On May 23, 2010 in my very first blog, Teaching Math; It’s All in the Balance, I shared my view that both the traditional and reform camps have something to offer math educators. Basically, traditionalists believe skills should be taught based on algorithms, formulas and step-by-step procedures; reformists support a more inquiry-based approach that emphasizes developing conceptual understanding and problem-solving skills. My contention is that a balanced approach is best. Also, I am an advocate for using engaging, interactive technology whenever possible to reach and teach this generation.
I shared a conversation I had with Grant, my oldest grandson, about adding two two-digit numbers. During our talk it was obvious his skill for adding single digits was developing nicely, but he lacked an understanding of place value concepts. Even though he could get the right answers, when I asked him the value of the digits, he had no clue.
In my second blog posted on June 27, The First Steps in Developing Conceptual Understanding of Place Value, I emphasized the importance of developing a foundation of understanding. I also shared ways to help children understand place value when first learning to count with non-proportional items (straws and money) and with proportional manipulatives (base-ten blocks) when adding and subtracting.
Grant is now in the 3rd grade. He tells me he “gets” math. He doesn’t need my help, thank you very much. That is… until this week. Monday, he called to say he had taken a test last week, and he wasn’t happy with his grade. “Can I come down, Gigi? Can you help me?” Smile. Gigi is back in the picture.
He brought the module test he had taken last week. “C.” Uh-oh. There were definitely some trouble spots. Addition, subtraction, and estimation. But this test didn’t display the numbers written out vertically or horizontally with addition or subtraction signs. All were real-life word problems. All were multiple-choice, and in this case multiple-choice was appropriate. Third grade? Impressive.
I started the session by asking questions….explain to me how you got that answer-type questions. I’ve seen that look and heard that sigh before. [If teachers only had the time to ask every student that type of question…. ]
The first few problems involved money. This kid knows his money. No worries.
There were problems that involved alternative representations for numbers with charts and graphs. One question asked which candidate represented 486 people. The chart listed:
Dorthy 280 + 100 + 6
Emily 200 + 200 + 80
Mark 300 + 180 + 6
Oscar 400 + 60 + 8.
Yep. He said Oscar represented 486 people. Common mistake. But with a quick discussion and base-ten blocks, this one was quickly understood and corrected.
The problems with how many more were also real stumpers. I understand. Sure seems like it should be addition, doesn’t it?
But then came about how many kind of problems involving estimation. Example: Jacob’s mother runs 77 miles each week. About how many miles does she run in 4 weeks? Most telling for me was his work written to the side and that he had tried to erase. Thankfully, I could still read it because it revealed so much.
28 + 28
Oh my. I have seen this before. No conceptual understanding of place value—everything in the ones place.
Back to my trusty base-ten blocks. There was direct instruction on my part, outstanding focus and attention on his part, kinesthetic activities, and representations with base-ten blocks alongside the abstract numbers. After we did that, he “got it.” He actually explained regrouping to perfection! We have more practice to do, but I am confident he is on the road to place value mastery now! Then, on to multiplication!
I also wanted him to work on these concepts using technology. He LOVES my iPad, so why not take advantage of that enthusiasm and use it to reinforce place value concepts?
PlaceValue version 1.0 by MontessoriTech was the first iPad app used. I first thought he might find it too easy and redundant, but he really liked it. The screen can be set up to 4 digits, with or without zeros, with a voice to read the number, or in manuscript or cursive. It really drove home the value of each of the digits. Worth the $0.99.
Another $0.99 app for developing place value sense is also aptly called Place Value, two words, not one, developed by Joe Scrivens. The first time I downloaded it, Grant touched the screen immediately causing it to skip the instructional video. Once on the game screen, we had no idea what to do because there was no way to get back to a homepage and there were no intuitive icons to follow. So, I deleted it, then reinstalled it. (iTunes does not charge again once it is initially installed.)
The homepage does have an instructional video that explains what to do. Here’s how it goes. A number is displayed. Let’s say 4806. It then asks “how many” for the ones, tens, hundreds, and thousands if you have set it to use 4 digits. The touch screen counts the number of fingers you place on the screen. The object is to answer the questions for 10 problems as quickly as you can. The program records the number of seconds used to answer the questions for all 10 numbers. It gives the fastest time, and it gives the option of playing it again.
I liked playing it. It was fun. But I am wondering if this app really does anything more than have the student select the numeral that is the placeholder for the ones, tens, hundreds, or thousands? It wouldn’t take much to expand it and turn this app into something that really would be a useful teaching device for developing conceptual understanding of place value. By the way, I learned more about the app by reading the description page at the iTunes Store than in the app itself.
Latest update: Grant retook the module test Wednesday. Grade: 95. Gigi is happy, and so is Grant. We’re getting there!
4 thoughts on “The Pursuit of Place Value Understanding, Part III”
In my opinion, understanding place value is a concept that many adults still don’t get. It is difficult for a lot of people. I don’t think it is difficult because it is a complex idea. I think its difficulty lies in its simplicity. It is actually so simple that it is amazing how we (teachers) can make it complicated in a pursuit to make it simple.
I love the way you started this post discussing the great divide in mathematics education. I agree that both is needed because it is the only way to truly understand math. I think it is great to have students first discover the relationship between an actual real life quantity and its numerical representation. Students can then move to step-by-step procedures by which makes putting various quantities together in a manageable fashion. During the entire process, students and the teacher can constantly explain thinking. Imagine small groups of students talking to each other about how they got an answer while the teacher pops in on various discussions. The teacher also leads a whole-class discussion to help deepen understanding through the discussion. It is a very sophisticated dance that is awesome to experience.
I would love to see an app that keeps the balance. It would be a difficult task but I hope a person smarter than me could figure it out :).
Ms. Norfar, I appreciate your detailed comment, and I’m looking forward to following you on Twitter so that I can gain more insight into your teaching methods. I do hope you can check out Elevated Math, an iPad app that I truly believe does keep the balance.
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