In preparation for the upcoming semester for my algebra 1 sections, I analyzed student performance using a subset of questions from their final exam. Select questions covered the primary content for first semester, linear equations, while others spanned various prerequisite knowledge in arithmetic and pre-algebra. I originally intended to use a weighted average of these two subsets for students’ final exam scores, however, this lowered scores as compared to earlier cohorts. So, I gave students the higher score from the weighted average or their score on the entire exam. While this raised the average score, it did not appreciably change grades in the course, as only one student migrated across a grade boundary from a D+ to a C; his score on the linear equation part of the exam seemed anomalous anyways, so this was fitting.

### Linear Equation Knowledge Compared to Arithmetic / Pre-algebra Knowledge

### Segmenting Outcomes

As expected, higher scores on linear equation questions tended to occur with higher scores on the pre-algebra questions. To highlight scores for students who scored at a below basic or higher level of understanding on both linear equations and pre-algebra, I created a quadrant chart overlay where green depicts passing scores on both aspects (linear equations and arithmetic / pre-algebra), yellow depicts passing the prerequisite questions only or the linear equation questions only, and red depicts failing both aspects.

Adjusted pass / fail boundaries at 50% for each dimension are shown as dashed lines in the figure above. Using the adjusted boundaries, the distribution of students in each quadrant were determined as: green (30%), yellow (42%), and red (27%). I interpret this distribution to show that just under a third of students demonstrated a basic or higher understanding of linear equations on the final exam, a little over 40% have the potential to pass the final exam on a retake after relearning the content (with 15-20% within striking distance), and just under a third require significant intervention in prerequisite knowledge and skills before they have any chance of passing the course.

### Grouping for Differentiated Intervention

To help decide how best to identify students for targeted intervention with differentiated content, I boxed in three groupings of student scores as follows.

I refined this first cut in the next figure, refining the grouping resulting in four different groupings.

Additionally, I coded scores by period in the first figure and by grade level in the second.

### Coded by Section / Period

The most advanced students are shown in the green box frame. Advanced is a relative term, however, as some of these students still struggle with basic arithmetic operations. At the same time, they have the greatest potential to master new, or more advanced, content much more rapidly than the bulk of students.

Students who are closer to demonstrating understanding of linear equations while possessing stronger prerequisite skills are boxed in the yellow frame. Hopefully, they can fill any gaps in their understanding in a short period and move forward with newer content.

Students in the orange boxed frame seem to need a little more intensive intervention in prerequisites before they solidify their understanding with linear equations.

Lastly, those boxed in the red frame seem to require intensive intervention in prerequisite skills in order for them to move forward.

None of my first period students are in the green group, although a couple are close. All but one student in the green group are 9th graders.

### Coded by Grade Level (8th – 12th)

### Coded by both Section / Period and Grade Level

This last figure depicts each score by period and grade level for future analysis and reflection.

While there is benefit to my investing time in constructing the preceding figures and contemplating how best to serve each student in improving their proficiency in mathematics, it took me several hours to collect, manipulate, graph, and interpret the data. All of this during my winter break. The time that most who have no experience teaching believe is truly spent on leisure activities…

Thank you for sharing such an in-depth analysis of your students. I would be interested in learning your plans for differentiation. My 7th grade classes have an extremely wide range of abilities and it is a challenge to meet the needs of all learners every single day. I’m finding that it takes an incredible amount of planning and preparation to have those resources readily available especially for those students who are the outliers where they need even more extension or remediation.

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Hi Mary. You are absolutely correct. Differentiating instruction is nigh impossible without access to a large pool of great resources easily associated with specific levels of understanding for a variety of topics. Whew! I do not have such as pool of material available in any easily accessible or usable form. However, I am going to receive a set of dedicated Chromebooks soon where I plan to use Khan Academy as my source for differentiated instruction. I hope this turns out to be an effective approach. Only time will tell. Stay tuned for posts about my progress with them.

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