Introduction: Assessment Philosophy
Please click link: Meta-Reflection
Academic Learning Standards
Targets-Students will be able to add whole numbers fluently and accurately with regrouping ones and tens. This learning target is based on WA state PE 3.1.C: Students will fluently and accurately add and subtract whole numbers using the standard regrouping algorithm. Many third grade targets are pre-requisites for students to conceptually understand addition and subtraction with regrouping. If students are able to fluently and accurately add whole numbers they have learned strategies for solving math facts and have a strong understanding of place value. These targets will continue to be important skills as they solve multi-step addition story problems. Even if they understand the story type and what type of equation to use, they will not be able to accurately solve the story problem without the ability to add fluently and accurately with regrouping.
Connecting Learning Targets to Instructional Design
- Activate prior knowledge: numeracy, tens facts, facts below 10.
- Apply knowledge to practice following all steps for addition without regrouping with place value mat (PVM) and t-chart.
- Scaffolded Instuction (I do, We do, You do)
- I do: Introduce learning target, model solving a problem with regrouping using PVM PlaceValueMat and t-chart, follow the same steps for addition without regrouping, stop after adding ones column, show that the number of ones is 10 or greater. Discuss why we can’t put a 2-digit number in the ones column, model regrouping ones on PVM and t-chart, model tens, solve starting with ones digit.
- We do: Repeat same steps with students.
- You do: Students practice problems with regrouping ones and tens using PVM and algorithm.
Connecting Learning Targets to Assessment Design
I designed assessments that connected with the learning targets and showed student progress. Each assessment modeled the format I used for instruction and guided practice so students were clear about the skills needed to meet the targets.
Pre-Assessment: Artifact Pre-Assessment was used to assess prerequisite skills. The pre-assessment matches the summative. Both expect students to line up the problem vertically, add accurately and regroup when needed.
Formative and Self-Assessments: I used formative assessment throughout each activity to determine if students were ready for the next step. This included observation of students using the materials to solve problems on their PVMs, asking questions: “How did you know to regroup? Why did you put the digit in the tens column?” and analyzing students’ independent practice worksheets to determine next steps.
After the initial instruction, each student received a self-assessment (Example: StudentAExitSlip) where they were asked to write the learning target in their own words and write any questions they had. The self-assessment included a checklist to check their understanding. The self-assessment shows students’ initial understanding of the learning targets.
Throughout instruction students also received guided and independent practice. First, students had to show the conceptual model on the PVM by lining up the blocks according to place and regrouping the blocks when needed. I used observation and interviews so students could explain how and why they regrouped. Students then applied this model to the algorithm and explained how they matched. This artifact, Formative1, shows students’ work and the interview notes I took as they solved the problems and explained their thinking. Once students were able to explain, they solved the algorithm without the PVM or place value blocks. Artifact, Formative2, connected to the target because they had to solve the problem correctly and explain if they needed to regroup. In addition, I checked for understanding using whiteboards, hand signals, questions, think/pair/share, name sticks, and observation. I shared the results with students and provided individual and small group instruction for specific skills students needed support with.
Summative Assessment: Each student completed the same summative assessment (Post-Assessment) similar to the pre-assessment and had to solve varied addition problems to show they knew when and how to regroup. These problems also included 3-digit and 2-digit to assess if they were able to line up the digits vertically according to place.
Evidence of a Variety of Assessment Strategies to Address Learning Targets Description of how the pre-assessments, formative assessments, and summative assessments (described above) match multiple assessment types.
My assessment design shows both formative and summative assessments used to determine student understanding. The assessments are connected with the learning targets and show student progress. Each assessment models the format I use for instruction and guided practice so students were clear about the skills needed to meet the targets. The pre-assessment matches the summative assessment. Both expect students to line up the problem vertically, add accurately and regroup when needed. Formative assessments were then used to determine next steps for instruction. Each student received guided and independent practice. First, students had to show the conceptual model on the place value mat by lining up the blocks according to place and regrouping the blocks when needed. Students then applied this model to the algorithm. This is an example of a performance assessment. Chappuis et al. (2012) states, “Performance assessment is used to judge demonstrations, and products, or artifacts, that students create” (p. 90-91) Students were asked to explain how the model on the place value mat matched the algorithm. Once students were able to explain, they solved the algorithm without the place value mat and blocks. After students completed the problem, they were asked to explain how and why they regrouped. This artifact, Formative1, is an example of students’ work and the interview notes I took as the students completed the problems and explained their thinking. The interviews are an example of a personal communication assessment. Chappuis et al. (2012) states, “Personal communication is when we find out what students have learned through structured and unstructured interactions with them” (p.91). This artifact, Formative2, is another example of a formative assessment that connected to the target because the students had to solve the problem correctly and explain in writing if they needed to regroup. This is an example of written response. Chappuis et al. (2012) explains that written response allows teachers to not only determine if students know the correct answer, but also how students know. This minimizes students getting the right answer for the wrong reason (p. 95). This often happens when teaching regrouping because students have learned a trick for regrouping but don’t conceptually understand how place value is used to determine where and why numbers need to be regrouped.
When teaching regrouping, it was important that both written response and personal communication were a part of the process to help students develop conceptual understanding. According to Chappuis et al. (2012), “Personal communication leads to immediate insights about student learning, they can reveal misunderstandings and trigger timely corrective action” (p. 92). When interviewing students, these types of questions help me understand students’ thinking: Why did you line the digits up that way? Do you need to regroup? Why or why not? When you regrouped your 15 ones how did you decide to move the 1 to the tens place instead of the 5? If students are not able to explain each step they may have difficulty with larger numbers and subtraction with regrouping. When students are asked to explain each step they are learning to problem solve. Students realize they made an error because they aren’t sure how to explain. They then go back to the place value mat to try again. Students use critical thinking to progress toward the target because they have to explain their thinking, what they learned, and what doesn’t make sense.
Evaluation Criteria for Multiple Assessments
These rubrics were used for the pre-assessment and post-assessment interviews with students. During the interviews, students were asked to solve two different addition problems (2-digit and 3-digit). Students were allowed to use the place value mat and blocks to help them but they were not required to use them. To receive a point the student had to: have a complete answer, line up the columns correctly, show the correct regrouping (ones to tens place in Problem #1 and both the ones to tens and tens to hundreds place in Problem #2), and explain the regrouping correctly. This assessment required both performance and personal communication. Even if students were able to show the regrouping correctly, they could have memorized the procedure. Therefore, students were also required to explain the regrouping. The prompts on the rubrics are there to guide teachers in asking questions that will determine whether students understand why they regrouped and the process of regrouping. For example, the question, “Can you explain why you put that there?,” is asked when a student regroups ten ones to make one ten and puts a “1” above the tens place. The student needs to explain that the “1” represents one ten and it was regrouped because there were 10 or more ones in the ones column.
When students practiced completing addition problems they used the StepsforSolving as a checklist to make sure they completed each step for solving addition problems. Students also had to assess their understanding of the steps in Formative2 by checking if they had to regroup, explaining how they knew, and writing the steps for solving. Students had practice working in pairs to assess each other’s work. Students took turns being the “teacher” and the “student.” The “student” would solve a problem on a whiteboard and the “teacher” would use a checklist, PartnerChecklist, that matches the checklist from the interview rubrics to check the student’s work and ask the student to explain how he or she solved the problem.
What I Learned from Multiple Assessments
- Analyzing data from multiple assessment artifacts
- Using analysis to differentiate instruction and determine next steps
- Multiple opportunities for students to master learning targets
- Examples of student work at various levels (Student A, Student B, Student C)
Pre-Assessment- Analyzing the pre-assessment showed me that all students needed support with place value. The pre-assessment artifact, Pre-Assessment, showed that Student A was able to line up the digits but added incorrectly, Student B needed more practice lining up digits and Student C needed practice identifying where each digit went when regrouping.
Self-Assessments- Student exit slips completed after initial instruction shows students restating the learning target in their own words, asking questions, and assessing their understanding. This has made a positive impact because students see the skills they need support with. The examples below are students’ self-assessments at three different levels.
Student A: Artifact: StudentAExitSlip
This artifact shows that Student A is unclear about the learning target. In the artifact she explains that the target was to write the 2-digit number on the side. When I modeled regrouping ones on the t-chart I showed students how to check and make sure the correct digit was regrouped when there is 10 or more ones. For example, when adding 9+3=12, I would write the number 12 on the side of the equation before regrouping. I would label each digit with the correct place and stress that since 12 has one ten it needs to go in the tens column. This helped students choose the correct place to put each digit. Student A didn’t make this connection to regrouping and was confused. She checked, “I am stuck and need some help.”
Student B: Artifact: StudentBExitSlip
This artifact shows that Student B was able to explain we were adding 2 digits and checked, “I can explain the learning to someone else and am ready for a challenge.”
Student C: Artifact: StudentCExitSlip
This artifact shows she was able to state the target because she writes “to regroup on mat and sheet” but she doesn’t know where to put the digits when she needs to regroup because she writes, “Where do some of the numbers like 0, 2 go?” and checks, “I almost got it but need more practice.”
Summary of Student Understanding: The exit slips showed me that Student B and Student C were able to write the target, however this is not an indicator that they are proficient. I continued to use additional formative assessments to target specific skills each student needed to show understanding.
Artifact: FormativePracticeA, Example 1
Analysis: I knew she needed more support with solving facts above ten because she used her fingers to count and was often off by 1 or 2 when adding larger numbers. In the artifact, Example 1 shows her solving 562+27. She incorrectly added the ones column (7+2=8). You can see off to the side she was trying to make a ten but she had difficulty adding the remaining part.
Artifact: FormativePracticeA, Example 2
Analysis: Example 2 (problem #1) shows her adding 18+39. Her tens answer was inaccurate (9+8=18). She needed more practice with the “Make a 10” strategy to quickly solve and recognize facts that are above 10. These are facts she needs to know when regrouping.
Artifact: FormativePracticeA, Example 3
Analysis: Example 3 shows Student A needs practice with place value because she was able to solve the problem but wasn’t able to explain why she needed to regroup. Her explanation says she needed to regroup because it’s an adding problem. This doesn’t show that she understands that 13 ones doesn’t fit in the ones column because it has a ten.
Differentiated Instruction & Evidence of Progress: Student A practiced solving facts above 10 first on the hundreds chart with the same colored strips she used to learn her tens facts and facts below ten. The artifact, FactsAbove10, Example 1, shows the next step. In this artifact, she is using the same model but coloring each addend. She then wrote the algorithm to match the picture. The artifact shows 9+5 is the same as (9+1) +4=14. The fact 9+1 uses the first addend 9 and part of the second addend 5 to make a 10. Since she knows her facts below ten she knows that 4 is the other part that goes with 1 to make 5. So her new problem is a teens fact she knows, 10+4 =14.
Student A had extra practice identifying the value of each place and regrouping 10 ones to make a 10 and 10 tens to make 100 with place value blocks. We then transferred this knowledge to solving an addition problem with regrouping. She could explain that when there were 10 or more ones she needed to move the ten to the tens place. We practiced building each number on the PVM with place value blocks. She then combined the blocks together starting with the ones. After adding each place she had to explain if she needed to regroup and explain why or why not.
Artifact: FormativePracticeB, Example 1
Analysis: Example 1 shows she needs support with place value because when she lined up a 2-digit number with a 3-digit number (562 + 27) she placed the 2 tens and 7 ones underneath the 5 hundreds and 6 tens and added a 0 in the ones place.
Artifact: FormativePracticeB, Example 2
Analysis: Example 2 shows she needs support with writing numbers in standard form because five hundred seven is written 570. She needed more practice with identifying each place in a number starting with the ones place.
Differentiated Instruction & Evidence of Progress: I had Student B label each digits place in the numbers she was adding before lining them up vertically to make sure each digit was lined up in the correct place. The artifact, FormativePracticeB, Example 3, shows how she checked off each digit in the algorithm as she represented the numbers with place value blocks to make sure the correct blocks were used to build each digit.
Artifact: FormativePracticeC, Example 1
Analysis: Student C was able to line up the digits correctly by place but she wasn’t sure how to regroup when she had 10 or more ones or tens. Example 1 shows she got 18 ones but left the 1 ten underneath the ones column instead of regrouping to the tens place. In the tens column, she wrote 21 tens underneath the tens column instead of regrouping the 2 to the hundreds place.
Artifact: FormativePracticeC, Example 2
Analysis: In Example 2, she had difficulty identifying the thousands and ten thousands place. I knew she needed support with the value of each place in order to understand regrouping.
Artifact: FormativePracticeC, Example 3
Analysis: In Example 3, in both problems, when she got an answer of 10 she was no longer putting both digits as her answer under each place. However, she was still not sure where to put a regrouped ten or hundred because she was not regrouping. Instead she was leaving the extra digit out of the equation. In Example 4, she added 4 tens and 8 tens and got 12 but she put the 1 above the tens column instead of regrouping to the hundreds and the 2 above the ones column. She then went back and added 2 to her original ones answer of 6 and changed it to 8. This is evidence that she knows two digits won’t fit under the tens column but she isn’t sure how to use place value to regroup correctly.
Differentiated Instruction & Evidence of Progress: I reviewed place value with Student C and she practiced placing correct blocks in each place (ones cube, tens rod, hundreds flat). I then gave her a number and she had to represent it in multiple ways on the place value mat. For example, 27 can be 27 ones or 2 tens and 7 ones. We then practiced addition problems without regrouping using the PVM only so she could see the ones cubes being combined with ones cubes, tens rods with tens rods, and hundreds flats with hundreds flats. I then eventually added the algorithm on the t-chart so she could see how the PVM and t-chart matched. This artifact, Formative1 (Student C), shows her lining up the tens and the ones correctly. In the 2nd problem (245+87), she was able to line up 87 underneath the tens and ones place. She also explained that since 87 isn’t a hundreds number it can’t be lined up under the hundreds place. I then modeled addition problems with regrouping on the PVM explaining aloud why tens ones can’t fit in the ones column, ect. After she practiced regrouping and explaining why she needed to regroup, she had practice matching the problem on the mat with the algorithm on paper.
Summative Assessments- Evidence of Student Learning
Student A- Artifact, Post-Assessment (Student A), shows her representing the problem in multiple ways with regrouping. This shows she understands place value and it is not a memorized procedure.
Student B- Artifact, Post-Assessment (Student B), shows that her digits were lined up and regrouped accurately.
Student C- Artifact, Post-Assessment (Student C), shows she is able to accurately regroup ones and tens because she was regrouping the numbers in the correct place. She explained that she needed to regroup because the number had tens and ones and she needed to move the tens digit to the tens place.
Summary of Student Learning
The assessment results show that continued practice is needed to develop fluency. These students need many opportunities to apply regrouping so they can solve algorithms fluently. I will continue to ask critical thinking questions so students are explaining how and why they are using certain steps. I will continue to use the PVM in the future to teach regrouping to the hundreds and thousands to develop conceptual understanding.