掌握等级for Future Elementary School Teachers


We-Emily McMillon和George Nasr-是内布拉斯加州大学的研究生。我们在春季2020学期期间为几何教师提供了基于几何学课程的两部分的掌握测试,发现了我们的学生

  • 看待评估的错误,以提高他们的理解,
  • felt less stress and testing anxiety,
  • experienced increased confidence in mathematics and greater growth mindset,
  • 将考试视为展示知识的机会,
  • reflected on the purpose of assessment in student learning.

In this post, we will discuss what led us to try mastery based testing for this student population, how we implemented mastery based testing in our courses, and some student survey responses.


We first heard about Mastery Grading late in 2019 when奥斯汀摩尔gave a talk at UNL on the topic. At the time, we were both teaching mathematics courses for pre-service elementary teachers. Hearing about Mastery Grading, we both, independently, thought this type of grading would be excellent for pre-service teachers. Hence, with permission from our department, we decided to implement mastery grading in two sections of the same course in the Spring 2020 semester.

Before describing what exactly Mastery Grading is, we would like to discuss some general learning goals we find valuable in a course for future elementary school teachers. Our first goal is to guarantee that our students fully understand most of the course concepts upon leaving the class. We feel that it is particularly crucial that students in an education program fully understand concepts, given that they are responsible for being able to articulate similar concepts to their future students.

A second goal is to encourage students to revisit and reflect on their previous work and mistakes. It is particularly imperative that future teachers understand that mathematical ability can be improved upon, as studies have shown that elementary teachers pass on their views of mathematics to their students.

我们的第三个目标是扩大学生的范围’ understanding of the purpose of assessments beyond a numeric score. As future teachers, it is important that they are at the very least aware of different styles of assessment, and, ideally, critically assess different styles of assessment to determine which is ideal for their own future students.

总的来说,我们认为重要的是,elementary education mathematics classes are designed in a way that encourages future teachers to continue working on concepts until they have demonstrated understanding. We want our students leaving these classes feeling confident that they have truly mastered the concepts that they may one day teach for themselves. We also want assessments to be seen as low-stakes opportunities for students to show us the progress they have made, while also incentivizing them to look back at their mistakes and try to understand what it is they have yet to learn. We believe this can be accomplished with Mastery Grading.



There are many variations on mastery based grading; our implementation as described below is but one example. Many additional resources are available online. We found the following blog very helpful and so pass it along to the interested reader:https://mbtmath.wordpress.com/

Another feature of mastery is that it only rewards students points for a problem once they have shown full understanding of the underlying concept. This incentivizes learning from mistakes and has the potential to help students cultivate a growth mindset toward mathematics. We also feel that mastery provides students with another perspective on how to run a class and assign grades.

Course Structure

Usual Course Structure

Geometry Matters is a required course for most elementary education majors at UNL. The course covers geometry and measurement and follows chapters 10-14 ofSybilla Beckmann对小学教师的教科书数学。本课程是三道菜序列的一部分,包括上述教科书的第1-14章。序列中的第一课程是数学问题,必须在几何问题之前进行,涵盖教科书中的第1-7章。序列中的其他课程,数学建模,涵盖第8-10章,可以在任何时候拍摄。




Modifications (Moving Online)

As these courses were taught during the Spring 2020 semester, we were forced to move the courses online in March of 2020. We chose to make some modifications to the course assessment structure to better work in the online, asynchronous format required by our university.
Before the move to online, we had given the first assessment as well as two mastery quizzes. The second assessment had to be taken online. We decided to eliminate the third assessment and instead replace it with weekly mastery quizzes that would each test a single new concept and offer an opportunity for students to reattempt up to two learning outcomes they had not yet mastered. Recall that quizzes were made up of exam-level problems—the only difference between these and exams was the quantity of problems. The final exam remained as previously scheduled, albeit online.

An Example of a Learning Outcome and Student Work

The following is a description of one of our 18 Learning Outcomes assessing areas of polygons other than rectangles, which spans sections 12.3 and 12.4 of our textbook.

  1. Be able to determine the area of triangles and parallelograms in various ways, including by making reference to the moving and additivity principles of area.
  2. 能够使用面积公式进行三角形和平行四边形来确定区域和解决问题。

That is, to earn points for this learning outcome, students would have to show mastery of both parts A and B.


Sample student work on area problem, part 1a

On part (a), the student was very close and would have earned most points for this part, but we would have liked the student to say that you can form a rectangle out of two triangles of equal area, and hence, half of the area of the rectangle is the area of either triangle. One can infer from the dashed lines the student drew on the triangle provided that they are thinking about this as two triangles forming a rectangle, but being explicit in their explanation was critical for us to ensure their understanding.

Sample student work on area problem, part 1b








Survey Results

At the end of the semester, we surveyed our students on their experiences in the course. There was no concrete incentive to complete the survey, but 41 out of our 42 students completed the form.

This survey consisted of two parts — a series of Likert questions, and a series of open-response questions.

Likert Questions


  1. I feel like mastery grading allowed me to demonstrate my understanding of the course content.
  2. Mastery based grading influenced me to look at exams and try to understand my mistakes.
  3. This course has made me more confident in my ability to learn math.

Below are the results.

Student responses to statement, 'I feel like mastery grading allowed me to demonstrate my understanding of the course content.'


Student responses to statement, 'This course has made me more confident in my ability to learn math.'


Free Response Questions


Working through the responses, we found several themes that were shared among many students, which we now discuss, categorized into expected and unexpected results.



  • “Even though we didn’t have a final, I think I would have been able to pass a final easily because I actually remember the learning outcomes. This is probably due to doing the homework and actually caring to learn what I did wrong and how I can fix it. In the past, I just did the word for an ‘A’ and didn’t really bother to learn it.”
  • “我对CRAM学习的压力较小,并是完美的。我觉得我学习实际上了解这些材料。“
  • “[Mastery grading] made me care more about my learning rather than stressing over a test score. I was more willing to put in the work and less motivated to use shortcuts.”

Learning from Mistakes:We found that mastery grading encouraged our students to look back at their mistakes on their exams. Of our survey respondents, 16 mentioned learning from mistakes as a positive takeaway of the course in their open survey responses. Many commented that they would have never looked back at mistakes they made on exams in other classes. The following two quotes are representative of the types of responses in this category. Some students mentioned specific learning outcomes they learned best, while others gave more general responses indicating that looking back at their mistakes benefited their learning.

  • “I felt I learned how to do [Learning Outcome 5] the best during this course. I learned this because I failed the first time and I had to go back and figure out what I was doing wrong.”
  • “我有多次机会表明我可以掌握概念,并证明我可以从错误中吸取教训,并在数学中更好地更好。”

数学信心和成长心态:As one may expect from our third Likert question, several students indicated feeling more confident in mathematics. Students mentioned how they were able to learn content they did not think they would have been able to learn at the start of the semester. We also found some encouraging comments about students’ development of their growth mindset. In total, 8 respondents explicitly mentioned math confidence or an increased growth mindset in their responses. A representative comment is:

  • “如果你不过第一次尝试,那就不是世界的尽头,你只需继续尝试”。

One of the interesting results was that some students even commented on growth mindset oriented toward their future students, as in the comment that follows.

  • “我会始终告诉我的学生继续尝试,他们会最终得到它,有时它只是需要更多的时间和精力!”

Results We Didn’t Expect

Stress and Anxiety:15 students indicated in their responses that exams felt a lot less stressful since they could redo their mistakes. Several of our students admitted to struggling with testing anxiety and said that this grading scheme gave them some relief to that. It should be noted that several students commented that at the beginning of the semester, the “all or nothing” nature of these exams seemed daunting. However, all these students said that things improved once they became more familiar with the grading scheme and started passing outcomes.


  • “I knew that my instructor was looking for key factors that indicated I knew the material [on assessments].”
  • “[掌握]是基于创造对内容的真实理解。我觉得传统的数学评估有时可以更加令人沮丧,而且这与具体的目标和目标更透明。“


Student Learning:我们未来的教师也表现出巨大的能力,以考虑他们未来的学生。看似掌握的评分鼓励一些人仔细考虑他们的具体目标是作为教练,例如与以下学生一起做的事情。

  • “[掌握]帮助我意识到作为一名教师,我想总是问自己,”我真正想要他们知道什么?我如何希望他们展示它?'......有些因素可能已经搞砸了他们的那一刻,但掌握了那种技能是我应该寻找的。“

他们还显示一个预先empath能力ize with their potential students. In particular, many students who admitted to struggling with testing and/or math anxiety commented on wanting to try mastery based grading with their own students as a way to alleviate their students’ testing anxiety.

  • “有时学生以不同的速度学习,[和它是]不公平地在[在哪里]如果他们不做,他们不能赎回自己。

Other students perhaps did not struggle with testing anxiety, but still saw the importance of giving students multiple opportunities to demonstrate their knowledge.

  • “我希望我的学生用数学考试斗争[to],尽可能多地受益!”



We believe we accomplished two of our three main goals. Students seemed to be successful in understanding to course content. In addition, students appeared encouraged to learn the content and felt motivated to understand their mistakes. We even saw that students felt more confident with mathematics and demonstrated a growth mindset. However, we are less confident that we broadened the scope of students’ understanding of the purpose of assessments beyond a numeric score, although based on some student comments, it appears that our students at least started thinking about this.


We feel that future teachers were the most amenable to this style of grading as they themselves tend to value the opportunity to grow and learn.



  1. 与学生定期对话。掌握很可能是新的,所以拥有这些对话可以帮助他们了解如何感受。是透明的。告诉他们为什么你这样做。
  2. Positive feedback may compensate instead of partial credit. While it is not the same as getting points, you at least send the message that you recognize the good work they did.
  3. 您可以讨论如何获得部分信用并不意味着您没有做任何事情,以及如何伤害练习您已经理解的技能。

In conclusion, we found mastery grading to be a rewarding experience both for us as instructors as well as for our students. This testing style felt like a perfect fit for pre-service teachers, and we would encourage any instructors of pre-service elementary teachers to consider giving mastery based grading a try in their courses.


我们要感谢Allan Donsig和Michelle Homp,以支持我们使用基于掌握的测试教授这类课程的愿望,为她的帮助设计我们的学习和数据收集方法,以及Yvonne Lai为她提供的有用的反馈和指导写作文章。最后,我们要感谢Austin Mohr向我们介绍我们的测试方法,并鼓励我们自己尝试。


1 Response to掌握等级for Future Elementary School Teachers

  1. Avatar Vicky Kennard says:

    Thanks for this. I really enjoyed reading it and now want to look deeper into mastery grading.

Leave a Reply

您的电子邮件地址不会被公开。Required fields are marked*

HTML tags are not allowed.

156,391 Spambots Blocked by简单评论