Students (and Teachers) Do Not Understand Math

Three weeks ago educators wondered if conclusions from the Response to Intervention (RTI) research might be applicable to mathematics.

Educators gathered here last week to discuss a recent federal “practice guide” on response to intervention for students struggling in mathematics agreed that applying the RTI approach to that subject is challenging. But they also suggested that doing so was worth the effort.

It is instructive to discover the reasons applying the RTI approach to math is challenging.

Response to Intervention involves six steps: Screen, Teach, Intervene, Probe, Chart and Adjust.

Screen
Valid screening measures predict who is, and who is not, at risk for future reading difficulty. These measures are administered to determine if a child is at risk for failing a state's "high stakes" end of year achievement test, by which the state measures a school's overall performance. Children considered to be "at risk" are expected to experience difficulty responding (not keeping up) in the core curriculum as traditionally delivered in the regular general education classroom. Note: Due to the desire to capture all children who are truly "at risk," the false positive rate of early screening may be as high as 50 percent. In other words, as many as half of all the children who are identified as "at risk" by early screening may not be truly "at risk."
Teach
Core curriculum in the regular general education class should be research-based and field tested. This means, based on evidence from converging research, that the core curriculum contains all the elements found necessary to effectively teach reading and has a known track record of success. Such curriculum is to be delivered by "highly qualified" teachers sufficiently trained to deliver the selected instruction as intended, i.e., with fidelity to design. My note: Notice the language says, “trained to deliver the selected instruction as intended.” I knew one highly competent first-grade teacher in California that refused to deliver the selected instruction (whole language) as intended. It was a good thing because her students ALL learned to read even as California fell to 49th place in reading during the whole language period. The fad lasted until 1995 when phonics was reinstated in the curriculum. The change in the role of the teacher as indicated by the language I have noted is problematic.
Intervene
Provide "at risk" children with enhanced opportunities to learn, possibly including, but not limited to, additional time exposed to the core curriculum in small groups (3-6 students), other supplementary instruction, or special education.
Probe (progress monitoring)
Progress monitoring tests are brief measures of specific reading skills that are administered to determine if the child receiving intervention is responding as intended. They are given frequently, at least once every two weeks.
Chart
Progress is regularly charted to provide a visual record of actual rate of gain in specific reading skills in relation to a specified goal. The goal of intervention is for the child to improve relative standing and perform at or closer to grade level standards and is individualized according to the unique needs of the child.
Adjust
Depending on whether the child is achieving a rate of progress determined by his or her individualized goal, the manner and intensity of intervention will be adjusted. The cycle of progress-monitoring and adjustment of intervention will continue, even if a determination for special education eligibility is made.

Math educators met June 10, 2009 to explore whether the same six steps would be just as effective in math as in reading.

Educators at last week’s event said that fitting math into an RTI framework is hard, but that they believe it is now vital to improving math performance for struggling students.

One of the main difficulties is finding suitable math education materials.

Judith Russ, the mathematics curriculum supervisor for the 134,000-student Prince George’s County district in Maryland, said for her part that finding the right materials is hard.
The instructional materials “are not looking at building conceptual understanding. That’s one of the challenges we have,” she said.

The first step is to screen.

Karen D. Cheser, the assistant superintendent for learning support services for the 20,000-student Boone County district in Florence, Ky., said her school system started using RTI in reading two years ago, and had initially planned to leave math for later. But indications that students were becoming weaker as they reached higher-level math classes, among other factors, pushed the district to act.

Ms. Cheser said the district created its own universal screening program, which allows teachers to dig into what was going wrong for many students. It turned out that many students needed to focus so hard on computation that they were unable to grasp more sophisticated concepts, she said.

The panel found that children's gaps in mathematical understanding were fairly predictable.

... remediation for students in grades K-5 should focus on the properties of whole numbers, like counting, addition, and subtraction. Older students, up to 8th grade, should learn rational numbers in depth, including the meanings of ratios, decimals, and percentages, the panel recommends.

Another recommendation is that all students who need extra math assistance should work on fluent retrieval of basic arithmetic facts, like simple addition and multiplication. Higher-level mathematics often assumes that students can quickly recall facts like “3 times 9” or “11 minus 7,” when such operations may be difficult for those lagging behind their peers, the panel found.

The current reliance on the calculator does nothing to promote fluency with math facts. In fact, some teachers say that, with the ubiquitousness of calculators, it is no longer necessary to memorize math facts. But I know that students who cannot readily retrieve math facts struggle with algebra, and the research is confirming my observations.

Mathematics instruction has emphasized procedural competence over conceptual understanding. You do not necessarily have to understand the mathematics underlying long division as long as you can perform the operation. Sadly, students who perform mathematical operations reliably are told (through test grades) that they understand math when the reality is that they may have no idea why they do what they do. They do not “understand” math. The math materials in our schools “are not looking at building conceptual understanding,” as Judith Russ, mathematics curriculum supervisor, noted. But at least in times past, students were expected to memorize math facts in order to complete the procedural operations. So the situation is that students are not understanding math concepts, nor do they have the basic tools, math facts, for mechanically solving problems.

Years ago we used to call screening “diagnosis” and the best teachers have always made diagnosis part of their teaching practice. I used to diagnose struggling algebra students one-by-one. I still diagnose one-by-one, but nearly always I find the same gaps. They do not understand the function of place value (even if they can name a digit's place). They do not understand whole number properties (and fail to apply properties to numbers that do not look like “numbers,” especially numbers containing variables. They do not understand the difference between one and zero (such as when the “cancel” and say something like, “That's zippo.” They certainly do not understand fractions. Once I address these four abysses of knowledge, algebra suddenly becomes straightforward and even beautiful.

Gifted Smifted

TAG (Talented and Gifted, or GAT, Gifted and Talented, or GATE, or just GT, depending on the school) teacher, Tamara Fisher, corrects common misunderstandings in her lengthy and detailed description of what “gifted” is and is not. I have a ton of experience with the gifted issue so I am interested in her take. I can tell you right off the bat that I got sick and tired of the Salem witch trials. Our family moved around a lot and it seemed every GT program was inordinately fond of the Salem witch trials. All told, my own children covered four topics in their years of GT participation: Salem witch trials, bridges, French, and Russian.

GT is NOT a reward for kids who behave well in class and turn in perfect work. Rather it IS an academic necessity for children who learn differently….
GT is NOT a program for kids with exceptional grades...
GT is NOT fun for fun's sake…
GT is NOT extra work to fill extra time…
GT is NOT for kids who are "better" or "more special” than other kids…
GT is NOT about fun and games...
GT is NOT a program only for good kids…
GT is NOT a test of what the kid does know…
GT is NOT NOT NOT NOT NOT a privilege!!!!! Rather, it IS an essential need for children whose pace of learning dramatically out-steps other kids …
GT is NOT a self-esteem booster for children who seem to need one. Rather, it IS a sincere validation of ability(bold added)...
GT is NOT about preparing kids to “save the world” someday..
GT is NOT a “club” to belong to. Rather, it IS a peer group where gifted kids can feel like they actually belong…
GT does NOT address only academic needs. Rather, it ALSO addresses social and emotional needs and validates gifts and talents. (bold added)…
GT is NOT about pressure to fit a label or stereotype…
GT should NOT be an experimental group led by whoever is available….
GT should NOT be an optional offering, if convenient. Rather, it SHOULD BE a high priority because there are kids who need it (bold added)..
GT is NOT an easy A…

I bolded several of the statements because they allude to a very specific issue of academic achievement in our schools. Regardless of all our fine talk about academic achievement, society does not value academic achievement. Students perceive the hypocrisy every day at school. In predominately black schools, classmates often accuse the high achievers of acting white. In some schools, students hide their achievements and engage in theatrical misbehavior in order to protect “street cred.” I have been known to participate in these little conspiracies with some students.

Every state has laws that mandate the education of every child according to the child’s needs. Practically speaking, what really happens is that only children who have been labeled into a specifically funded program get that extra attention. Normally, these special programs are for students who have been deemed “deficient” in some respect.

In my state, where both state law and state accreditation standards mandate that schools identify and provide services for gifted students, only about 40% of schools claim they actually do so. There is no consequence for the schools that do not meet that portion of the accreditation standards. The only consequence falls onto the shoulders of the gifted students who are at the mercy of luck that they will get a teacher who recognizes their learning needs and does something on her own to try to reach them. Educating kids should not be an optional convenience. It should be a high priority. We SAY it is a priority. But when it comes to our nation’s gifted students, are we really educating them if research shows that they already know, on average, about half the year’s material before the school year even begins?

One of my main themes is that US society does not really want a world-class education system. If we did, nothing would stop us. It may be a trivial example, but is there anyone in the US who DOES NOT KNOW that their analog TVs will not work in six months? The point is when we want people to learn something, we make sure they do.

GT is NOT a surplus offering for kids who have surplus knowledge. Rather, it IS an academic intervention...

Schools are in the Business of Creating Low Achievement

Currently education reforms efforts operate like cosmetic concealer. Reformers focus on superficial solutions when the problems are profound. Researchers often fail to identify unexamined assumptions, leading to fatal flaws in the research. A good example is test scores. Test scores are low so we need to raise them. So we mandate and implement all sorts of myopic strategies for raising test scores. Then if test scores go up, the cosmetic concealer has been successfully applied and we can declare the problem solved. These strategies are myopic because they do not begin to address the deeply embedded, underlying features of the system. As long as superficial features take up our time and attention, we hardly notice, much less address, the deep systemic problems.

The public accepts increased test scores as an encouraging sign of improving academic achievement, in part because powerful education stakeholders, via the media, have conditioned us to consider test scores an acceptable proxy for achievement. Whether test scores do indeed serve as a valid proxy is a major unexamined assumption. Once in a while, the public becomes aware of an incongruity in the assumption. But we do not stop to question it closely.

For example, in the case of Head Start, whose goal is to give kids the advantages and opportunities they might not be receiving at home, early gains are lost by the third grade. Why is this happening? What is going on in school that is undermining the progress that the kids have already achieved? Is their achievement leaking out? Was it never there to begin with? Or is there a problem with the tests? The public simply is not demanding answers. On the contrary, society seems resigned to accepting low levels of academic achievement. Why could that be?

Let's ask Judge Joe Brown, an African-American TV judge, on Fox TV. On May 9, 2008, speaking to an African-American defendant, the judge declared, “There are not enough jobs to go around. So society is depending on people like you to get themselves in trouble, get locked up, and take themselves out of the job market.” Judge Joe Brown did not originate this idea; sociologists have been saying the same thing for fifty years. The judge is right. The unemployment rate is about 5 percent right now. Imagine what the rate would be if all people were educated to their full potential and much fewer were warehoused in prisons. Think too of all the people the criminal justice system supports. According to at least one account, criminal justice is one of the biggest growing occupational fields today. People who have accomplished high levels of academic achievement expect to be rewarded with a relatively high level job. Society needs to create people to take what society deems to be low level jobs. A PhD does not want to drive a taxi.

So children are legally compelled to go to school where they are exposed day after day and year after year to being molded to the specifications of society. One of the ways school shapes children is through a combination of low expectations and mislabeling the curriculum. Here are some examples:


1.Critical Thinking

What passes for critical thinking is not critical thinking. A second grade critical thinking workbook (critical thinking workbook? a workbook?) asks, Do you like African elephants or Asian elephants? Curriculum tells students year in and year out that any kind of opinion is critical thinking. In college, students often misinterpret an evaluative essay question to be an opinion question. When they lose points, they complain that everyone is entitled to their opinion and therfore there are no right or wrong answers to opinion questions. If so, there would be no point in ever having essays questions which ask for an opinion.

They do not understand that an opinion may be well defended or poorly defended. They have become used to assignments for which they are asked to give opinions without basis. For example, after a lesson on global warming, they typically might be asked how they would solve the problem. Scientists themselves have not figured that one out. Students are also encouraged to consider responses that are tantamount to “I think because I think” to be excellent critical thinking because they get good grades when they write such nonsense. After a lesson on the features of viruses and the debate among scientists as to whether viruses are living things, students might be asked whether they think viruses are living things. The most obvious approach would be to compare viruses to the characteristics of living things and evaluate whether they measure up or not. A response like, “I do not think viruses are living things because I think only living things can make other living things sick” is worthless.

Critical thinking is not just another subject like math or art. Critical thinking is integrative and interdisciplinary requiring research, analysis, synthesis, evaluation and defense. It is not supposed to be just one more piece of an already fragmented curriculum. By mislabeling poor thinking as “critical thinking,” schools betray that they do not really expect students to exhibit good thinking.

2.Integration

There are lots of good reasons to integrate the curriculum. For one, an integrated curriculum reflects the integrated nature of real life problems and promotes critical thinking. In practice, integrated curriculum is nothing but themed units. For example, the teacher might pick “the ocean” as a theme. In reading class, the story will involve the ocean somehow. Science may involve a topic from oceanography. The story problems for math will add and subtract fishes or shells or whatever. In reality the curriculum is as disjointed as it ever was.

At the next level, if “the ocean” is the organizing concept, then maybe the teacher would show how to determine the age of clam shells, or how Fibonaci numbers work in the spiral of sea shells. At an even higher level, the teacher would choose math, literature, science, social studies, art, and music topics where the ocean is integral to understanding the connections between the topics. In “The Old Man and the Sea,” Hemingway describes the anatomical effects of the old man's battle with the fish. This would be a good time to study the anatomy Hemingway describes. The anatomy illuminates the novel and the novel illustrates the anatomy. It would also be a good time to study not only Cuba, but American expatriates in Cuba. I am not saying that elementary teachers should teach Hemingway, but how about a fifth grade book, “To Kill a Mockingbird.” What an opportunity for a compelling social studies unit. By mislabeling themed units as integration, schools betray that they do not really expect students to synthesize knowledge.

3.Grades

Grades are intended to be a reflection of understanding, but they can be more like a fun house mirror with a distorted reflection. The problem is students do not know they are looking at a distorted reflection. A good grade is taken to indicate solid understanding. Students who get an “A” in math thus "know" they are good at math. However, the “A” may only mean the student has a good memory for recipes. The student is able to remember and execute accurately the mechanical recipe for solving a particular type of problem. A student can go a long time like that, but sooner or later, very often in algebra class, they find out they never really had a profound understanding of math. I cannot tell you how many struggling algebra students have complained that they do not understand why they are having so much trouble with algebra because they have always gotten an “A” in math. I often find they are harboring basic misunderstandings of place value, fractions, and other topics.


I surprise college students on a regular basis by telling them that they will not be able to earn an “A” or “B” by relying on partial credit. Too many of them respond as if I were breaching some implicit unwritten social contract. By mislabeling good grades as evidence of understanding, schools betray that they do not really expect deep levels of understanding because such understanding is not essential to earning a good grade.

Many educational buzzwords mislabel, resulting in a kind of Newspeak that allows society to maintain a structure for perpetuating and reinforcing the status quo. Here are few for the dear reader to think about: empower, diversity, self esteem, mentoring, student-centered, collaboration. I will leave it to you to consider how such words obscure more than they elucidate. Perhaps some of you will be able to add other words to the list.