Do you ever get the feeling that most of the test is just a waste of time for some of the pupils because the only mistake they can make is a numerical one? Are the problems too easy for them? And on the contrary, are the three included problems unsolvable for some of your pupils?

Imagine there are tests that pupils actually enjoy!

If you have ever played a computer game where you proceed from easier levels to the most difficult “villains”, you know what I am talking about. Who would not enjoy a test in this form? And at the same time they know at every moment what level they are at and what must be achieved to proceed to the next “villain”.

I used to hate tests as a child but also as a teacher

First to prepare the test, then to sit it (that’s the easy part J), then to mark it and finally, the worst part, to give those who did not perform well bad grades. All this hand in hand with the frustration if the test is a disaster for a diligent pupil who has no talent or simply has had a bad day. Some parents take grades very seriously and a D from me may mean a week of punishment for the child.

I have always hated tests!

Eventually, I got rid of one of the toils connected with tests. My colleagues gave me their older tests. However, these tests were designed for a different group of pupils, with a different level or different pace of work. This made me realize that tests from my colleagues would provide information neither to me nor my pupils. We only got a grade.

I started to suspect that my pupils did not enjoy my problems and took them just as the necessary evil they would have to put up with

A typical test that I was given consisted of four problems. You probably know how to mark a test. All four problems were of approximately the same level. Having used these tests for some time I realized that the most gifted pupils rarely got an A. They would always make some silly mistake in one of the four problems, although they were really easy for them. I started to suspect that my pupils did not enjoy my problems and took them just as the necessary evil they would have to put up with. That is why they would always make some silly mistake.

But a much more serious problem was that I could not tell the level of my weaker pupils. What actually happened was that I could hardly distinguish between a D and an E because the pupils were not able to solve any of these four intermediate problems.

Now I only construct graded tests. My pupils can choose the level of difficulty themselves.

I usually prepare several topics on three levels. The sixth graders are still anxious, they are wary and cannot tell their level. They rather try to solve all the problems.  It takes time. And I always give them enough time in tests. Mathematics is not a race, this must always be born in mind. Many tough propositions in mathematics needed centuries to be solved. And our problems may represent similar tough propositions for our pupils.

A frequent parents’ question is when my pupils learn to work under stress for entrance exams.

They will, do not worry. We simply focus on it before entrance exams and pupils put up with this external motivation easily. Because then they understand the purpose. They have discovered the principles on their own in my lessons and thus remember them well and can apply them at the right moment. In higher grades, having worked with graded tests for a long time I could see the pupils made a lot of progress in their choice of levels of individual problems in the test. Pupils only need to go through a number of problems and get some experience with work with them.

It is crucial to admit that a major part of the final grade can be based on other things than tests.

We are afraid of the school management, of parents, of the school inspection. I am sure that if we convince ourselves that we should primarily assess the pupil’s progress and work, we will manage to win over everyone else or at least to defend our work and avoid anyone’s intervention.  

Sitting tests is actually very questionable. I do not oppose sitting tests as such but I strongly oppose the stinking odour of fear that spreads out. 

Research confirms that human brain cannot be creative in stress

It can only repeat (source). If a child is anxious because of a test or fears getting bad grades, they are so stressed they are condemned to mere repetition. Thus advice of the type: “Check it” makes no sense because checking answers requires creativity.

Surprise tests. – Often justified by the following comment: “Thanks to them pupils learn continuously”.

I was utterly relieved when these tools of power disappeared at university level. We keep saying we try to teach pupils to plan their work. OK, so let us really do it. It means that we tell them: “The test is planned for Friday, voluntary homework for Wednesday. On Thursday, we will discuss the homework with those of you who do it”. This helps them take over responsibility for their learning and planning of revision and preparation.

A test should consist only of closed problems and questions. Open problems cannot be assessed.

Do you know how I get to know that a pupil has mastered the content? When they are able to prepare a series of graded problems for their classmates and then to evaluate them with the class. This is how my class prepare warm-ups. Everyone on the topic and in the environment they like and where they feel strong. And this is creation, not only answering. It gives much more diagnostic information to the teacher. Do I have to carry on? :)

There are a lot of myths and half-truths about graded tests.

It is true that their preparation is very time demanding and needs expertise. Also graded tests are often not used well. That is actually the reason for having written this text.

Myth n. 1: Pupils cannot assess themselves, they are not able to tell their level.

Yes, this is true. Because they were not given the chance to do so at school. It just needs time. In the first couple of years, do not stress them with a time limit. If they want to go through all three levels, let them do it. It will not take long before better performers realize that doing A is a waste of time and poorer performers realize that B is more than they can cope with. They must learn this, they must discover it on their own. Stressing them by time limits is counter-productive.

Myth n. 2: A teacher is not able to prepare a good graded test.

Don’t worry. You can do it. The truth is you will have to solve all the problems in the given chapter. It helps if you think of your two weakest pupils and think about problems these two will be able to solve but will not be too easy for them. Then take two average pupils. And again, two brilliant pupils. In their case, you can combine more parameters of gradation (see below) or get inspiration from final letters in the textbooks.

It is crucially important that the teacher should not be lazy. They must solve all the problems on their own and read through the teacher’s guide. The fact is that they were taught different mathematics. H-mat is here for you and we try to make your work easier. That is why you can use our depository of graded tests on H-edu.

And remember, if one test does not turn out really well, do not hang your head. We keep telling children that: “we learn from our mistakes”. So, a test that does not work well but you make conclusions about why it did not turn out well is an inevitable step in your teaching career. 

So what is a graded test?

A graded test is a specific type of test that helps to solve the issue of differentiation. Usually it contains several problems, each of which is offered at a different level of difficulty.

In case there are three levels, the easiest variant is a reasonable challenge for the weakest pupils, the intermediate problem is a moderate challenge for average pupils and the most difficult problem targets at above average or gifted pupils.

An important principle when constructing graded tests is that pupils may solve only one of the problems. The one that they choose themselves. In case they solve more levels of the same problem, the teacher assesses the most difficult one.

Level A should be solvable for every pupil who has grasped the main idea of the topic and is able to carry out basic calculations, considerations or constructions.

Level A problems are solvable for weaker pupils, whereas intermediate and difficult problems are often unachievable for them. If weak pupils are to solve only more demanding problems, they will get frustrated because they will always fail. If the teacher selects the level of difficulty inappropriately, they will learn nothing about how the pupils have grasped the topic and cannot help them.

Level B is the intermediate level of difficulty. The pupil is expected to carry out something extra, more steps that are needed for a successful solution or to handle another gradation parameter (see below)

An average pupil usually cannot solve the most difficult problems but can try to and if they fail they are able to solve B level problems.

Level C is an advanced level of difficulty. To solve problems at this level, the pupil must have full understanding of the topic and an ability to work with ideas.

This level is really challenging for every pupil and thus is very tempting. It is the top the pupil can achieve. Everyone’s dream. Imagine you are presented with three problems of an increasing difficulty. You will do your best to solve the most difficult one. And? We are absolutely oblivious of grades at such moments. We have created our own motivation. It is all a game and a game is what has made people advance since the beginnings of humankind.

Graded tests allow each pupil to be successful in the test.

The underlying principle is that an above average pupil is not motivated by too simple problems. Also too easy problems do not allow the teacher to see if the pupil understands the problem in its full complexity. Analogically, the same problems are too difficult for a weaker pupil.

Our goal in H-edu when creating materials is to help teachers diagnose the level of how a class or an individual has grasped the particular topic. Whether they should go back to a simpler model, keep working at the current level or progress.

How should we use tests for the maximum benefit of the teacher and, above all, their pupils?

  • Your class should always be informed about a planned test. (see above: time management, self-management, responsibility, ...)
  • It is essential that before pupils sit a graded test they have had enough experience with solving graded problems in lessons. Before you have your class sit a graded test, explain to your pupils the idea and purpose of graded problems and graded tests. If you manage to communicate the idea that such problems have the potential to develop the pupils individually, you have won. And your pupils will solve them with enthusiasm. A test without the aura of anxiety can become a parallel to a race or a competition in the positive sense.
  • When solving the first graded tests, your pupils need enough time to try to solve more than one level of the same problem. They must acquire the ability to assess their disposition for solving a particular problem. It is far from easy and needs time!
    • I had very interesting experience with a class that I started teaching using the Hejny method in the 6th grade. They had no prior experience with this method. First of all, I could see almost immediately that the pupils really enjoyed this type of tests. They actually said so. Secondly, a thing that was gradually improving was their ability to assess their performance, in better case also to describe it. “I was good at this but could not cope with that.” The third aspect was the choice of levels. Some of the pupils, especially self-confident boys, realized already during the first test they did not have to waste time solving problems that were too easy for them. Girls tended to be not so self-confident and always solved all the problems. Some of them to be sure. Other because they could not get rid of the feeling that leaving something out in a test was wrong. Thus they rather tried to solve everything.
  • It worked really well to set preparatory problems (the so-called pre-tests) for voluntary homework. This means graded problems similar to those that would be in the test. These were then discussed with the pupils who had tried to solve them. The test was sat after this consultation. What happens is the pupils get a preparatory problem very similar to the one that will be in the test. They have a chance to prepare for the test and thus to influence their grades. This is in line with the main idea that should be underlying the whole education process: “The pupil owns their learning and is the architect of their success”. By providing the chance to try the voluntary pre-tests, the teacher hands the burden of responsibility over to the pupil. Each pupil is given the chance to succeed.
  • Pupils should understand how many points they get for each of the levels and how the final grade is constructed. Then the pupil learns to state their test strategy. For example to get an A they can solve two problems on level C, or three problems on level B. Of course, learning how to solve tests is not the most important goal but it will be useful in mathematics competitions, job interviews or entrance exams. If the criteria are not officially given in advance, the process of getting a grade is somewhat blurred, which lowers the teacher’s credibility and objectivity in the pupil’s eyes.
  • It is up to each teacher how they define the score for each of the levels (the following worked for me: easiest - 3p, intermediate - 4p, most difficult – 5p or easiest – 3p, intermediate – 5p, most difficult - 7p).
  • It is not enough just to check the results. If we want to diagnose the level of a pupil’s understanding, it is necessary to go through all the solving procedure. That is why we should teach our pupils not to erase wrong results and procedures and also to record the whole thinking process on the paper. Then the teacher can follow what was going on. In case the teacher is not clear, I strongly recommend they should discuss the solution with the pupil individually.

What are the benefits of graded tests for teachers?

  • They help the teacher map “strengths” and “weaknesses”. I.e. what has worked out, what the pupils understand and what the teacher can build on in the future.
    • For example, it happened to me in the last test on the topic Kickstart – Decimal numbers that the majority of pupils could only solve level B problems, which is much worse than in the topics Coins or Cubic solids. Thus, I learnt that they did not have a clear idea of the concept of decimal numbers, so I would have to go back to the topic. Fortunately, it would happen very soon in the chapter Decimal numbers.
    • Many teachers believe that if a pupil is not good at some topic, they have to revise and revise. But what if the pupil has nothing to build on? What if they forgot or did not understand the topics prerequisite for the current subject matter? Graded tests help us assess the current level of knowledge in the class. In case that many pupils struggle with levels A and B, we have clear information that it is not time to move forward yet.
  • The teacher is able to assess the level of understanding of a particular topic in the whole class, as well as the level of understanding of each individual. Thus they can adjust the plan for the following lessons.
  • The teacher gets a strong diagnostic tool for assessment of how the pupils perceive themselves. If they overestimate or underestimate themselves. This is a big topic. To put it brief, pupils whose self-confidence is excessive need challenges that they may not manage to solve. Then have them assess their performance. It is never the teacher who should evaluate. The pupils must arrive at conclusions on their own.
    • I recommend that pupils who lack self-confidence and underestimate their abilities should explain the topics to their weaker classmates. The weaker classmates will really appreciate that and be thankful. The teacher should always commend them for good performance. The feedback should be as descriptive as possible. It will not do to say “Well done!” or “Good job!”.
    • If the teacher has time for it, they can help to focus on the pupil’s source of frustration and try to help them in this area. If the teacher cannot, it may a critical parent, classmate, brother. It is always important to stress that these people should not assess the pupil at the cost of their own self-assessment.

What are the benefits of graded tests for pupils?

  • When we have to sit tests, it helps if they are attractive. Pupils like sitting graded tests. They represent a challenge. Boys sometimes compare them to multiple-level computer games.
  • Above average pupils learn to assess their abilities and not to lose time and energy solving problems that will not make them progress.
  • Even weaker pupils have a fair chance to solve problems adequate to their level successfully. Thus they stand a real chance to experience success. And that is something these pupils often miss at schools. I have known pupils who were absolutely demotivated but thanks to reasonable challenges bounced off from the bottom, regain will to learn and joined in again. I do not say it was only thanks to graded tests. But they played an important part in this change.
  • Pupils learn to assess their strengths and weaknesses and gain sound self-confidence. This can help them considerably not only in entrance exams, at A-levels, when making career choices or during job interviews but in their whole lives. 

If you decide to create your own problems, consider using the so-called gradation parameter

A gradation parameter is a factor that increases the level of difficulty of a given problem. It may have a variety of forms and becomes fully known only when the whole problem is solved by the class.

A note from the authors of graded tests on H-edu: any feedback on the chosen gradation parameters in the available graded tests is more than welcome. Also because the level of difficulty of the same test may be perceived as very different in different classes and a variety of gradation parameters will have to be used. 

The following have been used most often as gradation parameters on H-edu:

  • more demanding work with numbers (use of greater numbers, decimal numbers, fractions)
  • more demanding thinking procedures (e.g. the pupil must make more steps leading to the solution in the problem Bus)
  • more demanding scheme (e.g. a more complex spider web – it is more difficult to find the solving strategy)
  • discovery of isolated models followed by generic models (e.g. wooden sticks – double windows)
  • more complex wording of a problem
  • more questions
  • more numerical operations
  • more solutions of one problem
  • problems with no solution

A sample graded test with coins: 

In the presented test, the level B problem was constructed from the level A problem using the following gradation parameters:

  1. adding an unknown to the other side of the coin equation
  2. increasing the number of unknown coins
  3. increasing the value of used coins (using 50p)

the level C problem has the following extra parameter in comparison to the level B problem:

  • the difficulty parameter of non-solvability of the coin equation, which should be written by the pupil (following a previous agreement made in the lesson) – the result is 6p
  • the parameter of the use of a coin with a higher value was removed; that is an easier parameter then the need to solve the problem and come to the conclusion that it cannot be solved in the given domain

When I decided to start preparing graded tests, I was taken aback by the high demandingness of their preparation.

Furthermore, I was uncertain whether the individual problems really corresponded to the three levels. Having prepared a number of tests, the time needed for their construction is getting shorter and the certainty of estimation of the level is increasing. However, the whole process is still very time demanding and I am still not quite sure about the levels. We believe H-edu can help you with this.

If your pupils were up to know used to working with one variant, remember it can take long before they acquire the new skill.

Most pupils enjoy sitting these tests. Only two girls expressed their dislike for such tests. These two girls fit the category of “nerds”. They tend to learn everything by heart. Since the most difficult problems often require innovation and creativity, they do not always manage to solve everything, which they naturally do not like.

I recommend that even if you use the H-edu depository of graded tests you prepare your own tests.

The teacher becomes a pupil here. A pupil who is able to pose a problem on three levels of difficulty (gradation) has obviously mastered the topic. Since all of us have experienced different way of doing mathematics, we also have to learn.

If you start to prepare your own graded tests or if you think about gradation of problems presented in the textbook, you will grow more sensitive to every pupil’s level and progress. Also, if the pupil gets “stuck” you will be able to help them by posing a problem that is one level lower and whose solution will help the pupil solve the current one.

Sitting tests lost its sinister atmosphere (thanks to e.g. sitting it as pair work, follow-up correction), all stress has disappeared from the class. And that was my main objective.

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Tomáš Chrobák

Tomáš Chrobák

My name is Tomáš Chrobák and I am 34.

While at elementary school, I attended a mathematical class. Mathematics lessons were so good there that I benefited from them virtually until the end of my studies (that is why consider primary and lower secondary mathematics crucially important). At university level, I studied Teaching Mathematics and Physical Education. University studies of mathematics were about nothing. ... I was running a hiking club and each year was organizing summer camps, which I consider the best experience for work with children.

I live in Frýdek Místek with my wife and our two little children – Vojta and Rozárka.

I am into any sport. I used to take part in Ultra competitions (jogging, inline, etc.) but nowadays I have no time to train. Hiking, reading or just enjoying life always make me happy.

I win the Global Teacher Prize CR 2019.

"Only those who are naive enough to think that they can change the world usually do.”

Steve Jobs