A Mathematics Problem: How to Help Students Achieve Success in Mathematics Through College and Beyond

   Guest Blog Authors:
   Malbert Smith III, Ph.D., MetaMetrics President and Co-founder
   Jason Turner, MetaMetrics Director of Professional Development
 

U.S. students are consistently outperformed in mathematics by their international peers. While 2011 NAEP results show a modest increase in mathematics performance, only about one-third of our eighth-grade students achieved the proficiency level. The truth is that many U.S. students graduate unprepared for the challenges they will likely face in college and careers, and this trend will continue to negatively impact our students’ and our nation’s ability to compete globally.
 
The Common Core State Standards, which nearly all states have adopted, provides policy makers, educators and parents with a road map for preparing students for postsecondary mathematics demands. But, as districts and schools begin to implement the Standards, they must also have access to the curriculum resources to make this aggressive drive toward real-world readiness a reality for all students. The Quantile® Framework for Mathematics—and the many free tools that support its implementation—is a unique resource for providing the targeted instruction that both striving and struggling students need as they ready themselves for the demands of their academic and professional pursuits.
 
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NCTM Recommendations for Common Core Professional Development

As a continuation of my exploration of professional development materials, here is a report released by the NCTM in November that provides professional development recommendations specific to the Common Core State Standards in Mathematics (CCSSM-PD). The recommendations help school leaders and professional development providers identify the best practices for teachers and how best to implement them. 

Additionally, a report entitled Gearing up for the Common Core State Standards, written last year by a group of math educators and experts, lists five initial content domains for professional development focus: (1) counting, cardinality, and numbers and operations in base ten (grades K-2); (2) operations and algebraic thinking (grades K-5); (3) number and operations-fractions (grades 3-5); (4) ratios and proportions (grade 6-7); and (5) geometry (grade 8). 

Much of the focus on the NCTM report surrounds the Standards for Mathematical Practice laid out in the CCSSM. These standards provide a framework for how math can be taught most effectively. The other recommendations adapt existing professional development practices to the CCSSM and facilitate communication between teachers and education leaders.

The five content domains identified in the Gearing Up report are useful in distinguishing the areas in each grade that will require the most training. While the CCSSM are intended to streamline math curricula, the overhaul will require teachers to update their syllabus and their teaching techniques. Some of the content domains have changed more dramatically in their treatment at various grade levels than others, and these deserve the most immediate attention. Focusing on these domains is the best way to implement the NCTM’s first recommendation: Emphasize the Substance of CCSSM-PD. The Gearing Up document provides the rationale behind each of the choices of domain. It includes the standards themselves as well as critical areas of focus for each target domain at various grade levels.

How are your schools preparing for the Common Core? Have you found any useful resources that you would recommend?

Connecting the Literacy and Math Challenge

There are distinctive differences between the cognitive structures for understanding math and those for learning to read.  These differences have implications for intervention with struggling math students.

Both literacy and math use symbols – letters and numbers – that children must learn and make sense of in order for them to understand the more complex topics in reading and numeracy.  In many ways, these symbols are similar.  Both are part of a finite set (26 letters, 10 digits), both are used in conjunction with other symbols (punctuation in reading, operational and relational symbols in math), and both have names that have nothing to do with the symbol’s actual meaning or value.  Most importantly, both are building blocks that students learn at the beginning of their education in reading and math. 

It’s important for students to understand the distinctions between the contextual usage of numbers and letters in order for them to develop strong number sense.  One of the most important big ideas in number is that numbers can be composed and decomposed.  A single letter cannot be broken apart – it’s the most basic component of literacy.  A number, on the other hand, can infinitely be broken apart into smaller numbers.  This is connected to students’ understanding that numbers represent quantity.  Understanding that 5 means ● ● ● ● ● will lead to the understanding that 5 is the same as 2 ● ● and 3 ● ● ● put together. 

Unlike numbers, stand-alone letters do not hold any meaning.  Only when put together do they begin to have purpose.  From letters we build words, from words we form sentences, sentences strung together create paragraphs, and so on.  On the other hand, when numbers are put together we are always left with another number. 

Finally, the many contexts in which we use numbers makes learning mathematics very complex.  When we combine letters, we create words.  This concept is definitive – the purpose of letters is to create words (and sentences, and paragraphs).  Numbers take on many different meanings.  Take the number 7.  Alone, it represents the quantity 7, ● ● ● ● ● ● ●.  In the number 75, however, the 7 takes on a completely different meaning, 70.  The same is true for the 7 in the fraction 1/7 and in the equation 7x + 3 = 31.

It is important to make the differences between literacy and numeracy clear, particularly for students who struggle with math.  Understanding the distinct qualities of number will help them unlock the structure of math and make sense of more complex topics.

Online Math Games

Many math educators are turning to free online games to supplement their curriculum and engage all types of math students.  Experts agree that many of these games have notable benefits, such as increasing curiosity and motivation, reducing math anxiety, differentiating learning, and building strategy and reasoning skills.

However, we know that many digital games, though engaging, often lack educational benefits and can waste precious classroom time. The National Council of Teachers of Mathematics (NCTM) recognizes this problem and provides a list of questions to help educators evaluate these online games.  Here are a few of them:

• Is there variety in the math tasks?
• Is there embedded scaffolding?
• Is the competition positive and non-threatening?
• Are there suggestions to integrate the game into the classroom?
• Was the math situated in a meaningful context?

In addition to a comprehensive list of evaluation questions, NCTM also provides a list of vetted online math games for elementary, middle school, and high school classroom use.  However, educators should be wary that some of these online games, because they are free, contain advertising that can be disruptive to the learning process. 

Online Teens and College Admissions

When I applied to college, I had never heard of Facebook, and I associated the term “social-networking” with lame mixers and “get-to-know-you” events that required attendees to adorn their shirts with a white name badge.  I created a Facebook profile when I arrived at college as a freshman in 2004.  By the time I graduated in 2008, I had 700+ Facebook friends, I was tagged in 600+ pictures, and much of my social life was wrapped up in online social networking. The career center at my college advised us seniors to delete our pictures posted on Facebook and to block our profiles from public consumption.  Apparently employers were using Facebook and other social networking sites to vet applicants. 

Kaplan Test Prep’s 2011 survey of college admissions officers illustrates that it is not just post-college youth that need to worry about the potential impact of Facebook and the Internet on their future.   Rather, college admissions officers are increasingly checking high school applicants’ social networking profiles and searching them on Google—about 25% of admissions officers reported checking a profile at least once and about 20% reported Googling an applicant.  Of the college admissions officers who used these online tools to vet applicants, about 12% said that what they found negatively affected the applicant’s chance of admission.  Some of the offenses included photos depicting underage drinking and other illegal activities, vulgar speech, and plagiarism.

According to a 2011 Pew Internet study, 80% of online teens now use social networking sites, which means that 8 in 10 online teens must consider how their online behavior may impact their future.  As teens spend more time on the Internet, we have a responsibility to educate children and teens in Internet safety, which includes issues of cyberbullying, online solicitation, and protecting one’s privacy.  Schools use computers – including online and networked applications – as never before for instruction and research.  Managing personal information in an online world is a critical concern for students, teachers, and families.

Overturning Theories of a "Gender Gap"

A new study in the Notices of the American Mathematical Society strengthens evidence that gender gaps in math ability do not exist. The study concludes instead that gender gaps in mathematics performance reflect unequal opportunity for girls and boys. Interestingly, these findings dispel hypotheses that boys have a greater variability in ability than girls, resulting in higher attainment for top performers. On average, girls perform at least as well as boys on the Trends in International Mathematics and Science Study (TIMSS) and Program for International Student Assessment (PISA). And in countries where there are observable disparities, these differences are mostly attributable to socio-cultural factors, especially gender stratification.

The authors used the results from 4th and 8th grade students on the 2003 and 2007 TIMSS as well as the scores of 4th grade students on the 2003 PISA. They analyzed a variety of factors that have been suggested as potential reasons for the traditional male-dominance of math. Some, including single-sex schooling, national income, and greater variability in male intellectual ability do not appear to impact math outcomes by gender.

The authors did observe, though, that women who are well-educated and have a high income are much more likely to ensure that their children reach a similarly high level of educational attainment. Thus, greater gender equity in the workforce correlates with higher math performance among both male and female students. Creating equitable employment opportunities and compensation promotes the development of higher math achievers. Fortunately, the United States has made strides toward eliminating gender stratification. For example, girls now perform equally with boys in math, even in high school, having closed the significant gap that existed in the 1970s. As we continue to expand opportunities for women in math and science, we will likely see even greater improvements in math performance across both genders.

New Models of Hybrid Professional Development

As technology becomes an increasingly important component in teaching, it is playing an essential role in professional development. In Education Week’s October publication, “Virtual PD Creates Connections,” Michelle Davis and Katie Ash discuss the benefits of a blended system of online and face-to-face professional development. This hybrid model allows teachers to receive small doses of PD digitally while meeting in person with PD professionals to synthesize the information in a more personal setting.

Online resources such as Edmodo and the PBS TeacherLine allow teachers to share ideas, view videos of expert teachers in the classroom, and take online courses, all according to their own schedules. The short length of these PD sessions allows teachers to handle topics as they need them. The blended approach can improve efficiency and effectiveness. Teachers can view video materials in advance to make their face-to-face time with instructors more productive.

Online resources may be especially useful as schools adopt the Common Core. As Ms. Ash points out, the total number of mathematics standards is small, only 28, but teaching them requires deeper exploration of each topic than most curricula currently provide. And since the same standards will be adopted by nearly every state, common core professional development will be applicable to teachers across state lines. With limited budgets, online forums are a useful tool for teachers trying to stay up to speed on the Common Core. The James B. Hunt Jr. Institute for Educational Leadership and Policy is one of several providers of PD resources on the Common Core. The institute has developed a series of YouTube videos featuring background on the CCSS. And many more teacher-targeted curricular resources are being developed to address the standards. What types of professional development does your school use? What do you think is the most effective model for professional development?

Basic Blocks with a 21st Century Twist

A recent New York Times article, With Blocks, Educators Go Back to Basics, explores the latest education trend in leading New York City public, private, and charter schools: the return of the block corner.  In the past few years, parents and educators have been calling for more unstructured play in early childhood education, moving away from the rigorous academic programs that have dominated early education in the past few decades.   The article attributes the “back in style” blocks to this recent resurgence in the critical role of play for healthy child development.

As this article from Sixty Second Parent suggests, there is no secret to why traditional wooden blocks can help young learners.  Blocks can help develop pattern recognition and spatial reasoning.  The tactile and visual stimulation from blocks reinforces the relationships between lines and shapes.  Blocks in different sizes let children practice comparison skills. 

In elementary grades, blocks are being used as foundational tools for math, science, social studies, and more.  However, these schools are putting a 21st century twist on this block play.  Some schools are using iPads and Flip Cameras to video block constructions and edit footage to create documentaries.  At other schools, children are learning critical web navigation skills as they search for images or videos to inspire their block constructions. 

Will the “blocks + technology” equation help us promote early interest in STEM learning?  How far can this formula take children in their acquisition of 21st century skills?  Or, do we need to simply use these blocks to bring play and discovery back into the classroom?

Middle School Transitions Affect Achievement

Harvard University recently released a study on Florida schools that investigates the impact of middle and high school transitions on school achievement. Though researchers have previously focused on the effects of high school transitions only, this study, as well as a 2010 study on New York City schools, reveals that the more critical school transition actually occurs during middle school. 

The study shows that students who move to middle school in the 6th or 7th grade show a substantial drop in both reading and math achievement scores and these students’ achievement scores continue to decline through the rest of middle school. These students have lower school attendance rates and are almost 20% more likely to drop out of school by 10th grade. In contrast to the transition in middle school, students exhibit a smaller achievement drop when starting high school (9th grade) and their math and reading achievement scores tend to improve in 10th grade. 

Why does this middle school transition cause such a significant achievement drop? The obvious assumption is that K-8 schools provide a higher quality education for students in 6th through 8th grade than do stand-alone middle schools. However, these researchers found little evidence to support this claim, instead suggesting that “being in the youngest cohort in a school adversely impacts student performance” and is “particularly costly” for middle school-aged children. 

If it is more difficult for middle school students to achieve in environments that do not include younger students, does this call for a massive restructuring of our public schools to the K-8 school model? Are there ways to support students effectively through these structural transitions? If older students benefit from the “leadership roles” they can adopt in schools that include young students, can buddy systems and increased contact between stand-alone elementary and middle schools provide the same opportunities for leadership as does a K-8 school? Certainly more research needs to be conducted on the effects of these structural transitions and ways in which we can help prevent this achievement drop.

Delaying Gratification for Academic Success

Many children have little patience to satisfy a want, especially when the rewards do not seem obvious. Math homework, for example, presents a challenge that stands between young students and their preferred after-school activities. Oftentimes, when the answer does not seem immediately apparent, students would prefer to give up or rush to the wrong answer rather than work through the problem. Walter Mischel’s famous marshmallow experiment in the 1960s demonstrated that some children have a stronger ability to delay rewards. Four-year-old children were placed in a room with a marshmallow and told if they waited for the experimenter to return, they could have two marshmallows. The children who were able to wait had higher SAT scores and academic results when tested several years later.

Recently in Newsweek, Sharon Begley and Jean Chatzky discuss the conundrum of spenders versus savers, people who would prefer to buy something now versus those who would save up for something big later. Scientists are starting to identify which regions of the brain connected to “saving,” specifically the prefrontal cortex (PFC). They have even figured out how to stimulate the PFC, which improves people’s ability to delay gratification.

While activating students’ PFCs might not be the immediate solution, there appear to be other ways to promote patience in exchange for a higher payoff. One of the challenges to both parents and educators is getting children to think into the future to see the benefits of motivation and hard work. A student that I tutor dreads doing her math homework because it does not come easily to her. She would always prefer to work on Spanish, a subject in which she excels. She struggles to see that focusing on math will improve her success in the subject and make it much more enjoyable. Fortunately, research by psychologist Warren Bickel of Virginia Tech suggests that improving working memory boosts people’s ability to develop longer time horizons. And with practice, children can learn that hard work in the present will result in better outcomes in the future.

Waiting for this higher payoff requires some practice and training, but the skill of delaying gratification can be developed, a hopeful sign for math education. While tricks and shortcuts get students to the answer more quickly, they often come at the cost of understanding core concepts. If we can improve students’ working memory and time preferences, they will become more successful math learners in the long term.