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Teamwork and Diversity

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How many jelly beans are in the jar? This looks like a typical Fermi problem.

MTMS_blog-2014-12-19.jpgFermi questions, such as found on http://www.nctm.org/publications/article.aspx?id=20696, are open-ended questions that require the solver to use common knowledge and back-of-the-envelope calculations to estimate quantities about which he or she knows very little. They are one of my favorite types of problem because they foster problem solving, estimation, and number sense. (I will discuss them more in my next blog post.) However, this type of prediction problem also gives us the chance to do a “social” experiment: What happens when we take into consideration the results of the “team,” the entire group of students who participate in the challenge? Now the Jelly Bean Jar problem (and similar Fermi questions) can become a paradigm for studying teamwork and collaboration

The NCTM Standards and the Common Core Standards, as well as other frameworks for teaching and learning (for example the Partnership for 21st Century Skills), mention collaboration and communication as well as problem solving as critical skills that students should master to function well and succeed in the twenty-first century. The issues they will have to face are complex: The simple problems have been solved; they have been left with the difficult ones. Matters such as freshwater scarcity, gender equality, and space exploration are not problems that can be solved by any one individual. Therefore, the value of teamwork becomes ever greater. It is well known that a team or a collection of people often makes more accurate predictions than the individuals in it. This phenomenon, called the wisdom of crowds, depends both on talent and on diversity. In fact “the collective accuracy of the crowd depends in equal measures on the accuracy of its members and on their diversity,“ as shown by the Diversity Prediction theorem. A diverse crowd will always be more accurate than its average member and sometimes more than any member in the crowd, as Scott E. Page explains in this video. Diversity here is intended as cognitive diversity, referring to the differences in how we think—the categories and models we use to encode problems, the tools we employ, and the diverse perspectives we apply to solve them. Cognitive diversity is connected with the process engendered by different training, experiences, interests, and outlooks.

So how did my middle school students fare at predicting the number of jelly beans in the jar? Well, there were about 80 individual guesses, ranging from 168 to 8,000, based on a large variety of assumptions and approaches. The average of the team was closer to the real number of jelly beans (1,817) than 85 percent of the individual estimates.

Although often the “crowd” does very well indeed, we do need to be aware of the limitations of group decision making for teamwork to be as powerful and effective as it can be. Fostering diversity and independent contributions and avoiding “group think” and overconfidence can lead to better decision making. Jelly beans provide lots of food for thought!


Allessandra KingAlessandra King, Alessandra.king@holton-arms.edu, studies mathematics with her students at the Holton-Arms School in Bethesda, Maryland. She has taught mathematics and physics at the middle school and high school levels and is interested in creative problem solving, critical thinking, and quantitative reasoning.

Enjoying Math in the Middle Grades

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“You teach math in middle school? You must have the patience of a saint!” This is the most common response I receive, often with an added look of commiseration, when I speak of my work. I usually try not to laugh out loud while explaining that I actually do it for fun.

There are many reasons why I enjoy teaching the middle grades: from the energy, enthusiasm, and sense of humor of the students to the fact that these years are somewhat removed from the high-stakes testing of high school and college admission, so that one can take a little time to have some fun (gasp!) with math. It is “during this time (that) many students will solidify conceptions about themselves as learners of mathematics—about their competence, their attitude, and their interest and motivation” (NCTM Standards for Grades 6-8). Thus, “it is important that [middle school students] be provided opportunities that foster the development of positive attitudes towards mathematics and positive perceptions of themselves as learners of mathematics” (MAA, Purpose of the AMC 8). In short, middle school is pivotal for inspiring students to see mathematics as an exciting, creative endeavor. We must therefore affect their approach to this subject and show them the many opportunities it offers. How many times have we heard about the need for more students in the STEM fields? Middle school is where it all can start—and this is for me the most inspiring aspect of being a middle school math teacher.

Middle school math students are no exception: Everybody is simply better at doing what they love, as social psychologist Paul A. O’Keefe argues in the SundayReview of The New York Times, which is also confirmed by research. “Interest matters more than we ever knew”; it “is crucial in keeping us motivated and effective without emptying our mental gas tank, and it can turn the mundane into something exciting.” In short, if the students learn to love math, they will be better at it!

It seems to me that middle school is the perfect environment to emphasize learning for learning’s sake, to foster intrinsic motivation, and what psychologists call mastery orientation as opposed to performance orientation. Because grades in middle school are not yet the all-encompassing preoccupation of students, parents, and administrators, there is the opportunity to create and sustain a classroom (and perhaps a school) culture that highlights intellectual risk-taking and curiosity and develops a growth mindset that focuses on improvement and boosts motivation.

All middle school students are particularly engaged when our lessons involve hands-on experiences, creative elements, projects, and class discussions. My girls—just as the students in the co-ed environment I taught before—also enjoy the constructive relationship and communal atmosphere they build with their classmates through participating in group projects, presenting to and critiquing one another, and discussing and comparing ideas and strategies.

Students worked cooperatively to build Sierpinski pyramids with wooden sticks and marshmallows. Sierpinski Pyramids 


In my blogs I plan to show how one can spark, or keep vibrant, the enthusiasm of our middle school charges and talk about cooperative projects or group activities that enrich the math experience of the students. Read more in two weeks about other engaging math tasks.


Allessandra KingAlessandra King, Alessandra.king@holton-arms.edu, studies mathematics with her students at the Holton-Arms School in Bethesda, Maryland. She has taught mathematics and physics at the middle school and high school levels and is interested in creative problem solving, critical thinking, and quantitative reasoning.

Written and Verbal Mathematical Understanding—How to Get Started: Part 2

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Getting your students to write and speak mathematically isn’t as simple as it may sound. This post will explore how to make the process a little easier. In part 1, I wrote about modeling these traits and listed activities that I use each week to make mathematical understanding a classroom staple. In this post, we will look at the time needed for student success in writing and speaking mathematically, connections to the Common Core Standards for Mathematical Practice, and some final thoughts on what can be expected after implementation.

3. Provide Significant Amounts of Thinking, Writing, and Discussion Time 

Time always factors into instructional decisions. Many students cannot develop and record a complete explanation in a short amount of time. Instead of asking a question and expecting an answer in 5 seconds (probably alienating 90 percent of the class), give students an abundance of time so that you can have greater participation by students who have actually thought about the question. Students will be less likely to shut down if they know they have time to think and work. Here is how much time I give for each of the activities discussed in part 1.

A. Two Problems 

Give written explanations, discuss with two students, and save until the end of class for a whole- group discussion. 

Time allowed: No set time. Some students may take 2 minutes, whereas others may take 15 minutes. End-of-class discussion generally takes 3 to 5 minutes.

B. Task 

Time allowed: One full 60-minute class period for students to work in groups and discuss and submit answers and explanations. They are also given a full week outside class to finish anything that wasn’t finished during the class period.

C. Daily Question 

Time allowed: 10 to 15 minutes to brainstorm, write out an explanation, and select 3 students to discuss the daily question. Allow 5 to 10 minutes to discuss as a whole class.

D. Estimation 180 Questions 

Time allowed: 5 to 10 minutes to brainstorm and write a clear understanding of thinking, with an additional 5 to 10 minutes of discussion as a whole group.

4. Connections to the Common Core’s Standards for Mathematical Practice 

In all honesty, having students write about and discuss mathematics is an essential piece of what should occur in a mathematics classroom, according to the Common Core Standards for Mathematical Practice and the NCTM process standards. Let’s consider a few of the standards that I found to be the most related to the mathematical activities I have described. Since all my activities involve the same standards, I will refer to Dan Meyer’s Leaky Faucet task in each case. Note: In this problem, I changed the amount per gallon to $0.02, so be advised that the price will not match the original problem.

CCSS.MATH.PRACTICE.MP1 

Make sense of problems and persevere in solving them. 

When students begin the problem, they do not try to get an answer in a few seconds, but instead look for a starting point, perhaps an idea of what would make sense. In the students’ minds, they have formulated an idea of what does and does not make sense before the actual solving begins. This idea of “making sense” also occurs throughout the problem-solving process. One student multiplied the number of milliliters in 10 minutes by 10 to get 1 hour instead of 6. This student was skilled in making sense of the problem and catching her mistake. A second student divided by $0.02 instead of multiplying and conveyed that $4800 did not make sense if the price per gallon is less than $1.

CCSS.MATH.PRACTICE.MP3

Construct viable arguments and critique the reasoning of others. 

 Students often work together on these tasks and are asked to write justifications for each statement. Many times, group members will devise their own lines of logic and discuss the plausibility of each. This Leaky Faucet problem has many approaches, so communication flourishes. The first student’s approach was to find how many milliliters it would take to fill 1 gallon and then to multiply by 3 for the entire sink. Her partner took a different approach and found the capacity of the entire sink first, then converted to time. Both approaches were acceptable, but each student had to convince the other one that their process made sense and do it in an understandable manner. You cannot have an argument until you fully understand your own logic, and both students were able to prove their answers to themselves first.

5. Additional Thoughts 

Developing good written and verbal explanations does not happen in a day. Most students, including higher achieving and struggling students, are going to be blown away by these expectations. Anticipate a bumpy road initially. Expectations need to be modeled multiple times, and student frustration will occur. My suggestion is to stay out of the way as much as possible. Ask questions to help students clarify their thinking instead of directly answering them. Walk away from a group when it is clear that the students need more time to think through the problem. Direct those in need to other students, so that they have additional opportunities to explain. These techniques will help students to persevere, which is one of the greatest gifts that a teacher can instill in students, math related or not. You will be surprised at what students can accomplish when written and verbal explanations are expected!

Comment or question? Join the discussion by responding below.


  Clayton EdwardsClayton Edwards, @doctor_math and cedwards@spartanpride.net, is a middle level mathematics instructor at Grundy Center Middle School in Grundy Center, Iowa. He is interested in the mathematical learning of all students of varying ability levels through self-pacing, task-based instruction, and other methods.

Written and Verbal Mathematical Understanding: Part 1

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Mathematical understanding is paramount in ensuring that students actually achieve the standards being presented. Any student can nod his or her head, give a thumbs up, and/or write down a few correct numerical answers to indicate understanding. Chances are, within all three scenarios, a majority of the class is not understanding the content.

I have had the best math students tell me that they understand something, and then when pushed, I find that they, in fact, do not understand. Students may convey that they understand even when they do not because they are embarrassed to admit their lack of knowledge, or they truly do think they understand. If students can present a clear and concise argument for their answers to peers or teachers, odds are they understand. The process of developing mathematical understanding isn’t easy to implement in the classroom, but the results far outweigh the struggle. Here are 3 ways to get started.

1. Model Understanding 

When I first introduce the idea of mathematical understanding to my sixth graders, they immediately think of showing their work, which is not exactly what I mean. When I show them an example of a task or assignment, jaws drop around the room. What they are about to encounter has never entered their realm of possibility. 

One way we practice writing in mathematics class, for example, is through a whole-class-assigned question. Two times a week, we work on a question in which students are expected to show their understanding of the solution in writing and then explain it to three people. I give them options of how to show their understanding. These include, but are not limited to, a paragraph, labeled diagrams, and explaining the significance of numerical answers with labels. The progression usually goes from writing down numerical answers and slowly moves to the final product after weeks of work. Students then begin to understand how much I am expecting them to explain. Asking, “Why did you use a division problem?” is something most students have never thought about. They just “know” to use a division problem, but I ask that they be able to state why. After many examples and nonexamples, this idea is accomplished. 

We use this same process to model mathematical conversations. During the whole-class questions, students are expected to discuss explanations and findings with three other students. As these conversations occur, I stop the entire class, talk about positives and negatives that I see, and ask the class what could happen differently. One common problem with these conversations is that the students on the receiving end will nod their heads as if they understand when they do not. We discuss how these two-way conversations should involve everyone. If something is not understood, clarifying questions must be asked. The person who is listening to an explanation needs to leave the conversation understanding the presenter’s point of view, and the presenter needs to leave the conversation knowing if what he or she said makes sense. 

2. Give Students Opportunities to Write and Talk 

Students will not instinctively start writing and talking about math if the opportunity is not given. The more opportunities presented, the more this practice will become the norm. The examples below can be implemented in your classroom to start the process.

A. Assign Fewer Problems 

Students can turn in correct answers to a page of 100 problems, and the teacher may still have no idea if they really know what they are doing. Instead of 100, choose two problems and ask students to show their understanding in multiple ways. After the written part is finished, each student must find two other students to discuss their findings. At the end of the class period, the teacher picks a random student to explain one method of understanding. The simplest problem can be turned into a good discussion if that is consistently expected.

Here is a simple proportion problem:

Six students have 18 pencils. If the proportionality holds true, how many pencils would 14 students have?

Although it is not a very enticing question, you can still drive home the mathematical understanding.

MTMS_blog_2014-11-10_art01.jpg
 

B. Provide Problem-Solving Tasks 

This is an idea I have used since discovering Dan Meyer’s blog. I have taught every one of his tasks in my classroom. My students discuss problems with one another in groups, and they are each expected to write a detailed synopsis of the solution, how they know they are correct, and everything it took to get there. I would start with his resource first. You can always make changes or add/subtract questions and information to your liking. Here is an example of one of the tasks and what is expected.

http://threeacts.mrmeyer.com/leakyfaucet/ 

 MTMS_blog_2014-11-10_art02.JPG 

 MTMS_blog_2014-11-10_art03.JPG 

C. Use Whole-Class Questions 

I already described the process for using a whole-class question above, but here is a video example to depict more of the writing and the discussing.

D. Implement Estimation 180 Questions 

This website created by Andrew Stadel shows a picture in which students have to estimate something about an everyday object or person. We often use one of these images at the conclusion of class with the same goal: Show your understanding (written) and explain your thinking (verbal). Not everyone is always correct, but that is part of the process. If you can explain your thinking (even if you are wrong), fixing the misconception is a lot easier. Here is an example with student work

http://www.estimation180.com/day-7.html 

MTMS_blog_2014-11-10_art04.JPG 

These are only a few examples of how to get students writing and talking. It is hoped the message is clear that this should happen with every activity in your class. Mathematical understanding must become commonplace and not just for special occasions. 

Come back to Blogarithm in two weeks for the finale of this post. Additional topics will include the amount of time necessary for your students to have success thinking, writing, and speaking mathematically; the connection these justification skills have with the CommonCore Standards for Mathematical Practice; and some closing thoughts drawn from my own experiences in the classroom.

How to Get Started (Written and Verbal Mathematical Understanding: Part 2)

Comment or question? Join the discussion by responding below.
 


  Clayton EdwardsClayton Edwards, @doctor_math and cedwards@spartanpride.net, is a middle level mathematics instructor at Grundy Center Middle School in Grundy Center, Iowa. He is interested in the mathematical learning of all students of varying ability levels through self-pacing, task-based instruction, and other methods.

Standardized Mathematics Testing Success without Substandard Classes: Part 2

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Standardized testing. . . . Like it or not, these assessments are pivotal cogs in today’s educational system. How do you navigate these assessments and still operate a high-quality class? Part 1 of Standardized Mathematics Testing Success without Substandard Classes focused on mathematical understanding and the pursuit of improvement. This part 2 post will deal with holding all students accountable, as well as aligning testing practices to high-quality teaching.

3. Hold Students Accountable for Learning 

I could probably write an entire blog post on this topic (and maybe I will eventually), but if students are to succeed on standardized tests, they must be held accountable for learning during class. How is a student supposed to perform if he or she has been let off the hook for learning along the way?

  1. If you give completion points and don’t actually check your student’s work, you are probably letting students off the hook.  
  2. If you let a student move on to new material with an 80% assessment score, you are probably letting students off the hook.  
  3. If you’ve ever said that students will figure it out when they get an F, you are probably letting students off the hook. (You may also be steering them toward a hatred of mathematics.)
  4. If you ask questions in class, choose the first hand to shoot up, and give little time for everyone to think and analyze, you are probably letting students off the hook.  

These scenarios do nothing for the learning of all students. You must have a process in place in your classroom to better assess your students’ learning as you go and not wait until the final assessment or chapter test. I accomplish this by having students turn in work at the end of each class period so that I can assess what they have done so far and whether they are finished or not. I also check students’ work during class to make sure they are initially understanding the concept. Having an iPad® mini helps with ongoing assessment because I can quickly flip through all the answers to everything they are working on. The expectation is that students will fix and/or correct any misconceptions until the understanding is solid.

You must also have a process in place for providing remedial support in your classroom to fix problems as they occur. Much of this is taken care of through the ongoing assessment process during class, but I also make extra time by being available at 7 a.m. and as late as needed after school to work with students. I also forego my planning period each day to work with ten to fifteen students who need assistance. Making yourself overly available shows students that you are serious about helping them improve.

You must also have a way for students to engage in long periods of think time so that they can come to a conclusion and not rely on someone else each day. I accomplish this wait time not only through our daily questions and other blocks of class time but also through the use of self-paced instruction (see this presentation and article). These ideas may not work for you individually, but you need something established to hold students accountable and stop letting them off the hook.

4. Testing Practice Should Reflect Your Normal High Expectations 

Maybe you are aware of everything I have discussed above. You ask students to clearly explain their thinking on paper so that they can prove their answers: Check. You hold students accountable: Check. You make sure the students have ample think time in class: Check.  

Everything is seemingly perfect, and then the teacher (or district) tries to jam a 50-question test into 60 minutes. This is a problem. Some of my daily questions take students 10 to 15 minutes to lay everything out and explain. Now I want them to do 50 “daily questions” in 1 hour? Have students fill out a timecard similar to this while testing (timer example). Do whatever is possible to get multiple sessions of extended time to test. Explain this situation to the administration so that he or she understands the mathematics behind this philosophy. I understand that some tests are timed, but many tests are not (e.g., NWEA MAP and Smarter Balance). Make sure the same high standards of success that you are pushing in class are used when testing. 

Although standardized tests aren’t everything, they are a driving force in the perceived success or failure of school districts in the current culture. The good news is that these tests should not make your class time any less rigorous for the students. Expect a high level of mathematical understanding, and your testing results will be anything but substandard.

Comment or question? Join the discussion by responding below. 


 

Clayton EdwardsClayton Edwards, @doctor_math and cedwards@spartanpride.net, is a middle level mathematics instructor at Grundy Center Middle School in Grundy Center, Iowa. He is interested in the mathematical learning of all students of varying ability levels through self-pacing, task-based instruction, and other methods.

Standardized Mathematics Testing Success without Substandard Classes: Part 1

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In an era in which schools are fighting to stay off state watch lists because of standardized tests, many of these institutions revert to simplifyingcurriculum, eliminating courses like social studies and science for extra math,and extensively practicing multiple-choice material. As a middle school math teacher entering my 12th year of teaching, I’ve been on the wrong end of a few standardized test results early in my career. I’ve worked to become a better teacher who garners better results, as noted from this MTMS article co-written with Brian Townsend. I’ve learned from my mistakes and can offer a few simple ideas that should help your standardized testing experience be more successful and less stressful while keeping high mathematical standards intact.

1. Math Understanding Matters 

Whether you still use a textbook, employ a problem-based approach, or combine various materials: If students can prove their understanding, standardized tests will become much easier. Some teachers try to pick and choose specific items that they think will be on an assessment. My advice is to stick with teaching the standards and expect a lot from your students, as far as proof and understanding. 

We recently used the Dan MeyerSuper-Bear task. Even if students use a calculator, which I do allow, they must write out the entire process and prove the answers to the questions that they are investigating. Here are a few examples offering different approaches.

Example 1 

Example 2 

Example 3 

 Example 4 

Example 5 

This link will take you to a listing of DanMeyer Tasks. I have used them all in my classes with great success.

My sixth graders finished a unit on exponents. I always have them write everything out to prove their responses. Many middle school students like to jump to the fastest answer that pops into their heads (e.g., 5 cubed being 15 instead of 125). When they write down the steps used in their thinking process, it helps them understand the problem and possibly avoid careless mistakes, two vital elements of successful testing.

When I assign a daily problem for the class, I don’t choose questions with multiple-choice answers, but instead choose more robust problems with multiple entry points and various ways to show understanding. I then give my students 10 to 15 minutes to collaborate on and discuss the problem. I often use problems off the Smarter Balance PracticeTest (click on “student interface” and use the guest login), similar to the example below.

Example 6 

Other good daily question sites that I use include these:

Estimation180 

Would You Rather? 

Graphing Stories   

At the end of the 15 minutes, we come together as a group. I randomly draw a name, and that student has to explain the problem to the class. I also give time for others to share various explanations. 

Students 1<

 Students 2 

 The focus is always on the understanding. My students get tired of hearing “prove it,” but that is what I enjoy most about teaching. If students can “prove it,” it is hoped that they won’t forget important concepts for testing. 

2. Push Improvement  

Let’s face it, most students are not going to go from the 2nd percentile to the 50th in one year. A student could improve the equivalent of two or three grade levels but still get placed in the nonproficiency category. From day 1, I tell students that as long as they improve from the previous year on these assessments, I will be happy with their performances. Setting that type of goal at the start presents a scenario that is doable for every student in the classroom, no matter the ability level. This goal is posted around my room and discussed daily.

I always have an improvement timeline posted to show students exactly how much time we have to improve before the next assessment. This goal is interjected into everything we do in class and becomes a way of life for students. They know that everything we do is beneficial for their improvement on these assessments. Therefore, everyone is committed to the same, attainable goal, which helps keep everyone positive and working hard to succeed. If this type of attitude can radiate schoolwide, it is hoped that students will eventually become proficient and that higher achievers will not plateau.

Improvement 1 

Improvement 2 

Check back in two weeks for the conclusion of my post. The next entry will feature ways to hold your students accountable for learning and how to match your high expectations in the classroom with testing practices and procedures.

Comment or question? Join the discussion by responding below.

 


 

Clayton EdwardsClayton Edwards, @doctor_math and cedwards@spartanpride.net, is a middle level mathematics instructor at Grundy Center Middle School in Grundy Center, Iowa. He is interested in the mathematical learning of all students of varying ability levels through self-pacing, task-based instruction, and other methods.

Sadie’s Circles

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Is it okay to blog about something that is not my innovation or lesson idea? Certainly! Blogging includes sharing your practices and things you’ve learned from other teachers. In the Math Twitter Blog-o-Sphere (#MTBoS), I love reading how teachers share and implement one another’s ideas.

At Twitter Math Camp 14, I blogged about my desire to implement Counting Circles. I learned it from Sadie Estrella who has blogged about it (read the Blame Game first) and shared her materials. As Sadie points out, the Counting Circles activity connects to Jo Boaler’s Number Talks; she, in turn, cites Ruth Parker and Kathy Richardson.

A more few posts to read about Counting Circles: 


What Is a Counting Circle? 

MTMS-2014-09-29-Art1 

Image: John Golden 

 

I introduced Counting Circles to my preservice teachers and emphasized the following:

  • We’re going to count up or down by a specified amount from a given starting value.
  • You have plenty of time; there’s no rush.
  • Please don’t comment on other people’s counts.
  • If you make a mistake, it’s not a problem. We’ll go from what you say.
  • When we stop, I’ll ask you a question about what would happen if we had counted on. When you’ve got an answer to the question, put your thumb up by your chest. I’ll ask for volunteers to share their thinking.

My first Counting Circle with college math students was pretty stiff: up by 97, starting at 235. It wound up being a little uncomfortable for a few people: up by 99 might have been better. (If you ask her on Twitter, Sadie will help you figure out a good counting amount and starting point for your students.) After 1.3 times around the circle, at 2175, I paused and asked, “What would Amanda say?” (5 people farther around the circle). When all students indicated that they had an answer, I asked for volunteers to share their thinking, recorded it as they spoke, and elicited details. The shared strategies included noticing that the units place went down 3 each time, from adding 100 and multiplying 97 ´ 5 and adding it to 2175. Even the computation can be interesting: 97 ´ 5 used partial products to get 485, and then one individual added the 5, the 400, and then the 80. My advice for the Number Talk at the end: push for elaboration on thinking and details. It’s rich material, helpful to students, and ripe for representation.

I recently asked if students wanted to squeeze in a counting circle. An enthusiastic yes ensued. With just a few minutes, it was worthwhile to highlight number sense, thinking, verbalizing, and noticing. The count: up by 3/4, starting at 11.

Using improper fractions

“Can we use improper fractions?” Sure. Other students went back to mixed numbers, where more interesting thinking occurred. “What is the third next whole number we would hit if we kept counting?” And who would say it?

Using 4s and 3s

Students’ thinking mostly centered on the pattern of whole numbers for every 4, and each number being 3 higher. Why would that be? There was an audible gasp when the student shared 25 1/4 + 27/4 is 25 + 28/4.

I am seeing the culture-setting power that Sadie described, including accepting mistakes, willingness to share, and valuing of process over answer. This practice has made an impression on students as well, becoming the topic of several of their first blog posts for class:

Now, form a circle. Start at 5 and count back by eleven hundredths. . . .

This is my last Blogarithm post. It’s been fun to share some of what I value from the online community of math teachers. I hope you, too, consider tweeting or blogging.


John GoldenJohn Golden, @mathhombre, is a member of the department of mathematics at Grand Valley State University in Michigan. He teaches math and elementary and secondary teacher preparation courses. At mathhombre.blogspot.com, he blogs about math games, geometry and GeoGebra, lesson ideas, and teacher prep. 

Head Tech

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Every day it seems like K–12 students have more powerful tech tools at their disposal. But what are they to do with these tools? And when is it appropriate for them to use these tools? If the answer is “not much more than to watch lectures and take quizzes,” then we are missing the boat. When I’m asked for tech recommendations for middle school and high school, there are two apps that I most want to get in students’ hands: Desmos and GeoGebra.

Desmos is what graphing calculators want to be (see fig. 1). This app is free to anyone with a browser and is built on HTML5. It is also on iOS, with an update coming soon. Desmos graphs in color, adds sliders seamlessly, and can be saved via a URL. It has free accounts and regular updates, often in response to teachers’ requests. (Check out https://teacher.desmos.com/.) One of the wonderful features of Desmos is some truly amazing support for assessing student understanding with real-time monitoring of the students’ work. Curious? Here’s a post on the Desmos blog by Lee Bissett about using it in a middle school, and there are several Fawn Nguyen posts on using Desmos in her classroom. There’s even a collaborative blog dedicated to challenging Desmos creations.

fig. 1 An actual graph on Desmos

MTMS-Blog-2014-09-15-Art01.png

As great as Desmos is, my heart belongs to another program.

Research on the impact of dynamic geometry programs goes back decades. Do a quick search of MTMS dynamic software and you can find a bevy of great articles, covering teaching ideas about geometry, algebra, and data. However, the cost of the software has often been a barrier for schools, and if schools had the software, students didn’t have access at home.

When I heard about GeoGebra (GGB) (version 2, I think), I was quick to give it a try. Instant mathcrush. Every object you make has a geometric and an algebraic identity; it’s Descartes’ fantasy world. Students took to it quickly and even started using it from home. GeoGebra is free and open source. Starting with version 4, the developers began a great support/feature/service: GeoGebraTube. Modeled on YouTube, teachers and students can upload their GGB work to save or share. Students can submit work this way or access activities selected by their teacher. The activities can be downloaded or opened directly in the browser. GeoGebraTube is searchable, so you can usually find an activity on your topic made by someone else as well as elementary school topics up through research mathematics. You can also make GeoGebra “books”: collections of activities that you can share with a single link. (See Jennifer Silverman’s book on straightedge and compass.)

As students are engaged in a GGB activity, they are exploring and making connections on their own. It’s one of the best tools I’ve ever used to get students to make conjectures, discover reasons for an argument or proof, or build experience to generate intuition. Examples from my blog include Percent Game Remixing and Flip Flop (about reflections). The ability to incorporate pictures into sketches makes for great modeling practice. (Here are several quadratic ideas.) Because the interface is often just click and drag, there’s no start-up time needed for students who are learning how to use the program. Read how Jed Butler uses the applets to promote area formula understanding.

But the most exciting opportunity is for the students to start to learn the program. For example, Audrey McLaren and her students produced astounding projectile projects. The program is powerful and a tool that will grow with students throughout the rest of their career. I’m literally learning new things every day through GGB, often through trying to “dynamicize” a classic theorem (like the Euclidean algorithm for the GCD; see fig. 2) or some intriguing art with mathematical properties (like Truchet squares; see fig. 3). (See, also, fig. 4’s Matheo on GeoGebraTube.)

fig. 2 Euclid’s GCD Algorithm on GeoGebraTube 

MTMS-Blog-2014-09-15-Art02.png

fig. 3 Truchet squares on GeoGebraTube

MTMS-Blog-2014-09-15-Art03.png 

 

 fig. 4 Mathio on GeoGebraTube

MTMS-Blog-2014-09-15-Art04.png 

Because GeoGebra is an open-source community, you will find support. We’ve just started #ggbchat on Twitter (every other Wednesday at 8 p.m. EST), and the GeoGebra forums are helpful to novices and masters. Or ask me, I love collaborating on GeoGebra. 


John GoldenJohn Golden, @mathhombre, is a member of the department of mathematics at Grand Valley State University in Michigan. He teaches math and elementary and secondary teacher preparation courses. At mathhombre.blogspot.com, he blogs about math games, geometry and GeoGebra, lesson ideas, and teacher prep. 

Yes, Playing Around

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I like to play games. Almost any type of game. I also like to play math. Left to my own devices, I’ll work with good problems from James Tanton, the Futility Closet, or Don Steward or try to make something interesting in GeoGebra. (GeoGebra is a free, open source, dynamic mathematics software program.) Or I will read about good math that other people are doing.

If you’ve known enough mathematicians, you may have noticed that this isn’t unusual. I’m not sure if a love of games and puzzles among mathematicians exceeds a love of music among mathematicians, but both are strong and intersect. Math in play is also a way of teaching mathematics. I think that as a metaphor, it best describes how I want to teach math.

Of course, games in education is a hot topic now. People talk about or implement gamification, grafting game rules, rewards, or structure into learning. The Math-Twitter-Blog-o-Sphere (MTBoS) has a wide variety of general purpose or review games that can be adapted to many different content areas. I try to keep an up to date index of many of these at my blog, such as trashketball, the block game, and Taboo game variants. (Several people collaboratively made a geometry block game recently.)

I am constantly seeking ways to get my students thinking about math as a verb. It is about doing, not just about having right answers or the end product. Games help set the culture I want to develop: Teaching students that multiple approaches and strategies are valued; trying is safe; and conversations about why, how, and discovery are the goals. MTMS has featured some good games over the years, allowing students to explore, use, or represent the math content that the games teach:

I like many of the games at Illuminations and Calculation Nation; several have free app versions as well (iOS and Android). My all-time favorite, the Product Game, is on the Illuminations site. What makes this game great? It is fun. An enormous amount of strategy is involved, so that even if students have the content, they are still engaged in playing.

 Game Board from Product Game
Any attempt to design a game that uses math for structure will have the benefit of getting students to play with the ideas involved. With this idea in mind, I designed Decimal Pickle (a decimal comparison) and Four Corners (an introduction to graphing). I gravitate toward strategy games, so I’ve had to intentionally try to develop more collaborative or creative games, such as Burger Time (decimal modeling) and Power Up (exponents). Usually I launch a game by playing against the whole class, then I review the rules by having students explain them, before breaking the class into teams. Team vs. team almost always results in better mathematical conversations. To conclude the game, we regroup, share stories of game play, recount what they noticed, and point out any strategies developed.

I’ve noticed, however, that some of my satisfaction in making or adapting a game is that the design process is itself about doing mathematics. Varying conditions, considering alternatives, play testing, and collecting data on the results are mathematical practices. I am now trying to involve students more in the game design process: talking about process and decisions, getting suggestions for improving games, modifying games themselves and ultimately designing their own.

Here’s to more game playing in your class! Pull out your tub of dice and get rolling.


John GoldenJohn Golden, @mathhombre, is a member of the department of mathematics at Grand Valley State University in Michigan. He teaches math and elementary and secondary teacher preparation courses. At mathhombre.blogspot.com, he blogs about math games, geometry and GeoGebra, lesson ideas, and teacher prep. 

Twitter Math Camp

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I spent July 23–27, 2014, at a camp. If this doesn’t sound silly enough for a nearly fifty-year-old, it was Twitter Math Camp (TMC). My spouse still hasn’t stopped making fun of it. It might bring to mind that Allan Sherman classic camp song, “Hello Muddah Hello Faddah”:

Hello colleague,  
Fellow teacher. 
Hope this letter,  
Still can reach ya. 
Twitter Math Camp,
Is fantastic. 
Despite that as a whole math teachers are quite spastic. 

Boys were dancing,  
Girls deriving. 
Algebra and stats,  
Math games all are thriving.
The learning was fun,  
The fun times funky. 
Meeting folks with twitter handles like Cheesemonkey. . . . 

“How was it?” FANTASTIC!!! All caps, multiple exclamation points. The best professional conference-like experience I’ve ever had.

TMC Logo

A lot of my personal professional development experiences now come from interacting online—mostly through Twitter and blogs—with an amazing group of teachers from around the world. These teachers have become the self-declared Math-Twitter-Blog-o-Sphere (MTBoS). Members of the MTBoS are deluged by interesting reads and intriguing conversations, including accounts of classroom practice, assessment dissection and analysis, activity development, and even discussion of research. Three years ago, a small group decided to meet in real life in the summer, and Twitter Math Camp was born. Last year, the group met at Drexel University in Philadelphia, home of the Math Forum. This year, 150 of us met at a high school with a stunning STEM facility in Jenks, Oklahoma (pronounced “jinx”) for three days.

Participants in Twitter Math CampThere were two-hour morning sessions that ran for the first three days, where we worked with the same group each day. This was no sit and git; it was an opportunity to work with gifted and dedicated teachers at length developing ideas in depth. Each day featured a whole-group keynote (Dan Meyer, Steve Leinwand, and Eli Luberoff); varied afternoon sessions; and flex sessions for people to expand on, extend from, or begin something different than what they had been working on. This structure worked really well, balancing variety and work, depth and coverage. One feature of TMC is the My Favorites presentations by participants when we’re in whole group. These short sharings hit on one or two ideas, practices, techniques, or resources that have made a difference in student learning in their classroom.

Probably as important as all of that was the out-of-meeting time. The majority of us had never met in real life. On the evening before camp, there was a game night in the hotel and a lot of “is that…?” and “who is…?” and “what’s their Twitter name?” (I’m one of the worst offenders, with a nonname Twitter handle, @mathhombre, and a nonpicture avatar.) I was in the awkward position of meeting people that I already considered friends. The dinners, after-hours math, discussion, and games had a bonding effect. I had observed the interaction (a little jealously, to be honest) the previous years and saw how working together in real life deepened and enriched the online relationships. TMC is available to anyone who can make it. A good share of the attendees were semilocal Oklahoma teachers who had little experience with the online community, and they were as much a part of it as the heroes of the community.

The MTBoS does have its heroes: Teachers who have galvanized the community, made multiple innovations that have been widely adopted, shared deeply and personally on their blog, put in time and effort to organize or grow the community, or have inspired and supported many of us to start writing or tweeting. These heroes are down to earth and just as interested in meeting you as you are to meet them.

One benefit of meeting in a community that exists primarily for sharing teaching ideas online is that the camp is very well documented. The wiki, twittermathcamp.pbworks.com, has almost all the information from the sessions and presentations, and the reflection blogposts have been amazeballs. (Some odd lingo gets picked up on Twitter.) My personal reflection is on my blog at mathhombre.blogspot.com.

I will close with an invitation: Consider joining some of your inspirational fellow teachers online by blogging or tweeting. You can always just observe until you feel ready to make the jump. One place to get started is exploremtbos.wordpress.com.


John GoldenJohn Golden, @mathhombre, is a member of the department of mathematics at Grand Valley State University in Michigan. He teaches math and elementary and secondary teacher preparation courses. At mathhombre.blogspot.com, he blogs about math games, geometry and GeoGebra, lesson ideas, and teacher prep. 

Complex Instruction: High-Quality Mathematics for All Learners

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Over the past several posts, I have been exploring how to make the mathematics curriculum more equitable. Another important aspect of equity is making sure that all learners have access to the mathematics being taught. Complex Instruction (CI) is one way of doing this.

CI looks at student engagement as an issue of status. Some students are assigned high status by their peers and teacher, whereas other students are assigned low status (through praise, listening to their ideas, body language, etc.). Low-status students rarely have their ideas taken seriously and are often excluded from group work, thus causing them to disengage with mathematics. CI teaching involves group-worthy tasks and pedagogical moves to support all learners.

Group-Worthy Tasks 

Group-worthy tasks draw on a variety of mathematical smartnesses. Most group-worthy tasks draw on several of the Standards for Mathematical Practice, and many teachers list the multiple smartnesses needed before beginning and then explicitly call attention to them throughout the task and during the wrap up.

Group-worthy tasks also make it difficult for one student to take over. In one task I created, each group member had his or her own set of shapes to sort however he or she wanted. As each person shared the sort, the other group members had to place a new shape in the appropriate place in their sort, which forced them to listen to one another’s ideas.

Pedagogical Moves 

One of my favorite pedagogical moves in CI is requiring all questions to be group questions. If someone calls the teacher over, the teacher can ask anyone in the group what the question is; if that person doesn’t know, the teacher can leave, saying, “It sounds like you need to discuss this as a group before calling me over again.” Another favorite is group quizzes in which teachers randomly choose a student who must explain a final solution/product after working on a task. This process holds all students accountable for learning the content, and it forces the students to support one another’s learning. If a student struggles, the teacher can leave, giving him or her a few minutes to consult with group members before the teacher returns.

Perhaps one of the most important pedagogical moves is assigning competence. The teacher watches for learners who are making mathematical contributions and then points them out publicly. Although this should be done for all students, it is of particular importance for low- status students because it will help them and their classmates see that this individual has something valuable to contribute. Assigning competence is made easier when tasks draw on multiple smartnesses—this makes it possible for more learners to contribute and it makes it less likely that any one student will excel at all aspects of the task.

Resources 

Smarter Together! Collaboration and Equity in the Elementary Math Classroom provides a wonderful introduction to CI.

• The NRICH website has a section explaining CI.

• I and several of the Smarter Together! authors are working to develop CI resources and lessons.

What do you do to reach all learners? Do you have experience with CI?

 


Mathew FeltonMathew Felton is an assistant professor of mathematics education in the department of mathematics at the University of Arizona and will be starting in the department of teacher education at Ohio University this fall. He is a coauthor of Connecting the NCTM Process Standards and the CCSSM Practices. His research focuses on supporting current and future teachers in connecting mathematics to real-world contexts and on teachers’ views of issues of equity, diversity, and social justice in mathematics education.

Bringing in the Real World

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I have been writing about how to use real-world contexts in the mathematics classroom. What are some practical ways to do this? Let’s consider three examples that explore how to integrate diversity, connect to students’ lives, and analyze social issues (three reasons for teaching mathematics).

McCoy, Buckner, and Munley describe how they used games from a diverse group of cultures to introduce probability concepts. Each lesson started with students learning about the game, “including the history and background.” Next, the students played the games and then were introduced to a relevant probability concept. The games and cultures could just be used as a superficial context, but if they are explored in greater depth, perhaps as part of a social studies lesson, then this form of mathematics could allow students to learn about other cultures. It also shows these students how mathematics, when viewed from a game format, can provide a new way of looking at and understanding something familiar.

Leonard and Guha explore how they allowed students to take photos of their neighborhood that were then used by students to write word problems, such as calculating the number of light posts along the road or the age of a local church. Although some of these problems might not qualify as “authentic” (based on my last post), they have the potential to be more engaging because they come from students’ environments and because the students wrote them. This scenario also engages students in posing problems. The real world is messy, and figuring out how to ask mathematical questions that provide insight into these contexts is a central part of mathematical modeling.

I have also explored income inequality with prospective and practicing teachers. I fill 20 Baggies™ with blocks, with each block representing $1,000 of annual income. I set up the Baggies to mimic the distribution of income in the United States. Everyone gets a Baggie (students can either pair up or we can pretend if the class is much larger or smaller than 20), and we line up from poorest to richest and break into quintiles (5 equal groups, so that there are 4 Baggies in each group). If I have enough time, I take a hands-off approach and allow the teachers to decide how to analyze the data. This can be particularly powerful because it allows teachers to think about how to analyze the data, which is at least as important as learning specific procedures (like calculating the mean or creating bar graphs). If I have less time, I take a stronger lead and guide the investigation in directions that are productive both mathematically and for understanding the context. I ask questions like those below. The first two questions introduce mean as a fair share, and I make connections between the standard algorithm and equally sharing the blocks. In my experience, this is a powerful way to introduce mean and median because it is set in a context that teachers understand.

  1. If every household at your table (quintile) made the same amount, how much would they make? Show how you can determine this amount by using your blocks.
  2. If every household in the class made the same amount, how much would they make? Explain how we could find this amount by using the blocks.
  3. Compare the median income with the mean income. Why are these numbers so different? Which do you think is a better measure of “typical” in this context? Why?
  4. Create bar graphs and circle graphs that show each quintile’s share of the total income in the United States.

How have you integrated real-world contexts into your mathematics teaching? Are there ideas from this post that you are planning to use in your classroom?


Mathew FeltonMathew Felton is an assistant professor of mathematics education in the department of mathematics at the University of Arizona and will be starting in the department of teacher education at Ohio University this fall. He is a coauthor of Connecting the NCTM Process Standards and the CCSSM Practices. His research focuses on supporting current and future teachers in connecting mathematics to real-world contexts and on teachers’ views of issues of equity, diversity, and social justice in mathematics education.

Mathematics and the Real World

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In my previous post, I argued that in addition to teaching mathematics for its own sake, we should also teach mathematics so that students learn to value diversity, see mathematics in their lives and cultural backgrounds, and analyze and critique social issues and injustices. These learn-see-analyze purposes require connecting mathematics to real-world contexts, which is emphasized in the Common Core’s fourth Standard for Mathematical Practice: Model with mathematics. What does it look like to connect mathematics to real-world contexts? I see two general approaches.

NCTM’s Principles to Actions: Ensuring Mathematical Success for All provides an example of using the real world as a stepping-stone for thinking about mathematical concepts. It describes a teacher engaging students with real-world problems involving proportional relationships; see the task below (NCTM 2014, p. 31):

 Figure 12 - The Candy Jar task 

Although this task is grounded in an out-of-school context, it is not a genuine dilemma that most students are likely to face outside the classroom either now or in their future lives. I am not advocating against this type of problem—these problems serve an important role in teaching mathematics. This real-world context is familiar enough and imaginable to students and can therefore serve as a stepping-stone for thinking about important mathematical concepts, like scaling up proportional relationships. This is similar to ideas from Realistic Mathematics Education (see here and here).

Another approach uses more authentic real-world contexts. These are either genuine problems that arise outside the classroom for which mathematics is useful or they are social issues that students can learn more about through mathematical analysis. Consider the following examples:

  • How can we redesign a neighborhood park that burned down?
    (Adapted from Turner et al. 2009)
     
  • How unequal is the distribution of income in the United States?
    (Adapted from Felton, Simic-Muller, and Menéndez 2012)

Problems like these are harder to use for several reasons. First, these problems are often open-ended and ill-defined. Although it is crucial for students to learn how to deal with messy real-world contexts, they will rarely encounter them in the classroom. Second, because of the nature of these open-ended problems, it is much harder to anticipate what mathematics students will use. The problems above can be approached with a range of mathematics, which is important for seeing the interconnected nature of mathematics. However, these examples can cause some teachers to shy away in an era of increased pressure to address particular standards in their lessons. Finally, because of the problems’ open-ended nature, students sometimes find approaches to these problems that use little or no important mathematics.

Despite these difficulties, I hope that teachers will integrate authentic real-world contexts in their classrooms. These contexts are crucial for engaging students in mathematical modeling and for preparing students to use mathematics beyond the classroom. Keep in mind that the problem must be authentic and that the teacher must encourage students to draw on their real-world knowledge and experiences and approach the task authentically.

Please share your thoughts below. What are your experiences with authentic real-world contexts? What concerns do you have? What opportunities do you see?

References 

Felton, Mathew D., Ksenija Simic-Muller, and José María Menéndez. 2012. “ ‘Math Isn’t Just Numbers or Algorithms’: Mathematics for Social Justice in Preservice K–8 Content Courses.” In Mathematics Teacher Education in the Public Interest: Equity and Social Justice, edited by Laura J. Jacobsen, Jean Mistele, and Bharath Sriraman, pp. 231–52. Charlotte, NC: Information Age Publishing.

National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

Smith, Margaret S., Edward A. Silver, Mary Kay Stein, Melissa Boston, and Marjorie A. Henningsen. 2005. Improving Instruction in Rational Numbers and Proportionality: Using Cases to Transform Mathematics Teaching and Learning. Vol. 1. New York: Teachers College Press.

Turner, Erin E., Maura Varley Gutiérrez, Ksenija Simic-Muller, and Javier Díez-Palomar. 2009. “ ‘Everything Is Math in the Whole World’: Integrating Critical and Community Knowledge in Authentic Mathematical Investigations with Elementary Latina/o Students.” Mathematical Thinking and Learning 11 (3) (July 8): 136–57.


Mathew FeltonMathew Felton is an assistant professor of mathematics education in the department of mathematics at the University of Arizona and will be starting in the department of teacher education at Ohio University this fall. He is a coauthor of Connecting the NCTM Process Standards and the CCSSM Practices. His research focuses on supporting current and future teachers in connecting mathematics to real-world contexts and on teachers’ views of issues of equity, diversity, and social justice in mathematics education.

Why Teach Mathematics?

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NCTM’s new Principles to Actions: Ensuring Mathematical Success for All aims to ensure high-quality mathematics education for all students. But what does high-quality mathematics education look like? Another way to come at this question is to ask, “Why do we teach mathematics in school? What do we want students to learn?” The most common responses I see to these questions, especially in policy documents such as Principles to Actions or the Common Core State Standards for Mathematics (CCSSM), are that students should learn mathematics—

  • for its own sake because it is a beautiful and amazing human accomplishment; and
  • to be prepared for college and future careers, especially in science, technology, engineering, and mathematics (STEM) fields.

Borrowing from Eric Gutstein, I call these reasons the “classical perspective” on mathematics education. The classical perspective, building on a strong research base about how students learn mathematics with understanding, suggests a particular vision of high-quality mathematics education. This vision generally emphasizes conceptual understanding, problem solving, making connections across representations and mathematical concepts, and engaging in reasoning and argumentation (in other words, engaging students in the Standards for Mathematical Practice). Within the classical perspective, equity is primarily seen as providing all students with access to this vision of high-quality mathematics.

I am a strong supporter of the classical perspective. However, there are other reasons that we might teach mathematics in school, which often receive less attention in major policy discussions. In addition to the goals listed above, I believe that students should study mathematics to—

  • learn about and appreciate diversity in human thinking and accomplishments throughout history and around the world;
  • see the role of mathematics in their daily lives, their community practices, and their cultural backgrounds; and
  • understand, analyze, critique, and take action regarding important social and political issues in our world, especially issues of injustice.

I call these goals the “equitable-curriculum perspective” on mathematics education. Equity is framed in the classical perspective as providing students with access to well-taught mathematics; in the equitable-curriculum perspective, equity is framed as teaching a form of mathematics that values and integrates issues of diversity and social justice.

Although the Common Core is not perfect (I recommend Usiskin’s excellent analyses here, see sessions 505, and here), I do think the Standards for Mathematical Practice—and especially Standard 4: Model with mathematics—provide an opportunity to integrate these goals into school mathematics (see Koestler, Felton, Bieda, and Otten for more on the practices). However, I am deeply concerned that the Practices will be underemphasized as new standardized tests are implemented and as they play an increasing role in student and teacher evaluation.

In the following weeks, I will unpack the equitable-curriculum perspective and will discuss Complex Instruction (see here and here) as one way to achieve greater access for all students in both the classical and equitable-curriculum perspectives.

What do you think? What other reasons are there for teaching mathematics? Which of the five goals that I described resonates with you? What potential concerns or challenges do you see with these forms of mathematics?


Mathew FeltonMathew Felton is an assistant professor of mathematics education in the department of mathematics at the University of Arizona and will be starting in the department of teacher education at Ohio University this fall. He is a coauthor of Connecting the NCTM Process Standards and the CCSSM Practices. His research focuses on supporting current and future teachers in connecting mathematics to real-world contexts and on teachers’ views of issues of equity, diversity, and social justice in mathematics education.

Teaching Students about Functions with Dynagraphs

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Introduction 

Function is a fundamental concept in mathematics; it is one that students explore repeatedly, at increasing levels of sophistication, throughout the early and middle grades (and beyond) (Steketee and Scher 2011). In the early grades, students may encounter functions as lists of inputs and outputs in classroom activities such as “Guess My Rule” (Huinker 2002). With that activity, students are given input values one by one as suggested in figure 1. With each additional input, students are asked to construct a single rule that transforms each value in the “input” column into the corresponding “output” value at right.

Figure 1: Students record their data 

Fig. 1 Students record their data. (From “Guess My Rule,” Huinker 2002, p. 320)

In subsequent activities, students may consider functions as “machines” that provide a specific output for a given input (Reeves 2005, p. 251). For instance, the tasks in figure 2 were developed for third graders studying functions.

Figure 2: Illustration of a third-grade version of a function machine task 

Fig. 2 These illustrations render a third-grade version of a function machine task.

In the middle grades (and beyond), students experience functions as (a) plots of ordered pairs on the coordinate plane and (b) as formulas to be evaluated for particular values of x (King 2002). Such representations are highlighted in figure 3.

Figure 3: Graphical, tabular, and symbolic representations of  f(x) = 2x + 1  

Fig. 3 Graphical, tabular, and symbolic representations of are available at http://bit.ly/demos-graph. 

Unfortunately, none of these representations adequately capture the dynamic nature of function; instead, each “portray(s) input and output in a discrete and static way” (Steketee and Scher 2011, p. 49).

Dynagraphs: An Introduction 

As an alternative to function representations presented in figures 1–3, consider the dynagraph sketch shown in figure 4.

Figure 4: Dynagraph sketch
Fig. 4 This dynagraph example is available at http://bit.ly/dynagraph1.  

The point along the bottom number line represents an input value of an unknown function. The upper point represents the corresponding output value of the function for the given input. As the bottom point is dragged, students make conjectures concerning the relationship between the input and output. For instance, looking at a dynagraph, such as that depicted in figure 4, students might guess that the following relationships hold:

  • Output = 2*Input (i.e., y = 2x)
  • Output = Input + 2 (i.e., y = x + 2)
  • Output = 6 – Input (i.e., y = 6 – x)
  • Output = 0.5*Input + 3 (i.e., y = 0.5x + 3)

By dragging the bottom point, students can test their conjectures against new input-output pairs. The image in figure 5 was generated by dragging the “input” point 4 units to the left.

Figure 5: Resulting graph
Fig. 5 This graph results when the point along the input number line is dragged 4 units to the left.
 
Of the original conjectures, only the last one holds for both input values. Clicking on the “Show Equation” checkbox, students see that the relationship between input and output is, indeed, defined as y = 0.5x + 3. As figure 6 suggests, teachers (and students) can type in new functions into the input box and explore new rules.
Figure 6: Graph showing results of clicking Show Equation checkbox
Fig. 6 Students can reveal the relationship between input and output by clicking on the “Show Equation” checkbox.
 
Instructionally, the approach is similar to the “Guess My Rule” game with several important differences:
  • The dynagraph emphasizes the continuous change of both the input and output values of the function.
  • The interactive sketch gives students and teachers a way to manipulate inputs directly through dragging, encouraging students to see variables as quantities that vary.
Dynagraphs were first envisioned by Goldenberg, Lewis, and O’Keefe (1992) as a means to bridge discrete, numerical representations of function (e.g., tables of values, function machines) with more sophisticated, abstract representations that students encounter in the later grades (e.g., formulas and equations, Cartesian plots).
 
From Dynagraphs to Cartesian Plots 
The output number line of a dynagraph can be rotated 90 degrees to yield a dynagraph that resembles the Cartesian plane (as shown in fig. 7). Constructing lines through “input” and “output” points perpendicular to the number lines, the x-y pairs are plotted by tracing the intersection of the perpendiculars (as shown in fig. 8).
Figure 7: A dynagraph that resembles the Cartesian plane 
Fig. 7 This dynagraph image, showing an output number line rotated 90 degrees, is available at http://bit.ly/dynagraph2.
 
Figure 8: Cartesian plot
Fig. 8 This Cartesian plot is available at http://bit.ly/dynagraph3.
 
Dynagraphs are a powerful representation to help students better grasp the idea of function. Traditionally, students in the upper middle grades explore graphs of functions wholly within the Cartesian plane. Too often, this approach doesn’t adequately connect back to experiences that students have had with functions in the earlier grades. By giving students a way to explore functions in one dimension, dynagraphs help students focus on relationships between inputs and outputs while considering functions as objects. Dynagraphs may also be used to help students transition to Cartesian graphing.
 
References 
Goldenberg, P., P. Lewis, and J. O’Keefe. 1992. “Dynamic Representation and the Development of a Process Understanding of Function.” In The Concept of Function: Aspects of Epistemology and Pedagogy, edited by Ed Dubinsky and Guershon Harel, pp. 235–60. MAA Notes no. 25. Washington, DC: Mathematical Association of America.
Huinker, D. 2002. “Calculators as Learning Tools for Young Children’s Explorations of Number.” Teaching Children Mathematics 8 (February): 316­­–21.
King, S. 2002. “Sharing Teaching Ideas: Function Notation.” Mathematics Teacher 95 (April): 636–39.
Reeves, C. 2005/2006. “Putting Fun into Functions.” Teaching Children Mathematics 12 (December 2005/January 2006): 250–59.
Steketee, S., and D. Scher. 2011. “A Geometric Path to the Concept of Function.” Mathematics Teaching in the Middle School 17 (August): 48–55. 
 

 
Michael Todd EdwardsMichael Todd Edwards is an associate professor of mathematics education at Miami University in Oxford, Ohio. He is the coeditor of Contemporary Issues in Technology and Mathematics Teacher Education, executive editor of the North American GeoGebra Journal, and codirector of the GeoGebra Institute of Ohio. His research interests focus on the teaching and learning of school mathematics with technology (specifically, dynamic mathematics software), ethical issues surrounding the use of free software and the free software movement, and writing as a vehicle to learn mathematics at all levels of instruction.
 
Jennifer NickellJennifer Nickell is working toward her PhD in mathematics education at North Carolina State University in Raleigh. Her research interests focus on preparing teachers to teach statistics and effectively incorporate technology into the classroom. 
 
 

A Critical Look at Movies

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Several weekends ago, my family suggested that we go to the local theater and watch a movie. As most any parent of a teenager knows, finding a movie that everyone can agree on is no small task. My wife and I have found online movie review sites, such as “Rotten Tomatoes,” helpful when deciding what to watch.

My thirteen-year-old daughter suggested that we go see Mom’s Night Out, a comedy that explores the many things that can go wrong when dads are left in charge of the kids. When my wife and I looked up the movie on “Rotten Tomatoes,” we found the low ratings by critics somewhat disconcerting (see the analysis at http://www.rottentomatoes.com/m/moms_night_out/).  

Of the 35 professional critics who viewed the movie, only 5 identified the movie as entertaining (for a 14% “fresh” rating). On the other hand, approximately 86% of registered “Rotten Tomatoes” users who rated the movie gave it 3.5 stars or higher (out of 4 stars).

When we showed our daughter the low rating and expressed our concerns, she was dismissive. “What do critics know anyway? They never like popular movies! They’re a bunch of snobs!”

As a teacher of mathematics, I found her comments intriguing. Her perceptions about critics were not wholly unfounded. Bloggers and movie critics, such as Vic Holtreman who owns Screen Rant, discuss the tendency of professional movie watchers to pan popular offerings (see http://screenrant.com/transformers-2-vs-critics-vic-14735/ for details). I wondered if these perceptions of critics were true. In particular, I wondered if data gleaned from the “Rotten Tomatoes” website would support (or refute) the contention that critics rate popular movies lower than the general public rates them.

The Data 

To explore this question in more detail, I gathered a list of the 200 highest grossing movies of all time from “The Numbers” website (http://www.the-numbers.com/movies/records/worldwide.php). With the list in hand, my daughter and I looked up each movie on the “Rotten Tomatoes” website and recorded critics’ and audience “freshness” ratings in an online spreadsheet (readers are encouraged to explore our dataset at http://bit.ly/movie-dataset). An exploration of such data is well aligned with the Common Core’s Standards for Mathematical Practice, in particular, SMP 3: Construct viable arguments and critique the reasoning of others.

Some Initial Findings 

After the data were compiled, my daughter used the spreadsheet’s built-in AVERAGE function to determine the average critic and audience ratings for the top 200 movies, noting that the critics’ average was 4 percentage points lower (namely, 71% to 75%). “See, critics rate popular movies lower, Dad!”

“Not so fast!” I retorted. Calculating the difference between freshness ratings for each movie, I used the spreadsheet’s built-in IF function to highlight instances in which critics’ ratings were higher than audience ratings (these instances were recorded as “1,” with noninstances recorded as “0”).

Figure 1 - Entries for 5 highest grossing movies 

Fig. 1 Entries for 5 highest grossing movies. Note that column H uses the spreadsheet’s built-in IF function to determine if the critic rating is higher than the audience rating for each movie.

Summing the critics’ higher column (i.e., column H in fig. 1), I determined that critics gave higher ratings than audience members for 112 of the 200 highest grossing movies of all time. In other words, critics gave higher ratings 56% of the time. “Ha! Critics aren’t snobs! They are less critical than the rest of us!” These two initial analyses seemed at odds with each other. Who was right?

What Do You and Your Students Think?  

Are critics snobs? Do critics rate popular movies lower than the general public rates them? You be the judge. Provide evidence supporting your contention. Graphs can be helpful for analyzing large amounts of information.


Michael Todd EdwardsMichael Todd Edwards is an associate professor of mathematics education at Miami University in Oxford, Ohio. He is the coeditor of Contemporary Issues in Technology and Mathematics Teacher Education, executive editor of the North American GeoGebra Journal, and codirector of the GeoGebra Institute of Ohio. His research interests focus on the teaching and learning of school mathematics with technology (specifically, dynamic mathematics software), ethical issues surrounding the use of free software and the free software movement, and writing as a vehicle to learn mathematics at all levels of instruction.

Creative Problem Posing in the Middle-Grades Classroom

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Earlier this year, I was introduced to Paul Lockhart’s remarkable text, A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. Lockhart persuasively advocates for mathematics teaching and a mathematics curriculum that provides students with opportunities to ask their own questions and devise their own methods for answering these questions. Taking Lockhart’s message to heart, I asked my eighth-grade daughter, Cassady, to construct a mathematics task that fosters creativity among students. My conversation with Cassady surprised me. Both she and Lockhart have encouraged me to reconsider the nature of mathematics and the way that we share it with students in the middle grades.

Cassady’s Task 

I began by asking Cassady whether mathematics was creative. An edited version of the hourlong conversation is highlighted in the following 10-minute video (http://bit.ly/mathcreative).

Video with 8th Grader
 

Cassady remarked that mathematics in the middle grades is “like a formula, and there’s one correct answer.” In response, she provided me with the following task:

John had 10 sticks. They were 12 cm, 2 cm, 4 cm, 9 cm, 20 cm, 6 cm, 15 cm, 5 cm, 10 cm, and 1 cm long. Can John make a polygon using these 10 sticks? If so, draw it. If not, explain why not.  

Remarkably, the task was not familiar to Cassady. It had not been posed to her in a previous class. The task was truly her own. In fact, she didn’t know if such a polygon could be built. When I asked Cassady if her classmates had opportunities to pose and solve their own problems, Cassady remarked, “We have kind of outgrown it. We don’t do story problems anymore . . . we haven’t done that since third grade.”

Our Solution 

During the next 30 minutes, Cassady and I worked collaboratively on her problem. Unfortunately, we had no hands-on manipulatives, so we attacked the problem using GeoGebra (http://www.geogebra.org/cms/en/), a free dynamic mathematics software program. A version of her sketch can be accessed on GeoGebraTube at http://www.geogebratube.org/student/m112159. 

Cassady began by plotting 8 of the 10 vertices as lattice points on a square grid. She arranged consecutive vertices to form right angles. This made it easier for her to satisfy the length requirements of the task. I’ve labeled these points consecutively as A through I (see fig. 1).

Fig. 1 We approached the solution to this problem using GeoGebra.

 Plot of first 8 vertices  

 (a) Plot of first 8 vertices 

 Constructing a circle to plot 2 remaining vertices  

(b) Constructing a circle to plot 2 remaining vertices 

With 2 lengths remaining (namely, 10 cm and 15 cm), Cassady constructed a circle with radius 10 units centered at the 8th vertex (point I). Point J was constructed on the circle. This step ensured that segment IJ, the 9th side of the polygon, had length 10 units. Next, she constructed JA as the 10th side, dragging J until JA had the desired length of 15 units.

The solution strategy also highlighted Cassady’s strategic use of technology (one of the Common Core’s Standards for Mathematical Practice is Standard 5: Use appropriate tools strategically). GeoGebra played an invaluable role in the problem-solving process. Using a compass and ruler to construct the location of point J would have been time-consuming and error-laden without the software. However, GeoGebra did not trivialize the task. Rather, the problem required a good deal of creative thinking with the software. The idea to use a circle was a “creative breakthrough” that involved considerable mathematics know-how and outside-the-box thinking. In the journey to the solution, Cassady and I discussed a number of interesting concepts, including the triangle inequality and slopes of parallel lines.

Math Is Creative 

At the conclusion of our conversation, Cassady noted that our mathematics work was “definitely” more creative than the activities she does in school. She noted that she “really likes problems where she doesn’t know if there’s a definite answer.” The experience has encouraged me to ask more questions of my students regarding their own learning needs, listening carefully to their answers, then acting on them. As Lockhart notes, “The only people who understand what is going on are the ones most often blamed and least often heard: the students” (p. 11). 


Michael Todd EdwardsMichael Todd Edwards is an associate professor of mathematics education at Miami University in Oxford, Ohio. He is the coeditor of Contemporary Issues in Technology and Mathematics Teacher Education, executive editor of the North American GeoGebra Journal, and codirector of the GeoGebra Institute of Ohio. His research interests focus on the teaching and learning of school mathematics with technology (specifically, dynamic mathematics software), ethical issues surrounding the use of free software and the free software movement, and writing as a vehicle to learn mathematics at all levels of instruction.

Towering Mathematics

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Towering Mathematics 

In a recent visit to a seventh-grade mathematics classroom, I was delighted to see students working collaboratively on the Skeleton Tower task as shown in figure 1.

Skeleton Tower
 

Fig. 1 How many cubes are needed to build a tower like this but 12 cubes high? (Source: http://www.illustrativemathematics.org/illustrations/75)

The task is well suited for exploration with middle-grades students. The problem lends itself to multiple solution strategies and multiple representations and can be explored with a variety of tools, including manipulative cubes, spreadsheets, and design software (e.g., see NCTM’s Isometric Drawing Tool at http://illuminations.nctm.org/Activity.aspx?id=4182).

Student Approaches 

In small groups, many seventh graders broke the tower into 5 pieces. They saw a center column 12 cubes tall and 4 identical “legs,” each consisting of 1 + 2 + 3 + … + 9 + 10 + 11 = 66 cubes. Ultimately, these students calculated that a Skeleton Tower with height 12 units would require 12 + 4 ´ 66 = 276 cubes. Others saw different patterns. For instance, one seventh grader told me that the problem essentially involved “finding the area of a rectangle.” Creatively rearranging cubes, she noted that “12 times 25” cubes were needed for the tower of height 12. Following up, I asked this student how many cubes would be needed for a tower 100 cubes tall. She quickly replied, “Easy! 100 ´ 199.” Then she showed a diagram similar to the one below.

 Student Approaches
Do you see what she saw?

The Skeleton Tower task is well aligned with the Standards for Mathematical Practice, in particular, 7. Look for and make use of structure. Note that both solution strategies described here highlight students’ use of structure, both numerical and geometric, to solve the task. In the first approach, students sum consecutive integers to find a solution, which is an approach intimately linked to triangular numbers and Gauss. For instance, students can calculate 1 + 2 + 3 + … + 9 + 10 + 11 by pairing addends creatively.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = (0 + 11) + (1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) = 6 ´ 11 = 66

In the second approach, students rearrange blocks to create an easier shape, a 12 ´ (11 + 11 + 1) rectangle, to analyze. Both approaches illustrate the suitability of the task for a wide range of learners. Both strategies hint at generalizations that may be explored with more advanced students. Meanwhile, the task’s concrete context makes it a meaningful problem for students to explore with hands-on materials.


Michael Todd EdwardsMichael Todd Edwards is an associate professor of mathematics education at Miami University in Oxford, Ohio. He is the coeditor of Contemporary Issues in Technology and Mathematics Teacher Education, executive editor of the North American GeoGebra Journal, and codirector of the GeoGebra Institute of Ohio. His research interests focus on the teaching and learning of school mathematics with technology (specifically, dynamic mathematics software), ethical issues surrounding the use of free software and the free software movement, and writing as a vehicle to learn mathematics at all levels of instruction.


 

 

Peace through Constructions

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Illustration from Euclids Elements.pngCompass and straightedge constructions have been in the geometry curriculum for a loooonng time. These constructions were around before Euclid included them in his textbook, The Elements, 2300 years ago. The painting shown here is an illumination from a translation of that book 1600 years later. I had to do constructions in junior high; I don’t remember being quite as unhappy about the endeavor as these monks or the woman teaching them. And now constructions are in the Common Core, too. Why are we eternally requiring students to construct things with these arcane and arbitrary tools?

Here is one answer: Plutarch tells a story of Plato and his buddy Simmias of Thebes coming back from a trip to Egypt when they encountered a delegation from the isle of Delos. The famous oracle at their temple had told them that the Delians and the other Greeks could put an end to their current troubles if they could figure out how to double the temple’s altar, which was a cube. In other words, they had to make a new cubical altar that was twice the volume. They told Plato that they had tried to double all the dimensions, but they ended up with an altar that was 8 times the volume, not double the volume.

Plato told them that this was not a problem for estimation but for precise construction. (It can be solved by finding two lengths in mean proportion, in other words, construct a length of the cube root of 2.) Although a few Athenians knew how to do it, the point was that the Delians should figure it out themselves. The oracle’s motive was that the Greeks would only end their troubles when they focused on cultivating the art of mathematical problem solving, including making logical arguments for their constructions. This would give them the ability to sort out their differences through reason rather than force. 

This classic problem of doubling the cube turns out to be impossible with only a compass and straightedge, but you can do it with other Greek tools (and you can do it with paper folding, too!). The work that occurred on this problem led to entirely new fields of mathematical study in ancient Greece.

I think maybe the reason we keep doing constructions is that it allows us to engage in at least two of the Common Core’s Standards for Mathematical Practice (SMP 1: Make sense of problems and persevere in solving them and SMP 3: Construct viable arguments and critique the reasoning of others). By forcing ourselves to figure out how to do something with a limited set of tools, we refine our problem-solving skills. By forcing ourselves to justify why we know our constructions will always work, we improve our ability to argue rationally, to show how ideas logically rely on other ideas.

Maybe the oracle of Delos was onto something. . . . 


Rob Ely, MTMS Blogger Rob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 

 

 


 

 

Bootstrapping Revisited: A Regular 3 1/2-gon?

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This is another weird example of the bootstrapping I mentioned in my first post. We were brainstorming a geometry class a few months ago when my friend Ian asked me, “Can you draw a regular 3 1/2-gon?”

After I mused about this Zen math koan for awhile, he handed me a ruler and a protractor and said, “Suppose you had only these and you needed to draw a regular 9-gon? What would you do?”

I told him I’d first figure out what each angle would have to be. The total angle sum in a 9-gon is (9 - 2) x 180° (because if you draw diagonals from one vertex of a 9-gon, you end up dividing it into 7 triangles, and each triangle contributes 180°). That’s 1260°. If the 9-gon is regular, then its 9 angles are all congruent, so they are each 1260/9 = 140°. To construct it, I would pick my favorite length, go 140°, and copy the same length. I would do this 8 times.

Then Ian said, “Okay, just do the same thing for a 3.5-gon.”

I said, “So. . . . How would I figure out each vertex angle when I don’t even know what it would look like?”

“Just use the same formula you used for the 9-gon: (n -2) x 180°, then divide by n to get each angle measure,” he said.

When I did this, I got 77.14°. Then I drew an inch-long side, measured 77.14°, drew another inch-long side, measured another 77.14°, and just kept going. Eventually I got this:

 MTMSPost3Pic 

Using this same method, for what values of n would you get a closed figure for an n-gon?

Here is the bootstrapping:

  1. We started with a definition of a regular n-gon: a polygon with n congruent sides and vertex angles.
  2. We discovered a rule and justified that it had to work: We proved that each vertex angle had to be (n - 2) x 180°/n, by putting in all the triangles.
  3. Then we generalized the rule to a new domain, namely, for fractional n. The only way was to extend the formula, which works fine if n is a fraction. But the formula can only be proved for whole-number values of n.
  4. This created new mathematical objects. Again, they didn’t have to be defined this way, but it nicely continued the rule we had gotten fond of.

Another example of the justification structure of the definition and the proven rule getting switched around and bootstrapped when we want to generalize to a new domain. This phenomenon occurs when we give ourselves the chance to express regularity in repeated reasoning (practice 8), giving us formulas that can then be generalized, weirdly, to new domains.

Want to try a bunch of other fractional gons?

RobElyRob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 

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