Finding the Beauty of Math in Nature

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In the spring when flowers and leaves come out, it provides the perfect opportunity to get outside and learn some math. One of the most abundant examples of math in nature is Fibonacci numbers. The first published work about the Fibonacci sequence appeared in the early 13th century, written by Leonardo Fibonacci. The sequence, which starts with 0 and 1 and consists of the sum of the previous two numbers, has some surprising applications in the real world.

Many flowers, such as daisies, buttercups, marigolds, and asters, tend to have a consistent number of petals that is a Fibonacci number. Even clovers more commonly have three petals than four because four isn’t a Fibonacci number. There are plenty more examples of flowers for which this pattern applies.

The Fibonacci sequence often appears as a spiral. The pattern can be illustrated in a series of radiating squares where the length of the sides of two adjoining the squares make up the side of the next square:

The seeds in the center of sunflowers, pinecones, pineapples, and artichokes also contain this perfect spiral. It is amazing to see how math can manifest itself so beautifully in nature. Students will love the chance to explore the outdoors and make real world connections with math. Here is a website I came across that further explains Fibonacci and provides additional resources.

How do you teach the Fibonacci sequence? What are some other nature-related math activities appropriate for spring?

Exploring Geometric Concepts in Two-Point Perspective

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Art has always been one of my favorite subjects (in fact, my dream job from ages five through seventeen was fashion designer). One concept we always came back to in art was that of perspective.

Perspective is something we witness in everyday life; the most obvious is the road vanishing in the distance (this is one-point perspective, a visual style I talked about a couple months ago). As I mentioned previously, there are many types of perspective. Today we’re focusing on the two-point kind.

Two-point perspective is when all horizontal parallel lines converge at one of two vanishing points and happens when you look at the edge of an object. With two-point perspective, you get a better sense of depth. Here you can watch a YouTube video on how to create a small street scene with two-point perspective.

Here are some additional resources for bringing two-point perspective to your class:

Sample lesson involving two-point perspective
Worksheets on two-point perspective

When creating perspective drawings with your students, it’s important to remember to erase all guidelines once the drawing is complete.

Lesson Plan Resources: NSA Sponsors Summer Institutes for Teachers

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Each year, the National Security Agency (NSA) sponsors Summer Institutes for Mathematics Teachers (SIMT) and Summer Institutes for Elementary School Teachers (SIEST). Teachers can submit an application to attend these paid workshops and several lucky teachers are selected to attend the week-long institutes. Fortunately, even teachers who don’t attend the institutes can reap the benefits of the attendees’ work. One of the tasks of teachers attending the workshops is to create Concept Development Units which are published on-line for any teacher to use. 

The units are organized by grade band: Elementary, Middle School, High School. Within each grade band, the units are further broken down by broad topics as shown below:

 
Within each subject area, the units are organized by the year in which they were created. Because the teacher-developer chose their topics, lessons in each topic are not created each year. Each unit includes a detailed document that provides an overview of the unit, standards that are taught, grade level, expected time frame, student outcomes, materials needed, and details about how to move through the unit each day. Users will also find suggested questions to ask and differentiating the lessons. And, the best part is that each unit is ready-to-go because all of the necessary resources (worksheets, assessments, etc.) are also included.

I’ve mentioned before the need for collaboration with your colleagues to create exceptional lessons. This site provides you with a plethora of lessons and resources created by teachers who understand what is needed to make a unit successful. The lessons provided are some of the most comprehensive that I have found, so I urge you to take advantage of this resource. Let us know how it goes!

Math Anxiety: Conquering Students’ Fear of Math

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Math is a subject that many students dread because they fear making mistakes. This anxiety really can paralyze students’ problem-solving ability by stimulating greater activity in the amygdala, the emotional center of the brain, which hinders the effectiveness of the prefrontal cortex, the home of working memory and critical thinking. Usually, a person processes a problem initially in the amygdala then sends the relevant information to the prefrontal cortex. When students feel stress about facing a math problem though, their brains will not allow them to access the working memory needed to solve the problem.

Math and education experts continue to research what may trigger this early math-related anxiety, as highlighted in Education Week’s article about a recent Learning and the Brain Conference. Math stress creates negative feelings about math and keeps many students from pursuing math in higher education or their careers. An especially interesting finding is that the students who experience the effects of math anxiety most acutely may be the ones who would otherwise have the most enthusiasm for the subject. Stress caused students who identified most with math to perform worse than other students. However, in non-stressful situations, the math-leaning students performed better. The students who identified with math were taken to be those who sought out further opportunities in the math program.

An important part of the research shows that parents and teachers can pass on their math anxiety to students. One way that Dr. Judy Willis, a neurologist, former middle school teacher, and author of Learning to Love Math, says that teachers can keep students from developing a fear of math is asking each student to answer every question. They can answer anonymously, either using electronic clickers or scratch paper, and then “bet” on answers. Participation is crucial to building solid foundations and confidence in math. What other strategies do you use to help students conquer their dread of math?

Project-Based Learning: Now in the Math Classroom

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Project-Based Learning is known for its use in a variety of disciplines, but not math. Andrew Miller’s article Tips for Using Project-Based Learning to Teach Math Standards shows us how this doesn’t need to be the case.

In the article, Miller identifies the challenges one can encounter with designing math-based PBL projects, such as creating a robust project when there’s pressure and emphasis on testing and how to make smart choices in regards to selecting a learning target. He lists three main tips when designing mathematical PBL projects.

• Reframing terminology. Most people immediately go to equations when they hear "math problem.” But in connection with the Common Core State Standards, educators should be able to redefine a more rigorous meaning of the word “problem”, which focuses on relevant and unique real-world applications for the student.
• Choosing the correct unit. It’s always best to select a unit with a longer time span when planning out a PBL project. Students will have more time to create a stellar project during a three-week unit than with a three-day unit.
• Picking standards with easy real-life application. Students will have an easier time creating a project if the topic is focused around something more concrete like right angle triangles as opposed to factoring.

For more information on Project-Based Learning, go to PBL Online.

The Art of Questioning: How to Gauge Actual Learning

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With such a focus on teaching content to prepare students for state testing, it’s often a wonder if students are actually learning content. It’s likely we were all introduced to Bloom’s Taxonomy during our schooling, and it is important that we really consider this hierarchy when developing questions. Asking straightforward knowledge and comprehension questions may show that students can find an answer, but do they understand the how or why behind what they are learning?

Recently, Elizabeth Stein published Teaching Secrets: Asking the Right Questions which provides suggestions for ensuring that the types of questions you ask are the questions that will gauge actual student learning. First, she suggests that you set the stage by using cooperative learning, encouraging dialogue, and observing student interactions. Once a collaborative learning environment is established, enhance learning opportunities by asking the right types of questions. 

• Open-Ended Questions: These allow multiple entry points because there doesn’t have to be just one answer. Ask students their opinion, or in math, how they see themselves using a skill in the future.
• Diagnostic Questions: These questions require students to have a deep understanding of a topic. Ask students what would happen if a particular number in a problem were changed.
• Challenge Questions: Require students to analyze, apply, and evaluate. Question why a particular formula works. Ask if they can come up with another way to solve a problem.
• Elaboration Questions: Require students to provide more information. What made you solve a problem this way?
• Extension Questions: These questions require students to think beyond the basics. How could you change this problem to make it appropriate for a younger student? What could you do to make this problem more challenging? 

In math class, we tend to think that there is always just one type of questioning technique. Students can do more than just find an answer. Try using some different questioning techniques and see if you can extend students’ thinking and learning. Tell me, what great questions have you asked in the past?

Math Activity: Practical Lessons in “Grown-up” Math

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Students inevitably want to be treated as though they’re grown up. They like to feel mature. So why not teach math through practical, real world lessons? My sixth grade teacher used an innovative curriculum that made math fun and accessible. The class invested in the stock market and purchased cars and apartments. We were excited to be engaging in the “real world,” and we were eager to learn math that we knew we would use in the future.

We each had $10,000 to invest in the stock market, either in a few stocks or several, and then we had to track the amount of money we made or lost based on the actual market. We practiced multiplication, addition, and subtraction as we avidly followed our stocks’ performance. Then we searched through car ads and classifieds to find our dream vehicles and homes. We learned what “APR” stands for (annual percentage rate) as we calculated interest rate payments. These calculations helped us master fractions and decimals as well as exponents. Lessons like this one could be adjusted in complexity to fit a variety of levels of math. What makes it so successful is the enthusiasm it generates. Not only does it place math in the context of the real world, but it also allows students to take on “grown-up” activities such as watching returns on stocks and investing in cars and homes.

It is important to find lessons that will grab students’ attention and encourage them to connect with the material. This lesson was one I still remember vividly because I felt so empowered knowing about these complex real world activities. What are some ways that you have brought practical and fun math lessons into your classroom? What other topics have you used to teach fundamental math concepts?

Music, Fashion, Video Games...and Math

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Sometimes we can find hidden educational gems in places we may not think to look. Recently, I came across a great set of resources that has been put out by the New York Public Media Provider, THIRTEEN. This group produced and aired a series of math projects titled Get the Math. These projects were designed to help middle and high school students relate math to areas of interest. The activities include a video segment to set-up an interesting problem along with activities and additional “challenges” related to the video.

Each of the three projects is based on a topic that is of interest to young students:

• Math in Music – Depicts musicians using proportional reasoning to produce a song.
• Math in Fashion – Shows how a designer uses Algebra to get the production price of a clothing item within a certain range.
• Math in Video Games – Explores the use of graphs and equations to create the movement of objects in a video game.

After watching a short video, students are presented with a challenge. The program guides students to complete the challenge and then presents additional problems that are similar so that students can practice their skills. The site also includes teacher lesson plans so that participating teachers have a clear path to follow when presenting the activities. 

Students are in constant need of fun, motivational activities that show them how the math they are learning may be used outside the classroom. Presenting three different contexts for activities that are engaging, interactive, and purposeful can help students recognize that many of their future careers may utilize math skills. You may even want to have your students come up with their own challenge by taping an introductory segment and developing activities much like those presented on this site. Then, have students present their work to the class so that everyone has exposure to the creative ways in which math can be used.

Can Students Maintain Their Math Skills over the Summer?

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In a recent post, I wrote about resources that can be used to help keep students’ minds sharp throughout the summer months. Use of these sites, however, is left up to the discretion of the students or parents, and many choose not to fill their summer months with math activities. Some districts opt not to give their students a choice, and they distribute work packets that students must complete during the summer months. In my district, all students entering Grades 5 through 8 are assigned a packet in June that is due at the start of school in September.

The purpose of our math packet is to help students maintain the skills acquired in the previous year so that they have a better start in September. The packet consists of roughly 50 questions based on content that should have been mastered in previous years. The topics covered are ones that are essential for success in subsequent years. Ideally, students would complete the packet gradually throughout the summer. However, realistically we can expect that most students are likely to complete the packet at the very end of the summer...right before the start of school. By this point, students may be struggling to remember skills from the previous year. To combat this problem, our packet includes a very brief explanation on topics that are covered. For example, a particular page may include an explanation and a sample problem showing how to multiply decimal numbers followed by 8 or 10 problems on the topic.  Students who are struggling can refer to the mini-lesson, and it might even assist parents who are helping their child with the packet.

I do believe that summers are meant for having fun and relaxing. However, it’s unfortunate when kids come back to school in September completely “cold”. It’s as if they haven’t thought about school at all since exiting the classroom in the spring. Requiring students to complete a small assignment over the summer, even if it is done the night before school starts, ensures that students start the school year with at least some recent exposure to math skills which will, hopefully, lead to a more successful year.

Lecture or Deliberate Practice? Different Styles in the Classroom

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A Nobel Laureate in physics, Carl Wieman, is heading up a $12 million initiative to improve instruction with research-backed methods. A finding from his team released in the journal Science this month suggests that a "deliberate practice" lecture style could boost student learning. Such a class actively engages students and allows them time to synthesize and respond to new information.

In the study, one undergraduate physics class was lectured in its normal routine while another integrated the following elements:

1. Pre-class reading assignments (three or four pages)
2. Pre-class reading quizzes (true/false quizzes based on readings)
3. In-class clicker questions with student-student discussion
4. Small-group active learning tasks
5. Targeted in-class instructor feedback

Though these techniques have been studied independently or in smaller clusters, Wieman and his colleagues combined them all in every 50-minute class. Both courses were taught as usual for the first 11 weeks, and the new style was introduced during the three lectures in week 12. Students taught in the "deliberate practice" style scored an average of 74 percent on a post-test, which was more than twice as high as the average for the comparison course.

Though promising, this study has limitations that others have pointed out. James Stigler, a UCLA psychology professor, said that it was not a good idea for the authors of the study to also be delivering the intervention. As reported by the New York Times, he explained that, "This is not a good idea, since they know exactly what the hypotheses are that guide the study, and, more importantly, exactly what the measures are that will be used to evaluate the effects. They might, therefore, be tailoring their instruction to the assessment — i.e., teaching to the test."

Have you tried those "deliberate practice" techniques before? Did you experience classroom change?