EDU 6524: Understanding vs. Knowledge (Blog 2)

An important component to implementing the backward design strategy is to differentiate between the words “understand” and “know.” If students are to understand a concept they must be able provide evidence by demonstrating they know specific pieces of knowledge. Understanding is different from knowledge because it requires thinking beyond the facts. Wiggins & McTighe (2005) state, “To understand is to have done it in the right way, often reflected in being able to explain why a particular skill, approach, or body of knowledge is or is not appropriate in a particular situation” (p. 39). Unlike facts or skills learned through memorization or drill, understanding requires students to apply what they learned to different contexts. The goal then is for teachers to design instruction that focuses on applying what students learn to larger contexts so they can transfer their knowledge. Otherwise, students will likely learn topics in isolation through memorization without understanding their relation to other ideas or the purpose for learning the content.

It is true that a lot of instruction in classrooms today is designed for coverage. This is a problem that exists in classrooms with pressure of state testing and so much content to cover. The problem with this approach is that it is not always engaging for the learner and often students forget content later because memorization does not stick. This is why a lot of re-teaching and review has to happen prior to the state testing and at each grade level. In order to determine if students’ understand, assessments must include evidence of transfer. Wiggins & McTighe (2005) state, “Getting evidence of understanding means crafting assessments to evoke transferability: finding out if students can apply their learning and use it wisely, flexibly, and creatively” (p.48). So students have opportunities for transfer, assessment that requires students to inquire and apply their knowledge is important. This type of assessment allows teachers to learn more about a students’ understanding. Otherwise students may have gotten the correct answer by accident or memorization.

Our math curriculum is designed to develop conceptual frameworks that will endure as they progress to higher mathematics. For example, students develop an understanding of place value prior to learning operations. In my own classroom, students are reviewing and building on concepts they learned previously until they can apply what they learned to a new concept or strategy. Students use many models to construct their understanding of math concepts. What is frustrating to teachers at the primary level is that students are developing strategies and concepts but are not prepared to solve an algorithm or the multitude of problem types on the MSP. As teachers feel the pressure of the MSP and revise the pacing guide they struggle with questions like, “At what point do we move away from the strategy and teach the content directly? Should we continue to spend time developing these concepts or ensure that all of the standards are taught prior to the MSP?” I struggle with the answers to these questions. It is understood that we might see a drop in scores in the primary grades but the conceptual models students are building will benefit them in high school. It is difficult to make decisions about what to teach next when we know we are ultimately responsible for our students passing the standardized test. In response to the question, “Do these innovative approaches produce better learners?,” Green  replies, “I don’t have a doubt in my mind, but it depends on what the measure is. If it is how successful they are going to be in the real world, definitely. If it’s how well they perform on standardized tests, not necessarily” (Diamond & Hopson, 2008, p.84). Our constructivist approach to math has allowed us to reach all learners at their level. The conceptual models have helped students who struggle to solve an algorithm or solve a fact. When comparing American and Japanese-style lessons Diamond & Hopson (2008) state, “The American lesson required mostly memorization and guesswork” (p.77). Green says she was a great memorizer and learned a process, but remembered nothing after the test. Her husband was a self-taught learner and read what interested him. Years later he is still able to carry on a conversation about his learning (Diamond & Hopson, 2008, p. 84). This validates the value of engaging all students in conceptual learning including students who can solve the algorithm quickly or have their facts memorized. They may continue to be successful throughout their education but will have difficulty transferring their learning beyond their schooling

Wolfe (2001) states, “One of the most effective ways to make information meaningful is to associate or compare the new concept with a known concept, to hook the unfamiliar with something familiar” (p.104). Accessing students background knowledge is one teaching strategy that can help students make connections and add to their schema. Some teaching strategies for accessing background knowledge may include KWL charts, graphic organizers, think-pair-share, preview of texts, and read alouds that connect to the content. This artifact, LearningSlip, is an example of one way I activate prior knowledge before a lesson. Students complete an exit slip before and after a lesson. Before the lesson students list prior knowledge that will help them with the new learning. This allows students to apply new knowledge to their existing schema in order to develop a deeper understanding of the learning.


Diamond, M., & Hopson, J. (2008). The Jossey-Bass Reader on the Brain and Learning. San Francisco: John Wiley & Sons, Inc..

Wiggins, G., & McTighe, J., (2005) Understanding by Design. Alexandria, VA: ASCD.

Wolfe, P. (2001). Brain Matters: Translating Research into Classroom Practice. Alexandria: Association for Supervision and Curriculum Development.

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EDU 6524: A Backward Approach (Blog 1)

Understanding by Design (Wiggins & McTighe, 2005) offers a backward way of thinking about curriculum that focuses on defining first what students need to know, rather than the activities and instructions that will be used. Design of assessment needs to take place first in order to clearly identify academic standards and the purpose for learning. This makes the learning evident for both the student and teachers. Teachers should first think about the evidence they will need to determine students’ learning. This can then guide the lesson planning and instruction. According to Wiggins & McTighe (2005), students should be able to answer questions during a lesson such as:

  • What are you doing?
  • Why are you being asked to do it?
  • What will it help you do?
  • How does it fit with what you have previously done?
  • How will you show that you have learned it? (p. 17)

Teachers can apply backward design to any curriculum or learning standards and at any grade level. There are three stages of backward design which include:

Stage 1- Identify desired results.

Stage 2- Determine acceptable evidence.

Stage 3- Plan learning experiences and instruction. (p. 18)

In backward design, the purpose is to teach to reach objectives, instead of teaching to cover content in a textbook. As teachers we need to think first about what students need to know and second about the instruction and activities that will help students meet the objectives. This way the design of instruction becomes more purposeful and students have a clear understanding of what they need to know and why. The backward design template serves as a way to guide teachers through this process to make sure their instructional plan aligns with the evidence they expect to see. Assessment evidence can take various forms including observation, tests, quizzes, journal entries, self-assessments or prompts (Wiggins & McTighe, 2005, p. 22). This artifact, FormativeAssessment, is an example of a formative assessment I have used in my classroom to assess the learning target: addition with regrouping. I started by identifying the desired results: students should be able to add with regrouping. Next, I created an assessment that included acceptable evidence that would tell me the student met the learning target. In addition to the student solving the problem correctly, he also had to explain if he needed to regroup and why and the steps for solving. Without this evidence I would not know if the student conceptually understood the regrouping process or if it was just a memorized procedure. If teachers can design their lessons based on what they expect students to be able to do by the end of the lesson, their instruction and learning activities will be purposeful and will align with assessment.


Wiggins, G., & McTighe, J., (2005) Understanding by Design. Alexandria, VA: ASCD.



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Standard 11 Meta-Reflection: Inquiry/Research

Standard 11 Inquiry/Research: Competently consumes and produces where necessary empirical data to guide educational practice.

Interpreting and Applying Educational Research I & II taught me the importance of looking at research and data with a critical eye. When conducting research to determine the effectiveness of curriculum or teaching strategies, there are several variables that must be considered to determine if the results are valid. There are two types of research: qualitative and quantitative. The type of research being used will depend on the question being asked.

Quantitative research is number-oriented. It works well with things that are concrete and can be converted into numbers for mathematical calculation and analysis. It disregards variables that are abstract such as emotions, perspectives, and beliefs. In quantitative research, the researcher tries to remove themselves as much as possible. Positivism relates to quantitative research because it is a position focused on objectivity. The belief is that researchers are able to prevent their own beliefs from affecting the data. Quantitative research is better suited for generalizability and cause/effect relationships. In quantitative research a hypothesis is already determined and variables are already clearly defined (McMillan, 2008) (Sprinthall, 2007). 

Qualitative research is based on people and their unpredictable behaviors, interpretations, and realities. Qualitative data is difficult to generalize because it is subjective. Because this data is based on feelings or perceptions it cannot establish a cause and effect relationship.  The researcher’s presence may also influence the group’s response. In qualitative research, the researcher gets involved and attempts to learn about the situation from the perspective of the participants/clients. Qualitative research connects with constructivism, which holds the belief that observers are apart of the system and that whether we like it or not our assumptions, beliefs, and values will effect our observations (McMillan, 2008) (Sprinthall, 2007).

It is important that teachers understand the threats to external and internal validity when conducting research and when using assessment data to make instructional decisions. Threats to internal validity are concerned with control of confounding and extraneous variables whereas threats to external validity are about generalizability of results. Threats to internal validity include history, selection, maturation, pre-testing, treatment replications, subject attrition, statistical regression, diffusion of treatment, experimenter effects, and subject effects. Threats to external validity include subjects, situation, time, interventions and measures. If students do not perform well on an assessment it is important to determine the assessments validity (McMillian, 2008). Were there extraneous variables that could have contributed to either high or low scores? Did the assessment questions match the standard being assessed? Was the testing environment or time of day a factor? Did some students receive different interventions than others?


Research Project Outline: I was able to apply my understanding of research to my own Research Project Outline to determine the effects of fluency instruction on students’ reading comprehension. This was applicable to research I may conduct in my own classroom when trying to determine if a specific teaching strategy or curriculum is effective. In this study I had to define the objective, connect my study to previous research, define variables, determine my research design, define internal and external threats, measures, and sampling procedures.


McMillan, J.H. (2008). Educational research: Fundamentals for the consumer (5th ed.). New York: Pearson Education, Inc.

Sprinthall, R.C. (2007). Basic statistical analysis (8th ed.). New York: Pearson Education, Inc.

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Standard 10 Meta-Reflection: Technology

Standard 10 Technology: Integrates current technology into instruction and professional communication/collaboration activities where appropriate.

In the course, Teaching with Technology, I learned multiple ways to integrate technology into instruction and professional communication/collaborating activities that connect with the ISTE standards. There are many digital tools and resources available that help with this process.

ISTE Standard 1: Teachers use their knowledge of subject matter, teaching and learning, and technology to facilitate experiences that facilitate experiences that advance student learning, creativity, and innovation in both face-to-face and virtual environments. 

Providing students with opportunities to collaborate with students in a virtual environment allows students to receive feedback and promotes student reflection. Examples of resources and strategies below:

Penzu -is a free web-based journal tool that allows students to create online journals to publish writing assignments. Each student has his/her own account and each account is linked to the classroom journal. This allows the teacher to access students’ work in one place. The teacher is able to edit and leave comments for students. The student then has quick access to the comments and can continue to revise their work prior to publishing. It also serves as a portfolio of students’ writing because the student has their work compiled all in one place. Penzu allows the teacher to share individual student writing samples with the class. This creates an opportunity for students to see model work samples and peer review or assess other students’ writing. This connects to Standard 1 because this tool allows the teacher to model collaborative knowledge construction by providing feedback. Students can then develop their knowledge by viewing student work samples and giving and receiving feedback from their peers. Hopefully, this interaction will help students reflect and make changes that will enhance their own writing.

Digital storytelling– a technology tool that will support student learning in the area of writing.  Sadik (2008) states, “Storytelling is a simple but powerful method to help students to make sense of the complex and unordered world of experience by crafting storylines” (p. 489). Tikatok is an example of digital storytelling that promotes creative thinking in a virtual environment.  It also connects with Common Core Standards, which addresses the concern of having limited time to integrate technology due to the amount of grade level standards that must be met.  Teachers can use student created stories as an assessment of students’ understanding of the standard. Tikatok motivates students to write because they are creating and publishing their own storybook. It makes the writing activities much more authentic and provides a great format for students to present their work. Sadik (2008) explains that digital storytelling offers a way for students to not only be the listeners but also learners who can interact and shape the story (p.490). This connects to Standard 1 because students are a part of the planning and creative process.

Toontastic– a resource that can motivate students who struggle coming up with an idea to begin the writing process. This app would provide my students with choices to get them started while following a story structure. For students who have difficulty completing a writing task, this app would allow them to feel a sense of accomplishment when completing a story. Hopefully this would motivate them to create stories in different disciplines.

ISTE Standard 2: Teachers design, develop, and evaluate authentic learning experiences and assessment incorporating contemporary tools and resources to maximize content learning in context and to develop the knowledge, skills, and attitudes identified in the NETS.

I learned how to use technology as a tool to provide differentiation for my students. Smith & Throne (2007) describe the need for differentiated instruction to meet the needs of all students and to help students achieve academically. Differentiated instruction requires the teacher to personalize learning activities that address students’ diverse learning styles. Technology can serve as tool for providing differentiation for students throughout the learning cycle. Digital tools can help students organize and keep track of observations. Some include outlines, graphic organizers, tables, and spreadsheets. Technology can also be used to meet the needs of multiple learning styles including visual, auditory, and social learning. Students can use computers to form learning communities that may help them share information, solve problems, and provide feedback.  Finally, technology also allows for project-based learning as an authentic tool for assessment (Benjamin, 2005).

Smith & Throne (2007) provide a list of digital tools teachers can use to enhance Marzano, Pickering, and Pollock’s (2001) effective instructional strategies. These strategies support differentiation in the classroom and are most likely to improve academic achievement:

  • Recognizing similarities and differences- Inspiration and Kidspiration Software, Word Software
  • Summarizing information and taking notes- Notestar, Read-Write-Think Notetaker, Cornell Note-taking Forms
  • Reinforcing effort and providing recognition- Kids Are Authors (Scholastic), Microsoft Publisher Certificates, Word feedback notes
  • Homework and practice- Word processing planners and organizers
  • Nonlinguistic representations- Digital cameras, Graph Club software, Excel, Paint Software, Kid Pix, Kidspiration, Micro Worlds, TimeLiner
  • Cooperative and collaborative learning groups by ability and interest- ThinkQuests, WebQuests
  • Setting objectives and providing feedback-Word journaling, Word learning logs, rubistar
  • Generating and testing hypotheses-Graph Club, Kids’ mysteries, Kidspiration and Inspiration hypothesis Webs, PowerPoint, Science Court, Word or Publisher reports
  • Questions, cues, and advance organizers-Cubing and ThinkDots templates, Word KWL charts, Inspiration and Kidspiration advanced organizers, word agenda and advance organizers (p. 14-15)

Although there are various tools that support the application of these strategies in a differentiated classroom, it is also important to focus on tools that will enhance student learning beyond just reporting information in a digital format. Porter (2010) states, “Educators need to help students rehearse thinking, creativity, and communication skills that go beyond repackaging existing information” (p. 15). Instead of simply using technology to make a project more appealing, the technology should enhance students’ understanding of the content. Examples of resources and strategies below:

Kidspiration– allows students to use higher-order thinking skills. It allows students to create graphic organizers to identify similarities and differences, use non-linguistic representations with pictures, illustrations, and pictographs, and generate and test hypothesis with hypothesis webs (Smith & Throne, 2007).

Prezi– Instead of using PowerPoint to present facts, students can demonstrate conceptual understanding. According to Orlando (2011), “The theory behind Prezi is that ideas are not linear, but rather bundles of interconnected concepts that are better captured as a whole with many parts. Prezi allows the user to illustrate the relationship of concepts to another” (p.7).

ISTE Standard 3: Teachers exhibit knowledge, skills, and work processes representative of an innovative professional in a global and digital society.

Teachers can use technology to communicate information to students and parents that will support student learning. Examples of resources and strategies below:– This resource allows teachers to post classroom information such as announcements, homework, learning activities and classroom events. It provides a way for parents to access information including links to additional educational resources and school websites. Parents can receive alerts when new information is posted. This site also supports the collaboration piece of ISTE Standard 3. It serves as a source for two-way communication because parents can send comments or questions to the teacher.

WordPress or blogger– Blogs allow teachers to post information and parents can leave comments that can be moderated by the teacher. One potential problem with class blogs as a tool for parent communication is that parents might not take time to read the information posted. According to Starkey (2011), “Learning in a digitally enhanced society is the ability to connect and collaborate with others beyond a constrained environment” (p. 21). In order to help students collaborate with their peers, students can use the class blog as a model for creating their own blog or can contribute to the class blog. Students can develop writing skills, share their learning in a creative way and receive feedback from their peers through the use of comments. Starkey states (2011), “Learners have to have connections with other learners or people with whom they interact, collaborate, critique, and gain authentication” (p. 21). Edublogs and Kidblogs are two resources available that allow students to engage in class discussion about a specific topic and reflect on their learning.

Twitter– a way to get information to parents in 140 characters or less. With just a quick note or reminder, parents are more likely to read it. They can quickly access the information on the go via their phone, while still using the blog as a resource for more information.

TeacherWeb– This resource includes several features for communicating with parents. It allows students to continue their study of a lesson at home with access to links that support their learning.

ISTE Standard 4: Teachers understand local and global societal issues and responsibilities in an evolving digital culture and exhibit legal and ethical behavior in their professional practices.

It is important that students are taught digital citizenship and safety so students are prepared to use the internet appropriately throughout their education. The resources and instructional strategies below provide teachers with tools to advocate, model, and teach safe, legal, and ethical use of digital information and technology. According to Ribble (2012), “Starting at a young age, educational leaders need to begin making parallels for students between being good to each other in the classroom and in the digital world” (p. 139). Students need to also follow online rules and protect themselves and their information (Ribble & Miller, 2013). Although my third grade students may have limited access to the internet or online programs, they will need the skills to navigate the digital world throughout their education. There will be multiple opportunities to use the internet for education and social media.  According to Lindsay & Davis (2010), we should begin educating students “as soon as they start using digital tools for communication, collaboration, and creation through connections offline or online” (p. 14). The resources available provide engaging ways to introduce students to digital citizenship so students are safe and use the internet appropriately. Examples of resources and strategies below:

Digital Passport-This resource is appropriate for grades 3-5 and provides online lessons, collaborative activities, games, and videos. Badges serve as incentives for students to meet learning goals and receive a digital passport. Students learn about topics related to digital citizenship, which include: communication, privacy, cyberbullying, searching online, copyright, and plagiarism.

Common Sense Media– The curriculum is “designed to empower students to think critically, behave safely, and participate responsibly in our digital world” (Stansbury, 2011, para. 1). You can search the curriculum by grade level so the topics are appropriate for each age level. The third grade lessons include: talking safely online, solving digital citizenship problems, privacy rules, cyberbullying, and stereotypes.

NetSmartzKids– Songs, interactive games, videos, and e-books teach kids about internet safety and digital etiquette.

ISTE Standard 5: Engage in Professional Growth and Leadership- Teachers continuously improve their professional practice, model lifelong learning, and exhibit leadership in their school and professional community by promoting and demonstrating the effective use of digital tools and resources.

I can also use technology as a tool to continue to grow as an educator, to model effective use of technology, and to learn new ways to integrate technology into the classroom. With so many digital tools available and professional learning networks, my goal is to continue to access these resources to grow as a professional. Hopefully, this type of professional development will improve my implementation of technology and enhance my instruction. In order to learn new ways to integrate technology and share information with others, ISTE Standard 5 recommends joining a local or global learning community. According to Koehler & Mishra (2009), “Teachers practice their craft in highly complex, dynamic classroom contexts that require them constantly to shift and evolve their understanding. Thus, effective teaching depends on flexible access to rich, well-organized and integrated knowledge from different domains” (p.61). With the amount of information available online, it’s important that teachers are seeking opportunities to connect with colleagues and create personal learning networks. This type of professional development is ongoing and can be accessed at any time. Utilizing digital tools allows teachers to be lifelong learners, take leadership in the use of technology, and enhance their instruction. Examples of resources and teaching strategies below:

Twitter– Twitter is one online resource teachers can use for professional development. Twitter allows teachers to share information in 140 characters or less and search for information related to their content area.

Social Bookmarking: Teachers need to learn how to organize the resources they learn about so they can be used efficiently. Social bookmarking sites such as Diigo, Evernote, or LiveBinder may be helpful in this process. Diigo allows teachers to search for information online about certain topics using tags. Teachers can join groups that focus on a specific topic, collaborate with educators, and learn new information that members have contributed. It also allows teachers to highlight and annotate any webpage they find online. They can send any webpage that has been bookmarked and highlighted to colleagues. Evernote allows teachers to annotate information from multiple web pages. LiveBinder allows teachers to organize multiple resources. Teachers can create different binders for different purposes such as professional development and teaching tools.


Digital Poster: This artifact is a digital poster I created with Glogster. This is a resource that students can use to create their own posters on a topic. This poster also serves as a learning tool to teach kids about digital citizenship in the area of plagiarism.

Digital Story: This artifact is an example of how I can model creative and innovative thinking. My digital story is also a learning experience that incorporates a digital tool to promote student learning.


Benjamin, A. (2005). Differentiated Instruction Using Technology: A Guide for Middle and High School Students. Routledge. Retrieved from

Common Sense Media. (2014). Reviews & Age Ratings – Best Movies, Books, Apps, Games for Kids. Scope and Sequence | Common Sense Media. Retrieved March 2, 2014, from

Koehler, M. J., & Mishra, P. (2009). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education, 9(1), 60-70.

Lindsay, J., & Davis, V. (2010). Navigate. Learning & Leading with Technology, . Retrieved , from

Marzano, R. J., Pickering, D., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, Va: Association for Supervision and Curriculum Development.

NetSmartzKids. (2001-2014). Retrieved March 2, 2014, from

Orlando, J. (2011). Teaching with Technology: Tools and Strategies to Improve Student Learning. Faculty Focus, , 7. Retrieved , from 

Penzu. (2014). Write In Private: Free Online Diary And Personal Journal | Penzu. Penzu | Classroom. Retrieved January 17, 2014, from

Porter, B. (2010). Where’s the Beef? Adding Rigor to Student Digital Projects. Learning & Leader with Technology, , 14-17. Retrieved , from’S%20THE%20BEEF%20-%20Adding%20Rigor%20to%20Student%20Digital%20Products%281%29.pdf

Ribble, M. Educational Leadership in an Online World: Connecting Students to Technology, Responsibly, Safely, and Ethically. Journal of Asynchronous Learning Networks,17(1). Retrieved , from

Russell, A., Amarasiriwardena, T., Clarke, A., Kim, W., Thomason, C., Rotondo, M., Clemmons, H., & McLawhorn, M. Launchpad Toys. Toontastic. Retrieved January 17, 2014, from

Sadik, A.(2008). Digital storytelling: A meaningful technology integrated approach for engaged student learning. Educational Technology Research and Development, 56, 487–506

Smith, G., & Throne, S. (2007). Differentiating Instruction with Technology K-5 Classrooms. ISTE. Retrieved from

 Stansbury, M. (2011) 10 ways schools are teaching internet safety. ESchoolNews: Daily Tech News & Innovation. Retrieved March 2, 2014, from

Starkey, L. (2011). Evaluating learning in the 21st century, a digital age learning matrix. Technology, pedagogy, and education,20(1), 19-39. Retrieved , from

 Tikatok. (2013). Tikatok – Everyone has a story. What’s yours?. Retrieved January 17, 2014, from


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Standard 9 Meta-Reflection: Cultural Sensitivity

Standard 9 Cultural Sensitivity: Establishes a culturally inclusive learning climate that facilitates academic engagement and success for all students.

In the course, Culturally Responsive Teaching, I learned that it is important that teachers are consciously aware of their own background and beliefs so they can appropriately respond to their students and create an environment that respects all students’ beliefs and cultures. Mvududu (2013) states, “Curriculum content and materials do not reflect multicultural perspectives of the populations taught” (p. 2). It is important that teachers are able to adapt the curriculum to meet the needs of all students. In order to do this, teachers need to be aware of their own cultures and backgrounds but also take interest in learning about the cultures of their students. This will be an ongoing process as teachers learn about the cultures and interests of their students. Teachers must seek opportunities for professional development, activate students’ prior knowledge, and model respect for individual differences. This will ultimately help develop multicultural classrooms in which all voices, backgrounds, and cultures are equally valued instead of Whites being the “norm” against which all other groups are measured (Mvududu, 2013, p. 9). 

Key Concepts

Views of Multicultural Education-

The views of multicultural education vary depending on the subjective and objective elements in which knowledge is constructed. There are different types of knowledge that should influence the curriculum taught in schools. These include personal/cultural, popular, mainstream academic, transformative academic, and school (Banks, 1996, p. 9). In a multicultural classroom, students should learn all types of knowledge so they are able to understand multiple perspectives and develop their own. If teachers understand different types of knowledge they can apply this knowledge in the classroom to help students see how the knowledge fits within a social context. It is important to remember that our own background and cultural experiences influence the way we construct knowledge. Banks (1996) states, “The goal of multicultural education is to create equal educational opportunities for all students (p.8).”

Culturally Responsive Instruction-

Culturally responsive teaching involves teachers designing instruction that meets the needs of all students. Mvududu (2013) states, “The general claim is that most children of color are socially acculturated to more communal living arrangements than White, middle-class kids, and thus they profit from more collaborative instructional activities than they usually experience in school” (slide 2). I think when cooperative learning includes: positive interdependence, face-to-face interaction, individual accountability, social skills development and group processing opportunities it provides a structure for students to learn from one another (Del’Ollio & Donk, 2007). It allows each student to engage and participate in the learning. Students are not only engaged in learning the content but are developing social skills and positive relationships with their peers. When students have the opportunity to problem solve and accomplish tasks they are learning to work productively with others. Cooperative learning is giving all students a chance to participate while building classroom community. Students need to feel successful and develop positive relationships with their peers. These social skills are skills they will take with them beyond the classroom and that will be applicable to real life. Transformative Knowledge, Curriculum Reform, and Action describes the dimensions of multicultural education which includes content integration, an equity pedagogy, the knowledge construction process, prejudice reduction, and an empowering school culture and social structure (Banks, 1996, p. 337). Content integration is when teachers use content from a variety of cultures in their instruction. An equity pedagogy requires teachers to use a variety of teaching styles to meet the learning styles of all students. The knowledge construction process involves teachers presenting multiple perspectives to help students understand how knowledge is influenced by multiple factors. Prejudice reduction is implemented when teachers develop lessons that will help students develop positive attitudes about all cultural backgrounds. Lastly, an empowering school culture and social structure requires the school to create a culture that supports all students (Banks, 1996, p. 336-339). I think the shift from content integration to transformation is the shift many teachers still need to make, including myself. Rather than adding units on different cultural groups, teachers can use the existing curriculum. This emphasizes the importance of creating an empowering school culture where all teachers are approaching the curriculum with a critical eye and making modifications that will allow students to see multiple perspectives. Since curriculums vary from school to school and do not always represent multiple cultural backgrounds, it is not enough to rely on simply delivering the content. It is important that teachers not only provide multiple perspectives but allow students to engage in critical thinking activities so they can learn to make their own interpretations. These are skills that students can apply beyond the classroom. Students need to be prepared to make decisions so they can make positive contributions to our diverse society.

Students’ Cultures and Background Knowledge- As our school populations become more diverse it is important that we seek opportunities to learn about students’ cultures and background knowledge that will facilitate us in adapting curriculum to meet the needs of all students. Mvududu (2013) states, “Multicultural entails knowledge of many cultures, which is impossible to posses in full” (p. 2). This reminds me that learning is an ongoing process and teachers may visit the area of  “distinguished” but often do not stay there. There are always opportunities for growth. Teachers need to constantly invest interest in understanding student backgrounds and lifestyles that will improve student learning. There are multiple factors aside from race that may affect the “achievement gap.” Mvududu (2013) states, “Understanding cultural issues associated with poverty is at least as complicated as race, and one has to deal with both of them to begin to make sense of the achievement gap” (p. 3). This stresses the importance of teachers taking time to understand the multiple factors that influence a child’s education. Mvududu (2013) states, “There is plenty of blame to go around, and the more important question is not who we blame but how we continue the slow process of incremental change in our schools and classrooms” (p.10).

Helping Students Academically-

Helping all students succeed academically is critical for improving the quality of life for students of color. “Forming better relationships and building self-esteem are important, but academic achievement is the best path” (Mvududu, 2013, Slide 3). Students will be motivated to learn if they are able to feel and celebrate their academic success. The other goals of multicultural education will greatly impact students’ pathways towards reaching academic success. Students must first be engaged in a curriculum they can identify with and that interests them. Part of our job as educators is to build our own understanding of our identities while learning the backgrounds and lifestyles of our students. This way we are delivering the content to address multiple learning styles and not just delivering content the way we were taught or learn best. Dr. Mvududu (2013) states, “All kids won’t meet the standards without significant changes in how curriculum is delivered to outside the main stream” (slide 15). I think it is possible to change the way the curriculum is taught while still meeting the state standards. I think the state tests are an accurate assessment of standards that can be taught through the curriculum without specially teaching to the test. I do think there is specific language and formatting that can affect students’ abilities to show what they know. So I think teachers do need to spend time teaching test-taking strategies and formatting. However, I do not think it’s necessary for teachers to teach to the specific content since the content is embedded in the curriculum being taught throughout the year. This is where there is flexibility in how the curriculum is delivered, while still ensuring that state standards are being taught.

Building Positive Relationships-

“Caring relationships are, and will always be, the centerpiece of effective teaching and learning” (Mvududu, slide 3). I am a strong believer in holding high expectations for all students regardless of ability level. Creating a learning environment in which all students are expected to take responsibility for their behavior and learning is a part of developing caring relationships with students. Dr. Mvududu (2013) states, “In its authentic form, it often requires what we call ‘tough love’” (slide 5). In the classroom, it is important to establish high expectations and hold students accountable for meeting them. In my classroom, there are some students who turn in work late or do not turn in homework on a weekly basis. The expectation is that the student will come in during his/her recess to complete the assignment. It can be very easy to excuse the student from having to complete the assignment during recess when the student gets upset, tells you they just forgot it, has worked hard during the day and needs a break, ect. Although there is some flexibility depending on the circumstance, I really think that not following through with the expectation is teaching the student that we as teachers don’t believe there is any value in having them complete it. We also run into this dilemma with behavior expectations. It can be difficult to make a decision about whether a student should be able to participate in our school-wide gold card celebration (for students who have modeled our school promises). It is easy to frequently give students multiple chances. However by not following through with the expectations, I think we are teaching the student that they do not have to model respectful behavior and they will still be rewarded. Sometimes it feels strict or uncaring, but I think students feel they are a part of a safe environment when there is accountability and structure. They know their teacher cares about their well-being and believes in their potential because their teacher will not settle for anything less than their best.

Knowledge Construction and Multiple Perspectives-

This course taught me the importance of understanding that knowledge construction is both objective and subjective. As teachers we need to not only be aware of the curriculum and information we present but how we interpret it based on our own background and beliefs. I think as educators, it’s important that we are knowledgeable in the content areas we teach so we can help students develop multiple perspectives that allow them to construct their own knowledge. I think it’s also valuable that we challenge knowledge that is presented to us or information that we read, rather than always accepting it as truth. I think working in a professional learning community is a great avenue for discussing how we can teach curriculum and content to help students construct their own knowledge.


The importance of communication is a valuable piece to culturally responsive teaching that may be easily overlooked in instruction. When I think about communication, I first think of language. For example, a culturally responsive teacher provides accommodations for English Language Learners in order to provide access to the learning material. However, communication includes both verbal and non-verbal. Students’ background and culture influence their responses and interactions in the classroom. Understanding how students’ cultures play a role in their communication is another component in designing instruction to meet the needs of all students. I think including multiple instructional strategies is one way teachers can provide students with multiple ways to respond to the learning. Since lessons are designed to focus on specific learning targets it can be easy to design questions that are convergent. I think allowing students opportunities to have conversations about the learning through cooperative learning or by allowing time to turn and talk gives students a chance to explain their thinking and understanding. This might support students’ development of inquiry skills and allow students to apply their background knowledge to the new learning. Mvududu (2013) states, “Teachers need to be able to read their students well enough to discern whether learning is going on or not right in the midst of lively discussion” (p. 4). I think this is a key component of effective teaching because the teacher is constantly using formative assessment to ensure that students understand the learning. If students do not have opportunities to actively participate in the learning it is difficult to assess their engagement and level of understanding. Teachers must be aware of students’ engagement so they can provide accommodations for students who need support. This may include asking divergent questions that allow students to respond creatively and discuss the learning with their peers.


A culturally responsive teacher takes time to learn about students’ background knowledge, strengths, learning styles, and needs. Culturally responsive teachers use what they learn about students to inform their instruction and provide multiple ways to access the curriculum. Teachers should establish a positive learning community by involving students in the learning process. Students should help create classroom expectations that contribute to a safe learning environment so all students know they are respected and cared for. Students also need opportunities to develop social skills and work cooperatively with others. This way, students know they are valued members of the classroom and their ideas are heard.  I think it’s important that teachers are seeking opportunities to learn about different cultures so they are able to apply students’ background knowledge to their instruction. Finally, I think culturally responsive teaching includes increasing parent involvement so teachers and parents can work together to meet the needs of each student.


Integration and Action Exercise This artifact is a reflection on what it means to be a multicultural teacher and includes instructional implications for the classroom.

Autobiography This artifact is an autobiography that describes how my background influences my teaching today.

Assessment Portfolio This artifact demonstrates my ability to design instruction based on students’ prior knowledge, experiences, and interests. It also demonstrates my ability to use a variety of assessment strategies and modify instructional strategies based on students’ needs.


Banks, J.A. (1996). Transformative knowledge, curriculum reform, and action. In J.A.

Banks (Ed.) Multicultural education, transformative knowledge, and action: Historical and contemporary perspectives (pp. 335-348). New York, NY: Teachers College Press (Chapter 18).

Del’Ollio, J., & Donk, T. (2007). Models of Teaching: Connecting Student Learning With Standards. Thousand Oaks: Sage Publications, Inc.

Hillis, Michael R. (1996). Allison Davis and the Study of Race, Social Class, and Schooling. In James A. Bank (Ed.), Multicultural Education, Transformative Knowledge & Action (pp. 64-87). New York: Teachers College Press.

Mvududu, N. (2013). Personalizing cultural diversity. Culturally Responsive Teaching Podcast. Podcast retrieved from

Mvududu, N. (Producer). (2012). Module 03: Multicultural education in the classroom. Podcast retrieved from

Mvududu, N. (Producer). (2013). Module 04: The twin pillars of justice and caring. Podcast retrieved from

Mvududu, N. (Producer). (2013). Module 06: Cross-cultural teacher-student communication. Podcast retrieved from

Mvududu, N. (Producer). (2013). Module 07: Authentically Multicultural Curriculum. Lecture retrieved from

Mvududu, N. (Producer). (2013). Module 08: Authentically multicultural instruction. Lecture retrieved from

Mvududu, N. (2013). The Need for Multiculturally Literate Teachers.  Culturally Responsive Teaching Podcast. Podcast retrieved from

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EDU 6613: Assessment Portfolio

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

Instructional Strategies:

  1. Activate prior knowledge: numeracy, tens facts, facts below 10.
  2. Apply knowledge to practice following all steps for addition without regrouping with place value mat (PVM) and t-chart. 
  3.  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 

Artifact: 3rd Grade Pre Interview Rubric 3rd Grade Post Interview Rubric

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.

Formative Assessments

Student A: 

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.

Student B:

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.

Student C:

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.

Artifact: StudentCExitSlip and Goalsetting

Analysis: On her exit slip, she explains she’s not sure where the numbers go when she needs to regroup. On her goal-setting form she  wrote she needed support with place value for her academic goal.

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.

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