Maintaining High Expectations
Knowing and understanding contemporary mathematics is a key competency for a highly qualified workforce and citizenry (SciMathMN,2007). It is our responsibility as educators to ensure that ALL of our students develop this competency. We can do this by employing an attitude of "warm demand," that is, maintaining high standards for all while supporting students to reach them.
Why is maintaining high expectations important?
There has long been a belief that learning mathematics is a function of a student's aptitude and talent. This belief directly affects expectations of students. If a student has aptitude, s/he can (and should) learn rigorous mathematics; if a student does not, that student can (and should) learn the fundamentals of mathematics but probably little else. Setting aside the question of whether this was ever acceptable or accurate, it definitely is neither today.
Grappling with a problem is half the fun, so [we] should never get discouraged. [My teacher] makes it interesting by saying, 'This is going to be surprising,' or 'This is going to be hard'
Audrey, first grade student interviewed by Koulimpilova, 2011
No matter what the post-high school and career goals are for a student, s/he needs a solid curriculum of mathematics throughout school. Because the study of mathematics is often sequential, with one concept building on another, this requires high expectations for all from the earliest ages. There can be no "throwaway kids," mathematically speaking. Not only must we maintain high expectations for all, we must provide the support needed by each student to meet these expectations.
In a study to determine how the demands of college readiness compare to those for work readiness, ACT, the organization that provides the college-readiness test taken by a majority of Minnesota students, concludes that "high school students need to be educated to a comparable level of readiness in reading and mathematics," whether they plan to enter college or a work training program after graduation. ACT finds that students need four years of rigorous mathematics to achieve readiness for college or a living wage career (ACT, 2006). Currently, 35% of the Minnesota students taking the test are prepared for this next step in all four core areas; 61% have taken four or more years of mathematics (Minnesota Office of Higher Education, 2011).
In The Toolbox Revisited, Clifford Adelman summarizes the results from a longitudinal study, begun in 1988, of students who expected to graduate from high school in 1992, and were followed until 2000. This follows a similar study of the class of 1982. Adelman examined the factors that are strong predictors of college degree completion, and concluded that "the highest level of mathematics reached in high school continues to be a key marker in precollegiate momentum, with the tipping point of momentum toward a bachelor's degree now firmly above Algebra 2." (2006) In fact, he declared that students who expect to complete a college degree need 3.75 or more Carnegie units of mathematics in high school. This generally means courses through precalculus, trigonometry, calculus, statistics or other advanced mathematics. Virtually all careers that provide a living wage for a family require such preparation for high school graduates.
Meanwhile, in schools, the students least likely to be seriously prepared for college are in those groups whose representation in the population is growing - students of color and those who are economically disadvantaged. These students are not well represented in advanced high school mathematics courses and perform below their peers on measures of achievement. These include state tests such as the Minnesota Comprehensive Assessments (MCAs), national tests such as the National Assessment of Educational Progress (NAEP) and the ACT test, and international measures such as Trends in International Mathematics and Science Study (TIMSS) (Minnesota Department of Education, 2011; National Center for Education Statistics, 2010; Minnesota Office of Higher Education, 2011; National Center for Education Statistics, 2007).
The Minnesota Mathematics GRAD Test is an assessment required of all 11th grade students, and measures basic proficiency on mathematics content accumulated in grades K-11. Results from 2011 indicate that while 67% of white students passed the requirement, fewer than half of several other groups met the requirement: 23% of blacks, 29% of Hispanics, 27% of American Indians, and 35% of those eligible for free or reduced price lunch, an indicator of poverty. Among Asian students, 54% met the GRAD requirement (Minnesota Department of Education, 2011).
The voluntary ACT college entrance test is taken by about two-thirds of all Minnesota students. However, the proportion of students whose scores indicate readiness for college is remarkably consistent with the results on the state administered GRAD test. A score of 22 or above is the minimum needed to have a 75% chance of obtaining a grade of C or higher in an entry-level college algebra course. For 2010, 65% of white students scored 22 or above, 21% of black students, 36% of Hispanics, 39% of American Indians, and 48% of Asians. No report is given for students eligible for free or reduced price lunch. This same report indicates that many students have taken advanced coursework, but are not able to produce test results consistent with their prior studies. ACT and the Minnesota Office of Higher Education conclude that "it is the rigor of the high school courses, rather than the number of courses, that best prepares students for life beyond high school" (Minnesota Office of Higher Education, 2011).
Students underrepresented in advanced mathematics may fail to participate for a variety of reasons. Many have felt or been told from the earliest grades that they are not capable of learning mathematics.
They might attend schools that do not offer advanced courses. They may lack sufficient academic preparation. They may not be encouraged to enroll, or even be discouraged from enrolling. They may be disenchanted with mathematics classes, or feel they would not be successful.
Teachers at all levels have the responsibility to maintain high expectations, so that all students will be confident and prepared to participate and achieve in advanced mathematics courses when they reach high school. As educators, we must own the problem that we are not currently preparing all students well for their futures, and work as hard as we want our students to work, providing classrooms at all grades in which there are both high demands and strong support to meet those demands for all students.
What are high expectations?
She really likes teaching so we can get smarter.
Jenna, first grade student interviewed by Koumpilova, 2011
Students in a demanding environment with high expectations are challenged, work hard, think hard, reflect often, recognize their own success, and are confident that their teachers will help them achieve their full potential.
Teachers in such a classroom expect a lot of their students, but also communicate to each student that s/he can meet the demands. These teachers understand the goals of each lesson, and carefully craft lessons to help reach the goal. The environment in these classrooms is what many researchers call one of "warm demand," where both the high demands and the positive relationships are apparent. Growth and independence is nurtured, higher order thinking is evident, and an atmosphere of friendly empathy surrounds students, who recognize and accept their responsibility to work hard.
Students in a high expectations environment have clear short-term and long-term goals, fostered by their teacher and others. They understand what is expected of them in the current lesson, know the goals of the ongoing unit, have a vision for the end of the school year, see how learning connects during a year and across years, and are realistically confident that they are on track for career or college after high school.
There are multiple well-known existence proofs that classrooms like this can happen, and probably thousands of lesser-known examples. What such classrooms all have in common is a combination of high demands and positive relationships, with students knowing that at all times the teacher will provide scaffolding as needed to help them achieve high goals. Famous examples include Jamie Escalante and his calculus students from East Central Los Angeles (Matthews, 1988), Geoffrey Canada with Harlem children (Tough, 2008), and Marva Collins' elementary students in Chicago. Collins said to her students "[T]oday will decide whether you succeed or fail tomorrow. I promise you, I won't let you fail." (Collins and Tamarkin, 1982, p.26).
Closer to home, reporters often ask what it takes to raise test scores for Minnesota students in an era of accountability. One reporter spent time in the District 197 classroom of an Eagan first grade teacher recently. The teacher, Ikhlas Abdelkhalig, a finalist for Minnesota Teacher of the Year, says her secrets are "high energy, high expectations, and a lot of help from parents." She has been recognized with a 2011 Teacher Achievement Award from the WEM Foundation for "routinely squeezing two years of academic growth into a single school year" (Koumpilova, 2011).
Changing the Odds for Student Success (Goodwin, 2010), summarizes what matters most in creating high performance classrooms. The report concludes that the factors that evidence indicates help all students succeed are the following: guaranteeing challenging, engaging and intentional instruction, delivered with strong student support, in a system that offers curricular pathways to success for all, and sustains teachers and students to rise to their optimal performance.
Challenging and Engaging Curriculum. There are many specific and accessible things classroom teachers can do to move toward such an environment where high expectations are maintained more systematically. An integral part of creating a classroom where students work hard to learn challenging lessons must be to convince each student that s/he is capable. For a long time, many people - students, parents, the public - have believed that "either you have it or you don't," especially in mathematics. And the "it" refers to innate aptitude or talent. But there is more and more compelling evidence that our brains are very malleable and able to grow, thus increasing our aptitude for learning, even for learning mathematics.
When you learn new things, [the] tiny connections in the brain actually multiply and get stronger. The more that you challenge your mind to learn, the more your brain cells grow. Then, things that you once found very hard or even impossible - like speaking a foreign language or doing algebra - seem to become easy. The result is a stronger, smarter brain.
From the Brainology Workshop, quoted in Dweck, p. 213
Professor Carol Dweck, from Stanford University, in her book Mindset, describes the difference between individuals with a "fixed" mindset, and those with a "growth" mindset. Fixed mindset people believe success or failure is determined by innate ability, while those with a growth mindset believe that, with effort, they can learn and succeed (2006). Dweck's research shows that when students are instructed on the physiology of the brain, and understand that it is possible to "grow the brain," they approach learning with a new fervor. She has developed a short web-based course called Brainology to help students understand their brains, and finds that this gives students a new determination and a positive outlook on their own learning.
You have to apply yourself each day to becoming a little better. By applying yourself to the task of becoming a little better each and every day over a period of time, you will become a lot better.
UCLA Coach John Wooden, in Dweck, p.200
Teachers can surely sympathize with students, since they exist in an environment with high expectations for producing high test scores, high value-added measures, good behavior, and all the other demands of systems and communities. Yet they often feel powerless, with little support, and minimal chance of success. They realize that without support they cannot succeed. Just as teachers can adopt a growth mindset and apply this to their own professional situation, they can help their students become more growth minded in their schoolwork.
When teachers are judging them, students will sabotage the teacher by not trying. But when students understand that school is for them - a way for them to grow their minds - they do not insist on sabotaging themselves.
Dweck, p. 194
When teachers instruct their students on the brain's workings, students can come to understand that they have control over their progress, and that they can create a "smarter brain" for themselves. Growth-minded students view failure as an opportunity to learn and grow. Failure is seen not as an identity, but as an event giving feedback, even though sometimes this is painful. This leads to an environment of cooperation, since everyone is able to succeed, and one student's success does not come at the expense of another's. Now, many students, especially older ones, use apathy as a screen to avoid failure, having experienced it too many times in the past. But when they believe that they are not "dumb," even when they lack prior knowledge or skills, they can begin to move forward.
Another result of Dweck's work concerns the use of praise. She has found that when praise focuses on ability, saying things like "You are so smart!", students work to maintain their "smartness" by avoiding risk and potential failure, which might expose their own flaws. This is again an instance of the "fixed" mindset. However, if the praise emphasizes hard work and effort rather than talent or innate brainpower, students are not afraid to accept new challenges, and in fact embrace them. Failure simply means that one must work harder. They typify the "growth" mindset, continuing to improve and grow (Dweck, 2006).
Another resource for teachers regarding students' self-identification is the work of Elizabeth Cohen. Her ideas for helping students develop confidence in their own efforts involves designing group activities that will demonstrate the ways in which each student is smart, and increase the status of all students, particularly those whose initial standing in the class is low (Cohen, 1994).
Challenging mathematics does not mean doing column addition, long division, or fraction computation with increasingly larger numbers. It means helping all students think every day, leaving a "residue" of new mathematical learning. This implies that the teacher needs to know where students are in their mathematical development. Carpenter, Fennema, Franke, Levi, and Empson in their Cognitively Guided Instruction (CGI) project structure elementary mathematics teaching around this concept of understanding where each student is developmentally and where to lead them next (Carpenter et al, 1999). This idea is demanding, both for elementary teachers, who have multiple subjects to consider, and for secondary teachers, who generally interact with 100 or more students during any given term.
Lev Vygotsky, the Russian psychologist, introduced the term "zone of proximal development," and defined it as
the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers (Vygotsky, 1978, p. 86).
In other words, teachers understand what students have grasped securely, and where they are ready to go next, starting out with teacher and peer support. Understanding such a zone can help teachers plan work for each student over time. Defining short-term and long-term goals for each student, with a realistic recognition of current status, helps make the challenge manageable.
Coming up with good questions before the lesson helps me keep a high level task at a high level, instead of pushing kids toward a particular solution path and giving them an opportunity to practice procedures. When kids call me over and say they don't know how to do something (which they often do), it helps if I have a ready-made response that gives them structure to keep working on the problem without doing it for them. This way all kids have a point of entry to the problem.
Teacher, quoted in Smith at al, 2008, p. 137)
All teachers want to help their students. This may tempt them to reduce the cognitive demand of tasks in the classroom. This is a particular difficulty for U. S. teachers. In the 1999 TIMSS video study, researchers studied videotapes of eighth grade classrooms in several countries, including the U. S. One of the aspects of classroom practice involved examining the types of problems students were asked to solve, and how they were approached in the classroom. Some problems were classified as making connections problems, which were defined as those that required students to "construct relationships among mathematical ideas, facts or procedures,", or engage in special reasoning such as "conjecturing, generalizing, and verifying." An example would be "Graph the equations y = 2x + 3, 2y = x - 2, and y = -4x, and examine the role played by the numbers in determining the position and slope of the associated lines." When these making connections problems were done in U. S. classes, fewer than 1% were solved by making connections. Instead, 59% were solved using procedures, with the rest solved by stating the result or a concept. Four of the high performing countries in the TIMSS video study, Hong Kong, Czechoslovakia, Japan, and the Netherlands, showed that between 37% and 52% of these making connections problems were actually solved in class by making connections. In these four countries, 20% or fewer of such problems were solved by using procedures. In U. S. classrooms, the teacher, in a misguided attempt to "help" students, frequently takes the mathematical thinking away from students (Hiebert et al, p.103).
An apt metaphor for this type of situation is the act of a teacher taking the pencil out of the hand of the student so that the teacher can do the problem. In one videotape, the teacher is heard saying "Here, let's show you," as she takes the pencil from the hand of a student who has asked for help. However, since the teacher already knows how to solve the problem, the result of such practices is to reduce the intellectual demand of these problems for students, in effect "dumbing them down" to problems that can be solved with less thinking, and hence much less learning.
Intentional Instruction. The intentional teacher continually examines the mathematics goal, clearly elucidating what s/he wants students to learn, and then thoughtfully selecting an appropriate instructional strategy to achieve that purpose. If a quiz is given, there must be a reason. How will it be used? Why? Who will use it? What will the next steps be - for each student? for the teacher? If students are asked to complete a task in pairs, why is that the best means to the goal? How will students be accountable for their work?
A lesson plan protocol that is rich with intentionality is Thinking Through a Lesson Protocol, TTLP, developed by Margaret Smith of the University of Pittsburg and colleagues (2008). The protocol is time-intensive and not practical for using every day. However, it models the kind of thought process that is recommended for intentionally planning lessons. When used with colleagues, and refined over time, it can contribute to a growing collection of successfully implemented lessons.
From the first day, you look students in the eye, and you make them believe they can do anything.
Ikhlas Abdelkhalig, teacher interviewed by Koumpilova, 2011
Student Support. The most obvious characteristic of classrooms with "warm demand" is the atmosphere of support and encouragement, in which every student believes that the teacher is there to help him/her succeed.
Support requires that teachers know the zone of proximal development for each student. This also relates to the idea of providing "scaffolding" for students, that is, building a frame for students in the early stages of learning something new, and then gradually removing it as they are able to stand alone, since the goal is ultimately to develop independent thinkers.
Growth-minded teachers tell students the truth and then give them the tools to close the gap.
Dweck, p. 193
There are multiple structured programs and practices to support students in a classroom of "warm demand." The idea of setting goals using the acronym SMART, as in SMART goals, has been used for some time in many fields. O'Neill and Conzemius have applied the idea to setting professional goals in education, with the following meanings (O'Neill and Conzemius, 2006):
Though O'Neill and Conzemius emphasize using SMART goals for teacher planning and collaboration, the same concept can be used with students, to engage them and help focus their efforts to learn new concepts and to fill in prior learning gaps. Sometimes reaching a goal seems impossible to students. Working with students on such SMART goals can give them a sense of being in control of their learning and future.
Advancement Via Individual Determination (AVID) is another type of plan that offers students support for meeting high expectations. It is a College Prep program in which middle school students prepare explicitly and consciously for higher education. There is ample evidence that such programs succeed. AVID expects all participants to enroll in advanced coursework throughout middle school and high school, while providing tutors to support them, usually college students who can also serve as role models. Students also have an enrichment class which ranges from study skills to homework help to college visits (AVID, 2011).
In Outliers, Malcolm Gladwell makes the case that to become expert in any field, a person needs to accumulate 10,000 hours of practice (Gladwell, 2008). Helping students realize that successful and creative people like Gauss or Mozart or the Beatles or Bill Gates or Lindsay Whalen all needed to put in this much work time might help students see their own hard work in a more favorable and productive light.
Curricular Pathways to Success. It is imperative that each student has the opportunity to learn the mathematics required to prepare for college and career. We have seen that both Adelman, in his longitudinal study of the class of 1992, and the ACT, in its analysis of Minnesota students' course taking and college/career readiness, found that for all practical purposes, college and career readiness means four years of rigorous high school mathematics, through the level of pre-calculus, statistics, trigonometry, or other advanced courses. The optimal situation for high school students is one in which all students have the opportunity to learn not only geometry, advanced algebra, trigonometry, and pre-calculus, either in sequential or integrated programs, but also have access to Advanced Placement (AP) Statistics and/or Calculus classes. Other similarly rigorous alternatives are the International Baccalaureate (IB) program of advanced classes and the College in the School program. In all three of these, students can earn college credit for courses taken during high school, and the teacher takes on the supporting role of a coach, helping students to learn thoroughly and do well on exams to gain college credit.
Having all students prepared for this rigorous high school program implies that at every earlier grade, students must have the opportunity to learn the content of the state standards, and that at the high school level there are choices for individuals beyond the fundamentals of geometry and advanced algebra. This is a challenge, especially in small high schools. There may be opportunities at nearby two-year colleges, or for online or distance courses with multiple schools collaborating. For students who have fallen behind, immediate intervention is crucial.
We must be impatient enough to take action and patient enough to sustain our efforts until we see results..
NCTM, 2007, p. 189
Since teachers are the leaders in the classroom, the chief responsibility for maintaining high expectations falls to them. With focused and strategic efforts and planning, they can turn students away from the "fixed" mindset toward a "growth" mindset, creating a learning community where each student is challenged and engaged, instruction is intentionally planned and executed, students are supported, and all students are able to participate in on-grade mathematics, ultimately yielding high school graduates that are well prepared mathematically for their futures.
TALK: Reflection & Discussion
- Organize a study group to read and discuss the book Mindset by Carol Dweck.
- With colleagues or a study group, watch the movie Stand and Deliver, based on the story of Jaime Escalante and his LA calculus students, or the Marva Collins Story, or Waiting for Superman which includes the work of Geoffrey Canada. How are high expectations and warm demand demonstrated in the film? What was the effect on students? Were there instances of low expectations? What was the effect on students?
- Conference with five students who are behind or discouraged learners to develop SMART goals. This will provide them with an attainable place to start and the opportunity to experience success when they meet interim goals.
- With a teaching partner, select a lesson at least once each week and exchange the answers to the following for your lesson: "What do I want students to learn, and why have I selected this particular instructional strategy?" Give each other feedback on the match between the goal and the activity.
- In your teaching journal, record your insights on the zone of proximal development for 10% of your students. Continue weekly until you have done it for all students.
DO: Action Steps
- Videotape yourself teaching a lesson, then view it from the perspective of a student who you know is struggling in that class. How do you maintain the challenge in the lesson but also provide scaffolding and encouragement?
- Use research on mindsets to explicitly educate students on the potential our brains have for continual growth and development. Record their reactions and any other results you notice.
- Enroll your school in Dweck's Brainology workshops to help students understand how they can become "smarter."
- Determine one small step that you can take tomorrow to move your students closer to becoming a high-performing class.
- Explore enrolling your middle school in a program such as AVID.
- Investigate the curricular pathways available to students in your school or district. If there are not sufficient offerings for students to pursue their individual interests, advocate on their behalf for more pathways.
- Ask your students to approximate the lifetime number of hours spent to date on studying mathematics, and how many hours per week they would need to study in order to reach 10,000 hours by age 25.
References & Resources
ACT. (2006). Ready for college and ready for work: Same or different? Iowa City, IA: ACT.
Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through college. Washington, DC: U. S. Department of Education.
AVID. (2011). AVID. Retrieved June 10, 2011, from AVID: www.avid.org
Carol S. Dweck, P. (2006). Mindset: The new psychology of success. New York, NY: Random House.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B.. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Cohen, E. G. (1994). Designing groupwork: Strategies for the heterogeneous classroom. New York, NY: Teacher's College Press.
Collins, M. & Tamarkin, C. (1982). Marva Collins' way. New York, NY: Tarcher/Putnam.
Gladwell, M. (2008). Outliers: The story of success. New York, NY: Little Brown.
Goodwin, B. (2010). Changing the odds for student success: What matters most. Denver, CO: Mid-continent Research for Education and Learning (McREL).
Haberman, M. (1991). The pedagogy of poverty versus good teaching. Phi Delta Kappan, 290-294.
Hiebert, J. G., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J.,...Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: National Center for Educational Statistics.
Koumpilova, M. (2011, May 30). "Eagan first-grade teacher's devotion to her students raises test scores". Saint Paul Pioneer Press, p. 5A.
Matthews, J. (1988). Escalante: The best teacher in America. New York, NY: Henry Holt.
Minnesota Department of Education. (2011). GRAD test mathematics. Retrieved June 10, 2011, from GRAD test Mathematics: education.state.mn.us/MDE/Accountability.
Minnesota Office of Higher Education. (2011). ACT scores. Saint Paul, MN: Minnesota Office of Higher Education.
National Council of Teachers of Mathematics (NCTM). (2007). Mathematics Teaching Today. Reston, VA: NCTM.
National Center for Education Statistics. (2010). Nation's report card mathematics. Retrieved June 10, 2011, from NAEP: nces.ed.gov/nationsreportcard/.
Nationa. l Center for Education Statistics (2007). Trends in international mathematics and science study. etrieved June 10, 2011, from TIMSS: nces.ed.gov/timss/.
O'Neill, J., & Conzemius, A.,(2006) The power of smart goals: Using goals to improve student learning. Bloomington, IN: Solution Tree.
SciMathMN. (2007). Developing world class students through world class mathematics standards: Do Minnesota's standards, students, and teaching measure up? Saint Paul, MN: SciMathMN.
Smith, M. S., Bill, V., & Hughes, E. K. (2008, October). Thinking through a lesson: Successfully implementing high level tasks. Mathematics Teaching in the Middle School , pp. 132-138.
Tough, P. (2008). Whatever it takes: Geoffrey Canada's quest to change Harlem and America. Boston, MA: Houghton Mifflin.
Vygotsky, L. S. (1978). Mind and society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.