"An exclusive reliance on a single type of assessment can frustrate students, diminish their self-confidence, and make them feel anxious about, or antagonistic toward, mathematics" (NCTM 1989, 2002).

If teaching is an art, than assessment is a science. Philosophies of how to grade students are as varied as an international school population. All teachers should have a grading policy that is anchored in research and is vindicated by their own action research on students in their class. The purpose of assessment is not to merely put a grade in the gradebook, but to indicate to students areas of their learning which they have to improve and communicate to parents their child's level of understanding. Category weighting will also vary greatly by subject area as educational outcomes and products can have many forms. My philosophy of assessment and grading has been influenced greatly by Robert Marzano and Tom Schimmer. The following gives an outline to the many facets of math education and our assessment of it.    

Practice Work/Homework 0% For starters, I use "flipped lessons". A flipped lesson is when a student will prepare for a lesson in class by taking notes prior to the lesson outside of class. We have online instructional videos, and online textbooks so students can prepare for lesson so when they come to class, I don't have to spend so much time introducing a new topic of study. What this translates to also in real time, is that I don't spend 40 minutes teaching a lesson, I spend only 10 minutes clearing up any misconceptions that they have. Furthermore, many math assignments start off easy, but towards the latter half of the assignment the problems take the form of more higher order thinking with less recall and more application, analysis and synthesis. This is a frustrating part of the assignment and many students just get stuck when working by themselves.

With a flipped lesson, they do all their practice in class with me, along with their peers and an answer key in the back of the classroom to check frequently to ensure understanding. My students are astonished that I would provide answers for them, but I find they it puts them at ease and also takes the burden off the teacher by not having to be to "keymaster of knowledge".   How much practice should a student do per lesson? I think the answer is about 40 minutes. With a flipped lesson, I am able to allow students 40 minutes of practice in class which they self themselves on a three point rubric. Here's the wonderful thing: they assess themselves. Students do self-assess their homework with regards to what percentage of the assignment they've completed, number correct, but it is considered practice work in this stage of their learning. Because of this, students are not graded on homework, but rather, they self-assess their work and the grade is weighted as 0%  

 This is contentious area of education. Some of my colleagues would disagree saying that they grade homework because if they didn't, students wouldn't do it. I don't find this problem. I communicate to my students that the purpose of homework is to develop new skills in domains that may be new to them and it is vital: they are the first rung on the scaffolded ladder to understanding. However, I do understand that they are a new skill and at that stage students will not have mastered the skill yet. For this reason, I feel that it is unfair to grade homework. It would be similar to teaching a kid how to make free-throw shots in basketball and telling them that after 5 minutes of practice that if they want to play in the next game then they had better be able to shoot 80%.   Learning takes time, and some events in the learning process should be deemed as practice. However, although in-class practice is such, I do mark student's appraisals of their work (in the three point scale) in their gradebook for reference, but with a zero % weighting.    

Quizzes 10% Like with homework, quizzes are generally early in the learning process and gives formative feedback to the teacher to re-teach foundational principles with mathematics before proceeding to the next level. Quizzes are designed as a "check-in" before moving onto more complex tasks. For example, a student cannot learn how to calculate surface area and volume of three-dimensional shapes if they cannot calculate the area of basic two-dimensional shapes first. Quizzes are done about half way through a unit and after taking a quiz, students reflect on the nature of the mistake: was it a simple mistake or larger conceptual understanding? Quizzes provide a snapshot into a student's learning up till this point and help diagnose weaknesses.   Statistically, students score on a mean average between 61%-64% mastery on quizzes so it is not their highest level of understanding. Because of this, quizzes are considered to practice on the path towards mastery and are not counted. Their score is recorded in the gradebook, but given a 10% weighting.    

Projects/Inquiry Based Investigations 30% I am a big believer in projects to demonstrate learned concepts. Many teachers are quick to spout research that supports "that through projects, students are more able to see real world applications of content material". I am in full agreement as well. However, as projects can span several periods and involve group work, there is no guarantee that the work a student has done has been completely without assistance from peers or parents so they have a higher susceptibility to not be an accurate reflection of a student's true abilities.   In Math, most culminating projects are late in the learning scaffold, but students are still developing their understanding with regards to math skills. Because of this, we still consider this rung in the learning ladder practice, so activities that fall under the category of projects/inquiry are to be considered continued practice for application of skills. In my experience, I've found that math projects only touch on a narrow swath of the curriculum and do not assess ALL content and skill standards within a unit of study. Projects are a vital complement to math education, but are considered to practice on the path towards mastery and are not counted. Their score is recorded in the gradebook, but given a 30% weighting.    

Summative Tests/Authentic Assessments 60% By this stage, students have had ample opportunity to master their skills by scaffolding through assignments, quizzes, projects, review and practice testing. Summative tests are the final point in the learning process when students have had the maximum amount of practice and are most readily able to demonstrate their math skills. Statistically, my students have the highest mean scores on summative tests (usually around 75%). The California curriculum framework also states that "summative assessments are the most reliable reflection of student growth."   To be graded so highly on summative tests can be a scary thing for some students. Some parents too are quick to say that "my child doesn't test well", and "I don't believe in high stakes testing.". To debunk the myths, some students don't test well because they've never had to. Also, this isn't high stakes testing. High stakes testing is when if your school doesn't pass a certain % of a certain test, your school is shut down. The fact of the matter is that testing is a necessary part of education and students will run the gauntlet of testing through high school and college so it makes sense to develop good test taking skills in the middle school.

Through summative tests, students can demonstrate their understanding through many different ways such as written response, tables, diagrams, or mathematical notation.   Regarding retesting, our policy is "Retesting is a privilege, not a right. We never deny the opportunity to relearn, but we may deny the opportunity to retest." In the past, we had up to 40% of the class sitting retests because, despite all indicators of what students should have reviewed, they simply didn't take the time to sit down and review difficult concepts. Many math programs feel that students are unconditionally given a retest and their grade would be raised to the level of "C" but I believe that this creates grade inflation, poor student skills and an inaccurate picture of what students can really do. Also, if a student is getting a "C" instead of an "F" their parents may think that everything is OK. A bad grade is meant to prompt action and intervention. If we provide the necessary resources to help students learn, they must intrinsically motivate themselves to take ownership and responsibility of their learning. With respect to differentiation, we may offer retakes for blue and black level test-takers, but it's on a case-by-case basis. If a students has been negligent with coming to class prepared with the necessary homework, they might be denied a retest. If a student made a poor choice with their test level based on earlier indicators, they may be denied a retest as well.  

 Since this policy has been in place, my students have really learned to take test preparation seriously. I've had so many parents compliment me on how their child for the first time in their life, their child is demonstrating and developing good student skills of preparing for tests and reflecting on their learning. Most importantly, this has had a positive correlation to standardized test scores and internal test scores with math education.


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