Assessment as an ongoing process
There are many reasons why we want to assess students’ work, from informing students about their mastery of the subject matter to informing parents and school staff about the level of achievement in mathematics education . Moreover, teachers want to get feedback on their own work in order to make the right didactic decisions. Assessment is an integral part of teaching and much can be learned by listening to students arguing about a problem or analyzing students’ work on a written test.
Most teachers make lesson plans for their classes. They plan their teaching for individual lessons, for a whole semester, for a whole school year. Why not, in the same way, develop an assessment plan? The format for this assessment plan will differ from school district to district and from school to school, but in our view it should not differ for groups of teachers working with parallel classes of students of the same age and at the same school. Moreover, the general plan should be shared with adjacent grade levels. Apart from sharing your workload, there is also a great advantage in sharing different views on assessment between different teachers and look at the assessment plans from fifferent viewpoints. For the sake of the students it should be stressed they have the best possibility of success when their school experiences are as consistent as possible.
They should be treated fairly by having equal opportunities in different groups.
Often standardized tests are to be given to the students at some time during the school year but since teachers themselves cannot influence either the content of those tests or the date they are to be taken, in this chapter no attention will be given to them.
Before even starting to design an assessment plan, a teacher will have to think about the kind of information he or she wants to collect. For example:
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How students are progressing in their knowledge and skills on the different levels of competence;
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Which evidence can be collected about their mastering of the mathematical subject matter that was taught during the school year;
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How well students work together with classmates and are able to express their views and use mathematical reasoning to support their views;
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The way students present their work, neatly done and accompanied by correct and not too elaborate explanations or sloppy and untidy without (understandable) explanations;
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The way students take part in classroom discussions and the way they are able to get engaged in solving a problem.
For each of these aspects of student work a teacher needs a different format for assessment, ranging from informal to formal. Moreover, each of these aspects of student work needs a different way of scoring and grading. Using these assessments to guide teaching decisions will be discussed in Chapter 12. It is hardly possible to present the appropriate assessment plan for a school year , so we chose to give an example, provided by an experienced Dutch 8 th
grade teacher.
This teacher told us she makes a plan each year but always finds she did not do everything she wanted to. Working as a teacher you know you can do your very best but even so things always turn out differently than you had planned, just as the students start working very neatly in their new exercise book but invariably make mistakes and many erasures.

1.
Assessment plan grade 8
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I will take notes on my informal assessment of a group of weak students I want to give special attention. I will discuss the results with their parents on parents’ nights. No notes on their report cards will be made.
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Informal assessment during classroom discussion is important for my didactic decisions. I will not take formal notes but of course I know my students very well after some time and have a clear impression of their capabilities. This year I intend to give special attention to classroom discussions and the way my students take part in these discussions.
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Students are responsible for their homework themselves in my class, but every third or fourth lesson I have a look at the homework of every student. I only make notes in my gradebook if the work was not done or insufficiently done. I especially have a look at the work when drawings have to be made. Some students do very sloppy work on those. If the work is very bad, I will ask the student to do extra and then assess the new work.
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Once or twice during the school year I will ask the students to design their own problems or even a whole test. I always use the problems that are handed in (altered if necessary) in a test I give the whole class. Some students find that task very hard to do while others design problems I did not think them capable of doing. Of course, they must provide me with solutions for the problems they design.
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Out of the ten units I need to teach next year, I selected two chapters where an investigation or group work assignment is possible. Each year I try to develop one or two new assignments but the ones for next year were done before in one of my classes and I decided to use them once again:
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Use the internet to investigate where Pythagoras’ Theorem came from.
I will expect students to present and show understanding of at least one proof of
the theorem. Students have to hand in an essay about their findings, which I will
discuss with them individually. The rubric I use for all of these assignments is the
same and is known beforehand by the students. They have done this before in
grade 7.
Do your own statistics project.
Students will work in pairs or in small groups on this two-stage task. In grade 9
I will teach statistics at a more formal level so I need this introductionary task to
be done in grade 8.
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There are two abstract subjects next year I know students always find difficult. For these two chapters, one on algebra and one on functions and relationships, I will give an intermediate test, a quiz. All chapters in my book have a diagnostic test that students will do in class and correct themselves. The group of weak students in this class I mentioned before, will be asked to hand in those diagnostic tests in order to give them feedback if necessary before they do the formal end-of-chapter test. By looking at this student work I get feedback myself on possible misconceptions or parts of the subject being taught that need more practice or that need to be revisited.
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At the end of each chapter, a written end-of-chapter test is given. This test is used in all our grades 8. My colleagues and I make different versions of the same test using different numbers and even different contexts but with the same mathematical content and the same degree of difficulty. Next year we will start to record students’ results

on the different levels separately; until recently we just gave one mark for the test as a whole. But we found parents would like more specific information about their children’s progress and by looking at students' results on the three levels of competence, we get more information about their prohress.
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We do not use end-of-year tests until grade 9 or 10.
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We do not have to give standardized tests in our school. At the end of their school career however, students take final examinations that are provided by our government of education. Students passing these central examinations at this school gain access to university courses.
The publisher of our math textbooks provides two different end-of-chapter tests for each chapter. We use those to some extent but each time we do so, we try to design one new level 3 problem ourselves for each test. As a general rule, for this level 3 problem, we use a context that is close to our students, for instance something taken from a local newspaper, or using data taken from our own students’ results. Students always find those problems interesting to do, though sometimes hard to solve. But recognizing the context from our own environment gets them really involved.
As is shown in this balanced assessment plan for grade 8, the larger part of the assessments consists of straightforward written and time restricted tests. But a large variety of other formats for tests and tasks is used and the teacher recognizes the importance of them in order to have students show other capabilities than just remembering facts and procedures. Larger tasks are not given as an activity for activities sake but are chosen to go with chapters where they can be used appropriately to show other skills. Because of the amount of time they take, both in performing them by the students and in scoring and grading them by the teacher, a limited number of two or three each school year is done.
The assessment plan shown is just one of many examples. Each teacher will have to adapt it for his own use or more likely, will have to design his own plan. But it is always good to see what others do and use those examples as a starting point for ones own plan.
After designing an assessment plan for a whole school year, plans will be made for assessing one chapter or unit and even one particular lesson. Since these depend very much on the lesson plan and the math book in use, we will not give examples but just stress the fact this needs to be done.
During the school year a teacher will have to design tests for the different units or chapters and decide how to score and grade them. Before discussing how to design a balanced test, we will give an overview of different formats for assessment that are frequently used. In the same manner one does not use all the different formats for questions and problems in one test, there is no need to use all of these formats for assessment in one year. Just make sure to use some different ones and not to stick to one particular format only.
An overview
1.
Observation.
During the lessons students in general or some students in particular are observed while working at their task, participating in group work or presenting their work for the whole group. Some teachers make notes of these observations while others use it as background information about their students’ progress.

2.
Discussion
During classroom discussions teachers make (mental or real) notes of the way a particular student or groups of students are able to express their views in a mathematical way.
Individual talks with students about their work also come under this heading.
3.
Homework assignment, group work
Analyzing (selected) student work provides an opportunity not only to monitor the way in which a student completed the homework but also makes it possible to give feedback to a particular student and to the group as a whole. Every homework assignment need not be assessed. An accurate view of student understandings can be gained with a prudent selection, and does not overburden the teacher, freeing time to plan and create materials.
Even if students are expected to correct their homework exercises, it is very difficult to decide for them whether or not the explanation given or their reasoning is right. Common mistakes that are shown may be discussed in class later. Moreover the teacher gets an impression of the way a student works. Making remarks about neatness or correct wording of the solution of a problem works a lot better if given individually.
4.
Investigation
Larger tasks are analyzed and scored that were given as individual assignments or group work. Students present their results as an essay, design poster presentations, videos or during an oral presentation in front of the class. Using computer skills may be an important goal of this type of assessment. As discussed in the previous chapter, other skills can be assessed than those needed to perform well on a written test.
5.
Quiz
Quizzes are often used to assess the mastering of basic skills and knowledge and consist mainly of level 1 questions and problems. They can be given at the start of a new topic, to assess whether each student possesses the background knowledge and skills the new chapter builds on. They can be used during the time the new chapter is taught in order to assess whether all students master the mathematical content and algorithms needed to continue with the chapter. Or they can be used as some kind of diagnostic before an end-of-unit test is given. When analyzing student results on this diagnostic test the teacher gets feed back on the mastering of the subject and can make didactic decisions based on these results. Marks can be given for these tests but many teachers prefer not to , or even have students correct these tests themselves, thus sharing responsibility for their learning.
6.
End-of-unit test
More often than not this is the most formal type of assessment task given in class. These tests cover a larger amount of subject matter and should contain problems on the different levels of competency. The content of the problems should as much as possible match the goals that were set for this unit or chapter. The tests are in general administered through one
(sometimes two) class periods of 40 or 50 minutes. Results are used for formal grades.
7.
End-of-year test
Especially in higher grades sometimes an end-of-year test is presented to students, covering all of the subject matter taught during the past school year.
This overview is certainly not complete but is only used to show a graduate shift from informal to more formal assessment instruments. Other formats for assessment, used by teachers on a regular basis at their schools are (in random order):
8.
Project work
9.
Daily (short) assessment tasks.
10.
Section assessments (since students do not retain very well and have to practice a lot)
11.
Portfolio

12.
Supplemental basic computational problems
13.
Student notebooks, assessed according to a rubric
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Journal writing
15.
Games
16.
Oral test for students that are weak readers or use English as their second language.
Language used for tests
Many students have difficulties in reading comprehension.
In context problems, as they are used in a balanced test, less mathematical language is used but more every day language. Even so , there is much more reading to be done then is customary in “bare” problems. Special attention is needed for the wording of problems. Avoid sentences that are too long, avoid ambiguous language. Never pose “two questions in one” as in Draw the two graphs in one coordinate system and find the intersection point. More often than not one of the questions or even both will be forgotten. Whenever unfamiliar words or concepts are used, these should be explained either by making a drawing, using a photograph or explaining them in words. It should be perfectly clear for a student what exactly is expected from him. For some students, it may even be necessary to allow them to ask the teacher to explain difficult words for them. Only in exceptional cases the teacher could read the problems aloud, or describe the context in different words. Note that by doing so the complexity of the task may change.
Developing a balanced end-of-unit test
It is easier said than done: develop a balanced test consisting of problems on the different levels of competency. Some publishers provide schools with tests but even so teachers may want to adapt those for their own use. The grade 8 teacher from the example given before stated that she and her colleagues will start designing at least one new level 3 problem for each test they will use during the school year. Not everything can be done at once, it is better to start small and get new things done each year. With each addition you will gain new insights, and with each new group of students you will observe different strengths and needs.
Teachers should accept that not all of the students will master the subject matter of the unit sufficiently or are gifted enough to correctly solve all level 3 questions in a test. This does not mean this student cannot start a new topic or a new chapter. New opportunities to show mastering on a higher level of reasoning will be given when the same topic is revisited later in the school year or even in the next grade. As a general rule, but depending on the specific chapter or unit that is assessed, we sometimes use this distribution of score points over the levels:
Level 1 55% Level 2 30% Level 3 15%
This means that a student who masters the basic skills taught in the chapter and has some capabilities at the higher levels, can reach a sufficient percentage of about 60% of the available score points and has shown sufficient mastery of the subject matter to start a new chapter or topic. Once again, this is just a general rule and many variatons may be found in different schools. In that case the base of "acceptable" can be changed accordingly.
The format for the questions and problems to be used in the test may differ but mostly open questions will be posed. The didactic decisions that can be taken after analyzing students' results will be discussed in chapter 12. The following example shows how a balanced test, meant for a
40 minutes class period, could be constructed but once again it is only one of many possibilities.

2.
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The test starts with about five single answer questions, level 1. The first question is a simple one, meant to reassure students they are capable of doing this test and at least give it a try. Each question takes about two or three minutes and is awarded with one or two score points, depending on the number of steps needed to find the answer. The questions are similar to those found in the unit or chapter but of course with different measurements and numbers.
Total time for this part: 10 minutes, number of score points: 10
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One short context problem. The mathematical content of the problem is the same as was taught in the unit or chapter but the context itself is different. One entry question at level 1 is preferable, to enable students to get acquainted with the mathematical content of the real life context followed by one or two questions at level 1 and 2.
Total time for this part: 10 minutes, number of score points 6 to 8.
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One larger context problem, preferably a so called super item, consisting of an entry question at level 1, one or two at level 1 and 2 and at least one at level 3.
Total time for this part: 10 to 15 minutes, number of score points 10.
At least five minutes are needed by the students to look their work over, finish drawings and so on and they should be encouraged to do so.
It is obvious that in one 40 minutes test not all of the topics addressed in the chapter can be assessed. But this is not the only moment of assessment. Other parts can be assessed in other ways. When you design a new test it may be a good idea to start by finding the larger context problems and fill up the gaps in the subject matter to be assessed with the short level 1 questions.