Keywords for today: Working memory, long-term memory, subroutines, chunking, structured programming, math
When you mention common core, most people who are opposed to it immediately mention the “crazy math” instruction created by the standards. They roll their eyes and say, “I never had to learn math that way and I turned out just fine.” Then they point to the statistic that shows American students aren’t able to do basic math when they graduate high school. Nationally, 25% of our high school graduates must take remedial coursework in math in college. Here in Missouri, that number is just upwards of 30%. From there the typical response is to blame the teachers and call for the elimination of tenure of these terrible teachers who can’t seem to teach math. Those who like the standards often respond with the pablum that kids won’t need to do math in the real world anyway. They can just use their smartphone or a computer. However, cognitive science is on the side of the old schoolers and, interestingly, so it seems is computer science.
The fundamental differences in the approach to teaching mathematics (in elementary and middle school) comes down to; do you focus on teaching processes and creative problem solving or, do you focus on mastery of basic skills and rote memorization. These camps squared off against each other in California in the 90’s and 00’s in what is now known as the Math Wars. Real world experience was not on the side of those in the first camp. California moved towards the 1989 recommendations of the National Council of Teachers of Mathematics which de-emphasized memorization of math facts (among other things covered later in this post). The result was that in four years (by 1996) their fourth graders ranked 42nd out of 44 states in math on the NAEP. Five years after adoption, the percentage of students in their state university system requiring math remediation rose from 23% to 54%. Is there science behind these outcomes? It turns out there is.
A meta-study by Harmtan and Nelson (Automaticity in Computation and Student Success in Introductory Physical Science Courses) looks at cognitive research, the misguided guidelines produced by the National Council of Teachers of Mathematics and subsequent changes to state math standards that all seem to be working against each other to produce children who are so ill-prepared in math that they simply cannot complete a STEM degree.
First brief lesson in cognitive science or, how the brain works. From their paper:
“The cognitive science model for reasoning is based on the interaction between a long-term memory (LTM), where elements of knowledge are organized, and working memory (WM) where elements are processed.”
Long Term Memory: “Procedures (sequenced steps for processing) and facts are stored as small elements of knowledge in LTM.” While LTM has the ability to hold thousands of facts, procedures and associations, those elements are stored slowly over time through focused attention, repeated exposure and retrieval practice.
Working Memory: “The brain thinks, plans, and solves problems in working memory.” WM can recall unlimited well-memorized information from LTM but, can hold only a few small elements (3-5) of novel information (that which is not-well-memorized), for a very brief period of time (~30 seconds).
Automaticity: “The fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory.”
During problem-solving, if the limits in WM space for novel information are reached, the result is a sense of confusion and a likely inability to solve the problem. Therefore,
your ability to solve problems depends nearly entirely on how much knowledge you have “memorized to automaticity” in LTM. Ericsson 1995
The example Hartman/Nelson gives is the calculation 8 x 7. If the answer cannot be recalled automatically from LTM, and instead the child uses a calculator to come up with the answer, the number 56 must be stored in WM so that it can be transferred to where the calculation is being written. That takes one of the limited 3-5 slots our brains have available for novel information in WM.
Three ways around the novel WM constraints are chunking, algorithms, and automaticity. All require thorough memorization. More on this later.
How the “experts” ignored these scientific facts.
The National Council of Teachers of Mathematics (NCTM) is an organization composed primarily of faculty from schools of education and K-12 curriculum specialists and instructors. They are not mathematicians. In a 1989 detailed position paper, known as the “NCTM Standards,” the council recommended the use of a calculator in all grades. In the ensuing years, over 45 states adopted standards for K-12 education modeled on the NCTM standards. By 2005, 30 states actually required students as young as first grade to be taught how to use a calculator. Their standards also called for “decreased attention” to be given to:
- “memorizing rules and algorithms
- finding exact forms of answers
- manipulating symbols
- relying on outside authority (teacher or answer key)
- rote practice
- paper and pencil fraction computation”
NCTM’s position paper got states to move students away from placing math facts into LTM in favor of working out real world problems on their own or in small groups, relying very heavily on WM. It should begin to make sense why our students struggle with more complex math in high school and require remedial coursework in college given the laborious, anti-cognitive science way they have been taught math.
In contrast, the National Mathematics Advisory Panel (NMAP), made up of mathematicians, recommended in their 2008 report
“[During calculations,] to obtain the maximal benefits of automaticity in support of complex problem solving, arithmetic facts and fundamental algorithms should be thoroughly mastered, and indeed, over-learned, rather than merely learned to a moderate degree of proficiency.”
Hartman/Nelson’s analysis of American’s scores on math assessments shows that we are making progress with conceptualized math, but we are tanking on actual computational skills. The reliance on calculators has deprived students of the ability to recognize when the answers provided by the calculators are unreasonable and input error should be considered. A nurse can conceptually understand how important it is to get the right dose of medication for a patient. However, if she cannot do the simple calculation of dose to weight and instead relies solely on a machine programmed to provide the dose per kg instead of per lbs, into which pounds have been entered, she won’t recognize the problem with the machine’s recommended dose.
Calculators to Computers
In 1989 students had reasonable access to calculators. Today they have reasonable access to computers and once again we have “experts” calling for more use of computers in the classroom to be in line with “21st-century skills.” The existence of computers, which can calculate complex math problems often faster than humans, begs the question; do humans even need to be able to solve those problems themselves?
Cognitive experts respond that, even in the age of computers, “automaticity in support of complex problem solving is crucial for students because complex problems have simple problems embedded in them.” (Willingham 2009b) The example above of the nurse provides anecdotal evidence that we still need humans to understand the math.
The interplay between data sets and working memory is true not only for humans but also for the computers that are solving these problems so quickly. They are able to do that because they pull answers from their own stored data sets. They use common subroutines linked together, in a process called structured programming, to perform complex calculations and complex processes.
Structured programming takes advantage of the solutions offered by cognitive science: chunking, algorithms, and automaticity. “Chunks” are elements that have been memorized as a group (think keywords or tags). Algorithms are self-contained step-by-step set of operations to be performed. Automaticity is rapid retrieval from a known data set.
For example, say you want to move a virtual car through a simple maze.
Older coding would create a string of code that says, for example,
Advance 1 space forward
Advance 1 space
Advance 1 space
Advance 1 space
Advance 1 space
In this case, you would have to know exactly what to do in each square in advance and write out all the directions to get the car out of the maze. Make a mistake like telling it to turn left rather than right, or telling it to advance two spaces instead of one and your car (program) gets stuck. This is a relatively small simple maze. Larger mazes would require much longer code. To techies this is known as spaghetti code. It is long and prone to errors which are hard to find.
Structured programming relies on common tasks that are repeated and would look conceptually more like this.
1 Directive 1: Purple square stops advance
2 Sub1: Advance one space
3 Sub2: turn right 90 degrees
4 Sub 3: turn left 90 degrees, turn left 90 degrees
5 Test Directive 1
6 If false, sub1
7 Repeat until Directive 1=true
8 If true, sub 2, test Directive 1
9 If false, sub1
10 If true, sub 3, sub1
11 Resume line 5
Now, no matter how large your maze is, the program is only this long. The program combines known tasks and conditions to make minute but rapid repeated decisions to complete a more complex process. Programmers know that value of small discreet knowledge that is used frequently to speed up processing. Not only does this make programming faster, it also makes it easier to find where the code has gone wrong. This is very similar to how the human brain works and makes the case that children should be embedding those blocks of knowledge in their long-term memory for future use in more complex operations.
Fortunately, the young mind is primed to do this type of LTM storage. By the time they get to school they have been doing this with language for 4-6 years. It is relatively easy to get them to work on math facts by repeated exposure and retrieval practice. This will provide them with the necessary bank of knowledge to pull from LTM to use in more advanced math later. This will speed up their processing and make finding errors in their processing much quicker as well.
The significance of ignoring what cognitive science tells us and eschewing things like rote memory, as NCTM did, is captured in the abstract from Hartman/Nelson’s paper. The affect on a student’s ability to obtain a degree in the much lauded STEM field is impacted by their ability to do computational math.
Between 1984 and 2011, the percentage of US bachelor’s degrees awarded in physics declined by 25%, in chemistry declined by 33%, and overall in physical sciences and engineering fell 40%. Data suggest that these declines are correlated to a K-12 de-emphasis in most states of practicing computation skills in mathematics. Analysis of K-12 “state standards” put into place between 1990 and 2010 find that most states directed teachers to de-emphasize both memorization and student practice in computational problem solving. Available state test score data show a significant decline in student computation skills. In recent international testing, scores for US 16-24 year olds in numeracy finished last among 22 tested nations in the OECD.Recent studies in cognitive science have found that so solve well-structured problems in the sciences, students must first memorize fundamental facts and procedures in mathematics and science until they can be recalled “with automaticity,” then practice applying those skills in a variety of distinctive contexts. Actions are suggested to improve US STEM graduation rates by aligning US math and science curricula with the recommendations of cognitive science.
What we are teaching, or not teaching, today will take almost 20 years to show up in the workforce so these are not just academic debates among elementary and middle school teachers. There is some urgency to get it right.
To review. Reasoning is the interaction between LTM and WM. Without a bank of memorized facts ready for use in WM, our ability to reason is severely impaired. If basic math facts (+-x÷) are not available from LTM, students will struggle with more complex math as WM becomes overloaded. Without more complex math they will not be able to complete a STEM degree. Computer science is a STEM subject. Using computers early on in education reduces banking of information into LTM. Computer scientists rely on smaller more manageable known datapacks to do complex processing, much like the interchange of LTM and WM. Thus computer science makes the case for not using computers in early grades to do computational math.