Math scores on the SAT have fallen to the lowest level (511points) not only since the college admission test was overhauled in 2005 (519 points) but in four decades (see recent articles in Inside Higher Ed and the New York Times). And stagnant results from high school students on state, federal and international tests are adding one more reason to worry about the nation’s high school mathematics teaching and achievements.
Elementary school children have made steady progress on mathematics on many state and national tests, including international tests such as: The Trends in International Mathematics and Science (TIMSS) and The Programme for International Student Assessment (PISA), but, a great deal of disparity exists between the achievement of minorities and poor students at all grade levels compared to those with privilege—economic, social, geographical, and educational. More importantly, improvement in scores is not translating into high school students’ higher achievements in mathematics.
Policy makers, educators, and social scientists identify a plethora of social and educational challenges that continue to pose hurdles in lifting high school mathematics achievement. Among them are social conditions: poverty, growing economic inequality, language barriers, low levels of parental education, and social ills that plague many urban neighborhoods. Others relate to educational conditions: poor quality of teaching and teacher preparation, diversity of teaching strategies and assessments, and lack of uniformity in marshaling resources for implementing common curricula such as Common Core State Standards in Mathematics (CCSS-M).
Per capita expenditures in urban schools may not be lower; however, unsatisfactory educational conditions—inadequate physical and administrative infrastructure, poor teacher quality, low accountability of services, lower expectations of students from Kindergarten to high school—plague schools.
There are other reasons for the decline in SAT scores in mathematics. It may very well be due to more students taking the test in response to new state and federal laws requiring or encouraging testing. For example, several states and school districts require that students take the ACT or SAT to graduate. It is possible that the types of students who weren’t previously taking the test don’t have the same level of ability on standardized tests as the students who consider selective four-year colleges. However, I believe that the most important factors are: low expectations in mathematics from students by parents, teachers, and administrators and lack of emphasis on mastering the material taught.
In the last twenty years, fourth graders have been doing substantially better in mathematics and eighth graders also have made some progress in mathematics; however, tenth graders have shown no progress and even some decline. Many elementary school principals point out that they are sending better students to middle school, but something happens to them there. Actually, it is a false sense of achievement because the improvement is not deep enough. A child can do well in elementary mathematics by just being a good counter and having no generalizable strategies and understanding.
To do well in high school mathematics, a student should have strong preparation in mathematics ideas. That means mastery of important components of mathematics ideas: (a) linguistic—rich vocabulary and understanding of the structure of mathematics language, (b) strong conceptual schema and flexibility of thought, (c) mastery of skills/ procedures—understanding, fluency and applicability.
At the elementary school level, methods of teaching mathematics that emphasize only the coverage of material, using inefficient strategies such as sequential counting methods and without acquiring real mastery (understanding, fluency, and applicability) and efficient strategies do not result in lasting effects. Without generalizable mastery, students’ achievements from the elementary grades do not translate to future achievement. The growth in mathematics at the elementary level, therefore, does not translate into increased interest in mathematics and achievement at the middle and high school level.
Teaching should result in learners demonstrating the expected skills and behaviors: from recall of information to performing tasks—routine and non-routine, applied in solving problems, demonstrating higher order thinking and creativity. With this demonstration of skilled performance, students will be able to transfer the skills to other domains of mathematics.
Research shows that characteristics of expert performance (mastery) are acquired through experience and that the effect of practice on performance is larger than earlier believed possible.
Elements and Conditions of Practice
Skilled performance, in any domain, is the result of interaction between environmental factors (instruction), genetic endowment (student characteristics), and sustained supervised and unsupervised practice. To understand expert performance, teachers need to know the nature of this interaction and the conditions that maximize it. Mastery of any content is individual, but there is a close relationship between practice and mastery. A fair amount of practice is necessary for learning and mastering concepts, skills, or procedures.
Many teachers and parents believe that practice makes perfect. To some extent, practice does show improvement. However, people may spend time and effort in working at a skill with little to no improvement after a certain point; in fact, they quickly reach a plateau in their skill level and performance. The pay off of just practice after that point is limited. The difference between those who plateau and those who go on to higher level of mastery with fluency depends on the type, frequency, and the intensity of practice used. One benefits from practice significantly when:
- initial methods of learning a concept are efficient and strategy-based,
- practice focuses on the elements that contribute to faster growth in expertise and proficiency,
- practice is driven towards the improvement of a specific area of weakness or learning a particular skill or process, and
- practice profits from the expertise of an effective coach/teacher.
Practice with immediate feedback from an expert/coach or the taste of a successful outcome makes that practice a means to mastery. Practice without the analysis of performance and only the process of practice does not lead to significant progress. Ordinary extra practice will not take students beyond their plateau.
Effective practice results in higher achievement in mathematics. For example, practicing addition and subtraction facts by sequential counting is not going to result in mastery—understanding, fluency, and applicability. On the other hand, practice using decomposition/recomposition strategies will result in mastery of facts, not just whole numbers, but also integers.
A focus on process means that you are not just trying to output work, but you are trying to improve strategies. This is the difference between basketball players practicing their skills through playing games and doing drills. The focus of a game is on winning; the focus of a drill is on process. It is this latter focus, which is crucial in sustained improvement. Practicing a skill alone (drill) and solving problems (game) are the mathematics counterparts.
Finally, practice should be intense, beginning with comfortable to moderately uncomfortable to uncomfortable, and be focused on particular skills. Children should not just practice the skills they are good at. The teacher or the student should identify the trouble spots in the process and drill them intensely until the problems are resolved and the weaker points are remedied.
Mathematics For All 