**Two 5th grade teachers, Alena and Anezka, are speaking about their experience with introducing a divisibility test by 9.**

Alena: “As soon as the 3rd grade, when pupils learn to divide, they get dozens of tasks of the type What can digit X be so that the number 4X2 is divisible by nine? Then in 4th grade they solve the task again, for

example, Find a three digit number XYZ divisible by nine so that X + Y + Z = 8.

Pupils find out that such number does not exist but if X + Y + Z = 9, then there are a lot of such numbers. At this point, some pupils formulate the divisibility test by 9, so far only for three digit numbers. And only

now, in the 5th grade when one half of the class have discovered the rule for three digit numbers, can some pupils discover the same rule for four and five digit numbers. At the end, they will state the general

rule and prove it with the help of a calculator for numbers with up to 10 digits.”

Anezka: “**Your way is interesting but far too long**. First for three digit numbers, then for four digits, then five digits − it requires a lot of time. Moreover, I doubt that weaker pupils understand what it is about.

They do not have time to practise the rule well enough. Of course, , it is inspiring for two or three best pupils, but most pupils, at least in my class, would ask for a clear rule which they would acquire easily. By

the way, you speak about dozens of problems, do you have a collection of such problems?” When asked, Alena is able to show dozens of tasks which pupils solved in the 3rd and 4th grades 3 and for

some, she also shows erroneous pupils’ hypotheses and examples from the follow-up discussions in the class.

Anezka’s last sentence shows that mathematics educators have an important task: to create sets of sufficiently varied (in terms of content and difficulty) problems In topics of school mathematics, which would enable pupils to discover generic models via isolated models. This can easily be done by didactic mathematical environments (Hejný, 2011b).

**What is your teacher’s role in Scheme oriented education then?**

Your teacher’s work within scheme-oriented education is guided by the following principles. If you want to achieve

the desired results, you should:

**1. create optimal climate for learning:**

No pupil is frustrated, no pupil is bored, the teacher shares the pupils’ successes, the teacher encourages pupils who might give up mathematics, the teacher builds the pupils’ selfesteem;

**2. leave pupils enough space for their considerations:**

Do not impose your procedures in their minds even when the pupils’ ones are clumsy, do not orient pupils towards a quick solution by overly leading questions, do not interrupt pupils’ thinking processes, when a

pupil asks a direct mathematical question, the question “It is an interesting question” and ask the class to look for answers.

**3. Lead pupils towards discussion:**

When hosting the discussion make space for erroneous ideas and weaker pupils. For example, when two pupils discover two different correct algorithms for written addition, let each one choose which one they

will use.

**4. Do not point out mistakes to pupils:**

Let pupils discover the mistakes, or provide them with a suitable task where they can see them. The mistake is understood as a way towards a deeper understanding of the investigated situation. Teach pupils to analyse mistakes, mainly by analysing their own mistakes.

**5. Provide pupils with adequate tasks:**

Each pupil, both of a high and low ability, solves the task which corresponds to their abilities and thus can experience joy from success. Problems assigned which do not allow for differentiation are frustrating for weaker pupils

and boring for high ability pupils. On the other hand, tasks which allow both “quick” and “slow” solving strategies are suitable (see story C);

**6. Lead pupils (by your own approach to mathematics) to the need to understand not only mathematics as such but also the way it might be understood by their classmates:**

When a problem is solved by various solving strategies, pupils broaden not only their horizons in mathematics, but also their understanding of other people’s thinking processes and opinions. In this sense, mathematics contributes to critical thinking and cultivates pupils’ democratic awareness.

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