Category Archives: Assessment

Formative Assessment

Assessment always seems to be such a broad, hot topic  There are rubrics to help create assessments, rubrics for reviewing assessments, and tons of reading about the benefit of assessments. While I agree assessment is an important topic of conversation and all of these things can be helpful, I just lose a bit of interest when it becomes so cumbersome. I feel the longer the rubric and steps to create an assessment, the more detached the assessment becomes from student thinking.  This could be completely be my short attention span speaking, however the way assessment is discussed feels either like data (a grade or number-type of data) or a huge process with tons of text in rubrics that I really, quite honestly, don’t feel like reading. Not to mention, I just love looking at student writing and listening to student thinking when planning my immediate next steps (formative) or checking in to see what students have learned over a longer period (summative). This is why I find the work we are doing each month in our Learning Labs such a wonderful way to think about formative assessment in an actual classroom context, in real time.

This passage from NCTM’s Principles to Action really captures how I feel about the work we are doing in our Learning Labs:

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In this most recent Learning Lab in 3rd grade, we planned the activity together using the 5 Practices model and reflected after the lesson. Since this blog is always my thoughts about student work, I thought it would be great to hear what the teachers took away from the activities we are doing in terms of the students’ understandings and impact on their future planning, formative assessment.  

The teacher mentioned in the blog said, I was surprised by how quick many of the students defended their responses that 1/2 will always be greater than 1/3, and then proving this response using visual representation of the same whole ( which is an idea that we have made explicit). I was impressed with “skeptics” in the crowd that were looking to deepen their understanding around the concept by asking those “What if” questions.  Going forward, I want to create opportunities that push and challenge my student’s thinking. I want them to continue to question and explore math – especially when it uses the word “always.”

Another teacher who taught the same activity after watching it in action in another classroom said, “I learned that almost half of my students assumed they were comparing the same size wholes.  They agreed with the statement, and each student gave at least two different ways to prove their thinking (area and number line model were most common).  The students that disagreed almost all provided their own context to the problem, such as an example with small vs large pizzas, or a 2 different-length races being run.  I found it so interesting that almost all students confidently chose one side or the other, and were able to defend their thinking with examples (and more than one-yeah!)  I was excited to see that they could be so flexible in their arguments as to why they felt as they did.  Three students responded that they were unsure, and gave reasons to support both sides of the argument. This impacted my instruction by giving me such valuable formative assessment information with a simple, non-threatening prompt.  It took about 5 minutes, and gave me tons of information.  It was accessible and appropriate for all.  Students were comfortable agreeing or disagreeing, and in some cases, saying “unsure-and here is why.”  I was most excited about that!”

She also said, From this activity, I learned that I really needed to revisit the third grade standard to see what is actually expected.  It says they should recognize that comparisons are valid only when the two fractions refer to the same whole.  My statement didn’t have a context, so how cool that some were at least questioning this!   This impacted my planning and instruction by reminding me how thinking/wondering about adding a context to the statement would influence their responses.  I am also reminded that I need to stress that students must consider the whole in order to make comparisons accurately.”

Earlier in their fraction unit, the third grade teachers used the talking point below to hear how her students were talking about fractions. (This work is actually from another teacher’s class, but you get the idea;)

A teacher who did this activity reflected, “From this activity, I learned my students had only ever been exposed to a fraction as a part of a whole (and wanted to strictly refer to fractions in terms of pizza). This impacted my instruction by being sure to have the discussion that fractions can represent parts of a whole, but we can also represent whole numbers with fractions.”

To me, these reflections are what assessment should be….the teachers learn about student thinking, the students think about their own thinking, and what we learn helps us plan future lessons with our students’ understandings in mind!

More examples from different grade levels where the teachers and I learned so much about student thinking that impacted future instruction:

Kindergarten: Adding

Kindergarten: Counting

1st Grade: Fractions and Adding

2nd Grade: Counting and Leftovers

4th Grade: Division

5th Grade: Fraction Number Line

What Are They Really Thinking About Decimals?

Understanding student thinking is so hard. I make assumptions. I read into things. I SO want to believe there is understanding behind everything they write on their papers. However, it is so much more difficult than that and my most recent difficulty is addition of decimals.

We have talked about decimals in one frames, shaded grids, and I am confident that every student can compare decimals with understanding of place value and magnitude. They understand decimals independently. Then, enter decimal addition. What is it about computation that sends students right back to not thinking about the numbers themselves and straight back to “lining them up” and adding? I know it is not that they CAN’T think about the numbers, so then my wheels start turning…. is it just ease of use? Great. But is it ease of use with understanding? Or is it ease of use without understanding but just gets them the right answer? This is where teaching is so hard!

We do number talks at least 2-3 times per week and given a problem such as 38 + 47, the majority of the students would say 40 + 45 = 85 using a compensation strategy. Today, given 6.8 + 4.7, I got “I lined them up and added 8 and 7 and got 15, carried the one…” You can hear the rest. Wait, what? Where are the tenths? Where is the place value? Why didn’t I ask them to give me an estimate first (ugh, hindsight)? I ask for any other strategies, nothing emerges. I am left to wonder what they truly understand about addition of decimals. Is it the decimal place value that takes away from thinking about the numbers or is it simply that they see how the decimals operate like whole numbers in a base ten sense. After doing a contextual task the day before, with pencil and paper, I was excited by the outcome, there were numerous strategies. However, if pushed to solve mentally, the students reverted back to an algorithmic feel. I am not saying that it means the students do not understand the place values they are adding, but trying to bring to light how hard it is to interpret their understanding on my part.

I then gave them a problem involving three decimals and asked them to solve it two ways. I was trying to get a better feel of their understanding. The two ways would push those “liner-uppers” to work with the decimals in a different way and also allow me the time to walk around and question students about their work.  I was not shocked to see that the majority went to lining them up as their first strategy, however I was very excited by their second strategy that showed more understanding of place value.

Here are some examples of the students solving 0.98 + 0.05 + 1.06


I loved the number line in the first example and the breaking apart of the 0.05 in the second example. I was starting to see the flexibility and thinking that I want to see in my students.

I gave my second class a different problem involving two decimals, both in the hundredths that were not as “friendly” as the decimals in the previous example. I was happy to see the variety of strategies, including my student who starts assigning letters for each digit. He said he is ready to start doing some algebra 🙂  I love it!

IMG_9607_2IMG_9605I had a few who finished fairly quickly, so I gave them the problem 0.8 + 0.75 + 0.625, and then they started getting creative! This is one answer that was so interesting and will be the way I kick off my class tomorrow. His reference to columns and boxes are the hundredths grids we used for the Fill Two game.

IMG_9606Today was a day that really showed me how hard it is to understand student thinking and how important it is to push students to explain their understanding in more than one way. I could have very easily assumed that every student could add decimals by place value because they lined them up and added to get a correct answer.  However, if not given the opportunity to show another way to think about the problem, how would I truly know? I still have a few students who are getting the correct answer but are not able to articulate their process, so I am going to do a lot more estimating to get them thinking about the numbers before the operation. Going to be a fun day in math tomorrow!


What DO they know?

I love reading and giving feedback on my students’ journals,  I make time for it every day. But the mere thought of grading papers (feedback or not) makes me world’s biggest procrastinator. Unfortunately, my reality is that I need grades for progress reports and report cards, so I make the best of it. I try to make the assignments valuable for both the students and myself in their learning, however  I always wonder why I don’t approach the papers the same, they are both student work right?

I had a realization yesterday while I was grading, as to why I make time to read their journals vs the aversion I have to grading papers. While I was grading, my mind was focused on what the students DON’T know, what they aren’t getting, aggravation at the careless mistakes, aggravation that I didn’t “reach” that child and why they don’t all have 100%. As I graded, I was busy making notes in my own journal of the students who were missing certain items so I could make my plan for next week to help them better understand the material. And while I know this is invaluable in planning to better teach my students, I realized I was completely glancing over what they DO know. I was checking off the problems they were getting correct and focusing solely on the wrong. Don’t be mistaken, I LOVE mistakes in math, I love analyzing what students could have been thinking, misconceptions and/or misunderstandings, but when grading, the feeling is still not the same.

This focus on “wrong” wasn’t the only thing that bothered me though. I also wasn’t “feeling” my students’ voices in the assignments, like I do their journals. Maybe it is because I love hearing them talk about math so much, their journals are the next best thing when they have left class for the day.  Maybe it is the freedom for them to take more chances in their journals or simply say, “I don’t understand it from this point on..” that makes them so much more enjoyable. Or maybe it is the mere fact I don’t have to put a grade to their thinking. As I read their journals, I am looking at everything they DO know and how that led them to where they are instead of the other way around.

Their journals feel more like the way we learn then grades do. We try, we make mistakes, people help us along the way with advice, we try again, we test things out, we look back at what we did to build on it….no number is attached to that, so why grades? I would like to think I try my best to not have grades be a focus in my classroom and instead be a snapshot of where students are right now in their learning, but those assignments still do not hold the same value that their journals do for me.

Maybe someday standards based grading will make its way into our district but until then I will continue to read their journals for things just like this…

IMG_8986_2– Kristin

Pre/Post Assessment Reflection

We started our 2D Geometry unit with Talking Points:  This was the ultimate pre-assessment in which I could hear what the students were thinking around mathematical concepts while at the same time, they had a chance to also hear the thinking of their peers. After the talking points activity, I had the students reflect on a point they were still unsure in their thinking.

We are now wrapping up our Polygon unit, and I thought it would be interesting for them to reflect back on what they were unsure about in the beginning, and get their thoughts now. I have a class full of amazing writings, but here are just two of the great reflections (the top notebook in each picture is the pre-unit and the bottom is post-unit)….

Photo Jan 15, 11 42 12 AM Photo Jan 15, 11 55 54 AM

Looking at the class as a whole, it was so interesting to see their math language develop and see them laughing at things they had written before. I loved that the student above wrote, “I am smarter!!!” How amazing they can see their own learning!  During their reflection time, it was so fun to also hear students exclaiming, “See, I KNEW I was right!”

This is the first pre/post assessment I have ever done where I think the students enjoyed it as much as I did! They were as proud of themselves as I was of them!