Tuesday, November 29, 2011

Fun Math Trick Video: I Can Guess Your Phone Number

Here's a fun math trick video that your students might enjoy.  It's a trick where the calculations will result in your phone number, excluding the area code.  Students may find it interesting that you have to enter your entire phone number as part of the trick, but a series of calculations are performed with the digits.

I'm sharing this just as a fun thing to use with students.  It would be perfect for a time when students need a short "brain break".  You'll probably have at least one or two students who are interested in trying to figure out how the trick works.

Viewing Tip:  You can use SafeShare.TV to safely share YouTube videos with students.  SafeShare.TV takes all of the ads and comments off the videos.  I've already uploaded this video.  You can view it in SafeShare.TV by following this link.

Monday, November 14, 2011

Inspiration Can Come From the Strangest Places!

Gotta love it when great math content just comes to your inbox!  The image below was part of a GoDaddy.com ad I received recently.  When I saw the ad, I was immediately inspired to write this post.   Guess you never know where you might get inspiration for math content or teaching strategies!!!

The subject line of the email was "Choose your savings, all the way up to 30%".  Here's a screenshot of the GoDaddy.com ad:

Why I like this graphic:

Often times in math, we concentrate our efforts on the topic we're teaching to the exclusion of other questions or topics that could arise from the same problem or situation. By doing this we unwittingly lead our students to become locked in to one way of thinking about the situation. This causes problems for students when they later (usually on standardized tests) experience problems/situations that are a variation of or the counter to the problems they're used to solving.  Or, if the question that accompanies a familiar picture is different than the typical question.  For example, when students see a picture of an aquarium, they automatically think that they're going find the volume.  But, the question could ask how much sand is needed to cover the bottom of the aquarium.  If students have only found volume when given pictures of rectangular prisms, they won't even consider that the question could be asking for the area of the base.  If we don't give students experience in generating questions from given situations, they'll continue to jump to the wrong conclusions even if they do read the problem!  That's the big problem with formulaic teaching rather than teaching students to think and problem solve.

The graphic above is a prime example of this type of formulaic teaching.  Most students would look at the graphic and automatically think they need to find 10% of 50, 20% of 100, and/or 30% of 130 in order to answer the question.  Did you notice that there really isn't a question?...At least not one posed by a math teacher!  "How much do you want to save?" was part of the graphic in the email, so I'm not counting that as the question right now.  But, students may think this is the question.  If so, they'd probably still jump right to the calculations mentioned above.

The beauty of this graphic is that there are many questions that can asked about this sale offer. And, some of them lead to some interesting mathematics and considerations.

Generate Possible Questions:

Have students brainstorm questions that could be asked about this image.  There are many possible questions.  It will be interesting to see what students come up with.  You may learn a bit about what student's know, or don't know, about percents and interpreting information by the questions they generate.  You might want to check out the first post of my Brainstorming series to see some benefits of brainstorming.  Here are a few possible questions that came to mind:
• How much will you save if you choose the 30% option and only spend \$130?
• How much more will someone save if they choose the 20% option over the 10% option?
• Why would Go Daddy make this offer?  What's in it for them?  Which offer do you think Go Daddy wants people to choose?  Why?
• What's the most I can save if I choose the 20% option?  What's the least I can save if I choose the 20% option?
• Which one is the best deal?
• Are the offers proportional?  Why would this matter?
• If I choose the 30% option, am I saving 3 times what I would if I choose the 10% option?
• If I'm planning to spend \$115, should I just go ahead and spend \$130 to get the 30% discount?...At what \$ amount, would it make sense for someone jump to the next level of savings?
Some of the questions above have a correct answer and some are great for discussion because they don't have one correct answer.

There are also some added benefits of having students generate multiple questions for a given situation.  These are a just a few of extra benefits:
• Natural Differentiation --- As you can see from the questions listed above, there are questions at various levels.  Once you've created the list, you can assign students questions based on their level of readiness.
• Built-in Choices --- This is a great way to provide students with choices.  Students can choose which question or questions they find most interesting and really want to answer.  Providing choices is a key component in effective teaching because it gives students a sense of control.   It's also a way to differentiate by interest.
• Teachers Gain Insight About Students Level of Understanding --- When you have students generate questions, you get see where they are with their understanding of concepts and problem solving skills.  In the beginning, some students will only be able to come up with surface level questions.  As they gain more experience with problem solving and generating their own questions, you'll be able see and document their growth.
• Student Attitudes Improve --- When students become more involved in their learning experience, their attitudes generally improve.  If you combine this with differentiation and giving choices, you're likely to see an even greater impact on student attitudes about math and learning.
• Opportunities For Problem Solving Arise --- From the questions above, you can see that some of them would involve problem solving as opposed to rote computation.  And, I would venture to say that many students would much rather answer some of these problem solving questions than the more basic questions.  The best part is that the skill practice we want students to have is automatically built-in to the problem solving experience!
Whether or not you use this particular scenario with your students, you may want to begin having students generate their own questions with other content.  Give it a try and you might just see some unexpected growth in your students.

Hopefully, you've found this post useful.  If so, please pass it on to someone else who may find it helpful.

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Wednesday, November 9, 2011

Questions That Cultivate Mathematical Thinking

I just had the privilege of presenting at two of the 2011 NCTM Regional Conferences.  While attending both conferences, I noticed that a common theme in many presentations related to creating/developing mathematical thinkers.  This got me thinking about how the questions we ask, or more importantly, don't ask students on a daily basis can cultivate, hinder, or even prevent mathematical thinking. So, I've started a list of questions/question prompts that can/should be used regularly to help cultivate mathematical thinking and reasoning.

Questions That Cultivate Mathematical Thinking  (this list in no particular order):
• What if...? ---- "What if" questions allow teachers to change constraints on problems/situations.  This furthers mathematical thinking by having students see patterns and relationships beyond the initial problem.  "What if" questions also help students to see that their initial thoughts about the answer to the question or their problem solving strategy may no longer apply to the new situation.  This understanding helps students learn to focus on the context of a problem rather than what they perceive as "set guidelines" for solving a particular type of problem.
Example of "What if" questioning:  The two tables below represent the sales from Fu Do Chinese restaurant in Anchorage, Alaska.  The premise of the problem is that Fu Do, a family owned and operated restaurant, had to close four days during the week of 10/9 - 10/15 due to a death in the family.
The questions:
First Question:  Which measure of center (mean, median, or mode) best represents the sales of Fu Do on Oct. 2 - 8 (a typical week)?
"What if"Question:  What if Fu Do had to close 4 days for a funeral?  Which measure of center best represents the sales from the week of Oct. 9 - 15?
In Table 2, the constraints of the problem were changed resulting in the possibility of a different answer to the question.  This type of problem and questioning provides a basis for rich, interesting discussions about which measure of center is the best representation and what factors impact this decision (range, the context of the situation, etc.).  Students begin to see that they need to consider many factors when answering a question like this.
This problem/discussion/question can be taken even further by asking another "What if" question that again changes the constraints of the problem.
Another "What if" Question:  What if Fu Do only had to close 3 days during the week of Oct. 9 - 15?  Which measure of center would best represent the sales from Oct. 9 - 15?
Now the mathematical thinking and discussion can really get interesting!  By changing this one constraint on the problem (only 3 days closed), you've opened up new questions and factors that need to be considered when answering the question.  You can cultivate mathematical thinking even more by opening up the discussion to include comparisons between the 3 different scenarios. This may lead to asking other questions like "Under what conditions would mean be the best representation?...median?...mode?"
• What patterns or relationships do you notice?...How can you use this pattern to solve the problem?...How do these patterns/relationships help us to think about the problem? --- Mathematics is all about recognizing and using patterns to answer questions and/or learn more about a situation.  As students get better at recognizing and understanding patterns they begin to develop number sense, see connections between mathematical concepts, and become better problem solvers.

• Is this the most efficient way to solve this problem?...What's the most efficient way to solve this problem? --- It's important to have students explore various ways for solving a problem.  But then it becomes important to have students evaluate strategies for efficiency.  Some methods will always be inefficient and should be discarded as such. With other methods, efficiency may depend on the individual.  The method that's most efficient for you may not be the method I find most efficient for solving the same problem.  The key to this questioning is to get students to recognize that there are various methods for solving problems, that it's important to consider efficiency when choosing a strategy, that some methods are valid but inefficient, and that efficiency can depend on individual understandings and preferences.
In an effort to keep this post from getting too long, I'll stop elaborating on each question.  Below are a few more questions/question prompts that help cultivate mathematical thinkers.

Questions that Cultivate Mathematical Thinking Continued:
• What other ways can this problem be solved?
• How could you represent this visually?...differently?  In what other ways can this problem be represented? (tables, graphs, equations, pictures, etc.)
• Compare and contrast these two problems...How are these two problems different?...How do these differences affect how you would solve each problem?
• How could you define or explain this without using numbers?
• Based on ________, how would you approach this problem differently now?
Tip:  If you're just beginning to incorporate these types of questions into your daily practice, write some of the question prompts on posters and place around the room.  They'll serve as a reminder if you draw a blank.  And, students will think you posted the questions to prompt their thinking.

By no means is this an exhaustive list!  It's just a work in progress.  Let's continue to build this list together.  Leave a comment with your favorite questions that help cultivate mathematical thinking.

Tuesday, November 8, 2011

No Clickers, No Problem! Try Poll Everywhere

Want to create class surveys and get instant student feedback about math problems, but you don't have access to clickers?  No clickers, no problem!  Poll Everywhere allows you to create multiple choice and free response questions for your students.  You'll get instant results that can be shared with the class.

Poll Everywhere is committed to education and has many features that make it ideal for classroom use.  Below is a list of some of these features:
• Create as many surveys as you want with a Free teacher account...With a free K-12 account, you can have up to 40 students respond to each poll.  Just create a new poll or If you have a single class with more than 40 students, you can email them and they will adjust your plan.
• Polls are quick and easy to create...You literally could have a poll created and ready to use in a couple of minutes.
• Multiple ways for students to vote...Students can text their responses in or they can vote online if they have access to an iPad or computer.
• As responses come in, they automatically appear on the results chart...You don't have to refresh in order to see newest results.
• Results charts can be embedded into blogs, websites, Power Points, etc...This is a nice feature which would allow for comments about the poll or survey.  You could also use this as means to address student misconceptions if they were responding to a math problem.