This translation is in progress. Please excuse me for its poor quality. I will try to get help from a good english speaker to make it more accurate.
The Chinese Cycles Method is a teaching method relying on elements of traditional Chinese culture, Taoism and Confucianism. Although such premises may at first seem weird to Western readers, its many benefits make it truly a worthwhile practice.
Let us see some of them:
- Thanks to its routine and procedures it provides a reassuring framework to students who need a structured functioning,
- It gives meaning to all the notions taught and introduces them gradually and thoroughly in a natural order of assimilation,
- It provides an almost perfect balance between inquiry-based approach and step-by-step procedures. Between induction and deduction. Between real-world problems and theory,
- It significantly reduces behavioral problems,
- It generates a lot of enthusiasm and allows some students to work although they are usually known for their lack of motivation,
- It develops problem-solving skills as well as algorithmic fluency,
- Its implementation and its operation are relatively simple.
Brief description of the method:
This method relies basically on two cycles: the sequence cycle and the lesson cycle.
The sequence cycle lasts 8 hours.
Its timeline is represented on the diagram below (Read clockwise from the sector at the bottom left)
The first principle of the method, balance, clearly appears on that diagram:
- half of the time is spent on complex tasks, the other half on simple ones,
- half of the sequence goes from practice to theory, the other half from theory to practice,
- half of the lessons are teacher-directed and in the other half the students are autonomous in their research.
It may seem, at first, to be an insignificant detail but, in use, this timeline will reveal all its qualities. All the spirit, the « magic » of the method is concealed there, in this precise order, which is not anecdotal nor random.
I used the terms « complex and simple tasks ». A complex task is a task that entails more than one interaction or stage (complex does not mean complicated). A simple task is just a repetition of a learning that requires low cognitive thinking.
We could also use that diagram:
- Starting with problems: Start by one or more complex problems that can be easily solved by tools unknown to the students. The goal of the whole sequence is to discover, master and use these new tools.
- Feedback: Resume the previous lesson by presenting the different procedures used by the students then introduce new tools invented by the mathematicians ,
- Discovering new tools: Submit simple exercises using the new tools but without guiding the students,
- Lecture and practical exercises: Teach the tools and ask the students to use them in simple exercises,
- Lecture and practical exercises: Same as the previous lesson. Add new complementary tools,
- Training (drilling) : Students become familiar with the new tools by trying to make as many exercises as possible,
- Tutorials: Students start to solve problems again, but this time using the new tools. They are guided step-by-step,
- Ending with problems : It is the culmination of the whole sequence. Students know and master the new tools. They are now able to solve more complicated and complex problems.
A lesson is divided into five steps of equal duration, about ten minutes, in that precise order:
Start with a short test on the notions recently taught. During the test one or two students are writing on the board their own solution of the homework.
Some students come to the board and correct the short test. The teachers give their opinion on all the corrections, interviewing the students, correcting any errors and inaccuracies. They then introduce the next step by telling the expected objectives or by specifying some techniques.
All the work of the lesson is presented on the board. Students work alone, in silence, in the order presented, and at their own pace. During this phase the teacher moves inside the classroom to answer or ask questions, help, solicit, praise, encourage.
Now the students have understood the problems, they have found tracks that will complement each other. This time the teacher speaks to each group as a whole. During this step some students goes to the board and write their own solutions, without comments at this stage.
The groups come to the board and explain their solutions. They are interviewed by others and by the teacher. The teacher concludes the lesson and assigns homeworks (what exercises in the list should be finished for the next lesson)
Most of the method has just been presented. So far there is nothing typically Chinese. We will address the chinese origin of this method in a second part.
The interest of these two cycles is far from being obvious at this stage and it is necessary to present through several exemples how this method can be applied.
Examples of implementation
The examples have really been implemented in class in France. I hope that despite all their lacks they will make you feel the spirit of this method and measure its benefits.
All the following opinions are mine and have no other purpose than to illustrate the operation of this method. They commit only me and not the spirit of the method.
In China most notions are seen much earlier than in France, two to three years before, especially with regard to calculations. Equations, for example, are taught in primary school. The examples below only concern French students of about 13 years old and do not correspond to the way in which these concepts are taught in China.
Example 1: The Pythagorean Theorem / Pythagoras’ theorem
Lesson 1: Starting with a problem
Ask your students to:
Draw a square with an area of 1 cm², then another of 2 cm², and another of 3 cm², and so on.
It may not be useless to draw on the board a square whose side is 1 cm and to point out that its area is of 1 cm ².
Students are usually surprised, at first, by such an obvious question and thus start to draw, except in exceptional cases, a square with an area of 1 cm² then of 4 cm², 9 cm², etc.
They call you to check and you tell them that you are still expecting a square whose area is equal to 2 cm². It should trigger something in their mind and the problem appears harder than expected.
Most of them are now drawing rectangles with an area of 2 cm² and then a square who has a side of 1.5 cm. The area of the latter being 2.25 cm², a new approach should appear.
We are getting closer: 1.4 cm side is too small, 1.5 cm too big. Some students ask if they can write 1 point 4 point 5. « Have you forgotten everything about decimal numbers ? »
Some still hesitate in between 1.41 and 1.42 but the work has started.
Let us mention that Chinese students have a huge advantage over the westerners for the decimal numbers, their numeration being infinitely simpler. They do not say things as weird as the French « quatre vingt-treize » (four times twenty plus thirteen) but « jiu shi san » which literally means « nine ten three » or « nine tens three » and 1.45 is literally « a unit four tenths five hundredths » . Using the chinese system 1.45 is clearly less than 1.5.
Other students may have tried an other approach: they are drawing leaning squares. If no one is doing this, make an infrigement of one of the principles we will see later: the non-teaching intervention during the research phases and ask them to be more imaginative and to get free from the notebook grid.
And we logically get this square of 2 cm² as expected:
Fine ! They are now drawing square of 5 cm² and then 10cm² or 13 cm², etc. You will have to wait a little longer for a 3 cm² square.
At the end of the lesson what has been found or reviewed?
- Some integers (whole numbers) are equal to the product of another integer multiplied by itself: 1, 4, 9, 16, 25, and so on. They are called perfect squares or square numbers.
- For any (positive) number a, we can work out an approximate value with any precision of a number whose square is equal to a. We are not yet saying « square root ».
- The diagonal of a square of 1 cm on the side measures about 1.414 cm. « Was it really possible to guess it at the beginning of the lesson? »
- Simple areas (squares, rectangles, right-triangles) have been reviewed. Of course all this has been taught from the primary school, but it is sometimes no useless to see it once more.
- For some students, it was an opportunity to review some elementary properties of the decimal numbers that should have been well known: In between any two decimal numbers there is always another number. 1.5 is bigger than 1.41, etc.
- It is important to mention that the students have just been struck with an important epistemological obstacle: some situations are not proportional. Once more, this notion has already been seen, for example in calculations and especially area conversions but this issue was generally unnoticed.
- They also felt that there is obviously a difference between 2 cm² and (2 cm)². The meaning of the second being attributed to the first. An obvious thing for the teacher and far from being easily accepted by those who have just become aware of it.
As a conclusion just say that we will develop in the next sessions, a technique to calculate the length of a segment that joins two points in the grid of their notebook.
Lesson 2: Feedback
It is a complex task but this time the lesson is teacher-directed: the previous lesson is entirely resumed, now students are guided step-by-step.
Resuming the entire research session is usually never perceived as redundant. On the one hand, this allows those who had, or thought they had understood, to structure their approach, consolidate their reasoning and fill in any gaps. On the other hand, most students, although having enjoyed searching a 2 cm² square during the previous hour, and even if they finally understood how to draw it, do not necessarily want to push further investigations and are perfectly willing to forget everything. It will therefore be necessary to set everything in stone.
The aim of this session is to quickly present the different methods used by the students, to present those that have been developed over the course of history by mathematicians and whose interest is now becoming clear.
Exercise 1 is a discovery of the square root, whose interest was highlighted in the previous lesson. Those who were lost the day before will probably ask you again the same questions, for example if there are numbers between 1.4 and 1.5, if they can write 1.4.5. The teacher must remain patient. It is normal that some have not understood everything, that does not prove that they did not try to do anything .
Thanks to the spreadsheet and the trial and error possibilities it offers, all will end up at the approximate value of the square root of 2.
Then on Geogebra we have them draw squares with automatic display of the areas and lengths of the sides.
Without realizing it, students have calculated and visualized that they now have the means to calculate the length of any segment joining two points of the grid. The significance of this discovery does not often come to their eyes.
Touch on it during the fifth step of the lesson (Share)
Lesson 3: Discovering new tools
The third lesson is, despite appearances, a work on a simple task: the Pythagorean theorem.
You need to give some clues to the students to achieve this, but not too many. The third session is an unguided simple task.
ABC is a right-angled triangle.
AC=4cm and BC = 3cm. How far is A from B ?
Obviously it’s not enough. The students give all kinds of answers usually between 4 and 7 but provide only fanciful justifications. So we go to the next slide:
And now ?
And we leave them to fend for themselves. Some groups will succeed to demonstrate rigorously that AB = 5.
For these groups, we can go to the next slide:
Now you are able to work out AB in each case:
During the fifth and last step (Share) just to top off:
Let us rearrange the triangles:
The Youtube video « Pythagoras in 60 seconds » completes the lesson.
All the hard work is done !
Lesson 4: Lecture on square numbers and square roots
It is therefore the first lecture of the sequence. This is in fact the most classic situation: first a lecture then exercises.
For reasons of efficiency the lecture must be clear and simple. I usually break it down into small parts followed by immediate application exercises.
Lesson 5: Lecture on the Pythagorean theorem and its converse
This lesson is the same as the previous one: a lecture followed by exercises of direct application.
Add the converse.
Lesson 6: Training
During this lesson, it is important that no new notion is introduced.
What is expected now is that the students train on all the notions that have already been worked on for 5 hours.
Students must do as many exercises as possible to apply the courses of lessons 4 and 5 in order to better master the new tools introduced.
From lesson 4 to lesson 6 exercises should be very simple so that all students can understand and invest.
To avoid any confusion at this stage, it is absolutely essential not to introduce complexity (mixture of tools) and to exclude any form of tricky questions. Triky questions are useful from a pedagogical point of view to shed light on superficially understood concepts. Here they would be useless. The notions are fresh and under construction, do not add any confusion and generate unnecessary stress.
Students must acquire computational and procedural fluency, especially during lesson 6 and this must necessarily go through a repetitive routine. When doing simple tasks naturally, especially calculations, the students can easily go further. Poor numeracy skills related to lack of practice may hamper problem-solving performance.
Chinese students excel in that field and far surpass our Western students.
Three hours are not usually enough to acquire proficiency. In order to develop it, I assign students digital homework on Google Classroom, Socrative, Mathenpoche or Labomep platforms, which allows me to check at home the success rates of each student. Short tests are an opportunity to test the level of acquisition.
Lesson 7: Tutorial
At the end of these three lessons, students begin to master new maths tools (whose usefulness was clearly illustrated in lessons 1 and 2)
They are now able to implement them during the last two lessons dealing with complex tasks. During this lesson students have to solve complex problems with the Pythagorean theorem, these problems are directed step-by-step. For example:
Evariste and George are participating in a regatta
They leave from buoy A and must pass behind buoy B and then return to buoy A.
As they are “head to wind” they must take another bearing before turning towards the arrival buoy .
Evariste leaves buoy A, travels 5.1 miles southeast to point D, then makes a 90 ° turn, and travels 6.8 miles northeast to buoy B and returns to the buoy A.
George departs from buoy A, travels 7.5 miles to the northeast, then rotates 90 degrees to the southeast to buoy B and returns to buoy A.
We want to know which of the two has taken the shortest route.
- Using triangle ADB, work out the length of AB:
Triangle ….…………..… is a …………………….………..
so we can apply the ………………………………….……………….. :
- Using triangle ABC, work out the length of BC:
- How many miles has Evariste sailed ?
- How many miles has George sailed ?
- Who sailed the shortest distance ?
As you can see the exercises are directed step-by-step.
A designer wants to draw the plan of a children’s slide.
Due to space and safety reasons, they must respect these conditions:
- The ground length EG is 2.5 m
- The slide length HG is 2 m
- The heigth HS is 1.2 m
They want to work out the ladder length.
- Why is it not possible to calculate the length of the scale directly using the Pythagorean theorem in triangle GHE?
- Calculate the length SG in the right triangle HGS:
- Work out the length EG:
- Prove that EH=85cm:
- Prove that the angle <EHG is a right angle:
Now the students are ready for a guided proof of the Pythagorean Theorem:
Proof of the Pythagorean Theorem
All the triangles have the same measures
- Prove that FHCE, PNQJ, PMLI et LOQK are all squares
- Prove that the areas of squares FHCE and PNQJ are the same
- Prove that the area of the quadrilateral ABGD is equal to the sum af the areas of the squares PMIL and LOQK
- Prove that the area of the quadrilateral ABGD is equal to the sum BC²+AC² (Don’t use the Pythagorean Theorem, of course !)
- Prove that the sum of the angles in any convex quadrilateral is 360° (hint: Sum of angles of a triangle)
- Prove that ABGD is a square
- Conclusion: The area of the square ABGD is AB²=………²+………²
Lesson 8: Ending with problems
This is the conclusion of the chapter: complex, unguided problems.
Exercise 1: A ladder leans against a wall with one
end on the ground 4 metres from the wall. The other end of the ladder is on the top of the wall which is 7.5m high . How high is the top of the ladder if you step it back 1.1m to the right ?
The diameter of a 2 € coin is 25.75 mm and its thickness is 2.20 mm. Can a 2 € coin pass through a square hole with a side of 2cm ?
And then, we can now go back to the initial problem:
Draw a square with an area of 2 cm², 3 cm², 4 cm², etc.
For this lesson, taking into account that my students are in the national average:
- Every student is expected to solve the first exercise,
- The correction of the second is initiated by a group that comes to the board and give some tracks,
- The homework for the next lesson are: working out the the length of the diagonal of the square and make an accurate drawing of the introduction of a 2€ coin into the square
- The second exercise is a bit long if you require a totally rigorous demonstration that takes into account the thickness of the coin and I ask that it be done as homework assignement for a fortnight.
I almost always do this during lesson 8: a common part for everyone to be done during the lesson, a part to be addressed and more or less corrected in class and to be finished at home for the next lesson and a last part assigned as homework assignement to be collected within a fortnight.
I’m still waiting for a student to give me this solution which looks like the spiral of Theodorus
This first exemple raises questions:
In so doing, is it possible to complete syllabus in good time ?
The answer is yes, as long as you pay attention to the duration and order of every lesson and step. It is necessary, even if sometimes it seems impossible, to respect the spirit of this method: balance
It is sometimes tempting to resume a lesson because it went wrong. This is useless: each lesson are actually the reproduction in eight times of the same work, from eight different perspectives. If the first lesson of research did not go well, the second lesson « feedback » will allow to fix it. If the training part went wrong, the next lesson, although dedicated to problems, also contains training because the tools will be used again. Etc.
Similarly for the five steps of the lesson cycle. None of the steps should overflow, especially the second, because it would break the harmony of the whole. To lengthen one step necessarily forces to shorten another.
On the other hand, after several sequences, you will find that students have acquired many skills that will allow them to be progressively faster.
It takes a lot of work
At first perhaps. I can provide you with a link to a complete year of courses that covers almost the entire french syllabus in 12 chapters of 9h (8h of the sequence + 1h of assessment). Although designed for France, these courses can be used in many other countries. You can pick any element you like. This will help you get started.
And let’s not forget that the textbooks are well done and include just about everything you need.
Just do the work backwards: when you start a chapter you will find at the end of the chapter the first problems.
As for the exercises, lectures, tutorials, everything can be found in the textbooks.
It will therefore only be necessary to write the second and the seventh document for each chapter.
How to make group work succeed ?
Many teachers fear group work. Poorly managed, poorly designed, group work turn into a mess or an unmanageable heckling. In the best cases, it is common that many students do not work and rely on the others.
When dealing with unguided complex tasks some pupils are freezing. They ask how to do it, complain of not having had a preliminary lecture and do not want to start thinking.
In short, a huge waste of time that is pushed back at the end of the chapter, then indefinitely postponed.
The five steps of the lesson cycle, well managed, make it possible to overcome these difficulties and not to deprive oneself of this magnificent educational tool which is group work.
Step 1: Quiz
The short test are slightly late from the worked notions. Three or four hours late. They make it possible to return to what has been taught, reinforce it, assess assimilation and introduce what follows during the present lesson.
The first minutes of a lesson are crucial: it is essential for the smooth running of the whole lesson that this step takes place in the utmost silence. It is an assessment, no gossip can be tolerated.
This initial silence will be maintained, thereafter, interspersed with more « relaxed » moments.
Here are some examples of short tests:
Step 2: Discussion
This step is a little more relaxed than the previous one, but talking is not allowed and some ground rules must be established for the discussion.
Sending pupils to the board to correct and explain their approach has some benefits:
- The other students will certainly be more attentive and will take a more critical look for the work of their peers than for yours,
- While listening to the student on the board, you can easily watch the class, stand anywhere in the classroom, away from the board, close to those who would like to try chatting.
During this step, the teacher speaks as an expert.
After the correction of the exercises and of the quiz, all the work of the lesson is exposed simultaneously and the teacher introduces the next step (think = individual research).
In lessons 1,3,6 and 8: he says as little as possible. These are unguided sessions. Students must, in principle, fend for themselves.
In lessons 4 and 5 (lecture and exercises) the teacher update everything that has been seen before and then project and read the course on the board.
I do not write it on the board and students do not write it on their notebooks either. They have to listen and ask questions. They will copy this course at home. Why do I ask the student to copy the course at home ? Studies have shown that memorization is more effective, for many student profiles, through writing than by simply reading the course.
Before the next step I usually let the students take a little breath. they can chat a little while I hand out the documents.
Attention: it is extremely important that this phase does not exceed the expected duration (about 10 minutes) so as not to encroach on the following phases which are just as important. The whole lesson would lose all meaning. Balance is the spirit of this method.
step 3: Think
The third step starts with a clear signal meaning that total silence is expected (personally, I count 1,2,3 not without some shame), that this is very important and that any chatter will be sanctioned. You must be firm on this point. What follows depends on it.
Students must work alone and in silence. They will quickly want to interact with their neighbor especially if the task is complex. They should by no means talk to each other at this stage and you must be uncompromising on this point.
If this is the first time you do this, you may hear this kind of remark « Sir, you did not explain us how to do this. » After several sequences it’s a safe bet that you will not hear it anymore.
Students need to understand that you will provide only minimal assistance.
If the work is an unguided complex task it might happen that some student will not write anything. It does not matter, as long as you see that they are trying to work, and you have to tell them.
It would be a mistake to conclude that a student who did not write anything did nothing. Most of the time this student thought, tried to understand the problem, tried some tracks in their head but they did not dare to put them on paper. We will see later why and how to fix it.
During this phase, the teachers will have to help their struggling students. By constantly repeating that there is no shame in being wrong, that even the best mathematicians make mistakes. That without error there is no real progress, no research, no evolution.
After several sequences, the students all begin to gain real confidence and approach more serenely the expected skills: Searching, modeling, representing, reasonning, calculate, communicate
To satisfy the appetite of the most successful and do not disgust the others this is how I do it:
The work on the board is divided into 3 parts:
- 1: an easy part accessible for all, even the most refractory, those for whom we feel helpless as they seem lost whatever the level of simplification that we go down,
- 2: a part normally expected for most students,
- 3: a difficult part, for those who seem to be able to overcome any difficulty. There are many of them in every school.
If this is an unguided problem lesson, the problem statement should be simple to understand so that everyone dares to venture into it. The three parts can be (depending on the difficulty):
- 1: understand the problem and start a track even a false one,
- 2: find leads that will lead to the solution,
- 3: solve fully the problem (even if the writing is incomplete during lessons 1, with a correct writing during lessons 8)
I do not expect the same from every student. It sometimes offends some colleagues and I can only admire them to succeed in ensuring that all students reach the same level eventually.
Although it is a group work, I assess the students by questioning them individually and I tick the number of parts (from 0 to 3) for each of them with a margin of tolerance for each part.
It is possible to assess more than half of the class in one lesson (without necessarily telling students who was).
Step 4: pair
Now it is time to set the class on fire: the students may communicate with each other. If the previous step went well there is no reason that it does not continue in the same dynamic.
The students had time to understand the statements of the problems. They have assimilated the instructions and are now looking forward to finding the solutions. The pooling of different lines of research will produce its effect and many groups will find the expected solutions.
Step 5: Share
The teacher resumes control. Again this step must be initiated by a clear signal and no unauthorized intervention will be accepted.
Groups or students alone will justify their methods written on the board and then be interviewed by others.
The teacher runs the show and speaks from time to time as an expert.
Remind that students are much more attentive and inquisitive about the work of others than facing yours.
After these first clarifications, let’s look at a second example.
Example 2: Equations
Cycle of the sequence
Lesson 1: Starting with problems
Three problems are proposed:
Both bouquets cost exactly the same price. How much do they cost ?
On the left plate of the balance there are 12 balls and 2 weights of 50 g.
On the right one there are 4 balls and 5 weights of 50 g
The balance is well-balanced and all the balls have the same weight.
What is the weight of a single ball?
How much is a bottle ? (All bottles cost the same price)
Unlike most (all?) Textbooks, I begin with a sequence on equations before teaching algebra. As far as I know, algebra comes from the equations and not the other way around.
The students thus approach this session without knowing neither the equations, obviously, nor algebra. They have previously been confronted with letters representing numbers, through the formulas of area and perimeter calculations. They probably expanded some brakets and factorized some algebraic expressions but, let’s face it, for most of them these are simply procedures whose interest remains unclear, if they still hope to make sense of what they are taught.
And yet they will be able to solve the first problem, then the three, depending on the groups.
Once more these problems are easy to access. Here the utterances can be understood by primary students, but are ultimately much more difficult to solve than they appear at first sight.
Usually, after the first disappointment because the price of one of the categories of flowers is missing and which logically follows the expected and too hasty remark that the problem is too easy, the students starts from three different angles:
- Some will find the solution by trial and error. In general, they attributed to flowers whose price is unknown, a value, calculated as a result of false reasoning. They realized that the two bouquets did not have the same price and gradually corrected the shot in order to arrive by trial and error at the right solution.
You have to praise them for this method. First it shows, and it is very important, that all the data of the problem were taken into account, especially this one which escaped many: « The two bouquets cost exactly the same price »
Moreover, besides being a traditional process among researchers, sometimes the only way to solve a problem, taking the initiative to choose a random number is excessively rare among students. They have an image of mathematics totally distorted and imagine that this science gives everything, right away, as long as we know a formula or a technique. Many even think that all formulas and techniques have been discovered. To choose a random number would be a shameful method that would only reveal a lack of knowledge.
- Others will calculate the price of each bouquet without taking into account the flowers whose price is unknown, find that there is 4.20 € gap between the two bouquets. They will not necessarily see immediately that it is enough to divide this price by three. They divides usually by 4 then fall back on the first method,
- Others, finally, will have the idea to remove the same number of elements that are in both bouquets (the pot, three yellow flowers and a flower at the price unknown in each bouquet).
Lesson 2: Feedback
Let’s resume to the first problem with the method of removing the pot and the same number of identical flowers in each bouquet.
Exercise 1: Both bouquets cost exactly the same price. How much do they cost ?
a) Strip the pots and flowers found in both bouquets (scratch the same number of flowers in each bouquet)
b) What remains in bouquet 1?
c) What’s left in bouquet 2?
d) How much does bouquet 2 cost when you remove the same number of identical flowers and the pots from both bouquets?
e) Work out the price of flower C
f) Work out the price of bouquets
For now we have not talked about equations. We will introduce them gently pretexting to highlight one of the main features of mathematics: simplify all situations. Finally, is not this the essence of mathematics, contrary to the generally accepted idea?
2. On the left plate there are 12 balls and 2 weights of 50 g
On the right one there are 4 balls and 5 weights of 50 g
The scale is well balanced and all the balls have the same weight.
What is the weight of a single ball?
a) what happens if 4 balls are removed from the right plate ?
b) What can you remove next to keep the balance ?
c) If you remove the two weights of the left plate, what should you remove on the right board to keep the scale well balanced?
d) What’s left in the left end ?
e) What’s left in the right end ?
f) Work out the weight of a single ball:
g) Let us start to write:
- 12 balls and 100 grams weigh as much as … balls and ….. grams
- We remove 4 balls from each plate
- ….. balls and 100 grams weigh as much as … balls and ….. grams
- We remove100 grams from each plate
- ….. balls weigh …. grams
- Let’s divide the weight by the number of balls
- 1 ball weighs ……… grams
Isn’t it too long ? Mathematicians are effective people !
h) Let’s write:
« b » for « ball », « g » for « gram », « = » for « weigh as much as », « – » for « remove »
i) That’s what we get:
See ? Under the pretext of simplifying everything, we have smoothly introduced algebra which becomes, and this was the aim, a tool of simplification and not a new abstract notion without explanation.
For the last problem, we show the evidence of algebra:
Lesson 3: discovering a new tool
Let’s project on the board:
Students will solve the equations alone and, normally, without much difficulty:
The first equations are just a repetition of what was done the day before and normally easily solved.
Lesson 4: Lecture on equations and exercises
Lesson 5: Lecture on bracket expanding and collecting like terms followed by exercises
We can now introduce a new notion: « collecting like terms » giving it a concrete meaning.
« If you have 3 banknotes and 7 € then you add 5 banknotes and 5 € so you get a total of 8 banknotes and 12 €.
Let us write b instead of banknote: 3b + 7 + 5b + 5 = 8b + 12
That’s basically for collecting like terms.
So far the point is giving a meaning to collecting like terms, the accuracy will be introduced in a later chapter on algebra.
Show some simple examples of braket expanding:
2x (a + 5) = a + 5 + a + 5 = a + a + 5 + 5 = 2a + 10
3x (b-5) = b – 5 + b – 5 + b – 5 = b + b + b – 5 – 5 – 5 = 3b -15
By doing a repetitive series of exercises like the two examples above, students soon prefer to use the rule: a (b + c) = ab + ac
6th step: drilling
Expand brackets, factorise and collect like terms. As many as possible
Reminder: the first exercises are simple and the last complicated. Everyone should be involved.
7th step: Tutorial
Drilling alone is a task as futile as boring if the acquired automatisms are not re-invested quickly for problem-solving.
Kuo and Sophie are running on a stadium.
They must run the same distance.
Kuo has run 7 laps, he has 5km remaining.
Sophie has run 12 laps, she has 3km left to go.
We want to know how far they must run.
a) L=Length of the stadium tour. Kuo will run 7L+5km and Sophie will run ………………
b) Write an equation that means that they must both run the same distance
d) Solve the equation: 7L + 5 = …… L + ….
e) Check the solution:
Kuo: 7L + 5 = 7x …… …. + ………. = ……….
Sophie: … ..l + ….. = = ………………..
Anne Bonny, Mary Read, Jack Rackham and Peter Bosket are four famous pirates.
They want to share a treasure.
• Anne says, « We girls, we want half the treasure for us »
• Mary: « I want this necklace plus twice the share of Peter, he did not do much that guy »
• Peter: « I know this necklace well, it’s worth 200 pounds! «
• Jack: « I want this golden ring plus three times the share of Peter. He really did not do much »
• Peter: « This golden ring is worth 600 pounds! »
• Anne: « Peter really did not do anything, I want ten times his share! »
We want to know how many pounds is worth Peter’s share.
P = Peter’s share.
Mary’s share is: 2P + 200. (The necklace and the double from Peter)
What will be Jack’s share? …………. and Anne? ……………
What will be the total share of girls? ………….
What will be the total share of boys? ………….
Write an equation that means that boys and girls have received the same value and solve the equation:….
George and Sophie are on a ladder
At first George is 10 times higher than Sophie
They both climb up 2.4m
George is now only 2 times higher than Sophie
We want to know how high Sophie was at first
h = Sophie’s height at the beginning
How high is George at first? ……….
They both climb up 2.4m
How high is Sophie now? ……… and George? ……..
Write an equation that means that George is 2 times higher than Sophie then conclude how high was Sophie at the start (remember to check the answer)
Lesson 8: Ending with problems
Finish on a high note:
Evariste makes the following journey: CFBL-Watford-Hounslow-CFBL-Enfield-Bromley
Sophie makes the following journey: Watford-Enfield-Bromley-Croydon-Hounslow-Watford
Here are some distances:
Croydon-Bromley: 9km CFBL-Bromley: 19km
Bromley-Enfield: 28km CFBL-Enfield: 12km
Given that Évariste and Sophie traveled the same distance, what is the distance between the CFBL and Watford?
Henri is hiking for four days.
The first day he travels 37km.
At the end of the second day he has walked half of the hike.
The third day he travels twice the distance traveled the second day
The fourth day he travels triple the distance traveled the second day
How far did he travel on the second day?
Evariste is 3 years old and his father is 35 years old. How old will be Evariste when his father is three times as old as him ? and twice ?
For this lesson every student is expected to solve the first exercise. The correction of the second is initiated by a group that comes to the board and must be finished at home for the next lesson.
The third exercise is discussed together: it will be a homework assignment , to be done within a fortnight.
How to design a sequence with the method of Chinese cycles?
Let’s look at it through two examples: scientific notation and functions.
1. Choose the different knowledge of the official syllabus you want to teach, and use your personal knowledge and research to identify what it is useful for and in what areas.
The interest of the scientific notation is, for example, to simplify the writing of the numbers used in chemistry or astronomy. The functions are used to establish relations between several variables in all the fields of science and everyday life, for example for optimization.
2. Create one or more problems based on the expected knowledge.
The first problems must:
- Be easy to understand but not to solve. A primary school student should be able to understand the statement,
- Whet the class’s appetite for the subject at hand,
- Require for their resolution, the maximum, if not all, of the tools and knowledge of the program provided in the sequence, while being solvable by less powerful methods.
Here for example, for scientific writing:
Problems that could be done in primary school
A light year is a measure of distance equal to how far light travels in one Earth year.
Light in a vacuum travels at a velocity of about 300,000 kilometers per second.
a. How far is a light year ?
b. The milky way is the galaxy that contains our Solar system. Its diameter is about 150000 light-years. How long is it in kilometers ?
c. The Sun is at an average distance of about 150 million kilometers away from Earth. How long does it take sunlight to reach the Earth ?
d. How long does it take to cross this sheet at the speed of light ?
These exercises can be done in a primary school. Very quickly the calculations become huge and I take a lot of pleasure to pick up a notebook of which a page is entirely filled with hypertrophied multiplications and to present it to the class. Without humbling the student obviously but thanking him for having worked and giving me such a great opportunity to illustrate the interest of scientific notation.
Here, several students will notice that some zeros are superfluous and will invent personal techniques to simplify the multiplications, for example by writing on the right side of the operation the number of zeros that should have been added. It is absolutely necessary to boast this mathematician’s spirit.
Once again: what is the use of mathematics if not to simplify everything? Why do so many people think the opposite?
This problem, once again easy to understand, opens the door to many horizons: volumes, nets, functions, algebra, graphic representation, spreadsheets, etc.
The students are in a computer room. They have A4 sheets.
The lesson usually takes place spontaneously in this order (exceptionally I do not follow the 5 steps because the students would not have time to advance enough):
- Students draw a net of the cuboid and cut it out,
- They build a box and call me because they do not remember how to calculate its volume,
- They calculate the volume. I ask them to make a bigger box,
- Some increase the height, others the surface of the base and build new boxes. find that the volume is different. I ask them to do even better and I write the name of the group that built the biggest box to goad them a little,
- More advanced groups give up trying to build a box for each trial and begin to sketch nets and make calculations. Schemes appear with calculation of heights, length or width,
- I let these groups do one or two volume calculations and then take them to a computer and I show them all the interest of the spreadsheet to perform the same calculations much more quickly,
That’s what we get at the end of the session for some groups. Well done, isn’t it?
4. Create a feedback document
This document must:
- Resume the methods proposed by the students (it can easily be prepared prior to the first lesson, many methods are recurrent)
- Show new methods in application without entering the theory
- Be very directive: the students were completely free during the first lesson, this time they are guided step-by-step so that all those who were lost the day before can understand everything that happened there.
Let’s see an example of document about the scientific writing:
A light year is a measure of distance equal to how far light travels in one Earth year.
Light in a vacuum travels at a velocity of about 300,000 kilometers per second.
We want to know far is a light year.
a) Work out how far light travels in one minute (60 seconds) then one hour (60 minutes)
b) These calculations are too long. Let’s simplify them by using powers of ten:
Complete the calculations below:
c) distance traveled in 1 second: 300000 km = 3×100000 = 3×10 … km
d) number of seconds in 1 minute: 60 = 6×10 …
d) distance traveled in 1 minute (60 seconds): 300000×60 = 3×10 … x 6×10 … = 3×6 x10 …x10 … = 18×10 …
e) distance traveled in 1 hour (60 minutes):
18 x 106 x 6 x 101 = 18 x 6 x 106 x 101 = 108 x 10 …
f) distance traveled in 1 day (24 hours):
108×10 … x24 = 108x24x10 … = ………. x10 …
g) distance traveled in 1 year (365 days):
…………. x10 … x365 = 946080×10 … ~ 950000 x 10 … ~ 95×10 …
b. The diameter of the Milky Way (our galaxy) is about 150,000 al. How many kilometers does it represent?
We calculated that a light-year is approximately 95×1011 km (9,500,000,000,000 km)
Rather than ask and perform the calculation 80000 x 9500000000000 use the powers of 10: 80000 = 8 x 10 ……
80000×9500000000000 = 8 x 10 … x 95×10 … = …. x … x10 … x10 … = ………….. x10 …
Conclusion: The diameter of the galaxy is …………………. ………………… …… km
d. The Sun is 150 000 000 km from the Earth. How long does it take sunlight to reach the Earth?
e. To find out, you need to make the following division: 150,000,000 : 300,000
f. Once again, it’s too long. Let’s calculate use the powers of 10:
150,000,000 = 15×10 …
300,000 = 3 x 10 …
g. Simplify then calculate: 15×10 … : (3×10 …) = 15:3×10 … 10 … = … 10 …
h. Conclusion: It takes…………. seconds to reach the Earth
i. How long does it take for the light to cross your notebook (about 30cm) in the direction of the length?
And for the functions:
Reminder: Volume of a cuboid = Length x width x height
We first use the net below:
a) What is the height h of the box (in cm)?
b) Prove that the length of the box L is 23.7cm
c) Prove that the width of the box w is 15 cm
d) Calculate the volume of the box in cm3 using the formula: V = w x L x h = …..
Let’s try to get a bigger volume by increasing the height:
e) What is the height of the box (in cm)?
f) Calculate the length L of the box (in cm)
g) Calculate the width of the box (in cm)
h) Calculate the volume of the box (in cm3)
i) Fill in the table (note: this is not a proportionality table)
Let’s use a spreadsheet to speed up
j) Write 2 in cell A2 and « = 29.7-2 * A2 » in cell B2
k) What number do you get in cell B2 when you press Enter?
l) Why (write the calculation done by the computer)?
m) Write a formula in cell C2 that calculates the width of the box
n) Write a formula in cell D2 that calculates the volume of the box
o) Select cell group A2: D2 with the left mouse button
p) You can quickly copy formulas into adjacent cells by using the fill handle.
q) Drag the fill handle down
r) For what height (in cm) do you get the maximum volume ?
s) Let’s try to be more accurate:
Without modifying the spreadsheet formulas write 0 in cell A2 and 0.1 in cell A3 then
select the range A2: D3 and Drag the fill handle downs in order to obtain a table as below
Select columns 2 and 3 (left button) and hide them (right button)
Select the entire calculation area and insert an XY (scatter) graph:
And now, be much more accurate.
Then ask yourself the same questions with a box with a lid.
5. Create a tool discovery document
The interest of the new tools has been highlighted but you should not rush things by doing a lecture immediately after.
The peculiarities of these tools, the difficulties would not appear to them and they would not measure to their true measure all the recommendations that you would make during the lecture.
Let the students discover by their own some properties and make some conjectures.
For the scientific notation:
As designed, this document allows students to easily discover a number of rules by themselves and to ask themselves a lot of questions.
- for a positive power of 10 just write 1 followed by as many zeros as the power,
- for multiplying decimal numbers by a power of ten: Move the decimal point to the right as many places as the power. Example: for multiplyng by 104 move the decimal point four step to the right. For negative powers, what do we do?
- The rules of the product and of the quotient of the powers of 10 quickly become obvious and the pupils discover them quickly
- The issue of 100 and negative powers is the subject of a lively discussion during the fifth step. Some students think that 100 = 0, 10-1 = -10, etc.
The second and third columns will be a good argument for the assumptions 100 = 1, 10 -1 = 0.1, etc.
Let us see a last example:
Ask students to find the maximum of techniques to measure the height of their school. Leave them a fortnight to think.
The first lesson (starting the problems) will be a presentation lesson of the different techniques proposed by the students.
They usually find about twenty ways. from incomprehensible trigonometry found online, to the request of the plan to the head teacher.
Here is an example of document for the second lesson:
a) Using shadows:
Erect a broomstick onto the ground vertically. Now measure the height of the broomstick and the length of the shadow it casts. Immediately measure the length of the shadow cast by the building. By similar triangles, we can work out the height of the building.
b) Using a picture
On that picture, the heigth of the window is 6mm, and the height of the building is 45mm.
Given that the real size of the window is 1.2 m was is the real heigth of the building ?
c) Using a ruler
Ask someone to stand straight against the wall. Hold a ruler at arm’s length. Close one eye and measure the size of your friend: 3.4cm and of the size of the buiding: 32 cm
Given that the person is 1.7m tall, how tall is the building ?
d) Using a pencil
Hold a pencil at arm’s length. Close one eye and adjust the pencil up or down so that you can sight the very top of the tree at the top of the pencil. Move your thumb up or down the pencil so that the tip of your thumbnail is aligned with the tree’s base. Rotate your arm so that the pencil is horizontal (parallel to the ground) Ask someone move so that you can sight him or her “through” the point of your pencil.
Given that the person is 3.5m far from the tree. How tall is the tree ?
d) Using a piece of paper
Fold a square piece of paper in half so that it forms an isosceles right triangle.
Hold the triangle in front of one eye by holding a corner opposite from the 90º right angle, and point the rest of the triangle toward you.
Move back until you can sight the top of the building at the top tip of the triangle. Mark this spot and measure the distance from it to the base of the tree.
e) Using similar triangles
Points B,D,E are collinear. Points A,D,E are collinear
How tall is the building ?
Then work on proportionality and finish the sequence, once again on a high note by flooding the school of pupils equipped with geometry sets.
The students are now ready to understand the intercept theorem.
At the end of the sequence they should be able to solve alone and with no help that kind of problem:
How tall is this tree ?
What are the advantages of following this order?
The beginning teachers can find online many teaching tools. But as they lack experience they often feel helpless in front of their class. What tools? What methods to use? When and how do students work in groups ? When to do a lecture ? How much time does it take to correct exercises ? etc.
There is nothing new in the method of the Chinese cycles. It only uses existing practices and pedagogical concepts in their most rudimentary form (think-pair-share, flipped classroom, quizzes, lectures, complex and simple tasks, drilling, etc.) But it presents a precise frame to use them. It is in its precise timeline and in the balancing of these different practices that this method is so useful.
At the risk of being repetitive, let’s resume all the lessons and their respective benefits.
- Starting with Problems: By choosing interesting problems of easy access, you allow all students to get to work. All that follows will make sense: solve a motivating problem of the same type as these ones,
- Feedback : Resuming the problems gives a clearer meaning to what has been seen. The new mathematical concepts introduced by the teacher will not appear as an other overload of knowledge but as what they really are: symplifying and useful,
- Discovery of new tools: Giving a lecture is too early. Students have to come up against these newcomers and their peculiarities, and to discover by themselves some properties,
- Lecture and exercises: Now is the good time. The students can easily make a link with everything that has been seen before,
- Lecture and exercises: You can now introduce new tools, new concepts, new knowledge and writing requirements that will be used later. These new concepts are complementary to the previous ones and, consequently, easily accepted by the students,
- Training: Before going back to problem-solving (which is the only interest of mathematics) students must acquire fluency. The drilling must be simple and not require high cognitive thinking.
- Tutorials: Before releasing our students into the wild let’s show them how all this new knowledge is useful for problem-solving. This is an opportunity to define the expectations for the argumentation, the requirements of rigor and writing.
- Ending with problems: Students can measure by themselves all the progress made since the first lesson.
The interest of the method to put all the students to work
If this is the first time you propose complex unguided problems you will find that some students do absolutely nothing. It will be necessary to rid them gradually of bad reflexes, of rigid school habits, of confusions on the expectations of the teacher. The method of the Chinese cycles is indicated to achieve this.
Freezing can have multiple causes:
- The fear of making mistakes against which it will be necessary to fight, by emphasizing the essential importance of mistakes, by providing positive reinforcement to the struggling students as soon as they participate in a progress towards the solution,
- The fear of « serendipity ». Serendipity has played such a crucial role in science. To search you have to go ahead, try, guess, make mistake. Imagination is fertile ground for discoveries. I always congratulate a student who dared to try a random number for example,
- The abusive use of lectures. This mode of operation, the interest of which is not called into question here, can not be exclusive. Students could believe that mathematics is a compact block of knowledge to be absorbed and then restitute by applying an ever more impressive number of techniques whose meaning, apart from passing exams, has very little, if any, interest,
- The interweaving of unclear expectations of the teacher and totally counterproductive:
During the research phases (lessons 1 and 3) students are expected to search: spelling, errors in calculation, presentation, writing, vocabulary used are secondary and should not be obstacles. On second thought, we will put things in good order then.
During the guided phases (lessons 2,4,5 and 7) the students apply clearly explained techniques. If there are writing requirements they are presented in detail. Students do not have too much thinking to do. They are expected to apply techniques, learn a definite way to proceed, to argue, to present a solution.
During the drilling phase (lesson 6) the students apply, repeat, train, acquire automatisms: it would be counterproductive to introduce complexity.
During the last lesson (lesson 8) and only that one all the math skills are expected simultaneously: searching, modeling, representing, reasonning, calculate, communicate
In a lesson, during the third and fourth steps (individual research and group work) I check for each student each completed part (from 0 to III) with a little emphasis. For struggling students I am never stingy with praise. I try to be guided by these two words: requirement and benevolence.
My own assessment of the chinese cycle method
I started by applying this method from time to time, for one or two chapters in the year. Then gradually I used it more and more until making an exclusive use.
At first everything seemed to go too slowly, which is why I only used it sporadically.
The students were working slowly but they all worked, without exception, and I saw that progressively all were gaining in proficiency and the progress of the courses was faster and faster.
They developed fluency in number sense, reasoning, and problem-solving skills and I could measure the progress of each individual quite accurately by validating (from 0 to 3) the number of completed parts.
Thanks to its balance, the Chinese cycle method appealed to almost all student profiles.
During sessions 4,5 and 6 (courses / classes / training) struggling students enjoyed understanding how to do a task, although they didn’t understand its use, that they could thouroughly master and reproduce with ease.
Some students have proved positive in the group work. They were usually known to be rowdy, bored most of the time, and manifested it in disturbing behavior.
The champions in mathematics, natural individualistic, were so enthusiastic that I was quickly forced to provide a multitude of increasingly complicated exercises to keep pace and satisfy their appetite.
And, to think about it, is the acquisition of skills not the most difficult task and the longest to get, but also the most important ? Is not this the essence of mathematical culture, that culture defined by Ellen Key: Education is that which remains, if one has forgotten everything he learned in school.
« The only way to learn mathematics is to do mathematics. »
Parents’ reactions have been extremely positive. After the first worries logically aroused by a teacher who seems very different from what they are used to. The positive feedback of their children quickly came to dispel the fears of the first moments.
I have never presented this method as a method of Chinese origin, let alone a Taoist and Confucianist inspiration !
It is also useless to talk about the problem of origins. The method appeals to the parents. We can leave it without taking the risk of passing for an illuminated.
We will discuss it here because this presentation would be incomplete if we did not talk about the principles that led to the development of this method.
Chinese origin of the method
As strange as it might sound, the method of the Chinese cycles was not invented as a result of advanced scientific work but simply by the observation of ancient Chinese Taoism symbols: the bagua and the wuxing.
Let us first reassure the most secular, the most cartesian of readers, this method is actually not Taoist.
It is not an esoteric method that would seek to hide its total lack of intellectual rigor behind garments coming from so far back in time and space and try to convince our naive fellow citizens with any chakras or meridians.
It is simply a teaching technique inspired by Taoist and Confucian symbols and thoughts, and if the following lines make this reader laugh, let him not go so far as to throw the baby with the bath water. The method of the Chinese cycles requires no belief, not even in what allowed its conception. Its value has only to be judged by the its tangible results in the class.
So let us dream a little and roam freely over the fascinating wonders of Far Eastern Thought.
Taoism and Mathematics have many things in common. In the first place, perhaps, the will to discover similarities between phenomena of very different natures, to arrange, to classify, to discover laws that transcend different domains of knowledge. The same equations are used in a wide variety of technical fields. Taoism also establishes functional correspondences in seemingly unrelated fields such as the seasons, the human body and the functioning of society.
Taoism also uses many mathematical objects, the gnomon, the magic square, the Chinese version of the Pythagorean theorem. Leibniz, after having invented the dyadic (binary) arithmetic that he describes in his manuscript De Progressione Dyadica, was strongly impressed by the hexagrams of Yijing which he speaks with enthusiasm in his treatise « Discourse on the Natural Theology of the Chinese ». There are many more examples. By its own logic, still permeating the Chinese soul, Taoist thought, like mathematics, has been able to solve many of the problems encountered by humans during their History.
But the comparison stops quickly.
With these clarifications, let us see what are the concepts used as an inspiration for the development of this method:
WUWEI rule by non-activity
It’s up to the students and not the teacher to act !
BAGUA The Eight trigrams
The Bagua or Pa Kua are eight symbols used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Each consists of three lines, each line either « broken » or « unbroken », respectively representing Yin or Yang. Due to their tripartite structure, they are often referred to as Eight Trigrams.
The symbols assigned to each trigram can be read here as guides on how to teach during a given lesson.
In this method, the trigrams are followed in this order (starting from Northeast, bottom left, and turning clockwise):
If we look at the lower line of each trigram, the first four lessons are Yang: « They go from Earth to Heaven », in the method: from practice to theory.
The last four are Yin and go in reverse order.
This translates as follows: one starts from problems to introduce theories, theories then re-used to solve new and more complex problems.
The middle line indicates the complexity of the work: a line Yin (discontinuous) corresponds to a complex task and a line Yang (continuous) to a simple task.
The top line indicates whether the session is guided (Yang line) or not (Yin line).
Reading the trigrams thus we build the sequences according to this order:
- Unguided complex tasks
- Guided complex tasks
- Unguided simple tasks
- Guided simple tasks
- Guided simple tasks
- Unguided simple tasks
- Guided complex tasks
- Unguided complex tasks
The chart of symbols that Chinese philosophy has attached to each of the trigrams was used as a guide for the teaching practice in each of the eight sessions. The symbolism of each trigram can, with a little imagination, coincide beautifully with what happens in class.
Correspondence between symbolism and classroom practice
The Five Phases or Five Elements
In the same way that bagua is used as inspiration for the sequence cycle, Wuxing is the origin of the lesson cycle. Knowing and visualizing the nature of each phase and all the phenomena that are associated with each can help, once again with a little poetry and imagination, to cast a new light on what is happening in the classroom.
The Wuxing is a fivefold conceptual scheme that many traditional Chinese fields used to explain a wide array of phenomena, from cosmic cycles to the interaction between internal organs, and from the succession of political regimes to the properties of medicinal drugs. The « Five Phases » are Wood, Fire, Earth, Metal and Water.
- Wood feeds Fire
- Fire creates Earth
- Earth bears Metal
- Metal collects Water
- Water nourishes Wood
Correlation table (approximate translation):
The Five Phases apply to the division of the lesson into 5 periods:
1st phase: Metal
In Taoïst thought, Metal attributes are considered to be firmness, rigidity, persistence, strength, determination, firmness, obedience, organization, stability.
Is not this the state of mind that corresponds exactly to the one expected during an assessment ?
It is good to start a course from this angle a bit rigid: the Metal, well used will allow the Water, much more fluid to flow normally.
Ideally the correction of the quiz should be an introduction to the next phase just as Metal generates Water.
2nd phase: Water
Water is representative of intelligence and wisdom, flexibility, softness, and pliancy, associated with the character of the instructor.
It is a moment of discussion during which students may speak a little and where the teacher actually act as an instructor, corrector of errors and this is the moment when he explains the unfolding of the following essential phase: Wood.
The Wood only grows well when the Water has been properly distributed.
3rd phase: Wood
Wood heralds the beginning of life, springtime and buds. It represents birth, spring, growth. This is the real beginning of the lesson. The first two phases were only the conclusion of the previous one.
During the wood phase students work alone, in silence. The wood person is expansive, outgoing and socially conscious. As the wood element is one that seeks ways to grow and expand, students want to communicate with each other. You cannot allow this for now.
The next phase being Fire, if the Wood has grown like scrub you will get a bush fire, loud, noisy, useless and the class will become difficult or impossible to manage.
During this phase the teacher moves inside the classroom, available to all, answering questions, questioning, evaluating, in an extremely positive and benevolent attitude, as a gardener who watches with affection on the plants in full growth.
4th phase: Fire
Fire attributes are considered to be dynamism, strength, and persistence; however, it is also connected to restlessness. The fire element provides warmth, enthusiasm, and creativity, however an excess of it can bring aggression, impatience, and impulsive behavior. In the same way, fire provides heat and warmth, however an excess can also burn. It is therefore the most difficult phase to manage.
During this phase students work in groups. The Wood phase, of silence and personal reflection was necessary for the fire phase to unfold at best. The trees must be strong and well rooted: the students have appropriated the problems and want to be able to solve them, to progress, and to show their personal progress to others.
5th phase: Earth
Fire generates the Earth. The Earth is the element both Yin and Yang. It is the element of equilibrium and the one that logically concludes the session. Earth has stabilization and conservation energy. It is time to make a point and put a little order on everything that was seen during the two previous phases and to define what will be remembered. The Chinese think Earth is associated with the qualities of patience, thoughtfulness, practicality, hard work, and stability. The earth element is also nurturing and seeks to draw all things together with itself, in order to bring harmony, rootedness and stability.
The teacher retakes control. He synthesizes all that has been discovered. He is rooting the new knowledge.
The concepts highlighted and stabilized can be evaluated at the next session during the Metal phase. The circle is complete.
Finally, let’s add the other typically Chinese notions
Ren is the Confucian virtue denoting the good feeling a virtuous human experiences when being altruistic. Ren is exemplified by a normal adult’s protective feelings for children. It is considered the outward expression of Confucian ideals.
When we teach, we must keep in mind the nobility of this mission, the incredible responsibility that is ours is nothing less than to pass the torch of knowledge from one generation to the next . Teaching is a vital necessity for the well being of all. It is about taking it very seriously. We must teach with exactness, strive to make the best possible progress for each student entrusted to us.
The Confucian state of mind is rather the one prevailing during the guided phases. The work is clear, the students have to do it. This is very important for their future.
In China, teachers are highly respected by both children and parents. Their profession is one of the best regarded in Confucian philosophy. Mathematics enjoys a high regard.
The Chinese are proud to get better results than other countries with fewer means and much more crowded classes and attribute their success to Confucian values.
Emptiness is a central notion of Chan Buddhism, similar to the Taoist Wu.
We are always tempted to fill everything, Emptiness is scary. Beginner teachers are often filling the silence in the classroom by drowning students under a stream of uninterrupted words. By overwhelming their listeners with information they prevent them from entering into a thorough reflection. The information too many will be, anyway, quickly forgotten.
Taoism has been able to measure the virtues of emptiness. Listen to Laozi:
We join spokes together in a wheel,
but it is the center hole
that makes the wagon move.
We shape clay into a pot,
but it is the emptiness inside
that holds whatever we want.
We hammer wood for a house,
but it is the inner space
that makes it livable.
We work with being,
but non-being is what we use.
Let emptiness enter your classes. The students will fill it. Do not explain everything, leave voluntarily inaccuracies, students will take pleasure in identifying them and it will make them think a lot.
Trust the students. They all want to learn, they all want to understand, they all want to succeed. If they give the opposite impression it is that at some point something went wrong.
I often watched adults who were too impatient in front of students who did not find a solution to a problem. In seeking to snatch a result which they eagerly awaited these adults unintentionally built an edifice which was to become a frightful bulwark against the future progress of their pupils.
Be infinitely patient and accept that a pupil does not achieve the expected result rather than force his hand.
A Chinese proverb says that you won’t help the new plants grow by pulling them up higher. It is actually only a proverb.
Since I make exclusive use of this method I must say that I have the deep feeling that a peaceful harmony has settled into my classes. The adequacy of these old chinese principles with the effectiveness experienced on a daily basis never stops to surprise me.
I therefore like to believe, losing all rationality, that something mysterious and profound is hidden behind these old Taoist and Confucean concepts and acts without my knowledge for the greater good of all. It sometimes seems like a spirit flies in the classroom to appease the students and put them to work. This is surprising and opens the door to various spiritual ramifications if your nature is inclined to it.
Seeing me wondering, a person of Chinese origin simply asked, « Does it work or not ? » as I answered that it worked he just replied: « So what is your problem ? » and added that westerners use Chinese medicine and its mysterious meridians whose existence has never been scientifically proven. They use it because it seems to work in some cases and without scientific foundation. The criterion being only the success of the experience in class.
If you are interested in this method we could exchange a some ideas. If you want more information or if you find it delirious, please contact me or leave a comment.
Boliu Laoshi, The Lame Donkey.