Abstract to Concrete…

Which is the better deal? (cc licensed flickr photo by lantzilla: http://flickr.com/photos/lantzilla/75870909/)

Memories of my first year of teaching came rushing back to me last week. I vividly remember being quite distraught that my students were having difficulty understanding how to round numbers to the nearest 10, 100, and 1000. I had taught the lesson just as the book had suggested and assigned the page of problems to the students. The page consisted of all three types of rounding (in ONE lesson). For the life of me I couldn’t figure out what I had done wrong. The following year, it dawned on me! Just because the book suggested that I teach all 3 levels of rounding in one lesson, didn’t mean that is the approach my students needed. Over the course of the next several years I slowed down when teaching this concept by starting with rounding to the nearest 10 and building upon that. Yet, as I think back – I still wonder if my students really grasped the abstract task of rounding and how it is used in the “real world.” Did they understand what they were doing or were they just going through the motions of looking at the number in the one’s place to determine what effect it would have on the tens place, etc…

Last week a student was brought to my attention that was having difficulty with rounding. The task required students to find the estimated sum of two whole numbers. I spent my lunch break developing a plan that would take this abstract concept to a more concrete level. I started by making a 100’s grid using excel. I wanted the student to see the number placement and how that determined the estimated number. Yet, the plain 100’s chart was still confusing. When rounding a number I know I, and teachers, typically ask students to determine what the number in question is between. For example if rounding 54 to the nearest 10 we know the number 54 is between 50 and 60. That was hard to tell on the 100’s chart. So I added an additional column of to the left as a place marker, but not as part of the actual 100s chart.

I first explained my own thinking with the chart modeling how to use it with numbers ending in 0-4. For example, if the number to round was 54, I would locate 54 on the number chart. I would see that it was between 50 and 60, yet it was closer to 50. Therefore 54 rounded to the nearest 10 is 50.

We did several examples with numbers ending in 1-4. We then moved on to numbers ending in 6-9. For each number I had the student explain her thinking. She was feeling quite proud of herself that she was able to do this! Once I noticed the student was grasping both ends of the spectrum, we proceeded to the numbers ending in 5.

When she seemed comfortable with the process I asked her if she thought it would be okay if we covered up the chart to see if she could try some problems without it. She agreed. I left the chart within her reach and stated if she felt she needed to go back to it she certainly could. She was able to do the problems and explain her thinking without using the number chart.

Once we did several two digit numbers we proceeded to three digit numbers. I modeled my own thinking. I told her if I’m being asked to round to the nearest 10s I would look at the digits in the tens and ones. I may even underline them or circle them. Then I could use the same concept used with the chart above. Time only allowed for us to start on this concept.

While this still is an area that this student needs to work on, we made steps in the right direction. The concrete hundreds chart seemed to help her start to understand the abstract concept of rounding. She got quite good at explaining her thinking. Math language and concepts started to emerge in her descriptions. At one point, I had given her the problem 43+42. She immediately talked through how to round 43 to the nearest 10 and then proceeded to put a 40 by the 42 as well. I asked her to explain what she had done. She simply stated that 42 was one less than 43, so if 43 was rounded to 40, 42 would also round to 40. She beamed as she explained that – her self-confidence was shining through!

Many times when our students struggle with a concept, we need to create new ways to approach it in order for the student to grasp the new learning. The beauty of math is that there are many paths to the same answer.

What have you done to reach a struggling student? How have you changed an approach to meet the needs of a student? Let’s share ideas. It is in our sharing and collaboration that each of us becomes a stronger educator.

Disclaimer: The rounding idea mentioned was just something I thought of.  I’d love to hear other beneficial methods in helping students with this concept.


  • Aviva (@grade1) 22 Dec

    Kathy, I love your example here of how you took an abstract concept and made it concrete. I think that visuals help a lot with doing this. When I teach the concept of addition to my young students, I have some big laminated papers that have a big mitten graphic then a space followed by another big mitten graphic then another space followed by two big mitten graphics together. Under the first graphic is a line and then an addition sign and then another line (under the second graphic) and then and equal sign and then a final line (under the two graphics). When the students go to add 5+4, they put five objects on the first mitten (and write 5 on the line underneath), four objects on the second mitten (and write 4 on the line underneath), and the slide all nine objects over to the last two mittens, count up the total, and then write 9 (as this is how many they have ALTOGETHER). This really helps them understand the concept of addition. Thank you for reminding me about the importance of making these abstract concepts more concrete.


    • Aviva (@grade1) 22 Dec

      P.S. Sorry for the two typos: the “and” instead of “an” and the “the” instead of “then.” I didn’t know how to scroll up on this device first to proofread my comment before posting it. Hope the typos don’t cause too much confusion. My apologies again!


      • Kathy Perret 22 Dec

        I love your mitten idea and use of manipulatives. What a great way for students to understand the concept of adding. We can never do enough modeling for students. Revealing our thinking, using manipulatives and other forms of realia, or items is beneficial.

        As I think of my own rounding example, the challenge will be to then take the concrete and move to the abstract so that the student understand the use of the concept, rather than just rote problem solving.

        Our work is never done! 😉


        PS – Don’t worry about the typos. That happens to all of us. I haven’t figured out have to take the spell check off my phone. I sure has a habit of thinking it knows what I want to say – but is usually incorrect (and I don’t catch it until after I hit send!)

  • Robin Phares 22 Dec

    Hi Kathy, I enjoyed your post. I think, without even trying, you have defined a successful teacher. Wouldn’t it be great if all teachers would spend their lunch break developing new ways to reach our students?
    Recently, I was teaching excel to my computer application class. Many students were catching on and even, dare I say, starting to enjoy working with Excel. Then there was Matt. Matt, a reluctant learner in all his classes, just was not participating. He had made up his mind after out first experience with Excel that he could not and would not learn this program. The next day I was having a casual chat with his math teacher and she mentioned how sad it was that he was in 7th grade and still did not know his multiplication facts.
    That evening I went to sleep with Matt on my mind and somewhere in the middle of the night I got the idea of showing Matt how easy it would be to design a multiplication chart using Excel. I spent about 15 minutes with him and I could see he was starting to see the benefits of Excel.
    Our next class lesson was on Excel formulas. This is usually where I start to lose my students interest in Excel. I was so pleased when Matt stayed after class and said “Are you telling me that Excel can actually do my math problems for me?” I was amazed at quickly he caught on to formulas. He was one of the bright spots in the class.
    This experience taught me a lesson. I was approaching Excel all wrong. I was showing students how I used Excel. Matt taught me I need to teach it how they can use Excel. Don’t you love it when you learn things from a 12 year old?

  • Beverly Libell, Grade 3 ELANCO SD 6 Jan

    What a great idea, I think I will go and make some of these charts for my math students! I really like your idea of posting the tens on the left hand side of the chart as well for them to be able to visualize the concept. A friend of mine told me that she teaches the students that “0, 1, 2, 3, and 4 year old children get to stay in bed in the morning so they are ’rounded down’ and 5, 6, 7, 8, and 9 year old children get up to go to school so they are ’rounded up’. While this does not teach the concept, it’s a trick that works for some students.

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