Can Useful Graphics Help Close the Achievement Gap for ELLs?

  

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For English language learners, how does one tease apart their language and math abilities? It can be tough for ELLs in a curriculum with real world applications, word problems, and verbal instructions. Supplementing math problems with graphics, however, is a method that appears promising. Even though useful visuals help all students, ELLs benefit the most from them, according to research recently presented at the American Educational Research Association conference.

University of Georgia graduate student Albert Jimenez looked at 270 district test questions that were used for grades 3-8 in a large suburban school district. Of the 270 questions, 70 included a useful graphic such as a rectangle with measurements for a question about the area of a pool. The other 200 had no graphics or had ones that did not help with solving the math, such as a picture of a golf ball in a word problem about golf.

His findings, as published by Education Week, were as follows:

English-proficient students outperformed ELLs on questions without a useful graphic by 7.9 percent. Yet when ELLs had a useful graphic, that gap closed to only 2.8 percent, and ELLs outperformed English-proficient students on 28 graphic questions. Mr. Jimenez is conducting additional studies to determine whether the unconnected graphics help or hinder English-language learners' ability to solve math problems and if the useful graphics help without other accommodations at different grades.

These findings support the hunch that part of the math achievement gap between English-proficient and ELL students is likely a function of linguistic abilities. What other techniques do you use (or would like to see used) to accommodate English language learners in math classrooms?

Because Math Is Too Important to Leave at School

  

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A thank you to Jason Turner for contributing this guest post blog about maintaining math skills over the summer to the Math Hub. Jason is the Director of Professional Development at MetaMetrics Inc.
 

Summer is finally here. And for many students, an extended summer break means time spent with friends and family, summer jobs, camps, and well-deserved vacations. Unfortunately, for many students, a break in the school year also means a break from all academic activity, meaning that they could return to school with their math and reading abilities somewhat diminished from just three months earlier. This loss has been well researched, and many education reformers now consider fighting summer loss an important part of any serious education reform agenda.

The effects of summer learning loss in both math and reading have been well documented. Low-income students are especially susceptible to the corrosive effects of long interruptions in academic life. The reasons for this are complex, but it’s safe to say that in many cases low-income students do not enjoy the same academic opportunities (e.g. summer camps, academic retreats, tutors, etc.) as their high-income peers. Add in the fact that many low-income students may go home to text-free zones (lacking books, magazines, and newspapers) and it’s easy to see why the reading skills of so many students deteriorate over the summer months.

In mathematics the picture is even more dismal. Regardless of socio-economic levels, students experience a significant amount of learning loss in math. Though many states and districts offer multiple and intensive summer reading initiatives, too few have undertaken serious efforts to address the loss students experience in math. Admittedly, keeping students engaged in math activities during the summer is more difficult than engaging them in reading. There simply aren’t as many meaningful math resources available for students.

While it’s easy to dismiss summer slide as a fact of academic life, the consequences are profound. The learning loss that occurs each summer has an unfortunate cumulative effect. Add up the amount of learning loss over twelve consecutive summers, and the resulting gap is the difference between those that are prepared for the rigors of college and career and those that are not. It’s easy to see that any hiccup in the trajectory toward college and/or career represents a setback that can make a tremendous difference to where the student ends up.

One possible solution to summer slide would be to increase instructional time, which need not always mean more time in the classroom. Extending the school year can be done in a variety of ways, including providing students with resources that supplement and reinforce the skills and concepts acquired during the school year. In math especially, it is imperative that students continue to stay engaged in activities. Engagement does not necessarily mean learning new concepts and skills. During an academic hiatus, staying engaged in math may simply be brushing up and supplementing last year’s lessons.

For example, one specific solution is offered by MetaMetrics’ free online tool, Math@Home (full disclosure: I’m the Professional Development Director at MetaMetrics). Math@Home provides students (or educators or parents) with free, targeted math resources – like websites, worksheets, video tutorials, skill sheets, etc. – that support the textbook lessons studied throughout the year. Math@Home relies on a student’s Quantile measure as a way to target the student at just the right level of difficulty, though students can still use Math@Home even without a Quantile measure. Students and teachers have the ability to build specific lists of math resources to save for a later date. Best of all, Math@Home’s social networking features allow students and teachers to share multiple resource lists with others through e-mail and even post favorite math activities to Facebook and Twitter.

There are any number of ways to support year-round learning. Math@Home is just one way that educators and parents can keep students engaged in math activity throughout the summer months. As the focus shifts from proficiency to college and career readiness, it is critically important that educators and parents ensure that summer months are used to reinforce last year’s lessons and to prevent the pernicious effects of summer learning loss.

Understanding and Remedying Deficits in Numeracy

  

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In a recent blog post, Jennifer Chintala discussed how many students who experience math anxiety actually suffer from a condition called dyscalculia. On NPR’s Science Friday last month, Dr. Brian Butterworth, a professor of cognitive neuropsychology at University College London, further explained this disorder. He shared some of the signs of dyscalculia and methods he and other researchers have developed to teach dyscalculics how to deal with the number system. People with dyscalculia have trouble with numeracy; they may struggle to identify numbers in an array or to add or subtract without using their fingers. According to Butterworth, at least five percent of people suffer from dyscalculia.

In a separate study, Dr. Veronique Izard, a research scientist at Paris Descartes University, interviewed indigenous people in the Amazon to determine whether math, especially geometry, is an innate skill that humans possess. She found that they were able to understand geometric concepts, such as parallel lines. Furthermore, they have a sense of numerical quantity, but they were not as strong at exact calculations. These findings suggest that the parts of the brain that deal with geometry and numeracy are separate. Many people with dyscalculia are good at geometry, because the skills required don’t involve number sense. Understanding of arithmetic does not develop as naturally as geometry in humans and must be taught.

Dyscalculics can learn how to do simple calculations, but Butterworth says that they have to work with concrete materials for much longer than other students in order to understand the number system. Computer programs are also effective in place of using concrete objects to build proficiency in numeracy.

To read more about Dr. Butterworth’s studies of dyscalculia, see his article in the current issue of Science.

Differentiation and the Four Jungian Learning Styles

  

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I like to work alone. It’s not that I don’t often collaborate with my peers – I think that’s essential for all educators. But when I am studying something new or trying to get work done, I like to tuck away in my office and put my nose to the grindstone. According to an article written by Jane A.G. Kise, this makes me an introverted learner. I never think of myself as introverted, but perhaps when it comes to learning, that’s just what I am. Kise’s article highlights the four Jungian learning styles, one’s that I had previously not been familiar with. But, in my continuous search for how to best differentiate in the classroom, I’m always looking for a different perspective on how kids learn.

According to the article, a child has two ways he can gain energy and two ways he can process information. To gain energy, children who favor extraversion like to work with others and those who prefer introversion thrive while working independently. To process information, students may prefer intuition or sensing. Intuitive types focus on hunches and connections, whereas sensing types build knowledge in an orderly fashion. When combined, these create four learning preferences summarized below.

• Introversion/Sensing: Need certainty before completing tasks, like direct instruction and practice.
• Extraversion/Sensing: Learn through movement and interaction with others. Need to “see” the math.
• Introversion/Intuition: Process information internally and find creative solutions.
• Extraversion/Intuition: Process ideas out loud with others and easily transfer knowledge to different situations.

To help find a student’s learning style, use the checklist below:

 
Stay tuned for the next post for information about how these learning styles may be addressed in your math class!

Making the Most of Students' Test Score Data

  

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In my state, students’ standardized test scores usually arrive during the middle of the summer. While it would be easier if the scores arrived before the end of the school year, I should be grateful that we do receive the data prior to the start of the next school year. This mid-summer influx of data means that I get to spend a good part of August analyzing data and preparing reports with the teachers in my district. These reports are meant to help teachers identify students’ strengths and weaknesses so that the areas of struggle may be better addressed in the subsequent years.

As a 'math person', I enjoy this type of analysis and recognize the value in having a whole lot of data organized into teacher-friendly graphs and tables. Unfortunately, some teachers don’t recognize the value of such information and, perhaps that is because they haven’t been looking at it from the appropriate perspective. Just recently, Ronald Thomas authored an article where he shares his Nine Truths of Data Analysis. In the article, he shares his thoughts on how to analyze data in a meaningful way:

• Data needs to be organized into information. This transformation often involves summarizing and creating charts and graphs so that it is more useful for teachers.
• Data is more about improving instruction than just looking at the numbers. Teachers need to think about what clues that data is providing about how to help students.
• Data should be analyzed by a group of teachers who have a common goal.
• Teachers need to engage in frequent “data dialogues” to discuss how recent data can inform instruction.
• Data should be analyzed using an established protocol. For example, use data to identify strengths and weaknesses to help teachers recognize areas that need more or less instructional time.
• For data to be impactful, districts must align curricula, instruction, and assessments around the standards.
• After using data to inform instruction, teachers must reflect on the changes they have made and continue to alter their plans to increase achievement.
• Teacher leaders need to clearly articulate why time should be spent on data analysis. Teachers need to keep in mind a two-fold learning goal: increasing student achievement and eliminating learning gaps.

If you’re like me, most of this information has been engrained in your head for years. However, it may be helpful to share such ideas with your faculty and staff so there is a consistent philosophy regarding data-analysis.

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?