Series: Formative Assessment Practices to Support Student Learning

Math.Practice.MP1

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP1:  Make sense of problems and persevere in solving them.

    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, \"Does this make sense?\" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Download Common Core State Standards (PDF 1.2 MB)

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Math.Practice.MP2

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP2:  Reason abstractly and quantitatively.


    Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize--to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Download Common Core State Standards (PDF 1.2 MB)

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Math.3.NF.A.1

Common core State Standards

  • Math:  Math
  • 3:  Grade 3
  • NF:  Numbers & Operations--Fractions
  • A:  Develop understanding of fractions as numbers
  • 1: 
    Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.


    Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, 8.

Download Common Core State Standards (PDF 1.2 MB)

Formative Assessment: Understanding Fractions
Lesson Objective: Formatively assess understanding of fractions
Grades 3-4 / Math / Fractions
Math.Practice.MP1 | Math.Practice.MP2 | Math.3.NF.A.1

Thought starters

  1. Why does Ms. Romano decide to assess her fourth graders' understanding of a third grade content standard?
  2. How does Ms. Romano encourage productive struggle during the lesson?
  3. What instructional decisions does Ms. Romano make based on her observations during the collaborative work?
26 Comments
Nice way to use the cousinarie rods. Where is the actual student page located. Is it available so others can use that exact lesson as a model for bringing it to their classrooms?
Recommended (14)
I would like to see the same teacher deliver the same learning activity in a high poverty area.
Recommended (3)
Using color to support students' visual information is magical. The colors offer immediate vocabulary for communicating ideas. I use color for analysis whenever possible -especially on an interactive board. It's just as effective for ELA. Thanks for sharing a great lesson!
Recommended (3)
Also wondering where we can get a copy of the student worksheet . . . thanks
Recommended (1)
i would also like to get a copy of the student worksheet . very creative
Recommended (1)

Transcripts

  • Formative Assessment: Understanding Fractions Transcript

    Student: You're supposed to show which one is going to fit.
    Melissa: Well, tell me what

    Formative Assessment: Understanding Fractions Transcript

    Student: You're supposed to show which one is going to fit.
    Melissa: Well, tell me what you're thinking right now.
    Student: I think one thirty is going to be like this.

    Melissa: My name is Melissa Romano. I teach fourth grade at Four Georgians Elementary School in Helena, Montana. Your cuisinere rods are very colorful, right? The cuisinere rods on the page are not very colorful. When you and your partner are working through all of these problems, finding candy orders, I want you to use the crayons when you're finding the answer.

    Lower Third
    Melissa Romano
    4th Grade Teacher, Four Georgians Elementary School, Helena MT

    Melissa: This lesson is specifically addressing a particular third grade standard and in the Common Core standards, there's a nice progression of fractions, third grade through fifth grade. So this lesson was about unit fractions and making other fractions from unit fractions, which will then tie nicely into fourth grade lessons where students will be decomposing fractions. In order for you to be successful, here's your I can statement. At the end of the lesson today,

    we should be able to say, "I can use unit fractions and use them to make other fractions." Students are familiar with having an I can statement, and so they're often referring to that and at the end of the lesson, we try to go back to the I can statement, and they can then sort of self-reflect and determine whether or not they were successful with that. In the past, I know we've talked about how to share something equally.

    For example, I'm going to draw something like this. We've done this before. If this was a candy bar, how could you share this equally with six people? One way I clarify information for students, I try to use prior knowledge and relate the lesson of the day we've done previously and try to make connections explicit for students and get them to recall something that we did in the past. Come up to the board and show us how to do that, Jenny.

    Jenny: I'm putting the red dots, six of them into there.
    Melissa: Are you going to draw six lines?
    Jenny: Six blocks.
    Melissa: Okay, so how many lines did you draw?
    Jenny: Five.

    Melissa: Five. If I said to you, I want to name just this part, the fractional name, what would you say? What is that? What's the fractional part? Emmalyn?
    Emmalyn: One-sixth.
    Melissa: One-sixth, absolutely. One-sixth. Remember, we call this a unit fraction, because the numerator is a one. Everyone, we call this a what?
    Students: Unit fraction.

    Melissa: Unit fraction. One more part of clarifying that was really important to me was, once the student drew the five lines and divided that bar into six parts, really naming that that was a unit fraction, for one of the parts was a unit fraction and using the term denominator and numerator was an important part of the lesson, because that's vocabulary that I want students to use. I want you to imagine this:

    we all work at a candy factory. Yes, I know. And some customers are going to be placing some orders. Your job is to fill the orders. We're going to talk through the first one together and then I'm going to set you loose. Number one says, I want a candy bar that is one-third the size of this one. Ian?

    Card: You need to find which rod fits in the box, and then find out which cube would fit as one third of it.

    Ian: You need to find which rod fits in the box and then find out what size-- which cube would fit as one-third of it.
    Melissa: Okay. The most important part of this for me is the last direction. Everyone read the last part of the directions, when it starts with capitals W-R-I-T-E. It says...
    All: Write a brief explanation to justify your thinking.

    Melissa: Remember, mathematicians communicate their mathematics. All right, on the worksheet that I gave you, you're going to be working through this with your learning partner today. Everyone point to your learning partner. Good.

    Card: For more information about clarifying the intended learning for this task, go to the Toolkit section of this module.

    Student: That's the actual side, so one more bigger than purple would be... yellow's a little-- I think yellow.
    Melissa: Can you find what one-third would look like? Show me.

    Lower Third
    Melissa Romano
    4th Grade Teacher, Four Georgians Elementary School, Helena MT
    Melissa: So for the second part of the lesson, students were working with partners to complete the fraction task.
    Lower Third
    Keiran
    4th Grade Student, Four Georgians Elementary School, Helena MT

    Keiran: We did a paper where we had blocks, rods, and we put them and we had to measure like three-fifths of this block is what block?

    Student: So you're saying that because it would be three greens, if you had-- it would be-- so take these four and you'd have to take away one, so that would be one blue, right?
    Student: Yeah.
    Student: Okay.
    Melissa: And in this part of the lesson, it was important that I circulate around the room in order to elicit evidence of their learning. What's your next order?

    Student: I want a candy bar that is one-fourth the size of this one. So let's see what this one is. I'm guessing, it's not blue, but I think it might be brown.
    Student: Yeah, I think it's brown.
    Student: We'll measure how many, one-fourth. One, two, three, four. Okay, it's red.
    Melissa: How did you know to pick the red?

    Student: The reds look-- they're a little smaller than this one and the brown is just a little smaller than the blue. And so I knew if green is just a little smaller than red, then it couldn't be green, because green is like blue. So green goes to blue, so it couldn't be blue, so it was red.
    Melissa: Getting them to talk and think out loud about what they are thinking is a way I elicit evidence. And so you saw me doing that as I circulated around the room and said things like, "Tell me about that," or "What are you already thinking?" "What do you know?" "What are you wondering about?"
    "Where are you stuck?" "What don't you understand?" are all ways that I was eliciting evidence of their knowledge.

    Student: One-fifth is one five.
    Melissa: What one-third would look like.
    Student: I try it again.
    Student: I think it's the orange.
    Melissa: The note cards are just a way for me to keep notes on what's going on in the class. I write down notes about specific strategies students use or specific problems they're stuck on, or things that I just really want to make sure I explicitly draw out in some sort of a wrap up or conclusion. Tell me if I'm getting this right. She's saying, if this is four-fifths, you're saying this is one-fifth.

    Student: Mm-hmm.
    Melissa: Because this would be one-fifth.
    Student: two-fifths, three-fifths, four-fifths.
    Melissa: And so what are you writing for your justification? I know this because?
    Student: Because...

    Melissa: As an educator, it's really important to not jump in and rescue kids. And so having kids have that productive struggle and letting them sort of wrestle with the mathematics before jumping in is really crucial, and it's one way to gather and elicit evidence from students. What is this piece? What's this fractional name?
    Student: It's one-third.
    Melissa: One-third.
    Lower Third
    Katherine
    4th Grade Student, Four Georgians Elementary School, Helena MT

    Katherine: If she gives us the answers, then we wouldn't learn anything and if we're on a test and there's usually more than one question, she'd give us the answer, but then the answer to everything. But then once we do the test, we didn't know anything because she just gave us the answer.
    Melissa: I've found that if they need my help, they tend to pay attention to my questions and if they don't, they sort of ignore me and have this just ebb and flow and let the other partner listen to me. You're putting three on there, so which would be one-third?

    Student: That one?
    Melissa: Yes, so your partner's pushing them off. Why would that be one-third?
    Student: Because it's one out of three.

    Card
    For more information about eliciting evidence for this task, go to the Toolkit section of this module.

    Melissa: I'm certain you're going to get there. We're going to keep working at it.
    Student: This is the toughest worksheet.

    Melissa: We're going to go over the answers because I know you want to know if you got it right. But what do I care a lot about? How you got your answer, good.

    Lower Third
    Melissa Romano
    4th Grade Teacher, Four Georgians Elementary School, Helena MT

    Melissa: Interpreting evidence for this lesson is somewhat complex because they're using partners, and so I had to do a lot of talking and a lot of asking questions, besides just looking at their work, to make sure I understood that they knew what they were doing and that I also know that they were doing. So you're thinking because this one's the biggest one and this one's the next biggest one, this should be four parts out of five?
    Student: Yeah.

    Melissa: I was working with two students and they sort of said, "Oh, we're stuck." And so all I had to do was say, "So tell me what you know." Read the problem to me one more time.
    Student: The candy bar is one-fifth of the size of the candy bar I want. Which candy bar is the one I want? Oh.
    Melissa: Oh, wait a second. Why are you saying oh?
    Student: Now I get it.

    Melissa: It wasn't necessarily that he didn't understand the concept or what to do. If he were just to do a worksheet without the dialog or talk to me, he might turn that in and I might think that he doesn't know it, when in fact he does. And so the formative assessment process was critical for me to understand that he knew that and also for him to understand that he knew it as well.

    Lower Third
    Keiran
    4th Grade Student, Four Georgians Elementary School, Helena MT

    Keiran: So at the very end we went over it and discussed how we got our answers and different people answered, so it wasn't just one person. It was everybody.
    Melissa: Number two, raise your hand for number two. Which bar did you find that it fit? Jenny?
    Jenny: Red.
    Melissa: Red. Say oh yeah if you also found red.
    Students: Oh yeah.
    Melissa: Oh good, because that was the right answer.

    Lower Third
    Katherine
    4th Grade Student, Four Georgians Elementary School, Helena MT

    Katherine: My favorite part was explaining how you got the answer.
    Melissa: Number three, what did you find? Raise your hand. Finn.
    Finn: Red.
    Melissa: Say oh yeah if you found red for three.
    Student: Oh yeah.
    Melissa: A few people did. So was there another answer out there? I didn't hear as many oh yeahs. Padrick?
    Padrick: Orange.

    Melissa: Orange. Fin, I'm going to go back to you. You were the first one. What did you do? How did you solve that one?
    Finn: I used the orange, but instead of writing down-- coloring in the slots with orange, I used red, so it wouldn't be as hard.

    Melissa: So your answer is that the whole candy bar is the orange one? Oh, so you are agreeing. I thought we had two different answers. I'm not sure I would say 100 percent that all students met the learning goal. I know a few students met their learning goal for sure. I can use unit fractions and use them to make other fractions. Hold up your hand, one, two, three, four or five, a one being, "I think maybe I could do that," five being,

    "Yeah, I can do that now, I know how to do that." Show me with your fingers where you think you're at. Everyone show me. Hold it up. Nice work today. Give yourself an oh yeah.
    Students: Oh yeah.
    Melissa: That was good.

    Card
    For more information about interpreting evidence for this task, go to the Toolkit section of this module.

    Melissa: Hands off your bars for a minute. According to number eight, which one is two-thirds, the bar I want? Show me two-thirds of the bar I want.
    Student: Yeah, I think it's the orange.
    Student: Yeah, it's the orange.

    Lower Third
    Melissa Romano
    4th Grade Teacher, Four Georgians Elementary School, Helena MT

    Melissa: Acting on the evidence is crucial, so it's important that I take the information I learned from eliciting and gathering that evidence and then changed my instruction. There was a definite progression of question types in the worksheet that students were using, ranging from simpler to more complex problems.

    Lower Third
    Keiran
    4th Grade Student, Four Georgians Elementary School, Helena MT
    Keiran: Number five I have to say was the hardest for everybody, because we had to use two fractions, two different blocks.

    Student: This candy bar is one-sixth of a whole candy bar, but the bar I want is four-sixths of the whole candy bar. Can you find my bar? That one's confusing.

    Melissa: How about this? I want you to skip number five, skip number five, go to the back page and we'll do number five together as a group. I was kind of surprised that there were that many students that didn't get number five, but that's a great tool for me. I know I need to present some more problems and opportunities and tasks and students to work a problem like that. Number five, Shay is right. It's got two fractions.

    It gives you one-sixth and it asks for four-sixths. How do you figure that out? Keiran, what did you do?

    Keiran: I put an orange bar and measured, but the first time, I only got five reds on it, and then the second time, I thought, well...

    Melissa: Moving forward, those few students that really understood and can explain how number five was working, and that really you could just repeat the fraction over and over until you got what you needed, I need to make changes to further their instruction and push them deeper or to the next level. Ian?
    Ian: I have an easier way to do it.
    Melissa: Okay, what's the easier way?
    Ian: If it says this is one-sixth of a whole candy bar, you can just do four of them.

    Melissa: And then I also want to definitely utilize my document camera and have students who are really understanding the concepts and who were able to successfully complete the more complex problems to be able to use the document camera and talk through what they did. Sometimes those strugglers and kids who just aren't quite there, they really need to see it and then have another opportunity to do it. Does someone hear Ian? Shay, I want you to listen. Who can repeat what Ian said? What's his easier way? Navea.

    Navea: What he did was, he took one red one and he added more so that it ended up being four of the red blocks.
    Melissa: I don't think formative assessment is necessarily always easy, but once you can wrap your head around how to do it and how to make it part of your day, I would never go back. I would never do it any other way.

    Card
    For more information about acting on evidence for this task, go to the Toolkit section of this module.

School Details

Four Georgians School
555 Custer Ave
Helena MT 59601
Population: 516

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