The post Stanford Study: Brain Scans May Predict Students’ Math Capabilities appeared first on Education News.

]]>Researchers at the Stanford School of Medicine have reported that brain scans may predict which students are likely to improve their math skills in school and which ones won’t, with scans predicting more precisely than IQ or math tests.

NBC News’ Maggie Fox writes that the scientists worked with a group of students who began getting brain scans at the age of 8, and who followed up with scans into their mid-teens.

Even researchers were surprised when they found that certain patterns of brain activity when kids were not doing anything at all could predict how much they would improve in math skills. The accuracy of these predictions was higher than results from IQ tests, reading tests, or math tests, according to the report published in the Journal of Neuroscience.

The researchers were not suggesting that all children should have brain scans, but the information could result in new ways to identify the children who need intensive math coaching, said the authors.

“I can’t take one child, put the child in a brain scanner and say with certainty this is how the child is going to end up,” said Tanya Evans, a psychiatric researcher who worked on the study.

But the study does show which regions of the brain seem most important in developing math skills. Evans adds that the work is similar to studies done on dyslexic children. Dyslexia affects the ability to read printed words.

“A long-term goal of this research is to identify children who might benefit most from targeted math intervention at an early age,” said Vinod Menon, a professor of psychiatry and behavioral sciences who led the research.

The study included 43 students with normal intelligence who were asked to take tests and to sit still for functional magnetic resonance imaging scans and ordinary MRI scans. MRIs map the physical structure of the brain while fMRIs show the function of the brain. Researchers were able to see the parts of the brain that were more active in the children who improved their math skills over the next few years.

“Some of the kids started out really bad and ended up really good,” Evans said. “Some stayed average. Some started out good and got worse.”

This does not necessarily mean that math skills are hard-wired into the brain, says Evans. The idea is to work on ways to improve math skills to see if the brain structures also change.

“Practice makes perfect in everything,” she said. “We are looking at what types of math interventions are most effective.”

The children were studied for six years. The research showed that brain characteristics indicated which kids would be the best in math over the course of the research.

The findings will move scientists closer to their goal of assisting students who struggle with their math skills, writes Stanford University in the publication Bioscience Technology. Vinod Menon, who led the research team, says the work identifies a network of brain areas that provides a scaffold for long-term math skill development in children.

He reports that the next step is to investigate how brain connections change over time in kids who show large versus small improvements in math skills and then to design new interventions to help students improve their short-term learning and long-term skill development. Evans suggests that parents and teachers encourage children to exercise their mental math muscles.

Headlines & Global News’ Aditi Simiai Tiwari reports that the researchers hope to establish a baseline for understanding development that will, in turn, help experts develop and validate programs for the remediation of children with learning disabilities.

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]]>The post Math-Anxious Parents Affect Their Children’s Math Achievement appeared first on Education News.

]]>Parents who have math anxiety may pass that nervousness on to their children, says a new study published in the journal Psychological Science.

Erin A. Maloney of the University of Chicago led researchers in the analysis of math attitudes and abilities of over 400 first- and second-grade students through the use of information from a larger, unrelated study. Melissa Dahl, writing for New York Magazine, reports that the children were tested two times on their math skills, first at the beginning of the school year and again at the end of the year.

They were also asked on the test how nervous math, and everything to do with math, made them feel. Parents took surveys to measure their math anxiety and how much they assisted their children with their math homework throughout the school year. The results were that kids with parents who were anxious about math learned less math during the year, and they were more likely to become anxious about mathematics.

The catch is that the kids became anxious only if their parents helped them with their math homework.

“Notably, when parents reported helping with math homework less often, children’s math achievement and attitudes were not related to parents’ math anxiety,” Maloney and her co-authors write in their paper.

The takeaway is that if parents are nervous about math to the extent that their kids pick up on their anxiety, it is probably a good idea for those parents to step away from the math homework table and find a tutor who loves math to help their children.

Psychologists Sian Beilock and Susan Levine, who were part of the team that developed the paper “Inter-generational Effects of Parents’ Math Anxiety on Children’s Math Achievement and Anxiety,” explained that previous research had found that teachers who were anxious about math had students who learned less math during the school year, writes Susie Allen of UChicagoNews.

“We often don’t think about how important parents’ own attitudes are in determining their children’s academic achievement. But our work suggests that if a parent is walking around saying ‘Oh, I don’t like math’ or ‘This stuff makes me nervous,’ kids pick up on this messaging and it affects their success,” explained Beilock, professor in psychology.

As a control measure, the researchers also analyzed reading performance, but found that it was not related to parents’ math anxiety. The relationship between parents’ math anxiety and children’s math achievement, the research suggests, stems more from math attitudes than from genetics. One solution may be developing tools to teach parents how to help their children with math in the most effective way. This could include traditional board games, math books, computer games, or internet apps.

The Telegraph’s Javier Espinoza writes that math-anxious parents “are breeding a generation of innumerate children.” Levine put it this way:

“Math-anxious parents may be less effective in explaining math concepts to children and may not respond well when children make a mistake or solve a problem in a novel way.”

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]]>The post The Never-Ending Story: Procedures vs. Understanding in Math appeared first on Education News.

]]>**Finding the Balance between Procedural Fluency and Conceptual Understanding in Teaching Early Grade Mathematics to Students with Learning Disabilities**

By Barry Garelick

I went to school in the 50’s and 60’s when students first learned how to add and subtract in second grade. After spending some time memorizing the basic addition and subtraction facts and learning how to add and subtract single digit numbers, I was excited to hear my teacher announce one day that we would now learn how to solve problems like 43 + 52 and 95 – 64. In teaching the method, the teacher explained how the procedure relied on place value – what the ones place and tens place meant. I became bored with the explanation, began to daydream and missed the description of the procedure. The teacher then announced that we would now take a test on what we had just learned.

Faced with having to solve ten two-digit addition problems, I fell quickly behind the rest of the class. The teacher announced that she would not go on until everyone turned in their test. Students now put pressure on me as I desperately drew sticks on the side of my paper to “count up” to the answers. Finally, a girl across from me whispered “Add the ones column first and then the tens.” This advice made perfect sense to me and I finished the problems quickly. Although I had missed the explanation of why the ones and tens columns were added separately, it wasn’t long until I understood why after hearing the explanation again when the time came for learning how to “carry”. I was now receptive to what was going on with the procedure.

**Procedures as “Magic Corridors” to Understanding**

The issue of balance between procedural fluency and conceptual understanding continues to dominate discussions within the education community. The vignette above illustrates how procedural fluency may lead to understanding. This is true for all students, but is particularly relevant for students who may have learning disabilities.

Such students may find contextual explanations burdensome and hard to follow, resulting in feelings of frustration and inadequacy. It is not unusual in the lower grades for LD students – as well as non-LD students – to become impatient and wish that teachers would “just tell me how to do it.”

For many students, the “why” of the procedure is easier to navigate once fluency is developed for the particular procedure. The reason for this is given in large part through Cognitive Load Theory (Sweller, et al, 1994), which states that working memory gets overloaded quickly when trying to juggle many things at once before achieving automaticity of certain procedures.

An example of this is the plight of a visitor to a new city trying to find his way around. In getting from Point A to Point B, the visitor may be given instruction that consists of taking main roads; the route is simple enough so that he is not overburdened by complex instructions. In fact, well-meaning advice on shortcuts and alternative back roads may cause confusion and is often resisted by the visitor, who when unsure of himself insists on the “tried and true” method.

The visitor views these main routes as magic corridors that get him from Point A to B easily. He may have no idea how they connect with other streets, what direction they’re going, or other attributes. With time, after using these magic corridors, the visitor begins see the big picture and notices how various streets intersect with the road he has been taking. He may now even be aware of how the roads curve and change direction, when at first he thought of them as more or less straight. The increased comfort and familiarity the visitor now has brings with it an increased receptivity to learning about – and trying – alternative routes and shortcuts. In some instances he may even have gained enough confidence to discover some paths on his own.

In math, learning a procedure or skill is a combination of big picture understanding and procedural details. Research by Rittle-Johnson et. al., (2001) supports a strong interaction between understanding and procedures and that the push-pull relationship is necessary. Daniel Ansari (2011), a leading scholar of cognitive developmental psychology who studies brain activity during the learning of mathematics, also maintains that neither skill nor understanding should be underemphasized — they provide mutual scaffolding and both are essential.

Sometimes understanding comes before learning the procedure, sometimes afterward. The important point is recognizing when students are going to be receptive to learning the big picture understandings about what is really happening when they perform a procedure or solve a particular type of problem. Like visitors to a strange city, for many students, understanding comes after some degree of mastery of a particular skill or procedure.

For students with learning disabilities, explicit instruction on procedure should take precedence. A recent study (Morgan, et al, 2014) indicates that direct and explicit instruction given to first grade students with learning disabilities in math has positive effects. Conversely, student-centered activities (such as manipulatives, calculators, movement and music) did not result in achievement gains by such students. Of particular significance is that the study also found that direct and explicit instruction benefited those students without learning disabilities in math. To this end, we would add that an undue emphasis on understanding can decrease the amount of needed explicit instruction for students.

For many concepts in elementary math, it is the skill or procedure itself upon which understanding is built. The child develops his or her understanding by repeatedly practicing the pure skill until it is realized conceptually through familiarity and tactile experience that forges pathways and connections in the brain. But in terms of sequential priority, there is no chicken-and-egg problem: more often than not, skill must come first, because it is difficult to develop understanding in a vacuum. Procedural fluency provides the appropriate context within which understanding can be developed. It is important to note, however, that for some children, there may be certain procedures for which understanding remains elusive. It is even more important to note that such situation need not prevent such children from performing procedures and solving problems.

This is not to say that the conceptual underpinning should be omitted when teaching a procedure or skill. But while some explanation of the context is necessary to motivate the procedure, the issue is the degree of emphasis. Students with learning disabilities should be given explanations of how to proceed sooner rather than later. As discussed in more detail in the next section, after the standard procedure(s) are mastered alternative methods designed to provide deeper understanding of the concepts behind the procedure can then be provided when students are more receptive to such alternatives. It is also important to recognize that there will be some students who may not fully comprehend the concepts behind a procedure or problem solving technique at the same pace as other cohorts.

**Worked Examples and Scaffolding**

In teaching procedures for solving both word problems and numeric-only problems, an effective practice is one in which students imitate the techniques illustrated in a worked example. (Sweller, 2006). Subsequent problems given in class or in homework assignments progress to variants of the original problem that require them to stretch beyond the temporary support provided by the initial worked example; i.e., by “scaffolding”. Scaffolding is a process in which students are given problems that become increasingly more challenging, and for which temporary supports are removed. In so doing, students gain proficiency at one level of problem-solving which serves to both build confidence and prepare them for a subsequent leap in difficulty. For example, an initial worked example may be “John has 13 marbles and gives away 8. How many does he have left?” The process is simple subtraction. A variant of the original problem may be: “John has 13 marbles. He lost 3 but a friend gave him 4 new ones. How many marbles does he now have?” Subsequent variants may include problems like “John has 14 marbles and Tom has 5. After John gives 3 of his marbles to Tom, how many do each of them now have?”

Continuing with the example of adding and subtracting, in early grades some students, particularly those with learning disabilities, have difficulty in memorizing the addition and subtraction facts. On top of the memorization difficulties, students then face the additional challenge of applying this knowledge to solving problems. One approach to overcome this difficulty has been used for years in elementary math texts, in which students are provided with a minimum of facts to memorize and then given word problems using only those facts the student has mastered. Such procedure minimizes situations in which working memory encounters interference and becomes overloaded as described in Geary (in press). For example, a student may be tasked with memorizing the fact families for 3 through 5. After initial mastery of these facts, the student is then given word problems that use only those facts. For example, “John has 2 apples and gets 3 more, what is the total?” and “John has some apples and receives 3 more; he now has 5 apples. How many did he have to start with?” Additional fact families can then be added, along with the various types of problems. Applying the new facts (along with the ones mastered previously) then provides a constant reinforcement of memorization of the facts and applications of the problem solving procedures. The word problems themselves should also be scaffolded in increasing difficulty as the student commits more addition and subtraction facts to memory.

Once the foundational skills of addition and subtraction are in place, alternative strategies such as those suggested in Common Core in the earlier grades may now be introduced. One such strategy is known as “making tens” which involves breaking up a sum such as 8 + 6 into smaller sums to “make tens” within it. For example 8 + 6 may be expressed as 8 +2 + 4. To do this, students need to know what numbers may be added to others to make ten. In the above example, they must know that 8 and 2 make ten. The two in this case is obtained by taking (i.e., subtracting) two from the six. Thus 8 + 2 + 4 becomes 10 + 4, creating a short-cut that may be useful to some students. It also reinforces conceptual understandings of how subtraction and addition work .

The strategy itself is not new and has appeared in textbooks for decades. (Figure 1 shows an explanation of this procedure in a third grade arithmetic book by Buswell et. al. (1955).

The difference is that in many schools, Common Core has been interpreted and implemented so that students are being given the strategy prior to learning and mastering the foundational procedures. Insisting on calculations based on the “making tens” and other approaches before mastery of the foundational skills are likely to prove a hindrance, generally for first graders and particularly for students with learning disabilities.

Students who have mastered the basic procedures are now in a better position to try new techniques – and even explore on their own. Teachers should therefore differentiate instruction with care so that those students who are able to use these strategies can do so, but not burden those who have not yet achieved proficiency with the fundamental procedures.** **

**Procedure versus “Rote Understanding”**

It has long been held that for students with learning disabilities, explicit, teacher-directed instruction is the most effective method of teaching. A popular textbook on special education (Rosenberg, et. al, 2008) notes that up to 50% of students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction. The final report of the President’s National Math Advisory Panel (2008) states: “Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities, as well as other students who perform in the lowest third of a typical class.” (p. xxiii). These statements have been recently confirmed by Morgan, et. al. (2014) cited earlier. The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes memorization and other explicit instructional methods.

Currently, with the adoption and implementation of the Common Core math standards, there has been increased emphasis and focus on students showing “understanding” of the conceptual underpinnings of algorithms and problem-solving procedures. Instead of adding multi-digit numbers using the standard algorithm and learning alternative strategies after mastery of that algorithm is achieved (as we earlier recommended be done), students must do the opposite. That is, they are required to use inefficient strategies that purport to provide the “deep understanding” when they are finally taught to use the more efficient standard algorithm. The prevailing belief is that to do otherwise is to teach by rote without understanding. Students are also being taught to reproduce explanations that make it appear they possess understanding – and more importantly, to make such demonstrations on the standardized tests that require them to do so.

Such an approach is tantamount to saying, “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.” Forcing students to think of multiple ways to solve a problem, for example, or to write an explanation for how they solved a problem or why something works does not in and of itself cause understanding. It is investment in the wrong thing at the wrong time.

The “explanations” most often will have little mathematical value and are naïve because students don’t know the subject matter well enough. The result is at best a demonstration of “rote understanding” – it is a student engaging in the exercise of guessing (or learning) what the teacher wants to hear and repeating it. At worst, it undermines the procedural fluency that students need.

Understanding, critical thinking, and problem solving come when students can draw on a strong foundation of domain content relevant to the topic being learned. As students (non-LD as well as LD) establish a larger repertoire of mastered knowledge and methods, the more articulate they become in explanations.

While some educators argue that procedures and standard algorithms are “rote”, they fail to see that exercising procedures to solve problems requires reasoning with such procedures – which in itself is a form of understanding. This form of understanding is particularly significant for students with LD, and definitely more useful than requiring explanations that students do not understand for procedures they cannot perform.

Barry Garelickhas written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long term substitute in a high school and middle school in California: “Teaching Math in the 21st Century”.

——————–

**References**

Ansari, D. (2011). Disorders of the mathematical brain : Developmental dyscalculia and mathematics anxiety. Presented at *The Art and Science of Math Education, University of Winnipeg, November 19th 2011. *http://mathstats.uwinnipeg.ca/mathedconference/talks/Daniel-Ansari.pdf

Buswell, G.T., Brownell, W. A., & Sauble, I. (1955). Arithmetic we need; Grade 3. *Ginn and Company. New York. p. 68.*

Geary, D. C., & Menon, V. (in press). Fact retrieval deficits in mathematical learning disability: Potential contributions of prefrontal-hippocampal functional organization. In M. Vasserman, & W. S. MacAllister (Eds.), *The Neuropsychology of Learning Disorders: A Handbook for the Multi-disciplinary Team*, New York: Springer

Morgan, P., Farkas, G., MacZuga, S. (2014). Which instructional practices most help first-grade students with and without mathematics difficulties?*; **Educational Evaluation and Policy Analysis Monthly 201X, Vol. XX, No. X, pp. 1–22*. doi: 10.3102/0162373714536608

National Mathematics Advisory Panel. (2008). Foundations of success: Final report. *U.S. Department of Education.* https://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf

Rittle-Johnson, B., Siegler, R.S., Alibali, M.W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. *Journal of Educational Psychology, Vol. 93, No. 2, 346-362*. doi: 10.1037//0022-0063.93.2.346

Rosenberg, M.S., Westling, D.L., & McLeskey, J. (2008). Special education for today’s teachers. *Pearson; Merrill, Prentice-Hall. *Upper Saddle River, NJ.

Sweller, P. (1994) Cognitive load theory, learning difficulty, and instructional design. *Leaming and Instruction, Vol. 4, pp. 293-312*

Sweller, P. (2006). The worked example effect and human cognition.*Learning and Instruction, 16(2) 165–169*

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]]>The post US Wins International Mathematics Olympiad for First Time in 21 Years appeared first on Education News.

]]>For the first time in 21 years, the US has won the International Mathematical Olympiad.

The USA team, led by Professor Po-Shen Loh of Carnegie Mellon University, competed against more than 100 countries in Chiang Mai, Thailand. The USA’s last win came in 1994, notes Dominique Mosbergen of the Huffington Post. Since then, the competition has been dominated by the Chinese, who came in second in this competition. Other recent winners include South Korea, which came in third this year, and Russia.

NPR interviewed Loh, who suggested that the difficulty of the questions indicated how strong the field was:

If you can solve even one question, you’re a bit of a genius.

Students in the final competition solve six problems dealing with algebra, geometry, number theory, and combinatorics, among other topics, in 4.5-hour sessions over the course of two days. Answers are not given in simple numbers, but often demand long explanations. The team’s score is the sum of its member’s scores.

According to Michael E. Miller of the Washington Post, this year’s team had all male members: David Stoner, Ryan Alweiss, Allen Liu, Yang Liu, Shyam Narayanan, and Michael Kural. The fact that historically almost all the competitors are male illustrates the dearth of women who are mathematical high achievers and who go on to create careers in the STEM fields. However, progress is being made — this year, the Ukrainian team had an equal number of boys and girls, writes Natalie Schachar of the LA Times.

Loh said that gender balance is happening, albeit slowly:

That is actually something that one hopes will change. The top 12 people in the country on the United States Math Olympiad happen to have two girls in it. One might say, ‘Only 2 out of 12, that’s terrible.’ But I should say in many years, it was, unfortunately, zero.

Some feel that this victory is evidence that the math education situation in America is less dire than it seems. In 2015, 15-year-olds in the US were rated 35th in math and 27th in science out of 64 countries by the Program for International Student Assessment (PISA) test.

Loh said that their success shows that the United States’ best students are globally competitive:

At least in the case with the Olympiads, we’ve been able to prove that our top Americans are certainly at the level of the top people from the other countries.

Loh, who was himself a contestant in 1999, hopes that math education in America will be reformed to interest more students and involve more creativity. He said that making math more engaging could bring diversity of all sorts to the competition:

Ultimately, I think that as the mathematical culture starts to reach out to more people in the United States, we could quite possibly start to see more diversity. And I think that would be a fantastic outcome.

It could be that maybe the way math is sold, in some sense, is one in which it’s just a bunch of formulas to memorize. I think if we are able to communicate to the greater American public that mathematics is not just about memorizing a bunch of formulas, but in fact is as creative as the humanities and arts, quite possibly you might be able to upend the culture difference.

This year’s Math Olympiad was the 56th international competition.

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]]>The post Girls Perform Better in Math with Female Teachers, Data Shows appeared first on Education News.

]]>Who runs the world? Girls – especially when they are taught by fellow females, according to a study conducted at Texas A&M University.

Previous research has shown that females tend to perform better in the classroom than males. They rank higher across the board; from college graduation rates to test scores, they surpass their male counterparts in nearly every academic facet of education, writes Gabriel Fisher for Quartz.

Most recently, researchers Jonathan Meer and Jaegeum Lim found that there was a significant improvement in girls’ math test scores when a woman teaches the subject — an interesting finding, particularly since math is thought of as a male-dominated field.

Meer and Lim analyzed 14,000 test scores from middle school students in South Korea. They found that when a woman taught students math, girls’ scores were nearly 10% of a standard deviation higher than boys. They also found that when girls switched from a male teacher to a female teacher, their math scores went up by 8.5% of a standard deviation compared to boys’ scores.

“Female students outperform male students by roughly a third of a school year

when taught by female teachers than when taught by male teachers,” Meer explained.more

He feels that the increased performance is due to girls feeling more comfortable in class when taught by a female teacher:

“Female students report feeling that their female teachers are more likely to give students an equal chance to participate,” he writes, adding that “their female teachers are more likely to encourage creative expression.”

The researchers chose to conduct a study in South Korea because students are randomly assigned teachers in the country. In the United States, female math teacher have been known to be assigned weaker students, writes Julie Zeilinger for Mic.

This information may be good news for female students who have been shown to hold themselves back in STEM subjects. For example, female students generally underestimate their abilities and predict they will perform worse on tests, while boys overestimate their performance.

However, boys are being left behind academically, and Meer is, “personally deeply worried about male performance in schools.”

Researchers are divided on the specific reasons why male students are less engaged than female students, which makes it hard to diagnose the issue on a large scale. This is why it is important for parents to understand why their boy isn’t engaged and then to work with teachers to solve the problem, reports Shelby Slade for Deseret News.

“We must work equally hard to encourage boys to consider literature, journalism and communications. Boys are often pushed toward math and science, and receive inadequate social support. We need to recognize boys’ differences, and their social and developmental needs,” says educator Sean Kullman.

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]]>The post Study: Recognition of Abilities Helps Drive Interest In Math, STEM appeared first on Education News.

]]>According to a new study published by Florida International University Professor Zahra Hazari in the journal *Child Development, *interest and recognition can help increase a student’s enthusiasm for math and their willingness to pursue a career in STEM fields.

The study, “Establishing an Explanatory Model for Mathematics Identity,” suggests that an increased interest in the area of math is not something a person is born with, as was previously thought. It continues to say that students who have an increased confidence level in the subject do not necessarily become interested in it.

“Much of becoming a ‘math person’ and pursuing a related STEM (science, technology, engineering or math) career has to do with being recognized and becoming interested – not just being able to do it,” said Hazari, who specializes in STEM Education at FIU’s College of Education and STEM Transformation Institute, according to the NEA blog. “This is important for promoting math education for everyone since it is not just about confidence and performance.”

Participants included over 9,000 college students enrolled in calculus courses across the country. Researchers discovered that students who were enrolled in higher-level courses were doing so mainly due to an interest in the subject that evolved from some form of recognition of their abilities previously given to them, as well as finding the topic interesting.

The survey asked participants whether they felt family, friends, and math teachers viewed them as a “math person.” Those who responded positively to the question were classified as feeling recognized.

“It is surprising that a student who becomes confident in her math abilities will not necessarily develop a math identity,” Hazari said. “We really have to engage students in more meaningful ways through their own interests and help them overcome challenges and recognize them for doing so. If we want to empower students and provide access to STEM careers, it can’t just be about confidence and performance. Attitudes and personal motivation matters immensely.”

A separate study performed at Washington State University looked at 122 undergraduate students as well as 184 other participants who were all asked to complete a math test and then take a guess as to how well they did. One group received feedback pertaining to their scores prior to giving their guesses, while the other group had no feedback, and were also asked to state whether they held any interest in pursuing a career related to math.

While male participants repeatedly overestimated their math exam scores, women tended to predict their scores more accurately. However, it was found that more men wanted to pursue a math-related career than women, suggesting that the belief that a person is competent in a subject is directly related to the decision to follow a career path in that field, writes Dana Dovey for Medical Daily.

Researchers hope to use the study findings in order to get more girls interested in careers in math and science. To date, women in the United States make up almost half of the work force, although they account for only 24% of STEM positions, or those in science, technology, engineering or math jobs.

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]]>The post A Comparison of Common Core Math to Selected Asian Countries appeared first on Education News.

]]>Jonathan Goodman

Professor of Mathematics

Courant Institute of Mathematical Sciences

New York University

July 9, 2010

**Executive summary. **The proposed Common Core standard is similar in earlier grades but has significantly lower expectations with respect to algebra and geometry than the published standards of other countries I examined. The Common Core standards document is prepared with less care and is less useful to teachers and math ed administrators than the other standards I examined. I have reservations about the Common Core standards regarding statistics in grades 7 and 8.

**History and overview**. At the request of Bill Evers and Zeev Wurman, I examined the published math standards of (South) Korea, Singapore, Hong Kong, and Japan, and Taiwan, as well as the proposed Common Core US standard, which I simply call US below. Evers and Wurman expressed particular interest in the question of teaching of Algebra in grade 8. They told me that this might be a major difference between the proposed Common Core national standard and the California State math standard. I have not examined the California standard.

The foreign countries chosen were from a recent list of high performing countries. Other high performing countries whose standards I have not examined are Australia and Flemish Belgium (Walloon Belgium has separate standards and is not in the top group by performance). Neither of these standards was available to me. The Flemish standard is not translated into English.

The conclusions and opinions expressed here are mine exclusively. Evers and Wurman have their opinions, they emphasized that they are not mathematicians and did not try to influence me, which would have been futile. I agreed to do the comparison because I have a strong interest in math education. They did provide me with translations of the foreign standards.

In this report, I offer judgments and opinions. For example I judge the arithmetic standards for first grade to be similar across countries. I try to label opinions as such. For example, it is my opinion that the Common Core draft description of the “mathematically proficient student” is so unrealistic as to be detrimental.

My experience with mathematics comes foremost from my career of mathematics research and teaching in a department ranked among the top ten nationally (US News). I also have considerable interest in K-12 math education, and have been actively involved in mostly fruitless attempts to influence New York City Department of Education math curriculum. I am familiar with the NCTM standards and ongoing debates on the best ways to teach math to kids.

**Educational philosophy.** All six of the standards documents I examined offer statements of educational philosophy. These statements have important similarities, including the need for understanding as opposed to rote, the need for research based age appropriate curricula, and the importance of higher level skills such as judgment, strategy, resourcefulness, and the ability to communicate. All of them state that high level skills do not come without learning facts and practicing algorithms.

I go through the standards grade by grade with comparisons, opinions, and some recommendations. I refrain from discussing many things the six standards have in common or that I do not think are significant differences. For example, all six standards call for first graders to “know their shapes”, but the details differ from country to country. I also mention what, in my opinion, are some strengths of non US standards documents that the US would do well to adopt.

In my opinion, the US standards are the least informative of the six. This will make it harder for US teachers and administrators to determine whether a particular curriculum or test meets the standard. By contrast, the Japanese standards (for example) give examples for almost every item to clarify precisely what the expectation is. The US standards are a study in bureaucratic ambiguity. For example, the US standard for grade 1 arithmetic (page 16) includes: “Add within 100, including adding a two-digit number and (*sic*) a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and*strategies based on place value, properties of operations, and/or the relationship between addition and subtraction*:…”. In light of the math wars, it is important to state whether *strategies* means an exercise in student problem solving or practice with some form of a traditional algorithm. An example would clarify this.

**Grade 1**. The six standards have much in common [1]. A major goal is addition and subtraction within a limited range: 0 to 20 for the US, and 0 to 100 for the others. All but Korea include carrying and borrowing. All mention place value and abstract properties of addition and subtraction.

The US and most others include pre-algebra problems such as 4 + __ = 9. The US standard emphasizes students’ ability to explain in words the reasoning in the solution of a math problem or calculation.Â An example of a correct explanation would be most helpful. The Japanese standard includes working with a number line, which I think is a great idea (in grade 2 for the US). There are expectations in the other standards that are not included (at least not explicitly) in the US standard.

*– Money*. All the other standards call for students to learn to manipulate the local currency. Some only use coins, in view of the overall restriction to numbers less than 100. Many call for students to practice place value and arithmetic with money, for example by exchanging pennies for nickels and dimes. Many call for students to read price tags and count out payments. The US standard calls for more abstract manipulatives rather than money. My opinion is that money is more natural. The US standard puts money in grade 2, but with less emphasis on coins as manipulatives.

*– Mental arithmetic*. Most of the other countries call for students to be able to add and subtract “mentally” (without paper or manipulatives) up to about 20. The US standard does not. Singapore calls for memorization up to 9+9. The US standard puts this in grade 2.

*– Calendar*. All but the US call for some understanding of the calendar. At least days, including the names of the days, and weeks. Some call or understanding the yearly calendar, including months. In my opinion, there is value in this, but one could argue that memorizing the names of the days is not math.

**Grade 2**. The six standards have much in common. Almost every major topic mentioned on one is mentioned in all. The biggest exception is that Hong Kong has nothing about fractions. Multiplication is receives less emphasis in the US standard than in the other five. It is not clear whether the US standard calls for the operation of multiplication, and the multiplication symbol, to be defined.

The US standard for addition is ambiguous. Quoting from page 17 (italics mine): “…(students) *develop*, discuss, and use different, *accurate*, and *generalizable* methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations.” Page 19 has: “5. Fluently add and within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 6. Add up to four two-digit numbers using strategies based on place value and properties of operations.” A standard should be clear on this point [2] but is not. Are second graders responsible for the column-wise addition algorithm or not? The wordings have much in common with the 1989 NCTM standards that deprecated algorithms. Will students develop the methods, or will teachers develop (i.e. teach) them? The terms *accurate* and *generalizable* also are from the math wars and (in my opinion) are out of place here. I refer the reader to the Japanese standard, page 71, which has a beautiful, clear, and precise suggestion of how to teach column-wise addition.

**Grade 3**. Again the six standards are similar. The most significant differences are as follows. The US emphasizes area more than the others. Japan, Hong Kong, and Korea omit area in grade 3. The US catches up with the other five in calling for addition algorithms. [3] Japan, Korea, and Taiwan include decimals as fractions n.x = n+(x/10). Japan includes using an abacus and the Japanese names for the powers of ten up to 10,000. Taiwan has “vertical” multiplication of a pair of two digit numbers. The other five have more on angles than the US.

**Grade 4**. The six standards handle fractions and decimals similarly. Some operations are applied to numbers of arbitrary size, with warnings not to overwhelm the kids with huge numbers. Highlighting the main differences: The US standard for adding and subtracting (the standard algorithm) and multiplying (multi-digit times one digit) whole numbers is what the other countries call for in grade 3. The other five give more emphasis to approximation and estimation than does the US. They also mandate areas, as the US did in grade 3. The difference (in treating areas) is that the US explicitly says to ignore units, while the other five call for explaining units of area, such as square meters. In my opinion, the US is wrong to ignore area units. It means treating area as purely geometric (a property of plane figures) rather than physical (a property of things like rugs, roofs, etc.). The US and Japan are better than the others (in my opinion) in calling for placing fractions and decimals on a number line. Korea and Hong Kong call for calculators to check arithmetic.

The contrast in the styles of the standards documents becomes more pronounced in grade 4. The US document treats some topics, such as fractions, in pedagogical detail. Others are left obscure, such as (page 29): “…multiply two two-digit numbers, using *strategies* based on place value and properties …”. By contrast, Taiwan has a whole page (page 139) devoted to illuminating examples of what is and is not mandated for fractions in grade 4. The US call for the standard algorithm makes it less clear, not more, what the methods are intended in grades 2 and 3.

**Grade 5**. The six standards are similar in how they cover arithmetic with fractions and decimals. The US is unique in introducing plane Cartesian coordinates. The US also is uniquely weak in several areas: the greatest common divisor/least common multiple, areas on non-rectangular regions such as triangles, ratios of quantities with units (e.g. miles per gallon, meters per second, kilograms per liter), percentages, and graphical representation of data (pie charts, bar graphs, etc.). Other countries are doing more to prepare students for algebra. Hong Kong has what I consider a great topic, understanding large numbers such as 10,000,000.

There is a fundamental difference in the treatment of geometry in grade 5. The other five discuss interesting (my opinion) topics like tilings, similarities, and polygons. The US standard is what I consider rote memorization of properties of figures. You can appreciate the difference by thinking how you would test the material. One might be: “Give a way to divide a 1 X 3 rectangle into a collection of four triangles?” The other might be: “In what way is a square different from a rectangle?”

The US standards document continues to puzzle me. After paying little attention to estimation in grades 3 and 4 (while the other five standards constantly emphasize it), now estimation returns. The description of Cartesian coordinates seems to imply that people might be unfamiliar with the concept: “Use a pair of perpendicular number lines, called axes, to define a coordinate system …” Cartesian coordinates are great (opinion), but they should be described in a way that will make sense to fifth graders. The document suggests measuring volumes by counting cubes, including cubic cm, cubic ft (will classrooms have piles of foot sized cubes?), and *improvised units* (cubic cubits?).

**Grade 6**. All six standards finish arithmetic, including multiplication and division of fractions and decimals, factoring of integers, and LCM/GCD for manipulating fractions. All discuss ratios, percentages, constant speed, and general proportionality expressed symbolically (e.g. d = 5t, where d is distance and t is time). All standards advance algebra using variables and abstract manipulation, but to different degrees. But the standards are much more different in grade six than for earlier grades. I take these differences by topic.

In pre-Cartesian (non-coordinate) geometry, the US catches up by doing areas of triangles, while the others pull ahead by doing areas and circumferences involving p. [4] Japan discusses making scaled copies of figures.

Only the US and Singapore have coordinate geometry. The US calls for determining distance between points whose x or y coordinates are identical (not very interesting in my opinion). Singapore graphs linear functions, connects this to the algebraic idea of proportionate change, and discusses the slope of a line (more interesting).

Only the US discusses negative numbers. Other countries do negatives in grade 7. Much time is devoted to arithmetic with whole numbers and fractions of arbitrary sign, distances between signed quantities, absolute value, and the “rational number line” (see pet peeves below). In the examples of naturally occurring negative numbers (negative temperatures, owing money) there is “negative electrical charge”. I highly doubt (my opinion) that any sixth grader will be able to appreciate the difference between positive and negative charge.

The US standard has much more on statistics than the others, including multiple measures of center (mean and median) and variation (quantile differences, mean absolute value deviation). Korea has some probability, but that part of their standard is vague. Japan and Hong Kong have mean, but not median or measures of variation. Taiwan, Singapore, and Korea stop at data representation without doing summary statistics. The US has less practice in graphical data presentation than the other five.

The other five countries all have less material in sixth grade than they had in fifth grade or the US has in sixth. I can only guess that this is considered a consolidation year for them, with much time spent honing skills.

The US standard for sixth grade is (in my opinion) careless and vague.

– (page 39): “…*that a data distribution may not have a definite center and* that different ways to measure center yield different values.” The italicized part is incorrect whatever center means, a data set has one by that definition. If it were deleted, the rest would be simpler and more correct.

– The standard calls for formulas involving powers, but powers (squares, cubes, etc.) have not been discussed yet. Is this their first introduction?

– There are many references to “real world examples” but no examples. I wonder what they have in mind for real world examples of distances between points in the plane with the same x coordinate.

– (page 45): “Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.” As a mathematician conducting research in computational statistics, I wonder what this could refer to. Maybe it refers to questions of robustness in which fat tailed distributions call for using the median rather than the mean. If it is a *standard*, people should know what it means.

**Grade 7**. I leave out Hong Kong because combines grades 7-9. The standards from Japan for grades 7 and 8 are less complete than for earlier grades. All five countries now use coordinate geometry at least to place points and graph lines. All have some algebraic manipulation, though the US has the least. All have a more thorough discussion of rates and proportionality with a variety of practical examples and graphical implications. All expect a full mastery of the rational number system (ratios of signed integers) that the US specified in grade 6 (and repeats in grade 7). The US catches up in geometry, including p, for example. Beyond this, the US differs from the others.

Korea has the least algebra, somewhat more than the US, particularly functional notation and exponents. It has much more geometry than the US, both in breadth (lots about polyhedra) and depth (systematic reasoning bordering on Euclid style theorem proving). It is unique in covering binary representations of numbers.

Taiwan and Singapore have a most of a full “Algebra I” class, with much practice in general algebraic manipulation, including rational expressions and solution of two linear equations in two unknowns. Taiwan discusses exponents, particularly for representing powers of ten and scientific notation, which I strongly endorse (an opinion). Singapore has the Pythagorean theorem, probably without proof. Singapore also has basic probability and mean and median of a data set, but without measures of variability. Taiwan has no statistics or probability this year.

The US standard is distinguished from the others by its breadth and depth in statistics and basic probability. I have reservations about many of the specifics. The US standards document continues to be problematic. The grade 7 standard contains many helpful examples, which makes the many remaining vague items without examples less explicable. Here are some sample quibbles:

– I recommend removing the hydrogen atom as an example of charge cancellation. The background knowledge for this is missing: electrons, protons, atoms, charge, electric attraction (holding the atom together), …

– (page 50) suggests drawing “with technology”. Does this mean a computer drawing program that students would have to learn to use? If so, it would be a very large investment of student time for not very much educational gain. This kind of thing should not be ambiguous in a standards document.

– (page 50) states that someone (teacher, student?) should “give an informal derivation of the relationship between the circumference and area of a circle.” I imagine this might have to do with the area difference between circles of slightly different radius. Is that right?

– (page 51) Students are to choose words at random from a book. How? They are to use random digits to simulate a random process. Where do the digits come from?

– Most seriously, the standard calls for drawing “informal comparative inferences about two populations”. This amounts to testing the hypothesis of equality among sample means. The method suggested, comparing the difference between the means to twice the mean absolute deviation, has no basis in statistics. Any real test would depend on the sample sizes. This is an admirable topic, but the contents proposed are disinformation (a judgment).

– The notion of “probability sample space” is, in my opinion, not very helpful in this very elementary setting. I prefer the approach of Taiwan — discussing events as sets without taking time to define the whole space.

**Grade 8**. I am dropping Singapore because they combine grades 8 and 9. I compare the remaining standards (US, Korea, Taiwan, Japan) area by area.

Roots and irrational numbers: All have square roots, with the possible exception of Japan. Taiwan has more general rational exponents, but lightly. The US asks students to “know that the square root of 2 is not rational”. Does this mean being able to repeat that sentence or understanding the proof? The US standard asks students to tell that the square root of 2 is between 1.4 and 1.5 by truncating its decimal expansion. But where does the decimal expansion come from, a calculator?

Algebra: Both Korea and Taiwan call for generic operations of algebra such as factoring, multiplying polynomials, polynomial division (not in all cases), completing the square, etc. The US standard by contrast, supports a tightly circumscribed list of algebraic tasks centered on pairs of linear equations in two unknowns. The US catches up with exponents. It includes the mathematicians’ definition of function (set of ordered pairs so that …) but with an uncertain range of application. Included are general linear functions, at least one quadratic (with a proof that quadratics are not linear, as though a glance at the graph would not suffice), and possibly others that can be increasing and decreasing over different intervals of their argument (as functions are in calculus).

Geometry: Korea, Taiwan, and Japan call for significant work in the direction of traditional high school Euclidean geometric proof, but not as much as US high school geometry from the 1960s. This includes ruler and compass constructions (perpendiculars, bisections, congruences) and some of the easier proofs. The US calls for students to “understand” congruence of plane figures, but it is unclear what this means beyond the definition. US students are asked to know the Pythagorean theorem and its proof, but I question the wisdom of asking students to memorize something they do not have the background to appreciate. The US makes more of the geometric interpretation of pairs of linear equations than the other countries to.

Statistics: Only the US discusses statistical modeling and hypothesis testing. In grade 8, it discusses linear regression models and hypothesis testing on categorical data. But students have no systematic way to do either task. Linear fits are to be made “informally” from scatterplots. Goodness of fit is to be judged by eye. Categorical data decisions are made on (as far as I can tell) no basis whatsoever. If 60% of boys and 70% of girls pass math, is that a significant difference? How would one decide? In my opinion, this is negative education — giving kids the incorrect idea that they know something about regression and categorical data analysis. There is an AP statistics class in many high schools (both my kids took it). This has, for example, the ideas behind hypothesis testing, the role of statistical models, and the central limit theorem (informally). Making statistical inferences with less than this is dangerous.

**Algebra**. There are two styles of algebra, with the US on one side and most of the other countries on the other. Japan is in the middle. All ask students to understand abstract variables, such as *x* and *y*, and to know how to solve pairs of linear equations. All ask students to have seen manipulations such as dividing both sides of an equation by the same number to preserve the equality. The US (up to grade basically stops here. Japan includes multiplication and division of polynomials. The other countries go further to for example completing the square (solving general quadratics), manipulating rational functions, etc.

An item from the Japanese standard captures the difference: “Transform algebraic expressions depending on purpose.” General algebraic manipulation requires the student constantly to recall the mathematical principles and to plan strategy. Suppose, for example, a students wants to isolate a variable on one side of the equation. He or she must develop a strategy consisting of a sequence of operations. The strategy will be different from problem to problem. If manipulations are limited to pairs of linear equations, the student will simply memorize the operations required to solve them.

**Pet peeves**:

**–****Problem solving, rote, and patterns**. Most people believe that math education should involve some serious independent creative problem solving. It also is a sad fact that students want to be told how to do things, and teachers like to tell them. Items introduced into the curriculum as problem solving can evolve into rote. This is nowhere more clear than in looking for patterns in sequences of numbers. The sequences offered tend to be arithmetic, and students are trained to look for common differences. The grade 4 patterns of the six curricula are all arithmetic progressions. But students could be presented with a wider range of patterns. For example, 1,2,1,2,1,2,… (easy but not arithmetic), 1,2,1,1,2,1,1,1,2,1,1,1,2,… (slightly harder), 1,1,2,3,5,8,13,23,… (a real challenge, but probably some kids would get it).

**–****Mathematical exactitude**. Mathematicians have a bad reputation in the K-12 math ed world partly because of the “new math” disaster we helped create in the 1960s. My sixth grade math book from that period (found much later in my parents’ closet) had a discussion of sets that included (approximately) the sentence: “For any set and any property, you can form the subset of all members of the set that satisfy the property.” This probably was followed by an example such as picking out the apples from a set of pieces of fruit. As a recent math PhD, I recognized the *aussonderung* axiom of Zermelo Fraenkel set theory. I knew that this axiom has the purpose of ruling out Russell’s paradox. But how could it have helped a sixth grader? Mathematicians (in K-12 math ed discussions) must get used to things that are well enough understood even if not absolutely precise. We should avoid concepts that require more maturity than kids of a certain age have. We need not describe the coordinate plane in detail for fifth graders, or expect sixth graders to appreciate the distinction between a finite and infinite set of solutions. At the same time, we can prevent flatly incorrect statements such as (US, page 45): “Recognize a statistical question as one that anticipates variability in the data related to the question …”, which confuses a statistician’s model of random data variation with the fact of a single unchanging data set. Also (Taiwan, page 175): “For example, even though both 14 and 16 are composite numbers, they are coprime.” (Should have been 14 and 15?)

**Notes**

[1] The US is unique in having a standard for Kindergarten. I have combined the US kindergarten and grade 1 standards when comparing to grade 1 standards from the other countries.

[2] This opinion is stated as a judgment because I expect few to disagree.

[3] I do not know whether the plural is significant. The Everyday Mathematics curriculum adopted by the New York City Department of Education includes several variants of each of the algorithms of arithmetic. In my opinion, teaching multiple algorithms is a confusing waste of the students’ time.

[4] Hong Kong has a charming learning objective (page 44): “Tell the stories of ancient Chinese mathematicians discovering p.” I think this is time well spent.

The post A Comparison of Common Core Math to Selected Asian Countries appeared first on Education News.

]]>The post 75% of Oregon Community College Students Need Remedial Classes appeared first on Education News.

]]>A recent study performed for the national Institute of Education Sciences has found that 75% of the 100,000 Oregon high school graduates to continue on to community college needed to participate in remedial classes upon arrival.

Portland-based researcher Michelle Hodara, who conducted the study, said that the graduates were ill-prepared for college life. She added that race and income levels did not play a role in the need to take the high-school or middle-school level courses.

The most common remedial course taken was in math, with almost 75% of all graduates who attended community college taking a remedial math course. The majority of those students tested into an introductory algebra class. Less than one-third of the students who needed to take remedial math were found to have ended up taking and passing even one college-level math class. This could be attributed to three terms of remedial math classes needing to be taken and paid for prior to entrance into college-level courses, writes Betsy Hammond for *The Oregonian.*

Of the high school graduates in the state who enter community college ready to participate in a college-level math course, the study found that 54% will earn a two-year degree or a career-related certificate. However, less than one-third of students who enter community college needing to take remedial math will earn any type of credential within five years, according to the study.

In addition, it was found that more high school graduates of the state will attend community college than will go to a four-year university.

Hodara recommended that schools work harder to academically prepare their students for the work they will need to do in college. According to the study, the best way to do this is through the passing of the state’s reading and math tests.

Higher education leaders in the state have been pushing for additional funding, arguing that due to declining state funding they have had to rely more and more on extra funding coming in from rising tuition rates, writes Jonathan Cooper for *The Fresno Bee.*

The presidents of all 7 universities and 17 community colleges in the state wrote a letter to legislative leaders, saying that too many people in Oregon cannot afford the rising cost of higher education.

“Like never before, Oregon’s public universities and community colleges are aligned to advocate the imperative of improved funding for higher education,” the presidents wrote. “Beginning to restore funding after a decade of disinvestment is the right thing to do for students and, ultimately, it is the right thing to do for Oregon’s future.”

However, according to the new state education budget, little extra money will be available for new or expanded programs.

“We are going to be largely addressing base needs, and there’s only going to be a modest amount of funding out there for quote-unquote new initiatives,” said Sen. Richard Devlin, D-Tualatin. “In fact, I think very little, to be frank.”

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]]>The post Math Problems: Knowing, Doing, and Explaining Your Answer appeared first on Education News.

]]>by Barry Garelick and Katharine Beals

At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked “What do I do?” The proctor read the instructions for the problem and told the student: “You need to explain how you got your answer.”

The girl threw her arms up in frustration and said, “Why can’t I just do the problem, enter the answer and be done with it?”

The answer to her question comes down to what the education establishment believes “understanding” to be, and how to measure it. K-12 mathematics instruction involves equal parts procedural skills and understanding. What “understanding” in mathematics means, however, has long been a topic of debate. One distinction popular with today’s math reform advocates is between “knowing” and “doing.” A student, reformers argue, might be able to “do” a problem (i.e., solve it mathematically), without understanding the concepts behind the problem solving procedure. Perhaps he has simply memorized the method without understanding it.

The Common Core math standards, adopted in 45 states and reflected in Common Core-aligned tests like the SBAC and the PARCC, take understanding to a whole new level. “Students who lack understanding of a topic may rely on procedures too heavily,” states the Common Core website. “… But what does mathematical understanding look like?” And how can teachers assess it?

“One way … is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.” (http://www.corestandards.org/Math/).

The underlying assumption here is that if a student understands something, she can explain it—and that deficient explanation signals deficient understanding. But this raises yet another question: what constitutes a satisfactory explanation?

While the Common Core leaves this unspecified, current practices are suggestive: Consider a problem that asks how many total pencils are there if 5 people have 3 pencils each. In the eyes of some educators, explaining why the answer is 15 by stating, simply, that 5 x 3 = 15 is not satisfactory. To show they truly understand why 5 x 3 is 15, and why this computation provides the answer to the given word problem, students must do more. For example, they might draw a picture illustrating 5 groups of 3 pencils.

Consider now a problem given in a pre-algebra course that involves percentages: A coat has been reduced by 20% to sell for $160. What was the original price of the coat?”

A student may show their solution as follows:

x = original cost of coat in dollars

100% – 20% = 80%

0.8x = $160

x = $200

Clearly, the student knows the mathematical procedure necessary to solve the problem. In fact, for years students were told not to explain their answers, but to show their work, and if presented in a clear and organized manner, the math contained in this work was considered to be its own explanation. But the above demonstration might, through the prism of the Common Core standards, be considered an inadequate explanation. That is, inspired by what the standards say about understanding, one could ask “Does the student know why the subtraction operation is done to obtain the 80% used in the equation or is he doing it as a mechanical procedure – i.e., without understanding?”

**Providing instruction for explanations—the road to “rote understanding”**

In a middle school observed by one of us, the school’s goal was to increase student proficiency in solving math problems by requiring students to explain how they solved them. This was not required for all problems given; rather, they were expected to do this for two or three problems per week, which took up to 10 percent of total weekly class time. They were instructed on how to write explanations for their math solutions using a model called “Need, Know, Do.” In the problem example given above, the “Need” would be “What was the original price of the coat?” The “Know” would be the information provided in the problem statement, here the price of the discounted coat and the discount rate. The “Do” is the process of solving the problem.

Students were instructed to use “flow maps” and diagrams to describe the thinking and steps used to solve the problem, after which they were to write a narrative summary of what was described in the flow maps and elsewhere. They were told that the “Do” (as well as the flow maps) explains what they did to solve the problem and that the narrative summary provides the why. Many students, though, had difficulty differentiating the “Do” section from the final narrative. But in order for their explanation to qualify as “high level,” they couldn’t simply state “100% – 20% = 80%”; they had to explain what that means. For example, they might say, “The discount rate subtracted from 100% gives the amount that I pay.”

An example of a student’s written explanation for this problem is shown in Figure 1.

For problems at this level, the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious. As the above example shows, the explanations may not offer the “why” of a particular procedure.

Under the rubric used at the middle school where this problem was given, explanations are ranked as “high”, “middle” or “low.” This particular explanation would probably fall in the “middle” category since it is unlikely that the statement “You need to subtract 100 -20 to get 80” would be deemed a “purposeful, mathematically-grounded written explanation.”

The “Need” and “Know” steps in the above process are not new and were advocated by Polya (1957) in his classic book “How to Solve It”. The “Need” and “Know” aspect of the explanatory technique at the middle school observed is a sensible one. But Polya’s book was about solving problems, not explaining or justifying how they were done. At the middle school, however, problem solving and explanation were intertwined, in the belief that the process of explanation leads to the solving of the problem. This conflation of problem solving and explanation is based on a popular educational theory that being aware of one’s thinking process — called “metacognition” – is part and parcel to problem solving (see Mayer, 1998).

Despite the goal of solving a problem and explaining it in one fell swoop, in many cases observed at the middle school, students solved the problem first and then added the explanation in the required format and rubric. It was not evident that the process of explanation enhanced problem solving ability. In fact, in talking with students at the school, many found the process tedious and said they would rather just “do the math” without having to write about it.

In general, there is no more evidence of “understanding” in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way. How do we know, for example, that a student isn’t simply repeating an explanation provided by the teacher or the textbook, thus exhibiting mere “rote learning” rather than “true understanding” of a problem-solving procedure?

**Requiring explanations undoes the conciseness of math**

Math learning is a progression from concrete to increasingly abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities – entities like dollars, percentages, groupings of pencils. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically-relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. That is, information and procedures handled by available schemas frees up working memory. With working memory less burdened, the student can focus on solving the problem at hand (Aditomo, 2009). Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of “mathematical understanding,” in other words, impede the “doing” of actual mathematics.

**“I can’t do this orally, only headily”**

The idea that students who do not demonstrate their strategies in words and pictures must not understand the underlying concepts assumes away a significant subpopulation of students whose verbal skills lag far behind their mathematical skills, such as non-native English speakers or students with specific language delays or language disorders. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain – whether orally or in written words – how they arrived at their answers.

Most exemplary are children on the autistic spectrum. As autism researcher Tony Attwood has observed, mathematics has special appeal to individuals with autism: it is, often, the school subject that best matches their cognitive strengths. Indeed, writing about Asperger’s Syndrome (a high functioning subtype of autism), Attwood notes that “the personalities of some of the great mathematicians include many of the characteristics of Asperger’s syndrome.” (Attwood, 2007)

And yet, Attwood adds (ibid.), many children on the autistic spectrum, even those who are mathematically gifted, struggle when asked to explain their answers. “The child can provide the correct answer to a mathematical problem,” he observes, “but not easily translate into speech the mental processes used to solve the problem.” Back in 1944, Hans Asperger, the Austrian pediatrician who first studied the condition that now bears his name, famously cited one of his patients as saying that, “I can’t do this orally, only headily” (Asperger, H., 1991 [1944]).

Writing from Australia decades later, a few years before the Common Core took hold in America, Attwood adds that it can “mystify teachers and lead to problems with tests when the person with Asperger’s syndrome is unable to explain his or her methods on the test or exam paper” (Attwood, 2007, p. 241). Here in post-Common Core America, this inability has morphed into an unprecedented liability.

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers – from multi-digit arithmetic through to multi-variable calculus – doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?

**What is really being measured?**

Measuring understanding, or learning in general, isn’t easy. What testing does is measure “markers” or byproducts of learning and understanding. Explaining answers is but one possible marker.

Another, quite simply, are the answers themselves. If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child’s verbal abilities. But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way. At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.

As Alfred North Whitehead famously put it about a century before the Common Core standards took hold:

It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.

———-

**Katharine Beals** is a lecturer at the University of Pennsylvania Graduate School of Education and an adjunct professor at the Drexel University School of Education. She is the author of Raising a Left-Brain Child in a Right-Brain World: Strategies for Helping Bright, Quirky, Socially Awkward Children to Thrive at Home and at School.

**Barry Garelick** has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long term substitute in a high school and middle school in California: “Teaching Math in the 21st Century”.

———-

**References**

Anindito Aditomo. Cognitive load theory and mathematics learning: A systematic review, Anima, Indonesian Psychological Journal, 2009, Vol. 24, No. 3, 207-217

Hans Asperger. “Problems of infantile autism,” Communication: Journal of the National Autistic Society, London 13, 45-32), 1991 [1944]).

Tony Attwood. The Complete Guide to Asperger’s Syndrome, Jessica Kingsley Publishers, Philadelphia, PA, 2007. (p. 240)

G. Pólya, How to Solve It: A New Aspect of Mathematical Method, Doubleday, Garden City, NY, 1957

Richard Mayer. Cognitive, metacognitive, and motivational aspects of problem solving, Instructional Science, 1998, March, Vol. 26, Issue 1-2, pp 49-63

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]]>“Teaching Math in the 21st Century”

By Barry Garelick; Introduction by Ze’ev Wurman; 2015, $9.95; 164 pages

As reviewed by Matthew Tabor, Editor, Education News

I spend my days working, and primarily reading, in education media. When it comes to Common Core, everyone is an expert and everyone has a solution (and a not-so-short list of villains). Hardly anyone bothers to write about what it’s like in the classroom — and to be fair, those who could usually can’t because of the professional consequences. It leaves us with a dearth of what is arguably the most important testimony.

In ‘Teaching Math in the 21st Century,’ Garelick, who has written on math education for The Atlantic, Education Next, Education News and others, takes advantage of a very unique set of circumstances. He’s an experienced insider as a math teacher; he’s an outsider as a school’s long-term substitute. He knows mathematics, having studied the actual discipline (i.e., not a watered-down math education degree). Since he’s a second-career teacher after having retired from the EPA, he understands how math is, and is not, applied in the workplace.

In short, Barry Garelick is the guy you want teaching your kid algebra.

Garelick has an impressive resume which, thankfully, he doesn’t rely on to offer another series of pronouncements about Common Core in a terribly-saturated intellectual marketplace. Instead, he lets his experience teaching math in the current climate of a largely confused, disorganized transition to Common Core in a California school district speak for itself.

Interactions with Sally, a district official armed with Common Core sound bytes and little substance, show just how nebulous Common Core implementation is in the average school district. Garelick’s accounts of how he matched proven, effective math instruction with muddy, CC-inspired district requirements shed light on how good teachers approach an uncertain landscape with equal parts reluctant-but-genuine compliance and cunning.

More importantly, Garelick shows how Common Core’s implementation affects students. The school is intent on reserving 8th grade Algebra for the “truly gifted,” a designation the district does not define. They admit to making up the selection process as they go along, which is comprised of a mini-gauntlet of testing the district says they can’t evaluate until they see the results. It’s a paradoxical mindset reminiscent of Nancy Pelosi advising that ‘we have to pass the bill so that you can find out what is in it.’

Garelick’s students and their parents are confused by the whole regime. Vignettes of several students, from struggling to solid, show just how difficult it is for the average 13 year old to get a sense of what’s going on in their math sequence — a point demonstrated best by a student Garelick describes as ‘bright’ ripping in half a placement test of ambiguous purpose. Garelick does a capable job of leading them through the swamp despite encountering several hurdles, from classroom management to interactions with individual students, that bring a refreshing honesty to the book (I’ve read enough self-congratulatory, humble-bragging teachers-as-superheroes accounts, and since I’m familiar with the realities of the classroom, I’m not interested in reading another.)

That Garelick’s account is based on teaching Algebra and pre-Algebra courses — the foundation for one’s future study of math — gives weight to his testimony.

Teaching in the 21st Century does not lay out every problem with Common Core math or the effects it may have on public education. It doesn’t address how to influence legislators and policymakers, or even how one might approach a local school board. It doesn’t advise you on the best math curriculum for your algebra-saddled student.

Instead, the book offers a brief glimpse into the eye of the storm that matters to kids, parents and teachers: the classroom as it functions under changing curricula and mindsets and how stakeholders deal with it. The book shows how great teachers are desperate to deliver a solid education in spite of proclamations from disconnected, poorly-grounded leaders; it shows how students just want to learn math and parents want to feel confident and informed about the education their kids are receiving.

That honest, unvarnished perspective — what educators really do in the face of a curricular sea change after having been given faulty maps, crude navigation devices and high expectations — is what we need more of right now in the education debate. In Teaching Math in the 21st Century, Garelick delivers it.

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