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Qualitative growth in math, as showcased by work drawn out, is an important indicator of academic progress in the classroom. I provide numerous opportunities for students to prove they understand a mathematical concept by requiring them to show their steps and process used to get their answer. If students get answers correctly but do not understand the process to solving a problem and why it works, they will struggle when faced with more complex numbers in high school because they will have no knowledge of a process to apply. At the beginning of some of math units this year, I give students an entry ticket question (requiring all work shown) that gauges their understanding of forthcoming content in a unit. At the end of those same units, I give students the same question as an exit ticket to see how they were able to apply their new knowledge and skills.
Student Work Samples: Beginning of Unit
Student Work Samples: Beginning of Unit
At the beginning of Unit 3 in late September, I decided to start the unit with a question I hoped students would be able to answer effortlessly by the end of the unit. I strategically chose a question that tests students' understanding of two new concepts we planned to cover in the unit: the distributive property and solving two-step equations. I came up with the following mathematical equation for students to simplify and solve: 2 (x + 1) = 36. Another reason I thought this would be a solid question for students to try to solve was that students were briefly introduced to one-step equations and the distributive property in 6th grade, so they had some familiarity within the domain of algebraic reasoning and thinking.
As shown below, I presented students the rubric that would be used to score their entry ticket and emphasized to them that they would receive the same question as an exit ticket (with the same rubric) at the end of the unit. Their entry ticket scores were not submitted as a grade. After students handed in their entry tickets, I reviewed the responses and provided individual feedback to each student, but I intentionally avoided telling them the method most effectively used to solve for x because I wanted them to figure it out on their own during the unit. Therefore, my individualized feedback to them was rather vague, but I did praise them for correct practices in their initial responses.
This is what I wrote on the whiteboard the first day of Unit 3.
Let's look at 3 student sample entry tickets that I received that day!
These entry ticket samples were selected strategically. I selected students across grade-level placements according to the BOY diagnostic. Student #1 placed in 7th grade on the BOY diagnostic (high/Tier 1), Student #2 placed 6th grade (medium/Tier 2), and Student #3 placed 5th grade (low/Tier 3). Included with each student entry ticket is the rubric used to grade each student's work. The rubric contains the student's scores for each of the three ticket objectives (left column of the rubric). I provided individualized student feedback in the rubric itself, as shown in each of the boxes. Since the entry ticket was used to gauge student knowledge prior to instruction, the rubrics were not used for grading purposes.
Student #1
Student #1 placed in 7th grade on the BOY diagnostic. This student is strong in math, and it is shown here. This student's impressive guess and check skills enabled the student to come up with the correct answer. However, the student's lack of algebraic steps in the work demonstrates a lack of breadth and depth in the algebraic reasoning domain. In addition, the student was unable to provide evidence that they understand the distributive property.
Student #2
Student #2 placed in 6th grade on the BOY diagnostic. This student impressed me with a correct first step of dividing the left side of the equation by 2 to cancel out the multiplication but was unsuccessful in the follow-through for the problem. It is evident this student has seen equations before and started to do the inverse operations, but has not received enough practice solving equations algebraically since they multiplied the right side by 2 instead of dividing. In addition, this student failed to guess and check at the end of the equation because obviously, 68 would not be a correct solution when plugged back into the equation.
Student #3
Student #3 placed in 5th grade on the BOY diagnostic. This student demonstrated understanding of distribution as he knew to multiply the 2 by both terms inside the parentheses, but he demonstrated a lack of algebraic reasoning when he combined unlike terms together.
Student Work Samples: End of Unit
Student Work Samples: End of Unit
At the end of Unit 3 in late October, I gave students the same question (on the same pink paper) that I gave them at the start of the unit. Coming full circle like this was exciting to me because I thought students would be motivated to see how far they had come in their knowledge of algebraic problem-solving.
I projected this picture of my whiteboard from Day 1 on the Apple TV in my classroom so students knew it was exactly the same question with the same guidelines. The only difference this time is it was called an exit ticket!
Student #1
Entry Ticket 33% --- Exit Ticket 100%
Student #1 showed tremendous growth on this exit ticket. Student #1 demonstrated the distributive property with arrows, showed organization and algebraic logic as they listed out the equation and widdled it down, and showed depth in understanding inverse operations. An area of growth for me in my feedback to this student would be to remind the student to do a double check by showing me they know how to substitute 17 back into the equation to make sure it is correct. I should even add this criterion to the rubric for future students to ensure substitution is a part of our algebraic process.
Student #2
Entry Ticket 22% --- Exit Ticket 78%
Student #2 also showed growth on their exit ticket. Instead of multiplying and dividing the sides of the equation this time, this student knew to do the inverse to both sides and then continued doing the inverse to find the solution. The student still lacks the layered organization I expect throughout their work, so I tried bringing in some real-world feedback to motivate them to be more methodical and explicit in their presentation of work. Again, an area of growth for me in my feedback to this student would be to remind the student to do substitution at the end once the variable is solved. I should definitely add this criterion to the rubric for future students to ensure we are doing our due diligence when we solve for variables.
Student #3
Entry Ticket 22% --- Exit Ticket 89%
Student #3 also showed growth on their exit ticket. The student learned this time around that unlike terms cannot be combined, and didn't make the same mistake here. While it is evident the student could do a better job organizing the algebraic steps in the same format (in the format of an equation) throughout the problem, the student's algebraic reasoning improved tremendously and shows the student can identify the distributive property and solve for variables using inverse operations.
Conclusion
Conclusion
When comparing these three students, it is clear that all students made significant academic gains in the domain of algebraic thinking and solving. It is also true, however, that these three students could do a better job showing their work in an explicit, organized manner to effectively demonstrate their understanding that an equation gets simplified before it gets solved. Providing scaffolding and reminding students to use substitution to double-check their answers will be a focus of mine in my math instruction. I look forward to implementing both quantitative and qualitative asessments for my students.
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