers.
Take the time. The task is to calculate how much time they would
use running from Himo to Moshi if they were able to have the same speed all the
way.
I don’t know the exact
distance. Let us say it’s 40 km.
One of the students use 8
sec running 50 m: 4 km equals 40 000 m. 40 000/50 = 800 The distance
to Moshi is 800 times the distance they ran, so we also have to multiply 8 sec
with 800 to find the time needed. 8x800 = 6400 sec. 1 hour is 3600 sec. 2 hours
is 7200, so it must be more than 1 and less than 2 hours. It must be 1 hour and
2800 sec. 2800/60 tells us that it’s 46 and 2/3 minutes. 2/3 min is 40 sec. So
the answer is 1 hour 46 minutes and 40 seconds.
The student used 8 sec
running 50 m. How many meters per hour? Again 3600 sec equals 1 hour. 3600/8 =
450. The speed 50 m in 8 sec is the same as 50x450 meters in 8x450 sec=1 hour. 50x450 m = 22,5 km.
22,5 km per hour.
The fastest men on earth run
100 m a little faster than 10 sec. 3600/10 = 360. 100x360 = 36000. The fastest
man on earth have a speed of a little more than 36 km per hour.
The Cheetah is the fastest
animal on earth (except for some birds). Some people (internet) mean that it
can run 500 m with a speed up to 120 km/h. (A lioness can run 80 km/h. She is
faster than the male).
How long time would the Cheetah need to run 50 m?
120 km/h is the same as
120 000 m in 3 600 sec. 120 000/50
= 2400,
which means that if you run 50 m 2 400 times you have been running
120 000 m. To get the number of sec running 50 m we therefore have to
divide 3 600 with 2 400. 3 600/2 400
= 1,5
sec. Quite fast!
(I suppose it’s possible to
do the calculations in this task more pedagogically.) ‘
It’s possible to make a lot
of tasks based on production of for instance maze or beans. You can make
calculations connected to the production Godbless is organizing and the
students participate. If you know the production on 1 acre the students can
find out how much this is as an average per m2. They can ask and
find out how many days per year this production will support Bethel with maze
and how many more acres you need to cover a whole year.
You can also involve money
in the calculation. What is the cost per 100 kilo of maze? You need plants,
fertilizer, poison used to kill pests attacking the plants, people to do all
the work etc. How much do Bethel save compared to buy the maze?
You can do the same types of
calculations assuming a family has a field where they can cultivate. How big
must a field be to give enough maze for one year?
You can make calculations
about milk production, also involving money. The cows need different kind of
food, not only grass.
It’s better that you
formulate the problems and perhaps involve the students in this process. That
will make it much more relevant than I am able to and you and the students will
learn a lot in this kind of process. You will need information and must ask
questions to persons who know. These kind of problems solving may be very
relevant and important for a lot of the children as a preparation for adult
life.
6 See the math part of a text
We talked about this
example. Three girls/women Janet, Gloria
and Elisabeth. How old are they?
1.
In two years from now Gloria will be double as old as Janet
2.
Elisabeth is 13 years younger than Janet
3.
All together they are 81 years old
J is the age of Janet, G is the age of Gloria and E is the age
of Elisabeth.
The task is to formulate the
three sentences as equations and to find J, G and E.
1. G+2 = 2(J+2)
2. E = J – 13
3. J+G+E = 81
E = 10, J = 23 and G =48
The students can also be
asked to formulate the text given the equations. They can also be asked to
construct a new task like this, perhaps with four persons involved.
7 Kombinatorikk (Norwegian) Combinatorics
(English?)
In how many ways can five students stand in a row in the
playground?
Name the students a b c d e.
One student, a, can be
arranged 1 way.
Two students a and b can be
arranged in 2 ways. 2 = 1X2 ab
and ba.
Three students can be
arranged in 6 ways. 6 = 2x3 = 1x2x3
abc, cab, bca bac, cba, acb (You can see that for each of the two
possibilities ab and ba, the third student generates three new possibilities)
Four students can be
arranged in 24 ways. 24 = 1x2x3x4
Five students can be
arranged in 1x2x3x4x5 = 120 ways.
With math notation: 1x2x3x4x5 = 5! (this was the way we expressed
this when I was a student in the 1960ths.) You can ask the students to find the
correct formula for arranging n students - n!
You could also ask them, without calculation, if it would be possible
to let 10 students arrange in all possible rows within two hours.
Most students will probably
say Yes. But 10! = 3 628 800 possibilities. If you in average need 10
sec to arrange each possibility, you will arrange 360 in 1 hour. This means
that you need 10 080 hours – 420 days and nights – more than a year and,
that means slightly more than 2 hours, even if you need only 5 sec to arrange
every row)
8 Get a feeling about very big numbers
In English and in Norwegian
one million is 1000 thousands which is 1 000 000. In English one
billion is 1000 millions is 1 000 000 000. (In Norwegian we call
this one milliard and in Norwegian 1 billion is 1 000 milliards, but this
you can forget. 1 Norwegian kr. gives 256 T.Sh. 1 USD gives 1 936 T. Sh. Both
May 3th 2015).
Let us say that Jumanne in the future becomes extremely rich and
decides to give Martin 1 billion Tanzanian Shilling. He will do this by giving Martin
100 000 T.Sh. every day. How many days will be needed? Can you guess?
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