Let’s face it–data is everywhere. In a school setting, the data that drives our instruction comes from assessments, which we usually classify as either *formative* or *summative*. Typically, we define formative assessment as assessment *for* learning. It is more informal and usually comes in the form of quick, frequent assessments. Think Tickets In, questioning, journal reflections, and teacher observations or discussions from small group instruction. Formative assessment is taking the pulse of student understanding so we can plan our next moves. Summative assessment, on the other hand, is assessment *of* learning. This type of assessment comes at the end of instruction and assesses how well the students have mastered the objectives. Summative assessments can be, among other things, unit tests, district tests, or state tests. But are summative assessments the end of the road? Of course not! The data that is produced from summative assessments serves as the basis for our remediation efforts. You give a unit test, you analyze the data and remediate as needed. But we have to use the data in the right way if our math remediation is to be effective. What does that mean?

### Remediate the skill, not the grade

We have to resist the temptation to remediate based solely on the grade. Of course a child that makes a 60 needs remediation, but where do you start? Our test title might say Place Value Test, but in reality it probably covers several place value concepts and standards. The overall grade of 60 tells you little about the student’s actual needs. That student might have gotten every question about expanded notation correct, while missing all the questions about comparing and reading decimals. Likewise, a student who scored an 80 might not seem to need remediation. But what if he missed all three questions on expanded notation? If we remediate solely on the overall grade, that student would be moved right along without understanding an important concept. That’s how gaps begin.

### Remediate the skill, not the question

If you think this sounds a bit like the last section, you’d be correct. Just because students miss the same question, that doesn’t mean they have the same remediation needs. Look, for example, at this question:

### Forming math remediation groups

Once you have carefully analyzed your data, you plan remediation groups accordingly:

- The students who missed the question because of decimal placement most likely don’t have a solid foundation in place value. Go back to the base-10 blocks and pictorial support to help students better understand place value in general and, more specifically, multiplication of decimals.
- The students who don’t have a successful strategy for multiplication form a group learning strategies for multiplying, using concrete or pictorial support as needed. Strategies could include the area model or partial products leading to the standard algorithm.
- Have a conversation with the student who missed the problem due to a fact error. It’s important that students understand their mathematical strengths and weaknesses. Explain to the student that he did all the hard math correctly, but still missed the problem due to a fact error. Provide the student opportunities to practice his basic facts.
- The students who added clearly did not comprehend the problem. These students need strategies for understanding what the four operations look like in word problems. Work with these students to analyze problems and determine the operation. Learning to draw models to help them visualize the problem will be an important part of their instruction. Their emphasis is not solving the problems, just analyzing them to determine the operation.

Bottom line, our data is only useful if we use it correctly, and that means digging down to the skill level.

I hope you’ll add your comments about remediation. What works for you?

Honestly, you could go either way with 3rd. You can preview both sets, and that might help you make a decision.

If I teach 3rd grade which mathematical practice poster do you recommend? Primary or intermediate

What a great reflection!! I agree that finding the missing link in a students learning can be challenging.

I just discovered your post and am already spending hours exploring this amazing blog! I am so lucky and glad to have found this blog! I have been a classroom teacher for the past 4 years and this year I took a remedial math position servicing students in grades K-6. This is a new program for my district and well…..no one knows what they are doing. I have no one to turn to for help 🙁

I start pulling students Monday and have no idea where to start. Should I give students a diagnostic assessment to see what skills they need remediation in? Do I begin with place value with the upper elementary students? What are the best tools to use for progress monitoring? I’m so lost and have gotten upset almost every day since school started.

Any help is appreciated. I’m so close to throwing in the towel and I never felt this way in my life.

We, too, just added Math Interventionists at each of our buildings K-8. I accepted the position is one of our 3-5 schools. It feels a bit like building a plane while flying it, but posts like this help solidify my understanding. It’s so much more than just looking to see if a child got the correct answer. We need to be analyzing for exactly what they can and can’t do. I would encourage your district to look into PLC training. We just did this, and it was phenomenal. It’s important for all stakeholders (gen. Ed teachers, interventionists, SPED teachers and administrators to come together to have conversations about achievement AND growth.

Hi, Our district is currently struggling with finding a good math benchmark test. We had previously used MAP testing, but found that the data we got from it hard to use to help us set up intervention as it wasn’t specific enough. I was curious if you had any luck or knowledge of a math benchmark test that has been helpful in intervention planning?

This is honestly something I struggle with as well. I just ordered Kathy Richardson’s Assessing Math Concepts series. I’ve heard very good things.

I find your postings to be incredibly informative and useful in my math lessons. I teach overseas which allows for greater flexibility in how I teach math. I would, however, love to increase my understanding on how to create open-ended math problems and feel more comfortable with teaching math conceptually. Can you recommend any professional development workshops on teaching math conceptually? Thanks in advance.

Hi, Valeria! It’s so exciting to hear that you are looking to grow as a math teacher. You said you’re overseas, so I’m not sure if you are looking for something in the US or overseas. The NCTM conferences, both regional and national, are always amazing learning experiences. Since you choose the workshops you want to attend from a lengthy list of offerings, you can tailor the learning to your interests. And, honestly, I do a lot of personal PD just by reading new books on teaching math!

I love this article. One of the goals for our K-4 elementary schools was to share formative assessments at each of our faculty meetings. We have developed a file for useful assessments and articles. Thanks for the information; it will be part of our F.A. file!

Great post Donna Boucher,

I am an early childhood and special education dual major, and will be graduating in 2017. I am currently enrolled in a math class that focuses on teaching students mathematics. Your blog focused on a lot of the same issues that we discuss during my class time. This semester it has been apparent to me how important it is to include that assessment piece in lesson planning in order to have a mark of understanding for our students. However, the next step after planning the assessment piece is HOW you as a teacher, are going to use the results from the data or assessment to plan effectively. This blog focused on many key factors, such as, how it is easy to just view data/assessment quickly. However, it our job to use that data and dig to plan effectively for that student’s understanding. It was very interesting to me to read about how to form math remediation groups. Once you have carefully analyzed your data, you can begin to plan remediation groups. Your reasoning with the group formations matched the class discussions I have been having because you use the level of understanding among peers to shape the groups. Thanks for this post, it was a very interesting and well worth read.

Danielle

Thanks, Danielle! Nice to hear that my post is so well aligned with your courses. 🙂

I have followed you for a long time and I agree with every bit of this post. My issue stems from not having the time or personnel support to be able to do this. For example, 30 students in your class, 25 different needs on each of 10 different questions. I recognize the skill deficits, but I must continue with my one lesson a day. How do you suggest this be addressed?

That’s certainly a big issue, Anna, and it’s exactly why I think teaching with a Guided Math approach including small group instruction is pretty much essential! If you have flexibility, that type of structure makes it much easier to address the differing needs of learners than whole group teaching.

Does anyone have any tips on how to make time for remediation groups and/or what the other students do when you’re meeting with those groups? I already do guided math, but that’s for the general instruction of the skills. Tips please?!

If you’re already doing guided math for general instruction, set aside some blocks for remediation. So if you have time for three guided math group, maybe two of them are general instruction and one is remediation. Or maybe set aside one day and do only remediation groups on that day. To the kids, it will look exactly the same as usual–they will still be work in math workstations while you teach in small group.

I create a spreadsheet for every test and quiz I give. It’s a bit of work, but I do an item analysis for every student so I can see where they made mistakes. I keep track of how many points each student lost in each question. When I’m done, I can see what questions were most troublesome and know where to remediate for each student.

Your posts are so helpful and I learn much from the comment dialogue as well. I am looking for a record keeping form set up by CCSS to monitor student progress with specific assignments/ assessments. Any suggestions or examples?

I’m afraid I don’t know of anything like that for CCSS. I’m currently working on checklists for the Texas Essential Knowledge and Skills (TEKS).

Thank you for this most informative post. I have taught for many years, and am used to using data informally, although I didn’t think of it as such. It is new for me to do data mining and remediation with such specificity and depth. Interestingly, it occurred to me that I do this naturally in my Daily Five reading program. In Daily Five the children are taught to work independently while doing each of the five components, leaving me free to confer individually or in small groups and to give each child the strategy it needs to move ahead. In math, it’s not so much how to remediate, as it is when to find the time to do it. We are using a new program this year, which I’m not fond of – Math Expressions . (I loved Investigations, which I used for many years.) One problem is that because I am in second grade, for the first couple of months I was basically remediating the entire class because of the new vocabulary and the order in which new concepts were introduced. My big problem now is finding time to remediate individual and small groups in my day. The lessons are very long; in addition to a daily math routine, there are usually two big activities plus sometimes several pages from the workbook. There is no real workshop structure and most of the lessons need to be whole group owing to the level of difficulty. We are in the middle of a long unit on story problems, with a different kind of story problem introduced every day or so. With 27 children I’m just not getting to remediate enough to make a difference. Do you have any suggestions? I’m sorry for the long screed; I’m just very frustrated. My other second grade teaching partner whom, I love and have worked with collaboratively for many years, is under such stress this year that she cannot work on this with me. I loved teaching with Investigations because the children were encouraged to discover so much on their own. They loved math and I loved teaching it. I try to incorporate open ended questions and discovery. I use the strategies from (the book) Number Sense, but the lessons take so much time it’s very hard to do. As it is I am far behind because I have to keep breaking each lesson down into smaller pieces. Any help would be appreciated!

This is a source of frustration for so many teachers. Do you have the flexibility to use the program as a resource and not follow their pacing? I find that most textbooks have lessons and activities that are good quality, but the pacing guide is unrealistic. Do you have to use every part of each lesson? If not, pick and choose and tailor it more to your students’ needs. I’m sorry there’s not an easy answer to this one.

Thanks. Because it has just been adopted, we are expected to use it pretty much as written. I have tried to edit the lessons, but there is so much new content in each one, it’s hard to leave out lessons because either the children will miss the skill, or they will not understand the next lesson. There is a money routine that is supposed to happen every day. I do it every few days and substitute my own routines and those from the Number Routines book. One daily money routine is so complicated and odd I haven’t even attempted yet. There is a ‘quick revue’ activity every day that I don’t do, either. The lessons almost always have to be done whole group, and the student book pages are the same way. There is a homework sheet to send home every night, but I often can’t do that because the children don’t understand the lesson well enough yet and the parents don’t know how to help (it’s common core work that involves using various strategies, making a model, explaining your thinking). I end up getting emails from parents asking for help, which by email is not easy! I will keep trying to find ways here and there. Thanks for your insights and support!

Appreciate the intelligent analysis you suggest in the last section. It might take some work, but that’s probably what is required to cater to each student’s needs. Thanks!

I really like your idea of how little information teachers receive about the student work from Scantron tests. Many times students guess, or pick an answer choice closest it what they got without putting much thought into it. One way that my AP Calculus teacher tried to resolve this issue was that we had to turn in our work for each problem along with our Scantron answer sheet. By doing that, he was able to give students partial credit for those that they made a small mistake for, as well as see which problems the student made a lucky guess on. This method required us students to be more accountable for writing our work down because we knew he was going to look at our work.

This is a great article. Would it be possible to get it as a PDF? I’d love to share this with my colleagues as we work on math.

I’m so glad you want to share it, Karen. I just installed the PrintFriendly button that will convert any blog post into a PDF. Look for the little PrintFriendly button at the bottom of the blog post!

So now I can just print each post and turn your blog into a “book” so I can read, highlight and post-it!!!

Glad I saw this comment!

I tell my students that there are two kinds of math errors–arithmetic errors (multiplying incorrectly, dropping a negative, etc) and process errors (they make a mistake on the steps of the problem. I have them work to distinguish between the two, because that helps them take ownership of what they need to target.

I do the same but call them fact errors (FE), process errors (PE) and copy errors (CE). Copying a problem or answer incorrectly is a common mistake also. When I grade papers I “code them” with FE, PE and CE. The students understand the codes and then look to find and understand their mistakes. If students grade their own papers, they code their mistakes. Teaching students to take responsibility for their learning is a large part of remediation.

Laurie, thank you for explaining the coding of math errors. This will help me and my students.

I LOVE this! When I was in the classroom, I had students explain their errors, but I didn’t have an easy coding system like this. Thank you so much for sharing!

I think that’s great, Kari! Students need to be in charge of their own learning!

I am a data facilitator and interventionist. I would love to see a post on the role of interventionist. Often I feel as though I am more support or I take a group to reduce class size. How do you see the interventionist role?

I have been teaching for many years. In fact, Christine Moynihan was my student teacher. She loves that I follow your blog.

I’m in much the same role as Elaine, being a K-4 math interventionist. I facilitate twice monthly data meetings and have skill groups for each grade level. Finding the balance so that I am providing intervention and not just reducing class sizes is tricky. I’d love to know what others in similar roles do. This is a new position for the building and I’m new to the district so there is not a model in place and I am blessed with the opportunity to get to create this new role as we go.

I am the math intervention teacher for grades 1-8 at a parochial school. Since I’m at a private school, I’ve had much leeway in developing my program. My program has evolved over the years and is very focused on the individual needs of the students. I have grade level groups that I see twice a week. If I don’t feel this is enough for a particular student, I will add a third and sometimes even a fourth individual day for that student. I also teach some students in a resource room setting. Next year I will have a class of 3 third grade students and a class of 1 sixth grade student. I have also co- taught in certain math classes. This works well in classes where there are many students that need math support. As Amanda, I always try to “preview” and I also “review”. During “preview” I break down the new skill that they will be learning in the classroom. My older students really like this because when the teacher introduces it, they are familiar with the concept. I would say that my most effective strategy is scaffolding. As Amanda, I break down skills into prerequisite steps and provide lots of visual supports. I also communicate a lot with the teachers. I plan my lessons based on what I observe during previous lessons, input from the teacher and input from the students. I always ask the students what they need help with. I view teaching the math as a secondary goal. The ultimate goal is teaching students to be responsible for their learning. I’m going to end my comment although I feel like I could share much more. I am also interested in hearing how other interventionists structure their time and strategies they have found to be successful.

Thank you for your insights! I found your post very helpful!

I am a math interventionist in the middle grades. I have found a great deal of success in the process I use. I meet often with the core math teachers at my site and they share their lesson plans with me. Then I take some time to boil down those topics into the pre-requisite skills kids will need to know in order to learn that material successfully. Then I try to pre-teach the concepts at a basic level so that my students are seeing it for the second time in their regular classrooms. This gives kids an incredible boost in confidence! Remediation, a lot of times, simply does not work well at the middle levels. We have to empower kids to do grade-level work or they will never catch up!

Holy cow, Christine was your student teacher? How awesome is that! I’m a huge fan of hers. Thank you for the idea for a post. Have you seen this post on remediation? I don’t think it’s exactly what you’re looking for, but it still might interest you. I will definitely think about your idea!

Another thoughtful and thought provoking post Donna. Thank you. Can you steer me towards any effective protocols for looking at data with a team of teachers?

We use the ORID for looking at data. O stands for looking at the data objectively- identify the factual information. What factual statements can you make?

R stands for reflective level. This encourages participants to make connections and encourage free flow of ideas. What surprised you? What encouraged you? what discouraged you?

I stands for Interpretive level. This helps you to identify patterns and determine the meaning. What does the data tell us? What new insights do you have? What doesn’t it tell you? and what else might we need to know? What areas of need seem to arise?

D stands for decisional level. (the most imp. step) Propose next steps, develop an action plan. What are next steps? What is our action plan for moving forward?

I didn’t know there was a difference between formative and summative assessments. Would daily or weekly quizzes also fit into the summative category? Sorry, this is some new vocabulary for me!

I am trying to do this with my class this year. Before every unit, I am giving a diagnostic assessment and then looking at who needs what. The difficulty is in finding appropriate resources to get to those basic skills. Any suggestions would be greatly appreciated.

I love the preassessment approach, Fabienne! Would teachers in the grade below you be able to help with the prerequisite skills?

I love this post! I will be sharing with a number of teachers I work with. Thanks!