Increasing Rigor with Multimedia

Multimedia makes is so easy to create more rigorous math problems.  The example below shows how you can use and edit computer screenshots to add rigor to a typical math problem.

Rigor is increased using two different methods in the examples below.  In Example 1, rigor is increased by changing the constraints of the scenario.  In Example 2, rigor is increased by changing both the constraints of the scenario and the question.

The screenshot of the interactive spinner was taken from Glencoe's Virtual Manipulative website.  The problems are correlated to the 6th Grade Common Core State Standards.

6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

•  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Typical Question (Routine Procedural Problem):

Typical Question with Computer Screenshot (Routine Procedural Problem):

In this example, the problem comes to life because this a screenshot of an interactive manipulative.  The spinner has been spun 8 times with the percent of each color shown in the right column of the table.  The number of times each color was spun has been blacked out and the question is still the same.  The rigor of this problem has NOT be altered because the given information and the question remain the same.

Ex. 1 Typical Question Altered to Increase Rigor by Changing the Constraints (Non-procedural Problem):

 

The screenshot above is the same as the 2nd example, but this time the total spins and the number of hits for red, yellow, and purple were blacked out.  The question is still basically the same.  You're still asking how many times the spinner landed on certain colors.  However, the rigor has been increased because the total # of spins has been blacked out.  

By making this simple change, the problem goes from being a routine procedural question to one which involves problem solving.  Students now have to think about a plan for solving the problem based on the information given.  This problem also involves multiple steps with 2 different types of percent scenarios since students have to first determine how many times the spinner was spun before calculating the number of times each color was spun. 

Caution:  When making this type of change to a problem, it's only non-procedural the first few times.  If all practice problems are structured this way, they become procedural.  That's not always a bad thing if the goal is to practice the procedure.  Just realize that the rigor dissipates after the initial problem solving opportunity.

Ex. 2 Typical Question Altered to Increase Rigor by Changing the Constraints and the Question (Non-procedural Problem):

This time the same screenshot is used, but the total spins has been covered up and the question has been changed.  The rigor is now much higher and more problem solving is involved.  Students have to decide on a problem solving method(s) which would work for solving this problem. They'll also have to think about the percents given in order to determine what type of answer might make sense (a lot of spins or a small number of spins). 

Notice that the type of practice has not changed, but the level of rigor has dramatically increased.  In addition, if the goal is for students to practice finding percent of a number, this type of problem can provide a lot of practice for this skill as students guess and check possible answers.  The practice of the skill within the context of the problem is still procedural, but finding a solution to the problem requires students to think and make decisions which makes the process of solving the problem non-procedural.

Note: When the level of rigor increased to this degree, the problem became much more interesting and compelling.  This is a problem students might actually want to solve. This problem might be interesting enough to give students the motivation to persevere through the problem solving process.

What ideas do you have for increasing rigor?  Leave a comment and share your thoughts.

Mathematics and Multimedia Blog Carnival 18

Welcome to Mathematics and Multimedia Blog Carnival 18!

Before jumping into the blog posts, here are some Fun Facts about the number 18:

• the only positive number that is double the sum of its digits
• the third heptagonal number
• a semiperfect number because 3 of its factors (3, 6, and 9) add up to 18...All multiples of semiperfect numbers are also semiperfect numbers.
• the atomic number of Argon
• the legal age for voting for most countries
• Bobby Labonte's car number in the NASCAR Winston Cup Series when he won the 2000 Championship.
• the number of chapters Ulysses, James Joyce's epic novel, was divided
• 18:00 corresponds with 6:00 pm in military time

Now that we've enjoyed some trivia about the number 18, let's move on to some excellent blog posts related to mathematics teaching and learning.

Just for Fun:

Guillermo Bautista (founder of Mathematics and Multimedia Blog Carnival) recently wrote a fun post titled 25 Signs You are a Future Mathematician.  You can check out more of Guillermo's posts at Mathematics and Multimedia.

15 Items or Less, an image posted by Dave Gale, is sure to get a laugh.  Dave's blog is Reflective Maths Teacher on Posterous.

Wild About Math sometimes reviews math related books.  Review: Magical Mathematics was a recent post featuring the book Magical Mathematics.

Articles Related to Teaching Mathematics:

In his post, Calculus and Kobe Bryant, Dave Martin uses a video clip with Kobe Bryant to engage students in problem solving.  Dave blog is Real Teaching Means Real Learning.

Back to Back Shape Describing Game, another post from Dave Gale at Reflective Maths Teacher, requires students to know and use math vocabulary in order to draw geometric figures.

Patrick Honner presents A Quadrilateral Challenge at Mr.Honner Math Appreciation.  He recently posted a creative solution to his original challenge.

What's your problem Part III, is the 3rd in a series of posts by David Coffey.  In this series, David explores assessment and evaluation.  David's blog is Delta Scape.

In Mixed Up Mixture Problems, David Cox uses Geogebra  created graphics to demonstrate how proportional reasoning can be used to solve these traditional algebra problems.  You can find more of David's posts at Questions? Trying to Make it Matter.


Technology Integration:

Colleen Young shares some of her favorite online resources for teaching mathematics in Top 100 Tools for Learning 2011.  You can find more resources from Colleen at Mathematics, Learning and Web 2.0.

Using Popplet in the Math Classroom is one of the most popular posts from this blog.

For Students:

Colleen Young recently started the blog, Mathematics/Learning Mathematics - Resources for Students.  A Variety of Online Resources is one of her latest posts in which she shares some useful applications for students.

Sanjay Gulati shares many ready to use Geogebra applets at Mathematics Academy. His latest applet demonstrates Adjacent Complementary Angles.

6 Supurb Universities Around the World to Study Math was recently posted on the Tripbaseblog. 

Well, that's it for Mathematics and Multimedia Blog Carnival 18.  I hope you enjoy these articles as much as I enjoyed putting this Carnival together! 

Zootool: A Visual Social Bookmarking Tool

Oct. 23 was a sad day for me because my favorite bookmarking tool, SimplyBox, closed to the public.  I liked SimplyBox because of it's ease of use, organization and functionality.  But, most of all, I liked that it was visual.  You could collect screen shots of the sites you wanted to bookmark.  So instead of just seeing the name of the site, you would also see the screenshot.  SimplyBox was such a excellent tool, that I wasn't surprised with their decision to focus on Enterprise and close to the public.

So, that sent me on my quest to find another visual bookmarking tool that also had the ease of use, organization and functionality that I wanted.  I came across several candidates in my search, but Zootool was by far the best for meeting my criteria.  When you save a bookmark in Zootool, it takes a screenshot of the website.  You can organize your bookmarks into categories, which Zootool refers to as Packs.  You can also organize your bookmarks with tags.  The only real drawback I've found with Zootool is that the application doesn't allow you share an entire Pack.  You can only share a single bookmark or Profile Page link, which gives someone access to your entire collection of bookmarks. 

You can share your bookmarks and follow other users with Zootool .  If you find a Zootool user you want to follow, you can follow them through the application and/or through RSS.  The RSS option for following a user is located at the bottom of the users Profile Page.  You can find my Zootool Profile Page here if you'd like to follow my bookmarks.  You can follow and add your comments to my bookmarks.

Zootool also offers an iPhone app.  I put the app on my iPad too!  There was a small fee for the iPhone app, but it's well worth it! The Zootool app ranks very high among my favorite apps.

Another feature of Note is that you can place bookmarks into multiple Packs.  The iPad app allows you to select multiple Packs at the time of bookmarking.  If you're bookmarking on a desktop or laptop, you can select one Pack at the time of bookmarking.  Once the site is bookmarked, you can simply drag it into as many Packs as you want.

Calling All Math Bloggers Round 2

I'm pleased to announce that the Mathematics and Multimedia Blog Carnival is on its eighteenth edition and will be hosted here.  This is the second time I've had the privilege of hosting the Mathematics and Multimedia Blog Carnival, and I'm looking forward to making it one to remember!  The Carnival will be posted on Monday, December 19, 2011.  The deadline of submission is Wednesday, December 14, 2011. You may submit your articles here.

To increase the chance of your article of being published, read the Mathematics and Multimedia Carnival’s Criteria for Selection of Articles. To view the list of previous carnivals click here.

Note:  As of now the Blog Carnival Submission System is down.  In the meantime, you may submit your articles directly to me at kristi.grande@gmail.com.  

I look forward to receiving your submissions!

If haven't seen it yet, the Math and Multimedia Carnival 17  is now live at Mathematics for Teaching. 

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.

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.  

Added Benefits:

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.

If you enjoyed this post, you might also like:

Five Benefits of Brainstorming in the Math Classroom 
Questions That Cultivate Mathematical Thinking
Seven Ways to Go from On-task to Engaged 
Come On Down! Problem Solving on The Price is Right
Looking Beyond the Obvious to Deepen Understanding

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):

• Why?...Why did you do that?...Why do you think...?...Explain your thinking. --- In my opinion, asking "why" is one of the most overlooked, undervalued, and important questions a math teacher can ask.  This question helps teachers by giving them a better understanding of where the student is coming from and how much they really do understand about a given topic, concept, or problem solving situation.  Asking "why?" requires students to think about and verbalize their thought processes.  When you ask students "why?", you gather information to help you ask other questions that address misconceptions and further mathematical thinking. 

• 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? 
• How has your thinking about this problem changed? 

 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.