Tuesday, October 13, 2015

Rules that Expire: "Just add a zero!"

By Cassandra Hatfield, RME Assessment Coordinator

Many tips and tricks that we teach our elementary students as rules of mathematics, are introduced as a way to help students recall a procedure rather than truly promote their conceptual understanding of the content. However, many of these rules learned early on don’t hold true as students start to learn more advanced content in middle and high school.

An article in Teaching Children Mathematics, 13 Rules that Expire, by Karp, Bush and Dougherty addresses some of these common misconceptions. Let us know if you see these rules that expire in your classroom, and how you address them.

The first rule we are going to talk about is, "Just add a zero!"

When you multiply 4 by 30 what strategy do you use?

Consider these possible strategies for solving this problem:
Strategy A Strategy B
4 times 3 is 12.

Then add a zero and you get 120.
4 times 3 is 12. 

12 times 10 is 120.

At first glance one may think both of these strategies are appropriate. However, use the same strategies to multiply 0.4 by 30:
Strategy A Strategy B
0.4 times 3 is 1.2.

Then add a zero, so 1.20.
0.4 times 3 is 1.2. 

1.2 times 10 is 12.

The strategy of adding a zero to the right of the number when multiplying by a multiple of 10 only applies to whole numbers, and can’t be generalized. Additionally, utilizing this trick of “adding a zero” isn’t mathematically sound, and does not support students in reasoning and justifying their answer.

Let’s take a look at the mathematics behind Strategy B for each of the above problems.
4×30 0.4×30
4×3×10 0.4×3×10 Decomposition or Partitioning into Factors
(4×3)×10 (.04×3)×10 Associative Property of Multiplication
12×10=120 1.2×10=12

Elementary students can and do use the properties of operations when computing; it’s our job as teachers to help students see and understand the value of the mathematics behind each strategy.

Cluster problems are one way to support students with using facts and combinations they likely already know in order to solve more complex computations (Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M., 2016). Here’s a set of cluster problems that lead to 34 x 50. Consider how these problems are related and the rich discussion you can have with students about the properties of operations they used to get their final answer.


Karp, K.S., Bush, S.B., & Dougherty, B.J. (2014). 13 Rules that Expire. Teaching Children Mathematics, 21 (1), 18-25.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Boston: Pearson.

Friday, October 2, 2015

Creating New Learning Opportunities with FAQs

By Brea Ratliff, RME Secondary Mathematics Coordinator

One of the most important components of any informational resource is a section labeled “Frequently Asked Questions” or “FAQs”. The FAQ section is often easy to find, can be a very helpful tool when you need to quickly find the answer to a question you might have, or you need a reminder of a process or an idea you had seen previously and need a refresher. Whether you are a novice or an expert, the FAQs are helpful for everyone.

So, where are the FAQs in the classrooms? Would students know how to access answers to the pertinent and relevant questions they have about whatever concept they are learning? More important, as teachers, are we aware of some of the questions students might have which could be included in a FAQ section about our classroom?

Here are a few strategies for helping you establish an FAQ space in your math classroom.

1.  Have a clear understanding of your expectations. If we anticipate our students will rise to our expectations, we must be clear about what the expectations are. Many of our expectations are outlined in a syllabus or a letter that goes home to parents at the beginning of the year, but what about our expectations for learning mathematics in the classroom? Here are a few questions to consider:
  • What are my expectations for collaboration in the classroom?
  • What techniques will I use to ensure my students comprehend what they are learning?
  • What opportunities can I provide for students to communicate with me when they have questions about the math?
For the concept you are teaching, identify the major misconceptions or misunderstanding students might have. Understand the background knowledge necessary for being successful with this topic, as well as why the topic is foundational for future studies. Emphasize content vocabulary and mathematical processes. The FAQ is not only a resource, but can be used as an evaluative tool to help you identify what students do and don’t understand about a concept or unit of study.

Here are some example questions for a lesson or unit focused on dividing fractions:

2.  Get to know your audience. Most – if not all – people want to succeed in whatever they do, and can sometimes feel embarrassed to ask questions. I know many middle and high school students who would never raise their hands and tell their teachers, or the entire class, they don’t understand math concepts taught to them years earlier. I also know several highly educated adults who would rather “play it safe” and not ask questions, out of the fear of looking as if they don’t know something. An FAQ space can make learning accessible for everyone.

3.  Find your medium. So, now you’ve developed your FAQs, but where will you keep it? As I mentioned earlier, a syllabus can be a great starting point, but let your creative juices flow when selecting your medium. Try creating an FAQ bulletin board in the classroom, or maybe adding an FAQ section to your classroom website. Use Twitter as an FAQ space or create posters throughout the school so students can see and be reminded of these ideas outside of your classroom.

4.  Make it collaborative. One of the greatest rewards of being an educator is the gift of being a teacher and a student at the same time. To quote science fiction author Robert Heinlien,“When one teaches, two learn.” As you teach concepts, allow your students to draft and share questions to be added to your FAQs.

Please share some of your FAQ space examples with us on Twitter at @RME_SMU

Quote by Robert Heinlein. Think Exist http://thinkexist.com/quotation/when_one_teaches-two_learn/149371.html retrieved 23 September 2014