**Assessment in a Math Lesson**

Teachers can employ several assessment strategies before, during, and after a particular math lesson to collect data. These are meant to support teachers as we refine our lessons and think about reteaching a specific concept. More simply, the assessments before, during, and after a math lesson help us get an understanding of what our students know or do not know, essentially helping us understand where the breakdown is occurring.

To put things into perspective, the National Council of Teachers of Mathematics (NCTM, 2014) explains the difference between unproductive and productive beliefs when it comes to mathematics assessment. For example, an unproductive belief involves an assessment as a form of grading or report card mark. However, a productive belief involves using an assessment to inform and improve teaching and learning. Furthermore, the NCTM (2014) explains that assessments are ongoing and require multiple data sources to provide an accurate picture of both student and teacher performance. This ongoing process is important, as Suurtaam (2017) states, "Assessment does not merely occur at the end of a unit or course" (p. 1). Similarly, the National Research Council adds, "In the past, student learning was often viewed as a passive process whereby students remembered what teachers told them to remember. Consistent with this view, assessment was often thought of as the end of learning. The student was assessed on something taught previously to see if he or she remembered it" (p. 68). Therefore, the shift to include formative assessments throughout the unit instead of a summative assessment at the end of the lesson, is rooted in obtaining a holistic view of our students' learning. This also includes providing students with comment-based feedback on their work, as a way to communicate what students are doing well and what they can work on. We place an emphasis on a grade or the number of correct/incorrect responses, but there is no student growth or progress without feedback that is communicated with students in the form of a comment. Ultimately, such an approach also builds a growth mindset in our students while they develop mathematical thinking.

To put things into perspective, the National Council of Teachers of Mathematics (NCTM, 2014) explains the difference between unproductive and productive beliefs when it comes to mathematics assessment. For example, an unproductive belief involves an assessment as a form of grading or report card mark. However, a productive belief involves using an assessment to inform and improve teaching and learning. Furthermore, the NCTM (2014) explains that assessments are ongoing and require multiple data sources to provide an accurate picture of both student and teacher performance. This ongoing process is important, as Suurtaam (2017) states, "Assessment does not merely occur at the end of a unit or course" (p. 1). Similarly, the National Research Council adds, "In the past, student learning was often viewed as a passive process whereby students remembered what teachers told them to remember. Consistent with this view, assessment was often thought of as the end of learning. The student was assessed on something taught previously to see if he or she remembered it" (p. 68). Therefore, the shift to include formative assessments throughout the unit instead of a summative assessment at the end of the lesson, is rooted in obtaining a holistic view of our students' learning. This also includes providing students with comment-based feedback on their work, as a way to communicate what students are doing well and what they can work on. We place an emphasis on a grade or the number of correct/incorrect responses, but there is no student growth or progress without feedback that is communicated with students in the form of a comment. Ultimately, such an approach also builds a growth mindset in our students while they develop mathematical thinking.

Considering this understanding around assessments in math, I have provided a few forms of assessment that can be used on an ongoing basis before, during, or after a lesson to enhance student outcomes while supporting the teacher's planning and execution of a lesson. These are a range of suggestions that can be used based on the specific math lesson, as teaching and learning occur in a way that is instructed by a strong approach to the assessment

*for*learning process. Cards Up |
Do Now |
Exit Slips |
Hand In, Pass Out |

During the lesson's guided practice, have students independently work on math problems. Each student then has to put up one of their cards (e.g. A, B, C, or D) when they hear the teacher say, "Cards Up." |
Have students independently work on a question before a lesson to gage their prior knowledge. It should be a fairly brief and simple task, where the response(s) helps the teacher veer the lesson according to the students' existing understanding. |
Once students are taught the new/continuing material and they have had practice through the guided practice part of lesson, have students fill out an exit slip with a question relating to the topic. Use this to plan the lesson for the following day. |
This is similar to the exit slip, but students help "check" the answer(s). Students answer a question on a cue card without writing their names. Then, collect the cue cards and pass them out randomly to correct the answers as a class. |

Math Journal |
Partner Quiz |
Three Minute Quiz |
Whiteboards |

As an alternative to using mini whiteboards, journals are a central place where students can track their progress. I often use math journals as a way for students to take notes on a topic, in addition to using it as a space to work on math problems. While this is traditional in nature, it helps maintain a level of organization and easily helps students go back to think and reflect. |
During the lesson, pair students to work on a question. This one takes a bit of time to grow into because it requires students being taught how to solve a problem together, and then learning how to give feedback. However, it can be a useful strategy for middle school students, as a way to work on rigorous math questions together. |
At the end of a lesson, pass out a "quiz" with three questions, ranging from easy to difficult. Students have three minutes to work on it. This provides a brief overview of the students' level of understanding based on the different ranges of questions. For students who require an accommodation, additional time can be provided. |
This is probably the most useful one for guided practice. Students work on solving the problem(s) on their mini whiteboards, as the teacher circulates and provides individual or small group support. This whole to small class approach works really well because every student is working on something, show their steps at the same time. |

**Evaluation & Grading in Math**

In the grading section, I explain the difference between formative and summative assessments. In math, assessments are helpful for both the students and the teacher in learning (1) where the breakdown is happening and (2) how successful the lesson or unit has been in terms of student mastery and understanding. In the above section, we explored different ways to assess students to gather data, showing us where that breakdown is happening. Then, teachers can adjust their lessons to meet the needs of their students. The evaluation section focuses on collecting data around student mastery and understanding, although these are also helpful in adjusting future lessons or teaching methods.

Taking this into consideration, I begin by thinking about what makes most sense in terms of evaluation and grading for math. Here is a list of some ideas that I use for both evaluating and grading in math:

Taking this into consideration, I begin by thinking about what makes most sense in terms of evaluation and grading for math. Here is a list of some ideas that I use for both evaluating and grading in math:

Exams |
As a student, I always preferred an essay or take home assignment over an exam. That being said, math is a subject that really hinges upon our students answering questions in a timed environment. While I have my own thoughts around this, the curriculum and the way each province/state works does unfortunately focus on standardized testing. Therefore, exams are an integral part of the grading scheme to address such standards. The next section of this page covers how to make a math assessment that considers both how to retrieve data and ensure testing is helpful for our students. Personally, I believe students should not be graded on a right or wrong answer. Instead, teachers should take a more holistic approach and provide marks based on the students' thought process and steps to reaching an answer. |

In-class Evaluations |
These are really helpful in math. Although homework can be easily substituted for in-class assignments, these are more accurate representations of the students' understanding. For example, in-class assignments avoid the risk of other people doing the homework for our students, and it provides us with a more accurate assessment of what the student has grasped from a particular lesson or topic. I usually like to give several in-class assignments and count the highest towards the final grade. For example, if I give out 15 in-class assignments in the term then the ten (10) highest ones count towards the overall 25% grade for in-class assignments. This reduces the risk of students being weighed heavily on any one topic and also relieves the students of stress, providing multiple opportunities to do well. |

Quizzes |
If it were up to me, quizzes would never exist. However, I understand how quizzes really help us collect necessary information about our students and their understanding of a lesson or topic. In order to make it fair and stress free, I prefer to follow the same rule as in-class assignments and count the highest scores towards the final grades. Then the question becomes: do I have pop quizzes or not. That is up to the teacher and the students. I prefer pop-quizzes for math to keep my students on their toes and it helps me see how well they retain the information. However, I caution teachers to use their judgement and be fair when handing out pop-quizzes. |

Participation |
I used to be vehemently opposed to participation because my own learning experience has seen its flaws. However, participation can be really fruitful when teachers understand that participation for different learners means different things. In addition, I only recommend adding participation to the lesson for a classroom where engagement is usually low and when you want to guide the conversation towards more inputs. The Reflective Educator provides a few thoughts on how to encourage participation in math that I find helpful, including opportunities for turn and talk and wait time. |

Projects |
Are you serious? Yes, I am. Math projects are a thing, but far too many students never really get to work on projects in math. A lot of the above recommendations are more traditional forms of math grading ideas, so the project serves as an approach to really add something special to the grading scheme. The online forum has tons of ideas around math projects, especially with the popularity of STEM in the 2000s. However, I am less interested in the fancy names given to these projects and more interested in the actual project. As a result, I recommend spending a block of time with the students to list out a few project ideas. Come prepared with ideas to guide the conversation and then vote on five ideas that students can choose from to work their selected project. This way, students are a part of the decision-making process and it leaves room for ideas that you, as the teacher, may have not thought about for the project. Check out the math projects page for ideas to get started. |

Secondly, I think about how to weigh these different grading ideas/methods across the term. Personally, I prefer to use a 10-30-30-30 model or a 25 by 4 model. For example, 10% of the grade is for participation, 30% of the grade is for exams, 30% of the grade is for quizzes, and 30% of the grade is for a math project of the student's choice. In the 25 by 4 model, you can have 25% for in-class assignments, 25% for quizzes, 25% for a term exam (or three exams with one dropped from the grade; two for 12.5% each), and 25% for a math project of the student's choice.

25% |
25% |
25% |
25% |

In-class Assignments |
Quizzes |
Term Exam |
Project |

Of course, all of these can be substituted with anything that makes sense for your classroom and students. This type of grading scheme works for me not only in math, but in every other content/subject area. I prefer this method because almost every component is worth the same amount and there is room in each component for students to have some flexibility (i.e. having 10 out of 15 in-class assignments count towards the grade). It does make the process a bit tedious, but the long-run impact of our students' development is worth that effort.

**References**

- National Council of Teachers of Mathematics. (2014). Principles to Action: Ensuring Mathematical Success for All. Reston, Virginia.
- National Research Council. (1993).
*Measuring What Counts: A Conceptual Guide for Mathematics Assessment*. Washington, DC: The National Academies Press. https://doi.org/10.17226/2235. - Suurtaam, C. (2017). Classroom assessment in mathematics: paying attention to students' mathematical thinking. Retrieved on April 7, 2019 from The Learning Exchange.

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