**3. Assessment for learning versus assessment of learning in mathematics**

Assessment is generally broken down into three categories: assessment before instruction (pre-assessment), assessment during instruction (formative assessment), and assessment after instruction (summative assessment). It could be argued that pre-assessment is both assessments of and for (as) learning—that is, it assesses "prior knowledge" (as a pre-assessment) and that data is then used to revise planned instruction (making it formative assessment). Assessment of learning is used to determine what students have learned, while assessment for learning is used to determine what students are learning. It should be clear that assessment for (as) learning is a process of gathering information about students learning and provide qualitative feedback to support individual student learning and improve teaching practice in the classroom. However, there is a significant overlap between assessment of and for learning. Therefore, learning for assessment (summative assessment) and learning from assessment (formative assessment) are two complementary purposes of assessment. For example, the same test given in one circumstance would be considered an assessment of learning, while in another circumstance be considered an assessment for (as) learning. In short then, the difference between assessment of learning and assessment for learning is a matter of function and purpose [17]. Hence assessment that occurs during the lesson to continuously assess learning throughout instruction is formative assessment. For example, in teaching mathematics, I often use concept learning where students are given an explanation, examples, and non-examples after which they engage in working problems on their own or in groups. Periodically during the lesson, I stop students and have them share their answers. This allows me to know if everyone is on the task, if everyone has understood, and whether I have to revisit the instruction in a different way if students are making several errors. At the end of instruction, assess whether or not the instruction was effective and whether the students have gained the knowledge as per lesson objective, and if they have not, then the instruction is redesigned to better cater for the students. Therefore, "if the students do not learn the way we teach them, we must teach them the way they learn [1].' I shall not give an example of *assessment of learning* task because it is predominant in high schools. However, hereunder is an example of *assessment of learning task* anchored on discovery-based learning. The objective of the task is to help students derive and apply the distance formula for calculating the length of a line segment joining any two given points.

1.Given the diagram below:


**131**

*Formative Assessment in Mathematics Education in the Twenty-First Century*

a) (*x*2; *y*1) \*Rectangular coordinates

<sup>2</sup> + (*y*<sup>2</sup> − *y*1) 2

2

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (*x*<sup>2</sup> <sup>−</sup>*x*1) 2 + (*y*<sup>2</sup> <sup>−</sup>*y*1)

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (−2 <sup>−</sup> 3) 2 + (7 <sup>−</sup> 4)<sup>2</sup>

**Question Answer Concepts employed to arrive at the answer**

b) *x*<sup>2</sup> − *x*1 \* Distance moved from point P to point R (horizontal

c) *y*<sup>2</sup> − *y*1 \* Distance moved from point R to point Q (vertical

d) Right-angled triangle \* Horizontal and vertical lines meet perpendicularly

displacement)

displacement)

(at 90°); hence *PR* ⊥ *RQ*

\* Pythagoras theorem (result is a quadratic equation)

 \* Solution of quadratic equation (consider the positive solution because we are dealing with distance)

> \*Application of the distance formula \*Leaving answer in surd form

In Question 3, students will be assessed on how they organized their knowledge when finding the distance formula. Also it allows the teacher to establish how students understand concepts related to Cartesian coordinate system such as rectangular coordinate system, and methods of solving a right-angled triangle and algebraic processes such as solving quadratic equations. Thus the number of links of concepts tells the teacher the knowledge gaps inherent in students' connections between and among mathematical ideas, thereby using concept map as a teaching and learning strategy [5]. The concept map below provides a summary of the different concept employed

*DOI: http://dx.doi.org/10.5772/intechopen.88996*

= (*x*<sup>2</sup> − *x*1)

\_

(−5)<sup>2</sup> + 3<sup>2</sup> 

\_ 25 + 9

*PQ* = √

= √

= √

 = √ \_ 34

in the process of finding the distance formula:

**4. What are the assessment tools in mathematics?**

In the process of teaching and learning, teachers facilitate, observe, and assess student learning. This can be achieved by making learning practical through meaningful activities, embracing collaborative learning, using quizzes to engage students in

Solution strategy:

e) *PQ*<sup>2</sup>

2. *AB* = √

3.Generate a concept map in relation to concepts used to find the distance formula.


## Solution strategy:

*Theorizing STEM Education in the 21st Century*

joining any two given points.

1.Given the diagram below:

a.Determine the coordinates of R.

b.What is the horizontal distance PR?

2.Use the answer you found in 1(e) to calculate the length of the line segment

3.Generate a concept map in relation to concepts used to find the distance

c.What is the vertical distance RQ?

d.What type of triangle is PQR?

joining points *A*(3;4) and *B*(−2; 7).

e.Find the length of PQ.

(summative assessment) and learning from assessment (formative assessment) are two complementary purposes of assessment. For example, the same test given in one circumstance would be considered an assessment of learning, while in another circumstance be considered an assessment for (as) learning. In short then, the difference between assessment of learning and assessment for learning is a matter of function and purpose [17]. Hence assessment that occurs during the lesson to continuously assess learning throughout instruction is formative assessment. For example, in teaching mathematics, I often use concept learning where students are given an explanation, examples, and non-examples after which they engage in working problems on their own or in groups. Periodically during the lesson, I stop students and have them share their answers. This allows me to know if everyone is on the task, if everyone has understood, and whether I have to revisit the instruction in a different way if students are making several errors. At the end of instruction, assess whether or not the instruction was effective and whether the students have gained the knowledge as per lesson objective, and if they have not, then the instruction is redesigned to better cater for the students. Therefore, "if the students do not learn the way we teach them, we must teach them the way they learn [1].' I shall not give an example of *assessment of learning* task because it is predominant in high schools. However, hereunder is an example of *assessment of learning task* anchored on discovery-based learning. The objective of the task is to help students derive and apply the distance formula for calculating the length of a line segment

**130**

formula.

In Question 3, students will be assessed on how they organized their knowledge when finding the distance formula. Also it allows the teacher to establish how students understand concepts related to Cartesian coordinate system such as rectangular coordinate system, and methods of solving a right-angled triangle and algebraic processes such as solving quadratic equations. Thus the number of links of concepts tells the teacher the knowledge gaps inherent in students' connections between and among mathematical ideas, thereby using concept map as a teaching and learning strategy [5].

The concept map below provides a summary of the different concept employed in the process of finding the distance formula:

#### **4. What are the assessment tools in mathematics?**

In the process of teaching and learning, teachers facilitate, observe, and assess student learning. This can be achieved by making learning practical through meaningful activities, embracing collaborative learning, using quizzes to engage students in

reflections, or asking students to summarize lesson taught. The assessment tools they may choose to apply to assess student learning may differ depending on the stage of learning. However, assessment only *of* learning (summative assessment) and not *for* learning (formative assessment) is not enough to promote students' integrated understanding. They may use concept maps (connections between and among mathematical ideas), concept tests, examinations, oral and poster presentations (use different representations of mathematical ideas to support and deepen mathematical understanding), peer and self-assessment (introduce the peer or self-marking of home/ classwork in the classroom, and allow for discussion if there is a disagreement of an answer), portfolios, rubrics, or written reports. All these forms of assessment tools in mathematics allow for ways of assessment that motivate students to learn and thereby avoid damage to student self-esteem [7]. Besides, these different forms of assessment tools give helpful feedback to students in that they are guided on how to avoid making similar mistakes in the main examination. Furthermore, students are guided on how to improve their performance, and this impacts positively on student learning [7].
