Throughout my internship, I used a handful of formative assessments in my classroom. I am going to discuss 4 of them and choose 1 of the 4 to analyze.

PART I: FOUR FORMATIVE ASSESSMENTS

The first formative assessment I used I called “YMCA.” During the notes, students were given a question to try on their own after doing a few examples with me. The question had two possible answers- “Y” and “M” or “C” and “A.” I let students work on the problem on their own for a few minutes then I asked them to stand up when they had their answer. Then I counted down “3,2,1, GO” and on “GO” the students held up the correct answer with their arms in the way they would if they were dancing to the song “YMCA.”

I chose this assessment technique because I tried to do a similar one the week before and it was chaos. I asked students to stand on the left of the room if they chose one possible answer and the others to stand on the right of the room if they chose the other. There was too much moving around and people talking instead of doing the problems. I thought that I needed to adjust the activity and reduce the amount of movement while still quizzing the class and having them get out of their seats. I think that this was a good adjustment and made the classroom much easier to control.

I chose this activity so that I could see who understood the notes and the examples I completed. This assessment could really go with any content or standards, as long as the notes include quite a few examples. Specifically, the content objectives this assessment went along with were finding the degree of polynomials, simplifying monomial expressions, adding and subtracting polynomials, and writing polynomials in standard form. The standards addressed were “MA.912.A.4.1: Simplify monomials and monomial expressions using the laws of integral exponents” and “MA.912.A.4.2: Add, subtract, and multiply polynomials.”

The second formative assessment I used in the classroom was a “3, 2, 1 Worksheet.” I asked the students to fill out 3 new things they learned, 2 things they are still struggling with, and 1 thing that will help them in the next class.

I chose this activity so that I could see who understood the notes and the examples I completed. This assessment could really go with any content or standards, as long as the notes include ample examples. Specifically, this assessment went along with the standards “MA.912.A.4.1: Simplify monomials and monomial expressions using the laws of integral exponents” and “MA.912.A.4.2: Add, subtract, and multiply polynomials.”

The third assessment I used in the classroom was a “POMS & Muddiest Point Worksheet.” I asked the students to give the three most significant points or facts that they learned that day/have been good at (for example, one answer could be “I know the degree of a number is 0.)” I then asked the students to give the three points or facts that they still need help with today/problems they have been having with the class lately. If students did not have problems with anything I let them write, “I understand everything.”

I chose this activity so that I could see what students thought they were good at and were confident in. At the same time, I could see what students were struggling with. It is obvious that students need help with certain concepts if they can recognize and name what they are having trouble with without me giving them a test and a grade. This assessment could really go with any content or standards, as long as the notes include ample examples. Specifically, this assessment went along with the standards “MA.912.A.4.1: Simplify monomials and monomial expressions using the laws of integral exponents,” “MA.912.A.4.2: Add, subtract, and multiply polynomials,” and MA.912.A.4.10: Use polynomial equations to solve real-world problems.

The last formative assessment that I used in the classroom was called the “You Try Worksheet” and this is the assessment that I will be further analyzing. This worksheet was to be completed during the notes. Throughout the whole year, students get “now you try it” questions in the notes to complete on their own after seeing a few similar problems beforehand. I wanted to take this to the next level when turning it into a formative assessment. Firstly, they complete the problem on their own. Then I go over the answer and solution. Then students have to grade how well they did on the problem with a plus, check, or minus (“plus” means the problem was very easy for them, “check” means they got the problem correct but they are not perfectly confident in those types of problems, “minus” means they did not know how to do the problem/did not get the correct answer). Then I also asked the students to write down any additional comments or questions about the problem and to tell me if anything confused them.

Like I said, students have been given “you try it” problems during the notes the entire school year. I chose this assessment technique because I noticed that many students wouldn’t do the “you try it” problems and would sit and stare until I went over the answers. I wanted the students to actually have a paper they would turn in to me for a grade so that they could not sit there and write nothing for these problems. I also noticed that they don’t always put “plus,” “check,” or “minus,” on their problems to determine how hard the problem was for them to complete. Grading the problem based on its difficulty is a metacognitive skill, and seeing many of the students opt out on this is not a good thing. I gave them a box to put their grade on the left to ensure that they will do this.

I used problems from 8.5 for this activity because it was meant to go along with the 8.5 notes, which were about factoring. Specifically, this activity went along with the standard “MA.912.A.4.3: Factor polynomial expressions.”

PART II: ADMINISTERING THE ASSESSMENT

I assessed 43 students in 9th grade Algebra 1. I allowed students to take as much time as they needed to finish this assignment, so some students who usually need additional time could work into their homework time in the class to finish it. The space left over to make additional comments and questions gave students a chance to say if the method of factoring they learned didn’t make sense to them. Those students who usually need more time to understand concepts than others had a chance to tell me specific steps they had trouble with. Other than that, I did not make any specific accommodations for students with differing needs.

The assessment went along with notes for factoring trinomials. There is a worksheet that the students received that explained the steps to take while factoring trinomials. The worksheet states:

1. Multiply the coefficient of the first and last terms (a & c).

2. Find factors of that number that add up (signs and all) to the coefficient of the middle term (b).

3. Rewrite the problem by splitting the middle using the two factors you found in step 2.

4. Factor by grouping 2 X 2

The students were asked to perform every step of this method while completing the problems on the You Try Worksheet.

PART III: RUBRIC FOR THE YOU TRY WORKSHEET AND STUDENT RESPONSES

Levels

Explanation of the levels

Description of student responses at the level

Extended Abstract

Students generalize the structure to make it new and more abstract.

Students clearly performed every step of the method listed above and then checked their answer by multiplying the factors and seeing if it matched up to the original problem

Analytical

Students integrate the ideas to create a meaningful structure.

Students clearly performed every step of the method listed above and got the correct answer but did not check it

Quantitative

Students can identify mathematical ideas in a quantitative way but cannot integrate these mathematical ideas during the task.

Students messed up on either getting the correct factors to match up with the coefficient “b” (Step 2) or they split the terms into 2X2, but they added the factors instead of multiplying (step 4)

Transitional

Students focus on only one aspect of the solution.

Students had multiple errors in their solutions, not just one repeated error that occurred in multiple problems

Idiosyncratic

This level is based on subjective reasoning with unrelated data and is affected by subjective beliefs and personal experiences.

Students’ answers had nothing to do with factoring trinomials

PART IV: RESULTS AND STUDENT WORK

Levels

Explanation of the levels

Number of student responses at the level

(Out of 43)

Extended Abstract

Students generalize the structure to make it new and more abstract.

13

Analytical

Students integrate the ideas to create a meaningful structure.

15

Quantitative

Students can identify mathematical ideas in a quantitative way but cannot integrate these mathematical ideas during the task.

13

Transitional

Students focus on only one aspect of the solution.

2

Idiosyncratic

This level is based on subjective reasoning with unrelated data and is affected by subjective beliefs and personal experiences.

0

Student work for each level is described below along with evidence.

Extended Abstract

Students got the correct answers, clearly split the middle, grouped correctly, and followed all of the steps of the factoring trinomials method above, and then checked their answers to make sure that their factors multiplied out to the given trinomial. Those responses looked like the one below.

Or they checked their work using a half-foil (multiplying the inside terms and the outside terms of the factors and seeing if it matches the middle term of the polynomial). Those responses looked like the one below.

Analytical

Students got the correct answers and clearly performed all of the steps of factoring the trinomials, but there was no indication that they checked their work or took their understanding a step above giving the correct answer. All of the responses looked like the student sample below.

Quantitative

Students messed up on getting the correct factors to match up with the coefficient “b” (Step 2).

1 quantitative student got #3 incorrect because they used incorrect multiples of 20 in their answer. Their response is below.

5 quantitative students got #4 incorrect because they did not choose multiples of 21 that added up to -4. Their responses were similar to the one below.

3 quantitative students got #5 incorrect because they thought 72 did not have factors that added up to -1. Their responses were similar to the one below.

4 students split the terms into 2X2, but they added the factors instead of multiplying (step 4). Their responses were similar to the one below.

Transitional

Students had multiple errors in their solutions, not just one repeated error that occurred in multiple problems.

1 student added the factors in the answer instead of multiplying them. They also said that #4 and #5 were prime because there were not any factors that can equal the middle term.

1 student had a problem with #1 and did not have a matching set of factors after they split the middle. They also started writing x’s along with c’s in problem #4 and really mixed up the answer.

PART V: REFLECT ON STUDENT UNDERSTANDING

I actually was surprised that so many students got all 5 answers correctly, and many of the quantitative students got #1 and #2 correct on the worksheet. Seeing as how they just learned how to factor, I think these were great results. What this tells me is that the method of factoring that they learned makes a lot of sense to them. When I was learning factoring in school, I basically just looked at the coefficients a*c and had to find what numbers would equal b, but then the hard part was figuring out how to write those factors together in the answer. I really think a lot of it was guessing and checking. This method I pasted above is more straight-forward with clear steps to take when factoring.

I especially thought that students would have a problem with #5 and finding the correct factors of -72 that added up to -1. I noticed that many people used a horseshoe method of finding multiples of 72 and they correctly used the multiples 8 and 9. I did not know that students would have this skill coming into this class.

However, quite a few students did not understand that factors are multiplied together in the answer. A good amount of students added or subtracted their answer (like they wrote (x+3)+(x+2) when the answer is (x+3)(x+2)). I thought that this wouldn’t be a hard concept to grasp.

PART VI: CHANGES AND FUTURE INSTRUCTION

Overall, I liked this formative assessment and I thought it was useful that it went along with the lesson notes to ensure that students actually completed the “You Try” problems. 28/43 students found the correct answers for all 5 problems, which isn’t bad when they were just taught how to factor. Because of the success and the ease with which I integrated this assessment into the classroom, there are not a lot of changes I would make.

The first change I would make is to tell the class that they should not get “prime” for an answer on this worksheet. This would make those students who put prime for an answer think about the factors of the numbers to add up to the middle term more. I think a few of them gave up on finding the correct factors so they just put “prime” for the answer instead of trying to figure it all out.

The next change I would make would be with the grading and using the additional comments space. I noticed that barely any students made additional comments on this assessment. I think that was because for the “muddiest points” assessment I told the class that if they did not have trouble with 3 things then they could leave the rest of the spots blank. The students probably assumed if they didn’t have a problem finding the answer that they didn’t have to say anything in the comment space. After I already graded the papers and wrote all over them, I came up with a great idea to promote metacognition and to take use of this space. I thought that for students who got a problem wrong, I could put an X next to the incorrect problem. Then I would ask them to use their homework time next week to look back at the worksheet and write in the comments space where they think they messed the solution up. I would want them to re-do the problem and find the correct answer, and then I could write constructive feedback on their papers. I already messed this idea up because I wrote correct answers all over their papers and already corrected where they went wrong. However, this is a great idea to use in the future in order to get students thinking about why their answer was incorrect and what they need to do to change it.

As far as altering the way I teach the class now that I know how the students think, I believe that the above idea would help them understand their mistakes much more. To address the students who added or subtracted their answer

(like they wrote (x+3)+(x+2) when the answer is (x+3)(x+2)), I think linking this mathematical concept to a familiar previous concept would be beneficial. I could state that when we find multiples of 6, such as 3 and 2, we can state that 6=(3)(2). So the whole number equals a factor times a factor. This should be familiar to the whole class. So when we factor polynomials, we want to say that the whole polynomial equals a factor times a factor. So 6=(3)(2) in the same way that x^2+5x+6=(x+3)(x+2). I think this idea would help those students understand why the factors in their answer need to be multiplied.