Place Value: How to Assess Students’ Understanding

If you’re familiar with Cooks Illustrated magazine, you know that they use their test kitchen to tinker with recipes and improve on them. In each issue, they report on all of the attempts they make and then finally present a recipe they recommend. In a way, I feel as if my search for a way to assess students’ understanding of place value has gone through a similar math teaching test kitchen.

By the end of first grade, students are expected to understand that the two digits of a two-digit number represent amounts of tens and ones. In second grade, they extend their understanding to three-digit numbers and the meaning of the digits in the hundreds, tens, and ones places.

First and second graders are generally familiar with numbers up to 100. Many can write the numbers from 1 to 100. Many can compare numbers—for example, 54 and 45—and tell which is greater. Many can identify which digit in two-digit numbers is in the tens place and which is in the ones place. These used to be the assessments I relied on for checking students’ understanding of place value, typically using worksheet assessments on which students filled in missing numbers, circled the greater numbers, and indicated whether digits were in the tens or ones place.

But once I began to spend time asking students questions, face to face, I found that while worksheet assessments might be helpful, interviews often produced surprising and sometimes disturbing evidence of shaky understanding.

For example, while students could fill in missing numbers correctly, when I asked them to count to 100, some were confused when changing from one decade to the next—for example, stumbling when they reached 29, not sure what came next. When I asked students to write a number like 38, some incorrectly would write 308—writing 30 as they said “thirty” and then writing 8 as they said “eight.” While they could tell which digit was in the tens place and which digit was in the ones place, when comparing 54 and 45, for example, some could only offer the explanation that 54 is greater because “it comes after 45”—their explanations didn’t relate to the place values of the digits.

A More Reliable Way to Assess
Convinced that worksheet assessments were not sufficient, I collaborated with a team of colleagues (special thanks to Mallika Scott and Lynne Zolli) to create a litmus test for finding out whether students have the necessary understanding of place value. We wanted the interview to be short (because the information is important to have for all students in the class), efficient (requiring a minimum of materials), and informative (giving clear information about students’ understanding). I’m not including here all the interview options we tried and eliminated, as the writers of Cooks Illustrated do for their recipes, but instead to cut to our current final version.

PV.Paint

Last spring, when I spent a day in an elementary school in Brooklyn, NY, I used this as a baseline assessment to check second graders’ place value understanding with two-digit numbers. View how a few of the students responded.

First watch Anna. After asking me to repeat the question, Anna responds automatically that the 1 in the number 16 means 10.
 

Daryl responds correctly, counting out the remaining cubes to show that the 1 in the number means 10.
 

Jonathan also demonstrated his understand of the meaning of the digits.
   

However, other students’ responses indicated their lack of understanding. As an example, Gabriela shows one cube to represent the 1 in the number 16.
 

Dennis also responds incorrectly.
 

Now What?
My hope is that you’ll try this assessment with your students. My suggestion: Predict first for each of your students how you think he or she will respond. Then ask and see what you learn. I’d like to hear from you, and for all of you to hear from each other, so please post comments, results, surprises, and any questions you have.

After Interviewing, Then What?
Incorporate into your class instruction investigations that help students build understanding of place value. In the new fourth edition of About Teaching Mathematics, I include a collection of suggestions for teaching place value on pages 344–358. Also, I rely regularly on the ideas presented in Teaching Arithmetic: Lessons for Introducing Place Value, which I wrote with Maryann Wickett. And for students who would benefit from additional intervention, Do The Math modules Addition/Subtraction A and B are extremely useful.

24 thoughts on “Place Value: How to Assess Students’ Understanding

  1. Thanks, Marilyn, for sharing this Place Value interview assessment question. I am going to encourage our K-2 teachers so use it with their students. I am happy to report that some of them already use your AS Place Value book!

    Are you familiar with the work of Dr. Michael Battista, specifically his books on Cognitive Based Assessments and Teaching? He has 6 books, 1 per content area, including Place Value, based upon his 10+ years of research on how students understand math. We have been using some of his paper and pencil assessments 1-on-1 with our K-5 students. They help us to understand each student’s level of understanding of the content and then we incorporate his teaching ideas to help address the needs of each student. We have found them particularly effective with Place Value.

  2. I enjoy learning from these, but I teach kindergarten. Do you have ideas for me to help my students? I’m trying to do investigations but engage NY math has kinda taken over. They’re pretty good but I need some hands on stuff? Thanks. alice

  3. Yes, a very revealing interview question. I might be inclined to then move onto another teen number and have the student count out the correct number of cubes to begin with to see if that makes a difference to the way they answer. This may prompt them to view the number as a whole set before they work with its partitions.

    In the classroom I would be giving students such as Gabrielle and Dennis lots and lots of practice modelling two digit numbers with discrete materials (bundling sticks, beans in canisters, stacking cubes) and minimally structured materials (slavonic abacus, tens frames). It’s important that they are then able to talk about their models with peers and the teacher who should also model explicit explanations of the tens and the ones. This way students are developing the math concept alongside mathematical language to help cement the ideas and gain clarity.

    • THanks for the suggestions. Most often I have the students count out the cubes. Actually, I typically ask them to count out 10 cubes onto a sheet of paper. Next to the paper, on the desk, I place six cubes and ask them to count how many I have. Then I push mine onto their paper and ask, “How many cubes are there in all on the paper now?” That establishes the 16, but also lets me see if the student knows, without counting that 10 plus 6 more gives 16. I’ve found there’s no one best way to assess students and I continue to tinker.

  4. When you mentioned interview questions both Nina and I were curious and hoping to learn more. I like how this question can uncover misconceptions. It’s like detective work.
    I recently saw a Richard Feynman quote “The first principle is that you must not fool yourself – and you are the easiest person to fool.” It’s easy to use a pencil and paper assessment without an interview and think that students understand. Time is gold in a classroom so I’m equally enthused that you’re finding interview q’s that are just as quick as they are informative.
    Is there any place for a number line in an interview like this?

    This also makes me think about extending place value understanding from 0 – 100 to thousands, and beyond. I’ve seen some wild misconceptions with 4th graders and place value once the numbers get an order of magnitude or two larger. Think this assessment could scale up to larger numbers?

    P.S. Keep the posts coming about other interview questions!

    • I’ve seen the same issues when ‘scaling up’ to larger numbers. Possibly hiding a weak or flawed conceptual understanding of PV in the 1-10-100-1000s
      Misconceptions that bigger numbers behave differently

      • Where have you noticed it? I never demanded the traditional subtraction algorithm but students knew it and used it.
        The interesting thing isn’t that they’d use the procedure correctly. It was that they could explain what was happening in the ones, and tens, maybe even hundreds but then say something non-sensical about the thousands. These noticings were from students reasoning at a wide range of levels.

        I have my suspicions about the root causes but still much more to learn.

  5. I teach multi-age students, 7 and 8 year Olds (2nd and 3rd grade) and will try this interview. I have been trained to teach CGI, but always struggling with once students are understanding a concept where to move them next. I can’t wait to read your research to help with my uncertainties. Thank you!

  6. Thank you! I find this really interesting. I was listening to the language you used, and in Gabriela’s interview you said, “Show the 6 with 6 cubes… What the 6 means in 16.” Then with Dennis you said, “These 6 match the number 6. Can you show what matches #1?” I think that for many kids, place value is just about matching the numbers into 10s and 1s places and this interview can help us understand how they are thinking about the 10s and 1s. Many activities kids do to learn place value just have them identifying tens and ones, not actually building the numbers and seeing that they are composed of bundles of 10s and some 1s. It seems that having the child make the number themselves by counting out the right number of cubes by 1s would be important for the kids who are having a harder time holding on to both the 10s and 1s at the same time. For the purposes of giving us a quick sense of who is secure with this concept and who is shaky it seems effective. I might ask them to count out 26 next, and see if they can show that the 2 is both 20 and 2 10’s. Lots more packing experiences are needed in my opinion.

    • You’re correct that this is a suggestion for giving a quick sense of who is secure and who is shaky. Then I dig more deeply with students in several ways. As an example, see my response below to the comment from Michele Kire. For students who answer correctly to the assessment on the video clips, I also put 10 more cubes on the paper and ask students, “When I add 10 more cubes to the 16, how many will there be?” It’s interesting to see if they know or have to count. THanks for your post.

      • I like that idea. That way they’re not counting by ones. It’s such a good reminder that we have to watch young students do the math as much as possible in order to really know what they understand. The intuition that we get during an interview gives us a sense of a child’s understanding that a paper can’t show. If a child is secure with the concept, the exact words we say don’t matter to the same extent, (“show the 6, match the 6”) because I can see the confidence or lack thereof in his or her face.

  7. About 6 years ago I saw this questions asked for the first time on a beginning of the year second grade interview assessment. I thought it was way to easy for kids at that age and for sure they would all get it right. I was so surprised that only 2 kids in the entire class were able to answer it correctly. Since then we have placed a variation of this question on both our first and second grade assessment interviews and it has made such a difference in how well we understand what kids get and don’t get about place value.

  8. While reading the article about “Place Value: How to Assess Students’ Understanding” I liked the way how the writer uses metaphor of “Cooks Illustrated” magazine to describe students’ understanding of place value as teaching test kitchen. From the first grade to extensions of grades, learning about place value seems to be hard tasks for students. While reading this article, what I came to my mind is that the idea about “the multiplication of number 5 with even numbers disturb place value” which was covered in the lecture. As we see multiplication of number 5, it resulted in 5, 10, 15, 20, 25, 30, 35…… and so on. Therefore, when we get the equation like 5 times 60, some students make mistake of “extra zero.” To be specific, students think that there is 0 at the end, so they just put one zero without considering 5 times 6 is 30 which resulted in “two zeros”. By gathering information from example from the article and the example of article, I was able to gather information about how students can confused with math concepts and how can I adopt this in the future. Also, this also demonstrates the significance of core math concepts and teach them with conceptual understandings.

  9. I’ve been a big fan of Marilyn Burns for YEARS and have followed her thinking closely. Many years ago I was teaching third grade in an upper middle class urban school and decided to assess the third graders familiarity with place value by using this interview method. I jotted down how comfortable they were with their answers. I was shocked when about 18 out of 24 children showed me one tile to represent the ‘1’ in 14. I shared this with the other third grade teachers. They were quick to declare that THEIR third graders would most certainly be able to such a simple assessment and they were SURE their third graders had a very firm grasp of place value. Only one of the 3 agreed to also assess her class. Imagine her surprise when she got similar results to mine. I assessed the third graders again in January and again in May. By May I had only 1 child who insisted it was one tile, folding her arms and glaring at me. I’m not sure we were actually addressing her math conceptual knowledge! I want to add that once a child (or anyone) has assimilated a concept they can’t be budged. So, if you change the interview and do it several different ways you should get similar results each time. Simply using a different question, or using a different number of tiles etc. should not give you hugely different answers. Thanks for the great blog!!

  10. I’m currently working on a school-wide system to test fluency from Kindergarten through fifth grades. I’m really leaning towards the individual interview approach, but I’m having a hard time finding really good questions to ask at each grade level. I recently attending the 2017 KASA convention in Louisville, Kentucky and attending a session that consisted of a video conference with you. I remember these videos from that session, but I also remember you saying there were more videos, but I couldn’t remember where you said to find them. Any input would be most helpful, thanks!

    • You might be interested in the videos on Math Reasoning Inventory site. Click on the Resources tab, then scroll down and click on Video Library. Clips are sorted by questions and by students. Let me know if you’d like additional information.

  11. I gave this place value interview assessment to a second grade boy and he was successful in showing the ten cubes as representing the one in the number 16. So I added a task and gave him a card with 16 + 10 written horizontally on it and asked him to read it and then work it out. He said: “18”. I asked how he got that answer and rather than explain his answer, he proceeded to count-on from 16, but he got “25” because he began his count with “16” on the first finger and continued counting using up his ten fingers. Then I asked him if he was sure and he recounted-on with his fingers and this time started with 17 he ended on “26” as the answer. He said, “How come I keep getting different answers?” I asked which answer he thought was correct and he said “18”. I asked him to show me how he got 18. His response: “16 and 0 is 16, then you add this one, 17 (pointing to the one in 16) and then that one (pointing to the one in 10) is 18.”

    I know his teacher was teaching the algorithm for double-digit addition and this was evidence of a student trying to carrying out a taught procedure without any understanding.

    But what do you make of his understanding with materials but not being able to add 10 more? Was it because it was a written equation that he just worked with the digits? What would be a good next question to dig further into a (2nd grade) student’s place value understanding?

    • Wow, this is so interesting. Several thoughts: After he counts the 16 cubes, instead of showing the 16 + 10 equation, what about asking, “If I give you ten more cubes, how many will you have?” Then he can verify with the cube. Then ask, “What if I give you ten more cubes?” Again, he can verify. After doing this, you might talk about how to write down “in math” what you did: 16 + 10 = 26; 26 + 10 = 36. Then try another problem, like 18 + 10, or maybe 16 + 30.

      Also, I’m curious if you asked him, when he had the 16 cubes in front of him, “If you gave me 10, how many would you have left for yourself?”

      All of these are clues. If you try them, send me his responses and we can think some more.
      And thanks for writing.

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