# Qualitative Growth

I am showcasing my students’ improvement in quality of answer on word problems, specifically “analyze another student’s work” problems, to demonstrate that they have qualitatively improved over the course of the year.

## Introduction

For qualitative growth, I analyze several of the same students doing a work analysis task in November and in February. All students showed improvement in their ability to explain their reasoning. Analyzing another person’s work requires students “evaluate”, which is higher-order reasoning on bloom’s taxonomy. This is a common mathematical task that requires students not only identify an error in a problem, but explain why this is an error.

I chose these examples because they are demonstrative of “typical” qualitative growth in my math classroom. Students do not write extensive essays in 7th grade math, particularly when we are working with such a stark mathematical achievement gap. However, I did a substantial amount of work throughout the year on teaching students how to show their work and how to explain their reasoning. Error analysis is a very typical higher-order question that students need to be able to do in both school and life, and something we did regularly in the classroom. I intentionally chose two examples from exams because this shows students’ 100% independent work without teacher correction. I also intentionally chose students who began the year both low and high in mathematics to showcase that all students grew.

## Questions

When I had an error analysis be a question on an exam in November, I was surprised when no student gave an answer more complex than “Juan is correct because he followed the steps” or “both students got the same thing because they are both correct”. As a result, we spent a substantial time as a class going through what it means to explain your reasoning in mathematics, since many of my students were not used to writing complete sentences for a math class. By the time I analyzed student work where students had to explain error analysis in February, the vast majority of students improved their ability to explain their reasoning dramatically.

## Rubric

This standard rubric has been used throughout the year to grade student’s free response work. Students were initially surprised that a correct answer without explanation would still receive a 1/4, but quickly adapted to being pushed to demonstrate their reasoning skills. Through practice and clear communication about free response expectations, all students dramatically improved their quality of free response answers throughout the year. Students received feedback to both free response questions showcased using this rubric.

## Student A

Student A began the year on a third grade math level, and was on a similar level for reading when I spoke with their English teacher. Student A struggled with mathematics and reading, and began the year with a negative relationship towards both. They did not understand how to explain their mathematical reasoning in the written form, and earned the lowest score possible on the free response question on their exam in November. They got this score because “cause jessa and juan are both correct” did not indicate that the student knew any mathematical strategies to know their answer. I wrote on their rubric that they must explain their work in order to get any credit on a free-response question.

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Student response “cause jessa and juan are both correct”

I worked with Student A on breaking down word problems and explaining their reasoning with words throughout the semester. By the time they took an exam with a similarly styled question in February, their ability to explain themselves improved dramatically. They used complete sentences and structured their reasoning clearly. The student clearly explains that Matt did not multiply correctly, and proceeds to explain how Matt should have multiplied. The student concludes this lead Matt to the wrong answer. The student did not answer the second part of the question, so they earned a 2.5/4. Had the student also showed the correct answer, they would have gotten full credit, since they did provide a robust explanation for exactly where the sample student went wrong. This is still a dramatic improvement from November, when the student did not explain further from “because they are correct”. The student showed their work, used their words, and created a logical argument.

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Student response: “Matt messed up on the second question. How did he? Matt did not do the first problem right he did 4(x+2)=30 when he was supposed to do 4(x+2) [with arrows]=30 4 times x is 4x and 4 times 2 is 8 so its 4x+8=30. So that led him to the wrong answer [does not show work for correct answer]”.

## Student B

Student B began the year on a fourth grade math level, and lower in reading. Student B, like most of their classmates, was not accustomed to fully explaining their reasoning in math. They did not elaborate their reasoning beyond “because they both got the same answer”, so they earned the lowest score on the rubric. I made it clear in the rubric and my conversations with the student that there is no credit for work without any mathematical reasoning.

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Student response: “because they both are basically got the same answer”

Student B dramatically improved their written reasoning on their test in February. Student B showed every mathematical step on the left, then described what that mathematical step meant in words on the right. They explained how messing up on the first step meant that every step afterwords was incorrect, which is more than the prompt originally called for. However, like Student A, Student B did not fully answer the second part of the question. Had they shown the correct answer, instead of just saying the correct answer was not 7, they would have received full credit. They improved to a 3/4.

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Student response: “Matt did not distribute 4 among x+2. Madd  did not subtract 8 from 8 and 30. Madd did not divide the correct numbers to solve for x. X does not equal 7. [does not get final answer, but shows some steps]”

## Student C

Student C began the year on a fourth grade math level, but close to grade level in reading. They initially only explained their work in November by stating “Jessi because she was the one following the steps”, without further explanation. We worked on improving clarity throughout the year. They earned a one, the lowest score possible, on this question of their test in November. While this student was better than some others in indicating there should be some “steps” for a person to follow to get the right answer, their answer was too vague for me to know what steps were supposed to be followed or why they thought the steps were followed. The rubric and my conversations with the student urged the student to increase their clarity.

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Student response: “The one that was correct was Jessi because she was following the steps”

Student C dramatically improved their reasoning and explanation skills when they answered a similar question in February. They stated clearly “Matt’s calculations are incorrect”, then went on to clearly explain “He should have multiplied both x and 8 by the four”. They then show each correct mathematical step, and put the final answer back into the context of the problem, “Each sandwich was \$5.50”. While Student C still had room to grow with more precise mathematical language, as I noted to them on the rubric, they clearly and concisely answered the question and explained their mathematical reasoning. This earned the student full credit, although I did note on their rubric their answer could be even better by naming the distributive property.

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Student response: “Matt’s calculations were incorrect. He should have multiplied both x and 2 by 4. [Shows correct steps to solving problem]. Therefore, each sandwich was \$5.50”.

## Student D

Student D is an ESL student who began the year on a fifth grade math level. They initially explained their work in November by stating that “Jessa is correct because her diagram shows exactly points on the number line”. This received a 1.5/4 because the explanation did clearly reference a mathematical concept, but it failed to elaborate on how the strategy works or what “points on the number line” meant. I worked with the student and their ESL teacher to improve the student’s ability to fully explain their reasoning.

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Student response: “Jessa is correct because her diagram show exactly points on the number lines”

When Student D took a similar exam in February, they explained their reasoning much more clearly. The student clearly stated “the mistake she made  was her distributive property was wrong, she forgot to do 4 x 2”. This explanation demonstrated that the student understood mathematical vocabulary and identified precisely where the original was incorrect. The student went on to clearly show how to find the correct solution to the problem after fixing the error. The student earned a 4/4. This student’s growth indicated that work with his ESL teacher paid off, and he was better equipped with the tools to show the mathematical reasoning he clearly had in his native language.

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Student response: “The mistake she was was her distribute property was wrong she forgot to do 4 x 2 [shows correct math steps for answer, gets x= 11/2]”

## Conclusion

At the beginning of the year, all of my students were unaccustomed to being held accountable for explaining their reasoning in mathematics. Students initially struggled due to the newness of the exercise and reading difficulties. However, as the year progressed, all students grew dramatically in their ability to explain their reasoning both in how they showed their math and in words. Having a consistent rubric and going over this concept frequently in class helped substantially. All students had a more natural inclination to use their words and show their steps when answering math problems by the end of the year, which demonstrated substantial qualitative growth.

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