Knowledge of StudentS

KNOWLEDGE OF STUDENTS (KOS) refers to a teacher's understanding of students' ability to carry out educational tasks. KOS is particularly relevant to mathematics studies, and multiple studies have confirmed that observed that teachers who could more accurately predict their students’ mathematics performance (i.e., who had higher KOS) were those with higher value-added scores (i.e., had higher gains in students learning as measured by standardized test).

Heather Hill and Mark Chin's (2018) AERA paper (free) showed that a teacher's KOS was a more accurate predictor of teacher effectiveness on Value Added measures than: 1) teachers with higher mathematical knowledge and 2) teachers whose enacted lessons are better [as measured on an instrument Hill and Ball created with others called the Mathematical Quality of Instruction {MQI} ] (Hill et al, 2009). While KOS is easy to measure and can account for a lot of variance in teacher performance, the power of KOS has remained untapped for improving mathematical education.

Figure 1: Example from a teacher questionnaire used in Hill and Chin’s (2018) study that assessed teachers’ KOS. We refer to the constructs as a) KOS-C, b) KOS-P, and c) KOS-CWA.

This questionnaire does not give teachers KOS, but it primes them to consider how their students are thinking.

Figure 2. This is an example of an assignment that shows teachers’ generalized KOS data (compare to Figure 1). An instructional diagnosis of the given condition would tell the student "it looks like you multiplied the diameter with the rotations, but in doing so, you've confused circumference and diamter. Diameter is the distance across a circle, circiumference is the distance around a circle. You will need to calculate the circumference of the circle."

Table 2 provides a structured overview of the experimental process, discussed above. The No-KOS-Given column represents teachers that are not provided with any KOS. The KOS-Given column represents teachers that are provided information on problem difficulty and common wrong answers. The KOS+ID-Given column represents teachers that are provided the same information as the teachers in the KOS-Given condition as well as instructor diagnoses.

Testing KOS

Hill and Chin examined KOS on a a correlational basis, but KOS has yet to be studied on a causal basis. In order to demonstrate causality, Dr. Heffernan has proposed to run a Randomized Controlled Experiment.

Dr. Heffernan plans to have an experiment group receive KOS information before planning a lesson, and a control group to plan a lesson without KOS information. If the group randomly assigned to the experiment condition has higher levels of student learning, that would show that KOS causes more effective lessons. By randomly assigning teachers into both conditions, we hope to demonstrate that KOS can be effective on both novice teachers and experienced teachers who are adopting new curricula such as Illustrative Mathematics.

As shown in Table 1, teachers given KOS for a Class Topic will be informed of students’ general performance before they begin teaching their lesson. After the lesson, teachers will assign an “Exit Ticket” that will serve as Posttest 1. Immediately following Posttest 1, teachers given KOS will be told how their students performed, which may inform their homework development. The presence of KOS may lead teachers to create a different homework assignment than planned. In order to isolate the effect of homework, Posttest 2 will be administered when students arrive in class the next day. Then, teachers given KOS will be provided with information specific to their students’ performance, including CWAs. The presence of KOS-C may lead teachers to better orchestrate homework review and class discussion. Following this homework review and discussion, Posttest 3 will be administered to isolate the effect of this review. Finally, Posttest 4 will be administered with a one-day delay to assess student retention patterns. For the Class Topic on which teachers are not given KOS, they will follow traditional practices as shown in Table 1, with posttests administered in the same fashion.

A teacher’s report that includes KOS at the class-level and with regard to CWA. Using this report, teachers can quickly assess how hard each problem was and isolate the CWA to guide instructional strategies.

ASSISTments for KOS

ASSISTments is developing a Common Wrong Answer (CWA) feedback feature to the ASSISTments platform. This feature will increase teacher KOS, making it easier to clarify student misconceptions.

In the figure to the left, a teacher is able to view the CWA for an assigned problem. She is able to use this information to probe the class for an explanation of the misconception and to make remediation of the misconception a focal point of her homework review.