ILLINOIS SCHOOL BOARD JOURNAL
Figuring out a way to make math ‘cool’
by Linda Dawson
Linda Dawson is IASB director/ editorial services and editor of The Illinois School Board Journal.
Don Porzio has a collection of Games magazines that he’s collected for 18 years. He keeps them as a ready resource when he needs a new math challenge. As an instructor at the Illinois Mathematics and Science Academy (IMSA) in Aurora, he likes to use the puzzles with his students … as well as with their parents.
“On preview day, we have thousands of people on campus,” Porzio said. “Every day I start them off with a puzzle.”
One of his favorites is this:
Move one and only one digit in the following equation to make it correct :
62 - 63 = 1
(I’ll give you time to ponder the question and reveal the answer at the end of the article — no fair peeking. In the meantime, let’s move on …)
In a recent independent study commissioned by Raytheon Company, a leading U.S. Aerospace and defense contractor, 70 percent of middle school students said they liked math. In fact, they like it enough that 15 percent ranked it as their third favorite subject in school, right behind physical education (18 percent) and art (16 percent).
The students surveyed also found that 58 percent acknowledge that math will be important to them in their future, especially in the role it will play for careers in technology.
So, why is it that math achievement posts a noticeable drop off between fourth and eighth grades? The National Assessment of Educational Progress (NAEP) rates students tested as “basic,” “proficient” and “advanced.” In 2009, according to the U.S. Department of Education, 20 percent of fourth graders scored below the “basic” math level, but 27 percent of eighth graders scored below “basic.”
According to Illinois Standard Achievement Test results from 2011, statewide percentages were not as dire: just 11 percent of fourth-graders failed to score in the “meets” or “exceeds” categories in math, while 13 percent scored below “meets” or “exceeds” in eighth-grade math.
The failure of some students to increase math achievement and to grasp the importance that math may play in careers other than technology can be disheartening for instructors like Porzio, even though he sees the best of the best in his classes.
Many everyday jobs require math skills, from running a cash register and counting out change to tapping numbers into a calculator to come up with an estimate of how much new windows will cost for an entire house. Yes, these jobs allow the use of technology, but somewhere along the line, that person had to learn the functions behind the technology in order to be able to make sense of the numbers.
IMSA, where Porzio has taught for the past 14 years, attracts top-notch students who want to be challenged, especially in their math and science classes. The residential-style high school features a well-rounded, typical high school curriculum but the emphasis is on rigorous study and high student achievement. Almost all IMSA graduates go on to college, many to Ivy League and selective universities.
Even though there is no minimum grade point average or SAT score (needed for admission), the GPA average for incoming students (10th-graders) was 3.9/4.0 for the class of 2013. Their SAT scores averaged 607 on critical reading and 679 on math, out of a possible 800 points for each category.
Of the 600 to 950 students who apply each year as eighth- or ninth-graders, only 230 to 250 are accepted as entering 10th-grade students for the following year. And even then, maybe 40 or 50 students a year require a “bridge program” to address academic deficiencies in their background, according to Glenn W. “Max” McGee, IMSA president.
“We do well because the kids who come here had excellent teachers at the local level,” McGee said. “We’d like all teachers to be that good.”
Unfortunately, Algebra I looks different in different parts of the state, and the newly adopted Common Core State Standards (CCSS) won’t change that, he said.
McGee, who served as Illinois state superintendent of schools from January 1999 to December 2001, came to IMSA five years ago. He also serves on the group developing assessments for the CCSS that will be used for school year 2014-15 testing in Illinois.
Their biggest challenge is coming up with assessments that will be easy to score but at the same time require students “to produce rather than circle and solve.”
“There are lots of wrong ways to get a right answer,” Porzio said. “The answer is not enough. Show me your thinking.
“With assessments, we’re not talking about just designing good questions. The difficulty comes in finding a way to look at extended response questions to score with a machine,” he added.
Porzio knows all the mistakes that students can make trying to get to a right answer. He’s also aware that with multiple choice questions, whether in class or on standardized tests, you have to have a legitimate distractor for students who might just be guessing at the answers.
Leading up to CCSS
From his tenure in math education, Porzio can recall a number of different periods in math education now leading up to the switch to Common Core standards.
“The ‘new math’ in the 1960s was overrun by college and academia and was very theoretical,” he said. In the late 1980s, the first standards were developed by the National Council of Teachers of Mathematics (NCTM), “and that was when we had ‘fuzzy math,’ where it seemed like the answers didn’t matter.”
In A Winning Formula for Mathematics Instruction: Converting Research into Results, Judith Jacobs describes the differences between “traditional” and “standards-based” math. After a career in mathematics education professional development, Jacobs retired in 2008 from a position as math professor at California State Polytechnic University in Pomona.
She states that traditional math instruction has three components:
• Checking homework
• A teaching segment
• Practice by students on solving textbook problems
The first, she says, “can take half the time allotted” for the lesson. The second, while constituting the direct teaching portion of the class time, usually places the emphasis “on the teacher’s demonstration of how to solve problems of a particular type.” The third element of practice is self-explanatory, and then there is the homework that will be turned in the following day and the cycle begins again.
Other researchers, including Brian Rowan, Delena M. Harrison and Andrew Hayes in their 2004 article “Using instructional logs to study mathematics teaching in the early grades,” call this the “explain, practice, memorize” approach. Jacobs quoted the National Research Council as calling it “mindless mimicry mathematics” in 1989.
Could this be why some students lose their “love” for math after the elementary grades? These students see little connection to their everyday lives, even though many of those early story problems involve everyday items, like apples and oranges and how far it is to the store.
Standards-based math programs, however, “start from a clearly defined set of mathematics objectives usually organized by grade level …,” Jacobs states. Her list of standards-based program characteristics, drawn from research, include:
• Use good questioning to prod learning without telling students how to do the problems or providing answers;
• Have both teacher demonstration and guided opportunities for practice;
• Use challenging, interesting and complex questions and tasks that draw on prior knowledge;
• Promote student reflection on their mathematical experiences to explain their reasoning; and
• Create a positive classroom environment toward mathematics and students’ ability to do mathematics.
In a traditional math classroom, content knowledge moves through the teacher to the student and there is little other interaction. It’s often described as the teacher being “the sage on the stage.” With standards-based math, content flow has all three entities interacting: students with the teacher, the teacher with the content, and the students with the content and each other.
Standards, although a good move forward, cannot be the sole answer, according to Porzio, because if they are implemented poorly, the system will still fall apart. And that’s where he saw things in the early 1990s.
The first NCTM standards from the late 1980s were distributed, but there was no training with teachers on how to use them and teach to them. “They just said, ‘here, teach to these,’” he said.
The same thing happened when calculators, smart boards and now laptops and iPads started being more common in classrooms. Without proper training, many teachers are not “modeling” proper use of technology for students, Porzio said.
“School boards need to recognize the appropriate use of technology,” he added, and they need to provide money for professional development in its use. “If teachers don’t understand how to help students understand when technology is appropriate, you get into trouble.”
Pencils vs. calculators
In Porzio’s classes, students are more likely to use a pencil and paper or write on a whiteboard … not a Promethean board but a plain whiteboard … to show him their answers.
At the beginning of each year, he tells his students that they may not ever need a particular math skill that he is teaching them, but they will need to learn how to think in a particular way.
“My job is to make them better critical thinkers and problem solvers,” he said, “and I just happen to be using math to do it. They may not remember a particular formula, but what they did to solve the problem may help them think.”
As students file in for one of Porzio’s math classes at IMSA, they look like students at any other high school. They tote heavy backpacks. Some have pants that they’re tugging up. Some are in hoodies.
They take their seats at various tables around the room. Most of IMSA’s classrooms are arranged with tables that seat four or five, not with rows of desks.
Porzio asks if his students have “gifts” for him, and they all dutifully hand in their homework from the day before. Once the preliminaries are over for the 75-minute class period, students number off, then gather into new groups and write a solution to one of last night’s problems on the board. One member from each group explains how the answer was reached.
Porzio said before the class began that these students had been working with probabilities and how to predict election results. That day, students were working with Pascal’s triangle, talking about Pascal’s “hockey sticks” and working with binomial coefficients. Each problem, while illustrative of the concept, was slightly different and Porzio kept prompting the students for more information until everyone seemed to grasp the lesson and could make suggestions to the group.
IMSA president McGee said students talk a lot about working together in teams, and this illustrates how math needs to be a collective effort.
“There isn’t as much interaction when students are using iPads and laptops as when they are working in groups with paper and pencil,” McGee said.
As the IMSA students worked in groups, no heads were down on the desk. No one was staring into space. Everyone’s attention was on the board and whoever was talking at the time. These students were engaged and enjoying complicated math theories.
Where’s the dis-connect?
So where and how do some students lose interest in math? While acknowledging the many great math teachers in the state, Porzio partially attributes it to the way teachers are certified in Illinois.
Developing a mind for math actually begins in elementary school and it’s about a way of thinking rather than just the problems themselves, he said.
However, “at institutions that train and certify teachers, math is required for all elementary education majors,” Porzio explained. “Someone who hated math doesn’t learn much more in college than they need to pass.”
That teacher, especially in the elementary classroom, does just what’s necessary to teach math and probably not with the passion that a certified math instructor would have. The problem can continue in high school, McGee said, because “about one-fourth of teachers in high school math aren’t certified to do so.”
“Now we have a new group of kids who hate math,” Porzio said. “It’s a vicious cycle. Why can’t we have math specialists like we have reading specialists?”
In addition, teachers in Illinois are certified to teach grades K-8, but algebra is being pushed down into middle school. “We’ll have kids tell us that when they got to certain chapters, their teacher skipped those,” he said. So even some of the best and brightest in math haven’t always received all the lessons they might need.
As early as 1973, Morris Kline, in Why Johnny Can’t Add: The failure of the new math, criticized the way algebra was being taught. Kline, who was a math professor, said students were merely repeating a “bewildering variety of processes” by rote in order to master them. “They are like pages torn from a hundred different books, no one of which conveys the life, meaning and spirit of mathematics,” he wrote.
William McCallum, head of the math department at the University of Arizona and an author of the Common Core State Standards on math, believes it’s time to convey “the life, meaning and spirit” of algebra to students. In a 2008 speech he titled “Restoring and Balancing,” McCallum said that “traditional curriculum often left students unable to answer the simplest conceptual questions about functions” in algebra.
He advocates introducing the concept of a function — the central, unifying idea — rather than rote memorization. This approach, he said, motivates students by using realistic contexts and by helping them grasp abstract ideas and make them real.
The reality of math is what seems to be the point of disconnect for some students. Teachers stumped for lessons that are more reality based can go to various websites to find lessons regarding specific math concepts.
That might be fine, but there’s no way to know if those lessons are good or not, Porzio said. That’s why, as current president of the Illinois Council of Teachers of Mathematics, he and others are working to develop their website as a “members-only go-to” place for teachers.
A drop-off in school funding has meant fewer dollars dedicated for teachers to belong to professional associations such as ICTM, he said. Membership in Illinois, which had been about 3,000, has dropped by two-thirds over the past few years.
“We have to do things to make organizations like ICTM worthwhile,” Porzio said. “We need to say what the research base is with metrics and evidence to show that it works with students.”
Sometimes a little nudge from popular culture can help as well.
On the CBS drama, “Numb3rs,” Charlie Eppes was a mathematical genius who helped his brother Don, an FBI agent, solve crimes. Watching Charlie at the blackboard using sophisticated equations to show probabilities was fascinating.
If NBC’s “Bill Nye The Science Guy,” and Jamie Hyneman and Adam Savage on the Discovery Channel’s “Mythbusters” are enough to help spur kids to want to pursue science as a career, maybe seeing Charlie Eppes “model” math as fun and exciting will help mend the disconnect for teens and math … and make it seem like the “cool” thing to do.
And that in turn will make CEOs like Eric Spiegel of Siemens Corporation very happy. In an op-ed piece for the Atlanta Journal Constitution last December, Spiegel wrote, “… we need to instill the value of science, math and technology in our kids in their earliest years. It doesn’t matter if they are going to be engineers or not. It doesn’t even matter if they plan to go to college or not. Their future — and ours — depends on their ability to master a skill set they’ll need in the jobs of the future.”
Oh, and the answer to the puzzle is 26 – 63 = 1. Two to the sixth power is 64, minus 63, equals one.
Illinois Council of Teachers of Mathematics website, http://www.ictm.org/
Judith Jacobs, A Winning Formula for Mathematics Instruction: Converting Research into Results, Educational Research Service, Alexandria, Virginia, 2011
Morris Kline, Why Johnny Can’t Add: The failure of the new math, St Martin’s Press, 1973
William McCallum, “Restoring and Balancing,” speech, June 2008
Erik Robelen, “Middle-Schoolers Like Math, But P.E. More Popular, Survey Finds,” Education Week, www.blogs.edweek.org/edweek/curriculum2012/12/03/middle_schoolers_like_math_but.html
Eric Spiegel, “Get over your math-science aversion,” Atlanta Journal Constitution, December 5, 2011
Questions for board members to ask about math instruction
School board members should maintain their “balcony perspective” when it comes to district programs. However, board members also need to ask the proper question of administration to make certain that what is being implemented follows the direction that the board has set for the district.
The following questions from “Linking Research & Practice, The National Council of Teachers of Mathematics (NCTM) Research Agenda Conference Report” are appropriate for board members to ask:
• Do we have a curriculum that prepares our students to be functional in today’s society?
• Do our math assessments improve student learning?
• How has high-stakes testing affected the way our teachers teach math?
• Are our teachers using strategies that are specifically designed to close the achievement gap in math?
• How has technology been integrated into our math curriculum?
And one overriding question always exists: How much money have we budgeted for professional development for our math teachers?
Recognizing mathematics proficiency
Illinois is one of the governing states in a 24-state partnership developing Common Cores State Standards assessments, according to Glenn W. “Max” McGee, who serves on the committee developing elementary math assessments.
The Partnership for Assessment of Readiness for College and Careers (PARCC) has developed an eight-point definition of what it means to be mathematically proficient.
Mathematically proficient students:
1. Start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
2. Bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize (to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents) and the ability to contextualize (to pause as needed during the manipulation process in order to probe into the referents for the symbols involved).
3. Understand and use stated assumptions, definitions and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counter examples.
4. Apply the mathematics they know to solve problems arising in everyday life, society and the workplace. In the early grades, this might be as simple as writing an addition equation to describe a situation.
5. Consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package or dynamic geometry software. They are sufficiently familiar with the tools to make sound decisions about when each might be helpful, recognizing both the insight to be gained and their limitations.
6. Try to communicate precisely to others. They try to use clear definitions, state the meaning of the symbols they choose and calculate accurately and efficiently.
7. Look closely to discern a pattern or structure. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.
8. Notice if calculations are repeated and look both for general methods and for shortcuts. They continually evaluate the reasonableness of their intermediate results.
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