Smarter Balanced Item Preview Makes it Apparent That Students Need to Be Able to THINK!

I’ve finally gotten a chance to check out the recently released SBAC item preview. I am impressed with the rigor of the problems that they have created.  The tasks make it clear that students will no longer be successful simply by memorizing procedures. Rather, they will need to have a deep understanding of concepts.

Below is a list of the of sample problems for the different grade levels.

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So far, I’ve only looked at the Problem Solving tasks. For some of the problems, students are asked to choose the correct answers.  Yes, answers is plural. There may be one correct answer, but there may be more.   Some have students choose from a list of answer options and move them to correctly fill in the blank.   After each of these types of problems, the participant can check to see if the answer is correct and receive a score.

Other tasks are free response items and are not scored, but a rubric is provided. In this post, I will look at two related HS problem solving tasks: Circle 1 and Circle 2.

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My approach to coordinate geometry has always been to have students apply the Pythagorean Theorem when finding the distance between two points and when deriving the equation of a circle with a given center and a given radius.  Students who understand that the Pythagorean Theorem is the foundation for finding a distance between two points and who understand that the distance between two points is the foundation for finding the equation of a circle (set of all points that are a given distance from the center) can easily approach this problem.  Although the problem description suggests that trig can be used, I did not approach it in this way.

In my approach, the essence of the problem is creating congruent right triangles, in which the hypotenuse is the radius of each circle.  There is not even a need to find the length of the hypotenuse in order to answer the question. In fact, applying the distance formula would, in my opinion, cloud the issue. Once the vertical and horizontal legs of the initial right triangle are known, the student can simply copy those vertical and horizontal distances starting at the point on the initial circle.  The ending point will be the center of the second circle. However a student needs to understand that there is a unique right triangle that connects the center of the circle to every point on the circle. This requires a conceptual understanding of two important ideas: (1) the definition of a circle as the set of all points (x,y) that are equidistant from a given center and (2) the distance between two points is the hypotenuse of the right triangle whose legs are vertical and horizontal line segments on the coordinate plane.

Each problem has an explanation as well as a rubric.  Below is what is provided for the Circle 1 problem. Note that the student only gets 1 out of 2 points for the correct answer.  Also, the student can get an incorrect answer due to a slight error, but if the reasoning shows that they knew what they were doing, they will score 1 point. The reasoning is given as much weight as the correct answer! Hallelujah!

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Circle 2 uses the same original circle, but throws in the fact that the area of the second circle is 1/4 the area of the original circle.  For this problem, the student needs to understand that a radius that is 1/2 the radius of the original circle will be needed in order to result in an area that is 1/4 the area of the original circle.

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The Practice Standards that students needs to use to solve this problem include:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments. (explaining their approach)

5. Use appropriate tools strategically. (using the coordinate plane as a tool)

6. Attend to precision.

7. Look for and make use of structure. (use the fact that the structure of circles being made up of an infinite number of right triangles.)

Both of the problems are the type of conceptually-based problem solving that I espouse, so I am excited that the SBAC folks are are the same wavelength.  If this is what will be expected of students, it is clear that many high school teachers will have to change their approach from a procedural-based approach to a conceptual approach. I’m glad to finally have some rich examples to show teachers in the SBAC states. I’m anxious to peruse the other types of tasks.  I’ll save that for another post.

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Fawn Nguyen Rocks with Fraction Division

Ms. Win does it again! I’ve decided that Fawn Nguyen is my most favorite math teacher that I’ve never met! I look forward to meeting her and many other admired math bloggers at Math Twitter Camp 2013 in Philly at the end of July.

In this blog post, she presents a very conceptual way for students to visualize division of fractions.  I’ve tried different approaches using the area model, but none have touched this one for it’s efficiency.  Here’s why it’s better than what I’ve done in the past:

  • She uses graph paper, so the area models are neat.
  • She has students sketch the area model for both the dividend and the divisor.
  • She uses color to differentiate between the dividend and the divisor.
  • She gently leads students to discover that the dimensions of each whole are the most efficient when students use the two numbers that are  (1) the denominator of the dividend and (2) the denominator of the divisor.
  • She starts off simply and then gradually shows all cases such as mixed numbers, and finally a larger fraction dividing into a smaller fraction, resulting in a quotient that is a fraction less than one (this example is added in the comments).
  • She encourages students to check their answer using an online calculator with a fraction mode.

Here’s a peak at her first example, which asks students to divide 3/4 by 2/3:


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Although this lesson doesn’t show it, her comments indicate that she eventually introduces students to short cut numerical methods, first using common denominator and even perhaps (egads!) common numerator a la Chrisopher Danielson’s post, and finally showing the traditional invert and multiply algorithm that is easy and efficient, but leads to zero conceptual understanding.  However, after students have seen the concept unfold visually, at least the students have a chance of remembering the conceptual reasoning of fraction division.

If you were to poll students, or adults that are not math teachers, and ask them WHY 8 divided by 4 is 2, I bet that they would be able to explain it satisfactorily.  However, if you were to ask the same group (and even some adults that ARE math teachers)  and asked them WHY 3/4 divided by 2/3 is 1 and 1/8, I can almost guarantee a pained, puzzled expression.

Thanks, Fawn and Christopher, for adding to the conversation!

Both Fawn’s blog and Christopher’s blog can be subscribed to via email.  I highly recommend that you follow them for some great mathematical instructional ideas.






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Fawn Nguyen Molds Middle School Mathematical Thinkers

In her blog Finding Ways to Nguyen Students Over, Fawn Nguyen, aka Ms. Win, shows how she worked through a Dan Meyer 3-Act, Pyramid of Pennies,  with her Grade 8 students. I’ve been a Fawn fan for a while and this post illustrates every reason why I am a fan.

First, she makes the mathematics engaging and fun for the students.  The 3-Act task itself could be approached from a variety of ways.  However, once students got the answer, she realized that the problem itself was pretty straightforward;  no students “broke a sweat” figuring the answer out once they developed a plan. She decided to focus on the sequel.  Why does the summation formula work?  Students are provided with manipulatives to help them make  make sense of and model the problem.

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Fawn approaches the activity with the expectation that her students can think and figure things out.  She is the “guide on the side” rather than the “sage on the stage”, to quote a tired cliche’. The “letting go” required by a teacher can be scary. Any teacher who uses this open-ended approach must have a great deal of confidence in their ability to guide and also to in their willingness to say, “I don’t know, but I think we can figure this out together.”

In a previous post on this blog, I shared a chart in which I had shown how the Pyramid of Pennies  3-Act elicited all of the 8 Standards for Mathematical Practices.  This chart was created after I was in a class in which Dan Meyer presented the Pyramid of Pennies.  Recently, Dan Meyer revisited the 3-Act process in his blog, which encouraged Fawn to do this activity with her class.

In my work as a consultant with math teachers, I often hear some version of the following statement: “Yes, those 3-Act tasks and other problem solving ideas are all well and good, BUT… We can’t afford to take the time that it takes to work those problems.  We have too much to cover.” As the grade taught goes up, the tendency to hear this statement becomes more predictable.

High School teachers tend to be the most reticent to want to take the time it takes to help their students to learn to be independent, creative problem solvers. Yet the math thinking that Fawn’s students are undertaking will add more to their ability to “do math” than any memorized procedure will ever do.  The procedural memory will be eventually lost, but the habits of mind that are built when making sense of non-routine problems will become, as the phrase Phil Daro used to describe the CCSSM Practice Standards, “the content of a student’s mathematical character”. In another Phil Daro video, Against “Answer Getting”, he asserts that “a high percentage of what we teach kids is not mathematics, but answer-getting techniques.” (8:20) The video compares the different approaches to math problems by American and Japanese teachers.  American teachers first focus on teaching students the mathematical procedures they need to get the answer to the problem and then give them a problem to apply the skills and get the correct answer. Japanese teachers, on the other hand, choose problems based on the mathematics that can be learned by students as they work through the the problem. For American teachers, the focus is on the answer. For Japanese teachers, the focus is on the problem solving process.  My opinion is that teaching students to be independent critical thinkers trumps teaching them streamlined procedures that will lead to the correct answer as quickly as possible.

In the comments to Fawn’s post, she was asked how long it took her to complete this investigation.  She replied that it took her 3.5 days:  2 days for the original 3-Act and 1.5 days for the sequel.  My opinion is that these 3.5 days of making sense of the problem and determining WHY the summation equation works left the students with more tools in their math toolbox than 3.5 days of learning “answer-getting techniques.”

Thank you, Fawn, for sharing your students’ journey.




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Geoff Krall Makes it Easier to Mine Math Online Resources

My love affair with the mathtwitterblogosphere has gone up a notch.  After reading Dan Meyer’s post Geoff Krall Combs The Internet for Lesson Plans So You Don’t Have To,  I went to Geoff’s site Emergent Math to check out his curriculum maps.

Geoff has developed CCSSM-aligned Problem-Based Curriculum Maps for Grade 8, Algebra I, Geometry, and Algebra II.  In the works are maps for Integrated Math 9, Integrated Math 10, Integrated Math 11, and Grade 6/7.   Kudos to Geoff for taking the time to create these lists and share them freely!

BTW: In his post Math Blogging Retrospectus 2012, Geoff wrote a wonderful summary of posts to math blogs that inspired him in 2012.  It’s worth reading to get an idea of the richness of the discourse and resources available via the online math community.


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Star Polygons Take 2

The MathTwitterBlogosphere (MTB): Professional Development at its Best

I love to go to conferences and take workshops, but it involves packing, shlepping myself and my stuff to the airport, as well as money and time.  I will always continue to seek out “live” learning opportunities, but I have a PD source right at my fingertips 24-7 in the form of the MTB.  (Full disclosure: I don’t do twitter, but I more than make up with it by following many blogs.) Every once in a while, a blog post lights a fire under me, guiding me into to a line of inquiry that I can’t leave alone until I play around with the idea and write my own post to make sense of the idea.

3.5 gons? What the heck?

This post continues my pursuit of understanding of 3.5-gons and their ilk.  It all started with Dan Meyer’s March 27, 2013 post “Discrete Functions Gone Wild!” His post focused on what a regular polygon would look like when when the number of sides was not a whole number.  He used his understanding of how regular polygons with whole number sides behaved to determine the angle between sides, using the formula (n-2)180/n.  He let n = 3.5 and determined that the measure of each angle in a 3.5-gon is 77.1 degrees.  Then he constructed the 3.5-gon. and created a star shape. Then he left his readers with the following BTWs and went to bed, across the pond in Nottingham:

BTW. One of you enterprising programmers should create the animation that runs through continuous values of n and shows the regular polygon with that many sides. That’d blow my mind. I can only do the discrete values.

BTW. Malcolm Swan demonstrated the 3.5-gon on the back of some scratch paper in the middle of a design session here in Nottingham. That kind of throwaway moment (often before tea, of course) has been a lot of fun these last two months.

BTW. But where is the 3.5 in that shape? Maybe you see how the number 3.5 turned into the number 77.1 and how the number 77.1 turned into that star shape. But where is the 3.5 in the star? I’ll hint at it in the comments but I’ll encourage you to think this through. (It may be helpful to see 3.5 as the rational number 7/2.)

When he awoke, he found that several enterprising programmers had taken his challenge can created the goods.

2013 Mar 28. I love you guys. I fall asleep for a few hours and wake to find out it’s Christmas. Some interesting visualizations of rational regular polygons from Marc Garneau, Eric Berger, Stuart Price, and Josh Giesbrecht.

When I stumbled upon the post, I went to work on BTW # 3, trying to make sense to myself about why a 3.5-gon looks like this:

3.5-gon to 7/2-gon: Different representations of the same value open up new possibilities!

Looking at functions via multiple representations (graphical, numeric, algebraic, verbal) has always served me well. Some representations shine a different light on the function. Putting 3.5 into the form 7/2, a hint that Dan gave, opened up a new door for making sense of the shape.

I played with the different applets that were provided overnight by four readers of Dan’s post…another positive aspect of the MTB…great minds just waiting to use their talent for the betterment of mankind!  The 7 was pretty easy to see since there were 7 line segments.  The 2 took awhile for me to see a pattern.  What I finally noticed by trying out different fractions was that the denominator indicated how many “rotations” around the shape it took to “close up” the polygon.  These terms are used loosely, since a full rotation would take one back to the “starting point”, which would “close up” the polygon.

3 – gon, 3/1 – gon, or you may know it as a triangle

For example, in a whole number-gon, such as a 3-gon (triangle), which, by the way, could also be called a 3/1 – gon, follow this procedure:

Start at A. Heading counter-clockwise and draw segment AA’. Construct a 60 degree angle degrees at A’. Draw A’B the same length as AA’.  Construct a 60 degree angle at B.  Draw BB’ and B’ will coincide with A.  You have closed the 3/1 -gon after 1 rotation.

m/n – gons and partial sweeps or revolutions

When dealing with a m/n-gon, when n is not 1, it will take more than one sweep to get exactly back to the starting point  I am calling each “sweep around” a “partial revolution”, meaning that it gets close to the starting point, but doesn’t quite make it that time around.

Although my explanation was not mathematical precise…it was more like hand-waving and a “kinda” explanation…  I posted my comment to Dan’s post, putting that interpretation out there.

Here is part of my comment to Dan’s post:

Wow! I love math! I was blown away by this post and the idea of thinking of a 3.5 gon as a 7/2 “gon” that means that it has seven “sides” and takes two rotations to complete.

Out of the denizens of the MTB, a comment appeared from none other than Michael Serra, author of one of my favorite Geometry books, Discovering Geometry: An Investigative Approach. Of course, he has a better and more mathematically succinct way to describe how the “n” behaves. Before I explain his way, which is skipping vertices, and much less wishy-washy than my explanation, I will share the following comment that he made:

BTW: Expressing the n as an improper fraction opens the door to two ways of expression each star polygon. The star polygon 12/5 is equivalent to the star polygon 12/7. The numerator expressing the number of vertex points and the denominator expressing how many points to count from one vertex to the next vertex.

It is cool that Elaine Watson noticed that the denominator is also the number of cycles to complete the star polygon. I hadn’t seen that before.

This comment by Michael Serra served two purposes:

(1) it made me curious to find out more about 12/5-gons and 12/7-gons, and

(2) it fed my ego to have a mathematics educator that I highly respected comment on my post that I had noticed something that he didn’t!

So, in honor of Michael Serra, I am going try to make sense of my explanation of the role of the denominator and define more clearly what I mean by “it takes two rotations to complete.” While doing that, I am also going to refer to Michael’s explanation of the role of the denominator, which expresses “how many points to count from one vertex to the next vertex.”

Closer look at 7/n – gons

7/1 – gon

My explanation of the role of the denominator was influenced by how Dan created the original 3.5-gon.  He started with a line segment and the angle measure and added the sides one at a time, until the last segment met up with the starting vertex. On the other hand, Michael Serra’s more precise explanation required that the vertices needed to be placed first, followed by drawing in the sides using the rule for skipping the required number of vertices.

GeoGebra can conveniently create a regular polygon.  To do this, construct two points and a line segment connecting the points to represent one side.  Using the “polygon” tool, choose “regular polygon” and create a 7-gon. Below is a 7/1-gon.  The 7 means there are 7-sides. Using my revolution interpretation, the 1 means that it takes one revolution to complete the polygon. Using Michael’s vertex interpretation, the 1 mean that you go directly to the next vertex, without any skipping over vertices.  This forms the regular 7-gon, we all know and love.


In a 7/2-gon, use the vertices of the 7/1-gon as a skeleton.  The 7/2-gon will be represented by the black segments. Choose a starting vertex and a direction (I chose counter-clockwise) around the polygon to the 2nd vertex and draw the line segment between the two vertices. Repeat this method until you return to the starting vertex. Since you are skipping vertices, you will have to go around more than once.  In this case of a 7/2-gon, you will go around “about” twice to pick up the missing vertices before you get back to the starting one.

There is still one side to draw to close the 7/2-gon.  This is not a full sweep. The other two sweeps each created 3 sides. The final sweep will create 1 side, so we’ll call it Sweep 2 1/3.


What about a 7/3-gon? Start with the skeleton of the 7/1-gon.  Choose a starting vertex and count counter-clockwise to the 3rd vertex and draw the segment from the starting vertex to the 3rd vertex.  Repeat that process until you close up the polygon by returning to the starting vertex.  Each “partial  revolution” or “sweep” is illustrated below:

As in the 7/2-gon, we’re not quite there yet.  There is one more segment to draw. Each sweep filled in 2 sides.  This last sweep will fill in 1 side, so we’ll call it sweep 3 1/2.  Although the denominator doesn’t exactly define how many revolutions, it hovers around the ballpark.

Here is the final 7/3 – gon.

Both of the final sweeps here have required only one more line to be drawn. Questions to consider:

  • Will there always be one leftover side to complete the shape?
  • If not, what are the possibilities for leftover sides?
  • Explain and illustrate your conjectures.

Comparing and Contrasting 12/7-gons and 12/5-gons

My next project was to figure out what Michael Serra meant when he said “Expressing the n as an improper fractions opens the door to two ways of expression each star polygon. The star polygon 12/5 is equivalent to the star polygon 12/7.” What did he mean by equivalent?

As I worked through my ideas, I simultaneously began creating an investigation for students, Comparing and Contrasting Congruent Star Polygons.

This is a very rough first draft, so I’m not ready to post it.  I may have scaffolded it too much for it to be considered an investigation.  If anyone has any comments or suggestions, please pass them on.

Conclusion: Fan Letter to the MTB and All of the Math Nerds out there who Keep the Flame Lit

To conclude, all of this started with one post by Dan Meyer that intrigued me.  Because Dan has such a following, there were many more contributors who created applets, comments, and their own ideas.  I don’t think that professional development gets much better than this.  Being  actively involved in keeping up with math blogs and then continuing on with my own investigations sparked by the posts, has made me much more reflective about my work and better at what I do.  Books and articles have their place in my life; I’ll never be willing to burn my large math library.  However, reading is a solitary pursuit.  The MTB is a community pursuit.   I need both.




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What does a regular 3.5-gon look like?

Just when I thought I had no more to learn and discover about regular polygons, Dan Meyer draws me back in and wows me.  Once I know that the sum of the angle measures is (n-2)*180 and, if it’s regular, each angle has measure {[(n-2)*180]/n}, how much more is there to know? So the sum of the interior angles of a triangle is 180° with each angle 60° , a quadrilateral has 360° with each angle 90°, a pentagon has 540° with each angle 108°, etc. And, of course n must be a positive integer…right? Been there, done that.  Let’ move on!

Dan is currently in Nottingham soaking up the knowledge imparted by Malcolm Swan and others at the University of Nottingham Centre for Research in Mathematics Education.  There are a lot of names and organizations in the math education world that are all interrelated: University of Nottingham, Shell Centre, MARS, MAPS, UC Berkeley, The Charles A. Dana Center.  Before I continue here, I want to take some time to figure out a family tree of sorts.  Here’s what I came up with.  I may be wrong, so if anyone cares to comment and set me straight, please do.

No matter who begot whom, all of them have excellent resources for the mathematics educator. If you are not familiar with them, please visit the following:  MARS, MAPS, Shell Centre, CRME, Charles A. Dana Center

Back to 3.5-gons: Dan told the story of Malcolm Swan illustrating a 3.5-gon on the back of a scrap piece of paper one afternoon at a design course, which was probably a throwaway aside for Dr. Swan. However it provided the spark that generated an intriguing post by Dan. That spark, in turn traveled to  several programmers, Marc Garneau, Eric Berger, Stuart Price, and Josh Giesbrecht, to work their magic showing graphs of the n-gons. The same spark traveled to Vermont and I couldn’t let it go until I wrote a post about it.

Check out Dan’s post to see how he introduced the idea starting with a regular 3-gon (equilateral triangle), regular 4-gon (square), etc. Then Dan illustrated using graphing software what Malcolm Swan had shown him on the scrap paper for a 3.5-gon.  He challenged the reader to make sense of it by giving the hint to think of 3.5 being represented as 7/2.

With that hint, I watched his animation of a regular 3.5-gon being built one side at a time. Where is 7/2?  Oh…I see!  Then I got to thinking about how I could make this into an investigative unit for Geometry students.

Have the students look through the different ways that programmers showed the polygons.  Which was their favorite tool for visualizing the n-gons and why?

Have them play with their favorite visualization tool to answer the following questions:

  • What conjectures could students make about regular n-gons?
  • If given a regular 3.2 gon, what would the angles be?
  • How many sides would it have?
  • How many rotations would it take for the last side to meet the first side?
  • How are a 3-gon, a 3.5-gon and a 3.2-gon the same, different?
  • Add a 4.7-gon into the picture.
  • Why do some have many rotations and some have only 1 or 2?
  • What type of regular n-gon will have many rotations before the last side meets with the first side?
  • What type will have few rotations?
  • How do the angle measures change as n-increases? Why?
  • How does the graph of y = [(x – 2)*180/x] relate to the shapes of the polygons?
  • What is the domain of y = [(x – 2)*180/x]?
  • Can we build a regular ?-gon? If so, what would it look like?  If not, why not?

This is the type of problem that is not a modeling problem, but one that really forces students to think outside the box and look through a different lens of perception.  All of the CCSSM Practice Standards except the Modeling standard would be exhibited by students as they work through this problem.

So, Dan Meyer, thank you once again for sending that spark this direction.  And thank you Marc Garneau, Eric Berger, Stuart Price, and Josh Giesbrecht for sharing your gift for programming to give us tools so that we can analyze regular n-gons visually and discover some fascinating things.  And a final thank you to the Math Twitterblogosphere (MTBS) that provides so much free knowledge and expertise to so many people.


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North Country UHS 3-18-13 Inservice


Part 1 of 2 North Country CCSSM HS March Inservice

Part 2 of 2 North Country CCSSM HS March Inservice

The Lowell bowling ball images are not included in the powerpoint because they were too large.

Continue reading

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NEKSDC HS CCSSM Statistics Powerpoint

HS Stats & Prob ppt for Watsonmath

There are two videos that had to be taken out of the powerpoint presentation so that it could be uploaded.

Here are the links to the two videos:

Arthur Benjamin TED Talk: Teach Statistics Before Calculus!

Ionica Smeets TED Talk:  The Danger of Mixing Up Causation and Correlation


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Tilings,Tesselations, Math Munch, Vi Hart and Getting Carried Away on a Spiral of Mathematical Magic

It all started over an hour ago when I clicked on the latest Math Munch blog entry.  I was immediately intrigued because of this image:

I’m a sucker for tilings. The Math Munch post was about Marjorie Rice, who was influenced by Martin Gardner, who wrote about hexaflexagons, on which Vi Hart created a series of videos. Whew!  Some people may call this web surfing procrastination  I prefer to call it research. If you check out all of the links in the above two lines, you can decide whether your are procrastinating or researching.  Whichever you want to call it, I think you’ll have fun.

Martin Gardner, who died in 2010 at age 95, was famous for his math puzzles published in Scientific American in the mid-20th century and wrote over 70 books. His work is often described as recreational math. The word recreational has the connotation of “play”. I’ve always enjoyed looking at fractals, tesselations, tilings, origami, etc. However, my appreciation has always been more visceral and artistic than mathematical.

When I was a high school teacher, I used these recreational math activities as fillers, but never studied them in their own right.  Perhaps this is because of all of the other non-recreational (read dry and lifeless) math concepts that I was required to teach.  More and more, in my work now as a mathematical consultant, I am trying to figure out ways that this so-called recreational math can be used in a way that reinforces the important mathematical concepts while engaging students.

One activity that I developed based on the MARS Roman Mosaic Task.  The tasks calls for students to describe the mosaic verbally without the audience having the ability to look at the original mosaic. Describing the picture requires students to use many of the CCSSM Practice Standards: #1 Make sense and persevere, # 2 Reason abstractly and quantitatively, # 3 Construct viable arguments, # 6 Attend to Precision, # 7 Look for and make use of structure, # 8 Look for and express regularity in repeated reasoning.

I took this task one step further and asked students to create the task using geometric software such as GeoGebra or Geometer’s Sketchpad.  This provides great practice in understanding the effect of  transformations on the coordinate plane. It also adds CCSSM Practice Standards # 4 Modeling with Mathematics and # 5 Use appropriate tools strategically.

Roman Mosaic steps on GeoGebra  The previous link gives an example of ONE approach to making this shape.  In subsequent work with students and teachers, I have seen many different approaches, some more efficient than others.  However, the decisions that are made for any approach to solve the problem provide a rich learning experience for students.

As I was reading about Marjorie Rice, I came upon another idea that could be used with students to reinforce, among other things (1) their understanding of mathematical communication via labeling shapes, (2) their understanding of the angles necessary to create tilings, and (3) the idea that mathematics is a dynamic subject that is still under construction.

My idea was spurred by this organized list of the 14 tilings for pentagons that have so far been discovered. Here’s an example of two of them.  Notice how the pentagons are described using A, B, C, D, E  and a, b, c, d, e.

Students, working in groups, would be given the 14 pentagon tiling.

Idea for tasks, one with clear answers and the other open-ended:

  • sketch the pentagon unit that is tiled.  Label all sides and angles with the appropriate capital letters and lower case letters  to match the definition of the tiling that is given below each image.
  • Compare and contrast the tilings.  Group them into groups involving 90 degree, 180 degrees, 360 degrees, 120 degrees in their definitions.  How are they alike? How are they different?

Artistic extension:  Go to  Marjorie Rice Intriguing Tesselations to see the tesselations that Marjorie Rice created from the tilings.

  • Have students discuss the difference between a tiling and a tesselation.
  • Long term project: Have students create a tesselation of their own from one of the 14 pentagonal tilings.

In conclusion, there are so many ways to get caught up in spirals upon spirals of ideas that are available at a click on my computer.  I think it is time well spent.  Others may call it procrastination.  I think I’ll go play a game of Free Cell, which is my go-to procrastination tool…short and sweet…and I usually win!







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Modeling: Generating Creativity from Chaos

The Common Core Math Standards stresses modeling with mathematics. Modeling is a K-12 Practice Standard as well as a High School Conceptual Category. Developing a clear understanding of what is meant by modeling is the first step in being able to implement it in math classrooms. The next step is knowing which types of tasks elicit modeling practice in students.

Although most resources on modeling align it with real world tasks, I argue that a modeling problem does not have to be related to the real world.  It can be any well-designed task in which a student has to create a mathematical model (equation, graph, table, flow chart) to solve.  The connection to the real world does not by itself make a problem a modeling problem.  The depth of creative thought needed to solve the problem is, in my opinion, the marker of a good modeling problem.

The idea for this post was stirred up last week as I was working with a group of high school geometry teachers.  They had chosen tasks from the textbook bank of problems to include in a common assessment that would be administered to all geometry students in the school.  Here are two of the tasks:

The door on a spacecraft is formed with 6 straight panels that overlap to form a regular hexagon. What is the measure of angle YXZ?

A pillow is the shape of a kite. Heath wants to create a design connecting opposite corners from point B to point D, and from point A to point C. Show your work to find the amount of cording needing. Round your answer to the nearest tenth.


A spacecraft door?  A pillow in the shape of a kite? What is meant by cording?  Where does it go?  On the perimeter or on the diagonals? Or on both? Yikes!  Both of these problems were a lame attempt to dress up a routine task in a real world context. It’s not that these tasks are trivial. They require students to apply theorems about shapes that we would like a geometry student to know. In the first one, the students need to know either (1) the sum of the interior angles in a hexagon, or (2)  the sum of the exterior angles in any polygon is 360 degrees, (3) know that the angles in a regular polygon are congruent, and (4) know that each exterior angle and interior angle form an ordered pair.  In the second problem, they need to know (1) that the diagonals of a kite are perpendicular and (2) the Pythagorean Theorem, and it helps if they know the shortcut of creating a 3-4-5 right riangle .  In both of these problems, the diagrams give clues that help to scaffold the problem.  Both problems attempt to engage students by masquerading as problems that real people might solve. However, it would be a stretch to call these problems that would help students learn to model mathematically. Real world does not necessarily imply engagement.

Some good resources around modeling can be found in the California Mathematics Project website. Most of the resources found refer to modeling being around real world phenomenon. Both 100 ideas for modeling and four modeling problems  revolve around real life scenarios. In the CCSSM definition of modeling, the examples given are again focused on real life scenarios.The “How to Solve It” by G. Polya  definition closely resembles the Modeling Cycle described in the CCSSM definition of modeling.  The Modeling Definition by Henry Pollak introduces some examples of modeling that are not real world, but the final discussion focuses on modeling real world situations.

Here is one rather verbose, and perhaps too academic, definition of modeling found there.  But the last line   (bolding is mine) provided me with a great phrase to describe modeling. It is one of the few definitions that does not specifically require modeling to apply to to the real world. However, the use of the word “phenomenon” indicates that the author considers that the conceptual representation (mathematical model) describes an observable fact or event.  Can a fact or event be non-real world? I believe it can be.

Modeling refers to the process of generating a model as a conceptual representation of some phenomenon. Typically a model will refer only to some aspects of the phenomenon in question, and two models of the same phenomenon may be essentially different, that is in which the difference is more than just a simple renaming. This may be due to differing requirements of the model’s end users or to conceptual or esthetic differences by the modellers and decisions made during the modelling process. Esthetic considerations that may influence the structure of a model might be the modeller’s preference for a reduced ontology, preferences regarding probabilistic models vis-a-vis deterministic ones, discrete vs continuous time etc. For this reason users of a model need to understand the model’s original purpose and the assumptions of its validity. Models are basically known to generate creativity from chaos.
source: Elwin Savelsbergh

I love that closing sentence.  Our goal in “school math” is to give students plenty of opportunities to practice generating creativity from chaos. The ultimate goal is that students will transfer that knowledge of how to generate creative solutions out of chaos and be able to use it in their real life.   That doesn’t mean that every opportunity we give students to practice this skill has to relate to the real world.  There are plenty of interesting problems that build students modeling skills that have no immediate connection to a practical application in real life.

The paper Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics by Cynthia Nicol (University of British Columbia) and Sandra Crespo (Michigan State University) published in School Science and Mathematics, Volume 105(5), May 2005 gives examples of modeling problems that are in imaginative non-real world settings. Here is an excerpt from the introduction:

This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students’ everyday life experiences.  The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics….
Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.

The authors quote Sierpinka (1995) to point out the challenges of forcing real life contexts into mathematics:

“If one wants to make students’ activity more authentic and take children out of school into supermarkets and other kind of environment…their cognitive activity will become more authentically every day, but much less authentically mathematical.”

They quote Egan (1992) who suggested that “the more distant and different something is from students’ everyday experience and environments the more imaginatively engaging it is”. Egan also “challenged the assumption that to engage students in learning subject matter requires that they make connections with everyday experiences.”

Brown (2001), has a similar take as Egan:

“If we can speak of what is ‘real’ in a more vibrant sense than what ‘exists’ or what we can ‘touch’ and ‘see’ then we not only legitimize more interesting connections between mathematics and the real world but we also suppress the need to seek real world connections as a slave against an otherwise ‘unreal’ world of mathematics.”

The authors summarize Brown’s and Egan’s ideas by saying:

“Both reject the notion that making subject matter interesting and meaningful to students requires the need for it to be placed in real-life contexts. Instead they challenge mathematics educators to imagine alternatives that engage students’ mathematical imaginations.”

The study focused on students’ emotional and intellectual engagement with two mathematical contexts that they consider having no connection to the real world.  They call the contexts “mathematically imaginative”.  They studied students working problems in two contexts. The first group, who worked with a place value task, was comprised of 36 pre-service elementary teachers, so one could argue if the findings can be generalized to K – 12 students.  One could also argue that there is a connection in this activity to the real world. The second group was a class of 50 Grade 6/7 students who focused on ideas about space and shape in the book Flatland.

The place value activity was, interestingly, an activity that was included in an online history of mathematics grad course that I taught for a few years at Drexel University.  The activity involves decoding an archeological document in which Mayan symbols, in base 20, are converted to base 10.  The activity is from a book by Bazin and Tamez (2002) Math and science across cultures: Activities and investigations from the Exploratorium. 6_MayanCode. In the Mayan Code activity, “the exploration of Mayan symbols inspired student by presenting the context of human emotions and intentions (both the Mayan and archeologists) that helps to make the exploration of different numeration systems intriguing.” In the Flatland activity, “students were invited to imagine and visualize life on an infinite Euclidean plane – a space quite different from their own.”

The authors found that in both of these activities students generated questions not because the teacher asked them, but “out of genuine curiosity and wonder”.  In their conclusion, they do not suggest that all mathematical activities need to be “embedded in contexts that are less than real.  Our point here is that imaginative contexts, stemming from real or more imagined situations, can be important resources for mathematical engagement and thinking.”

I have recently stumbled upon a new blog Median by Don Steward. He has a wealth of rich tasks in which the student is asked to generate creativity from chaos. Here is an example of one such task, which is simple, but can be modeled in different ways.

An equilateral triangle and three isosceles triangles together make a rhombus.  What must the angles in the rhombus be?

When I gave this task to a group of six high school teachers, every one of them approached it a little differently.  Some of them created two equations with two unknowns.  I think one person created three equations with three unknowns, and at least one person was able to find the answer with one unknown. The task itself is meaningless in the real world, but the task starts out as chaos and the person solving it slowly makes sense of the issue, thinks back to what they know that can help them solve it, and creates a way to approach the problem.  A discussion following the solution could determine which solution was the most efficient, the least efficient, the most creative.  The purpose of doing the task and then discussing it is to help students build up their creative muscles.  In the next problem they are faced with, an insight they received by working and discussing this problem may help them to be more creative in their next solution.

Too often, we give students a lot of small problems that require no creativity [p. 135 (1 – 31 odd)].  Some of the problems masquerade as real world modeling problems.  (Usually those are # 27, #29, and #31.) There is a place for drill in math class, just as there is in sports a place for practicing skills. However, unless we are asking students to regularly generate creativity from chaos…real world chaos, imaginative chaos, or abstract chaos…we are not preparing them for the chaotic world they will encounter outside of school.

If you think about it, any “real world” problem that we put into a textbook has already been partly digested before we ask students to chew on it.  All of the needed information is given, or a diagram gives away a lot of information, or a table is set up that organizes the chaos for the students.  Dan Meyer has become famous in mathematics education circles for challenging the types of problems typically found in textbooks.  If you are one of the few reading this that has not watched his TED talk: Math Class Needs a Makeover, stop and watch it now. Dan, Andrew Stadel, and others have developed rich modeling problems in 3-Acts.  Act I presents the chaotic situation which students need to make sense of, where they generate their own question and determine what information they need to know in order to create order out of the chaos. Act II is where the teacher provides just enough information to solve the problem, and only when the student asks for it, but does not prescribe a specific way to solve it.  Act III provides the answer. I have found that students working a 3-Act problem not only build their modeling muscles, but also practice the other 7 CCSSM Practice Standards: make sense of the problem, persevere, reason abstractly and quantitatively, construct viable arguments, use appropriate tools strategically, attend to precision, look for and make use of structure, look for and express regularity in repeated reasoning. While the 3-Act tasks are based on real world scenarios, they are not pre-digested for the students.

Fawn Nguyen, a middle school teacher in California who hosts a great blog, has created a site, Visual Patterns, where she invites people to post growing visual patterns. The goal is to  find an explicit rule to for the pattern that will allow the “value” of any term to be found without knowing the value of the previous term.  “Value” is determined by the number of units in the pattern.  Some of the patterns have a single unit, while others that have multiple units. Although these are not “real world problems”, the rule that is created is a mathematical model that makes sense out of chaos.

I will close with a quote from Georg Polya, guru of mathematical problem solving, found in the preface of his book How to Solve It.

Modeling is problem solving. It is generating creativity out of chaos.  It is a life skill that reaches across all educational disciplines.  Problems that we choose for students to build this skill do not have to directly relate to the real world, although they can.  Most importantly, the tasks we choose need to engage students and create just enough disequilibrium or chaos to be challenging so the student can, in Polya’s words, “enjoy the triumph of their discovery.”












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