This is a course for preservice elementary teachers. The course was taught with a simple pattern of reading, lecture, and group-work following Mathematics for Elementary Teachers with Activities very closely. The only non-standard aspect of the course was a mastery based grading scheme. Students were assessed based solely on a Portfolio of their coursework. Students had to piece together evidence of mastery of a set of standards for each subject covered. The primary idea behind this is to ensure that students are reflecting on their own understanding and making an explicit connection between their coursework and the standards they will be expected to teach in the near future. This Portfolio (detailed below) was an idea that Michael Price shared with me in my time at University of Oregon.
The portfolio was divided into subjects. Each subject had assignments and a set of standards. These assignments were graded. Students then had to match these assignments to specific standards. If they successfully matched enough evidence to a given standard, it was considered mastered. The number of standards mastered determined a grade for each subject, which then determined their overall grade.
The subjects covered follow the structure of the chapters of the text.
Decimals
Fractions
Addition/subtracting
Multiplication
Division
There was also a bonus unit at the end of the term on coding using Code.org.
This class had three kinds of assignments
Homework: assigned weekly from text
Quiz: after each chapter
Activities: assigned in-class from text
Importantly, the class had no exams. The homework and quiz problems were both graded using the following scale, which was taken from the EMRN scale.
0 - Fragmentary
1 - Needs Revision
2 - Meets Expectation
3 - Excellent Example
Their total grades were never recorded anywhere, and students had to take it upon themselves to read feedback and decide on what homework problems to submit for regrade.
Each subject has 5 standards assigned from the Common Core State Standards (Indiana Academic Standards). For example, Addition and Subtraction had the 5 standards:
Students were not given any guidance on what problems corresponded to which standards. They were required to select evidence they felt matched a standard and give a brief written explanation. Evidence could be used for multiple standards if appropriate for both.
Each subject received a letter grade based on the number of standards mastered.
A - 4 /5 standards mastered
B - 3/5 standards mastered
C - 2 /5 standards mastered
D - 1 /5 standards mastered
F - 0/5 standards mastered
In order to prove mastery of a standard, students needed to provide two pieces of evidence. Evidence of mastery could be any of the following:
3/3 on homework problem
2/3 on quiz problem
oral defense of in-class activity
Students could submit homework for regrades as often as needed, and request a meeting to orally defend an in-class activity at any time.
Students were required to submit their portfolios before their midterm grade was due and before their final grade was due. In addition to these required submissions, there was optional submissions after each subject was completed. The Portfolio was graded as cumulative, so that the midterm grade was assigned using the same rules as the final grade. The Portfolio was a living document, that could be changed in its entirety up until the final due date.
After each subject received a grade, their overall grade was determined with the following rules.
Receive a grade of G if all subject grades were at or above the G level.
Receive a grade of G+ if 2 or more subjects were above the G level.
Receive a grade of G- if at most 2 subjects were one level below the G level.
For example, the following chapter grades would result in the given course grade.
B, B, A, B, B. Course Grade: B
A, A, C, A, B. Course Grade: B-
A, A, B, A, B. Course Grade: B+
This grading scheme was designed with the idea in mind that every subject is essential for a future teacher.
In addition to the intended purpose of the Portfolio, having students consciously match problems to standards also gave me some insights into students understanding of the subject.
Students read standards in a way similar that they read word problems: identifying only the key words without considering their relationship to one another in context.
The two standards 5.NF.3 and 6.NS.1 both involve division and fractions. The first standard is to identify the exact quotient of a division of two whole numbers as a fraction. The second standard is to interpret a division of two fractions. Many students were unable to distinguish between these.
The two standards 3.NF.3.D and 4.NF.2 both involve comparing and fractions. The first standard is to compare two fractions that either have common numerator or common denominator. The second standard is to compare two fractions that have different numerators and different denominators. Many students were unable to distinguish these.
Students did not pay attention to indicators of grade level when reading standards. Students frequently submitted the same piece of evidence for both standards, which indicates a lack of care when reading the standard.
The two standards 4.NBT.1 and 5.NBT.1 both involve the relationship between different place values. The first is to identify place values to the left as 10-times place values to the right. The second is to identify place values to the right as 1/10th place values to the left.
The two standards 3.NBT.1 and 5.NBT.4 both involve rounding numbers. The first is to round whole numbers to nearest 10 or 100. The second is to round decimals to any place.
Students have difficulty separating explanations from examples.
When giving evidence for 4.NBT.1 (understanding place values as 10-times the one to the right) students would provide examples of rounding or comparing. While both rounding and comparing rely on place value to explain why they work, giving examples that you understand the process of rounding does not show you know how to explain.
When giving evidence for 2.NBT.9 (explain why addition strategies work using properties of operations and place values) students would provide examples of correct addition strategies without explanations. Again, simply giving an example of the make-a-ten strategy does not explain why it works.
Students matched problem questions rather than problem solutions to standards. Some problems about operations are specific (requesting a particular strategy) while others are not. Students with distinct correct solutions could not use the same problem for evidence on the same standards. This is likely due to collaboration and carelessness.
It is clear to me that future iterations of this class will require more guidance for students to understand standards. This might be achieved through short activities at the end of each subject where students match sample solutions to standards or by having an activity at the start of term about carefully reading standards.