*艾米莉麦米尔隆和乔治纳斯尔（内布拉斯加州大学）*

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.

# 动机

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.

# 掌握等级

掌握评分是预计学生对课程目标的完全理解的评分计划。这是通过在整个学期提供多个机会来实现全部或全无的信用的重新尝试课程目标。学生们需要更长时间才能掌握课程目标没有罚款。目标是让学生最终表明他们了解材料，而不是学生一定是在评估时一定会展示对材料的完全理解。

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/。

我们认为，掌握评分有助于实现前一节中提到的三个目标，就我们的动机。掌握旨在鼓励学生重新审视概念，以获得学习他们的全额信誉。在基于点的课程中，学生可以获得部分学分，以部分学习某些东西，然后可能永远不必再重新审视该概念。通过这种方式，学生理想地将本课程留下强大地了解课程内容。

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。

该课程由教师，讲师和高级研究生教学助理教授，具体取决于任何特定学期的教练可用性。课程成绩通常由评估分数，家庭作业分数和书面项目评分（所谓的“心态”问题）决定。学生在课程中往往在过去六年中，通过的比率为79％至100％，大多数有超过90％的学生通过该课程。因此，等级和通过的价格不是我们决定实施掌握评分的原因。

## 我们如何应用掌握分级

我们将课程内容划分为18个学习结果。学生成绩掌握了这些成果的60％，在作业和项目问题上占剩下的40％。个人学习成果的评分是全部或全无的信用，而且家庭作业和项目问题与传统的基于点的系统进行了分级。我们的原始计划是在第一次评估中测试成果1-7，第二次评估1-13，第三次评估1-18。最终决定不会涵盖新材料，但将是掌握以前没有掌握结果的最终机会。此外，我们计划在机会出现时，偶尔会在课堂上作为“测验”作为“测验”。

## 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.

- 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.
- 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.

以下是评估这一学习结果的示例问题，以及一名学生的工作，这些学生并没有掌握他们第一次尝试的概念。学生知道他们可以使用标准形状区域的公式，例如矩形，三角形和平行四边形。学生还知道他们有望表达他们的结论，并证实了他们与面积原则等思想的推理。

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.

然而，部分（b）的工作是真正导致我们觉得让学生花更多时间审查这一结果至关重要。这部分的重点是为了学生识别阴影区域可以分解成三角形和平行四边形，并且添加这些形状的面积将产生原始区域的面积。学生的尝试仍然对如何解决和移动地区的努力来展示一些理解，以弄清楚该地区，这表明希望使用面积原则。但是，如果仔细检查，则无法符合学生在指定地区创建的三角形，这对于学生的工作方法至关重要。（虽然巧合，但他们确实得到了该地区的正确答案。）

回顾每个学习结果的价值15分。我们会说学生会在这个问题上获得左右8点，这已经用点分级了。然而，由于掌握了掌握的评分系统，这名学生有第二次展示他们的理解。下面，我们向这一学习结果和学生的回应显示第二个版本的问题。

我们认识到，部分（a）有一些非常规的符号选择，但我们觉得这是清楚的学生对被评估的潜在概念的理解理解。在第（b）部分中，我们看到通过将学生能够找到大形状的区域的第一次尝试来清楚地改善，通过将其分解成更小，熟悉的地区。该学生赢得了对此尝试掌握的认证。

# 学生表演

我们要简要介绍一下我们的学生所做的方式。到学期结束时，我们42名学生中的37名掌握了我们18个学习成果中的至少17名，没有学生掌握了少于14个学习结果。学生们首次尝试掌握了许多概念，大多数学生最多需要掌握一个概念。

## 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

我们要求学生按照1到5的等级回应以下三个陈述，其中1个意思是“非常不同意”和5意思是“非常同意”。

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

Below are the results.

我们注意到两名学生在调查的下一部分中写了绝大多数的积极的事情，而是用这些问题的“非常不同意”回应，因此我们推断对李克特调查的这些反应与他们的预期回答相反。

### Free Response Questions

我们也想让学生用自己的话语，他们的掌握分级的经验有机会分享他们的经历。我们向学生询问了一些关于他们掌握评分经历的问题。我们还要求他们将这些评估与他们在数学300中的基于积分评估进行比较（也是关于未来小学教师的数学的先决条件课程），并反思这些经验如何影响他们未来的教学。

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

### 我们预期的结果

内容理解：作为教练，我们注意到，与我们为未来教师教授的前期数学课程的学期相比，我们的学生所上的工作具有极高的品质。一些学生评论了他们的个人感受，他们在基于积分的系统中比他们在课程中更远的课程。

- “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.

**考试作为机会：**掌握分级也影响了至少6名受访者对考试的影响。特别是，他们认为考试是一个机会，以表达他们的知识和理解，而不是克服障碍。以下引用代表了这些答案。

- “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。“

在我们的经验中，学生有时将数学考试视为敌对，不公平，或者我们的目标是指导员是为了欺骗他们或让他们获得较低的等级。对我们来说，这些反应表明，学生认为这种评分计划是更友好的，更有利于让他们展示他们的理解。

**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！”

**Challenges:**我们非常重要，以确认并非我们收到的所有反馈都是积极的。那些发现掌握分级问题的人中有两个共同主题。首先，当他们认为他们只误解了一小部分结果时，学生并不喜欢重做结果，或者只是一个小错误。其次，学生们仍然希望他们可以为他们确实表现出良好理解的想法来获得一些部分的学分。少数学生评论说，拥有多次机会导致他们关心任何个人评估，所以他们少研究过。我们还指出了一种趋势，即持续“传统”数学课程的学生，即微积分序列课程，在他们做出相当小的错误时，似乎更令人沮丧。

# Discussion

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.

我们很高兴看到了一些预期的结果。学生们普遍报道对考试感到不那么焦虑，因为他们知道他们会有多种机会来展示他们所知道的。学生认为评估让他们有机会准确展示他们的知识。他们还反映了他们掌握的经历以及如何通知他们未来的教学。

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.

更多到这一点，我们已经看到了证据证明掌握影响了他们未来的经历。我们这里突出了一条证据。在2020年秋季，学期在我们实施掌握评分后，我们的一些学生为未来的师父们带着我们的同事，凯西Quigley拍摄了另一个数学。在学期的一个观点时，Kelsey为她的学生提供了一个有机会在他们的第一次考试中获得积分。正如她讨论与他们的后勤考虑因素一样，学生建议他们应该获得积分的方式是重做他们单独的问题并没有做好，而不是班级的问题，整体上没有做得好。他们说：“这就像你掌握了你错过的概念与回来并做你明白的概念。”Kelsey留下了这项学生通过他们在课程中基于掌握的测试的经验获得了这种观点。

前一节中提到的挑战对于解决重要意义。虽然学生呈现的许多挑战是固有的掌握评分，但我们觉得教师可以做一些事情来解决这些问题。

- 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。
- 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.
- 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。

我们所描述的不是方法掌握的唯一方法。一些实现有多个“级别”的掌握，所以在那个感觉中，学生赢得部分信贷。

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.

**Acknowledgements**

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。

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