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.