July 13, 2012 | 14

Contact Anna Kuchment via email.

Follow Anna Kuchment on Twitter as @akuchment. Or visit their website.

Follow Anna Kuchment on Twitter as @akuchment. Or visit their website.

* *

*When William Schmidt, an expert on math education at Michigan State University, moved his family from East Lansing to Charlottesville, Virginia for a year’s research leave, his work took a personal turn. He noticed that the public school his daughters would be attending outside Charlottesville was academically behind the one they had attended in Michigan. Back home, his 2 ^{nd} grade daughter would be learning multiplication tables up through the number 5, yet in Charlottesville, multiplication was not even part of his local school’s second grade curriculum.*

*His daughter’s experience, he explains in a new book excerpted below, is not unique. “ The [American] system of schooling represents a game of chance that few are even aware is being played,” he writes in “Inequality for All: The Challenge of Unequal Opportunity in American Schools,” co- written with Curtis C. McKnight. The inequalities pose a risk to every child, they write, regardless of socioeconomic background or race. They stem from differences in state education standards, in school funding, in curricula that districts choose to adopt and in the content that individual classroom teachers choose to teach. In this excerpt, Schmidt and McKnight focus on variations in how math teachers are trained and how that, in turn, affects student achievement.*

*The following is excerpted from Inequality for All: The Challenge of Unequal Opportunity in American Schools, by William H. Schmidt and Curtis C. McKnight. (Teachers College Press, 2012).*

One thing that most of us remember best about school is our teachers. Thus, when solutions are proposed for reforming American schools in response to critical reports or disappointing test results, teachers are always among the first to be singled out. Proposals often turn first to improving the teaching force by focusing on higher quality. For example, in the NCLB [No Child Left Behind] era, considerable emphasis has been placed on a “highly qualified” teaching force.

Districts must certify what percentage of their teachers is highly qualified. However, the states and school districts define what they consider “highly qualified,” resulting in a great deal of ambiguity.

What does it mean to be a highly qualified teacher? The definition of a high-quality mathematics teacher has never been standardized. Therefore, although improving the quality of teachers and teaching is a common cry when we seek to improve schools, there is little agreement and scant empirical evidence that indicates what characteristics define a high-quality mathematics teacher. Even an obvious definition, such as a knowledge of mathematics, is problematic, since there is generally no agreement as to what specific mathematics knowledge is needed.

The literature identifies two types of knowledge that are clearly related to providing opportunities to learn: mathematics content knowledge and pedagogical content knowledge. For mathematics, recent empirical work has advanced our understanding of what mathematics knowledge is necessary for teaching mathematics. The rest of this chapter lays out how this knowledge—in particular, mathematics content knowledge—is related to inequalities in content coverage, and provides data related to teacher content knowledge for a sample of teachers.

**What Teachers Tell Us About Their Knowledge of Mathematics**

We approached the question of teacher knowledge of specific mathematics content indirectly, by asking a sample of more than 4,000 teachers from the PROM/SE project [Promoting Rigorous Outcomes in Math and Science Education] to respond to the question, “How well prepared academically do you feel you are—that is, you feel you have the necessary disciplinary coursework and understanding—to teach each of the following?” We asked this question for multiple mathematics topics. The list of topics varied for teachers in primary and teachers in middle and secondary school combined.

By relying on teachers’ reports of their own feelings of adequate preparation, we only get at their knowledge indirectly. Fortunately, this approach is sufficient to demonstrate how much variation there is in teachers’ content-specific knowledge, or at least in their feelings of adequate preparation. Furthermore, the candor of the results suggests a degree of face validity and, hence, integrity in the responses. The overall tenor of the responses is very consistent with other data on the issue, some of which suggest that the pattern reported here might be a best-case scenario. All results reported in this chapter are based on the PROM/SE data.

**Primary Teachers (1st Through 3rd Grades)**

Primary teachers felt academically prepared to teach only the topics they taught to their students. Even for those topics, about one-fourth to one-half of the teachers surveyed reported that they did not feel well prepared. The teachers we surveyed were from 60 PROM/SE districts located in Michigan and Ohio.

Is it reasonable for teachers to focus only on the topics that they will teach? However reasonable such a position may appear, many of the more advanced topics for which teachers did not feel well prepared provide the mathematics background necessary to be truly well prepared to teach the more elementary topics at their grade level. To define a qualified teaching force, we adopted a criterion of 75% of teachers feeling well prepared to teach a given topic. We found that, over all sampled teachers, only two mathematics topics met this criterion: the meaning of whole numbers, including place value and operations with whole numbers.

Virtually all of the geometry topics (aside from the basics) are excluded by the 75% criterion. So are all of the proportionality topics and all of the algebra topics. These results imply that the quality of learning opportunities surrounding many of the mathematics topics taught in 1st through 3rd grades was not likely to be high. They also suggest that there is large variability in self-reported content-specific knowledge. For many of these topics, only a bare majority of 50% to 60% of teachers felt well prepared.

The other striking feature of the results was the large variability across districts. For example, for fractions, in some districts all of the primary teachers felt very well prepared, while in other districts only about half of the teachers felt very well prepared. For geometry basics (lines, angles, and so on), the results ranged from one district with only about one-fourth of its teachers feeling well prepared to another district in which about 90% of the teachers felt well prepared.

**Upper Elementary Teachers (4th Through 5th Grades)**

The results for districts for 4th- and 5th-grade teachers were quite different. For example, for eight different topics, all of the teachers in at least one district felt very well prepared academically for each of those topics.

At the district level, the results for whole number meaning and operations were similar to those for 1st- through 3rd-grade teachers. Further, fractions also had a median value around 75%.

However, the variability across districts remains a striking feature for 4th- and 5th-grade teachers, particularly for decimals, percentages, and geometry basics. These are all topics that were supposed to be introduced in these grades in Michigan and Ohio. For example, for decimals, in one district only one-fourth of the teachers felt well prepared, while in another district virtually all teachers indicated that they felt well prepared to teach decimals.

**Middle School Teachers (6th Through 8th Grades)**

We examined the pool of teachers from all of the districts taken together. From this perspective, there were no topics that at least 75% of the teachers felt very well prepared to teach. Only two topics came close. Among the whole pool of teachers, 73% indicated that they felt well prepared to teach the topic of coordinates and lines. Sixty-nine percent of the teachers indicated that they felt well prepared to teach the topic of data.

Eleven topics qualify if we relax the criterion to topics in which at least 50% of the teachers felt well prepared. This included the two topics just mentioned as well as nine others—negative, rational, and real numbers; exponents, roots, and radicals; number theory; polygons and circles; congruence and similarity; proportionality problems; patterns and relations; expressions and simple equations; and linear equalities and inequalities. The Michigan and Ohio standards call for including many of these topics at the middle grades. The fact that only about 50% to 60% of the teachers felt very well prepared to teach these topics suggested something of the magnitude of the problem that school districts face.

For example, there has been a strong national movement to include elementary algebra topics in the middle school, particularly in 8th grade. The Michigan and Ohio standards reflect this, as do the Common Core State Standards, which are in the process of becoming the new Michigan and Ohio state standards. Adoption of the Common Core State Standards brings states more into alignment with international benchmarks of what is expected in the equivalent of middle school.

The severity of the problems faced by these districts and, by inference, by the United States as a whole, was indicated by the fact that only about half of the teachers felt academically very well prepared to teach expressions and simple equations, as well as linear equalities and inequalities. Even fewer teachers (only around 25% to 40%) felt they had adequate content knowledge to teach other important algebraic concepts, including proportionality (41% of teachers), slope (38%), and functions (39%).

**High School Teachers**

The story for high school teachers is rather different, which is not unexpected given their typically greater preparation in mathematics. Almost 60% of the topics met the criterion of having at least 75% of the pool of PROM/SE teachers from the 60 districts indicating that they were well prepared academically. The areas in which high school teachers indicated that they felt less well prepared were number bases, three-dimensional geometry, geometric transformations, logarithmic and trigonometric functions, probability, and calculus. These findings are, however, still cause for concern. For example, there is an increasingly strong push for the inclusion of probability and statistics in high school, as is found in the Common Core State Standards, yet less than half of the surveyed mathematics teachers felt well prepared to teach it. Teachers’ self-perceptions of their preparedness seem likely, if anything, to overestimate what they know and how well prepared they are rather than to underestimate it.

Moving from the pool of all 60 districts to the district-by-district results for a large number of topics (16), at least 25% of the districts had all of their high school teachers indicating that they felt well prepared to teach those topics. However, there was still great variation across the districts, especially for geometry topics including transformations, three-dimensional geometry, polygons, and circles. There was similarly great variability in the percentage of teachers who felt that they had the coursework to make them well prepared to deal with calculus, probability, number theory, and logarithmic and trigonometric functions.

**Why Teachers Feel So Poorly Prepared**

We have surveyed how well prepared in terms of disciplinary course work teachers at various levels felt for teaching various mathematics topics in what is a fairly representative sample of 60 districts. In general, we would summarize the findings by stating that many teachers felt ill prepared to teach mathematics topics that are in state standards and in the new Common Core State Standards for mathematics. Why did these teachers feel so ill prepared?

There is perhaps a simple answer for the elementary and middle school teachers: They felt ill prepared because if we examine the coursework they studied during their teacher preparation, they were ill prepared. The new TEDS study results suggested this to be the case more generally, which clearly does not bode well for equality of learning experiences for students in these districts.

**College-Level Preparation**

In this section, we summarize what teachers have told us about their preparation in mathematics at the college level and as graduate students.

In 1st through 4th grades, less than 10% have either a major or a minor in mathematics. Teachers at this level are typically generalists—they must be prepared to teach many different subject-matter areas. They do not have adequate time in their preparation to get a major or a minor in each of those subject matters.

At 4th grade, the international data paint a different picture. Unfortunately, the definitions are not precisely the same, but the data do provide us with a benchmark of sorts. Including those primary teachers with either a mathematics major or a minor in mathematics or science, around one-third of 4th-grade students on average had such a teacher in the countries that took part in TIMSS, [the Trends in International Mathematics and Science Study].

Taking this estimate from the TIMSS data as an indicator of the percentage of teachers who had majored or minored in mathematics or science, this proportion was considerably higher than for the PROM/SE 4th-grade teachers, where the comparable percentage was 5%. The percentage was over 50% in Singapore and Russia. This suggests that, from an international perspective, other countries typically have around six times as many primary teachers who have a specialization in mathematics or in a related field of science.

The result is even more disturbing when we turn to the middle school mathematics curriculum and the higher level of mathematics offered there.

Three out of four middle school teachers in the PROM/SE sample did not have a specialization in mathematics. At 6th grade, the percentage was much like that for primary teachers—only around 10% had a major or minor in mathematics. In 7th and 8th grade, this percentage increased to around 35% to 40%.

These numbers indicate that a very large percentage of middle school students were being taught increasingly more complex mathematics, as called for in the Michigan and Ohio state standards, by teachers who lacked a strong background in mathematics. These results offer one explanation for why so many middle school teachers did not feel very well prepared to teach many of the middle school topics discussed in the previous section. This also foreshadows problems of implementation, at least as the newly adopted Common Core State Standards are put in place, in Michigan and Ohio.

What about high school mathematics teachers? We would expect that all high school mathematics teachers would have at least a minor in mathematics, if not a major. But the actual results for high school are quite surprising. Less than half of all high school mathematics teachers surveyed had a major in mathematics. Almost one-third did not have either a major or a minor in mathematics.

These numbers varied across the four grades of high school taught by the surveyed teachers. Almost one-half of the teachers whose major teaching responsibilities were at 9th or 10th grade did not have any specialization in mathematics. In 11th and 12th grades, over 71% of the teachers who taught primarily at those grades had some kind of specialization in mathematics.

Lest it seem too heartening that those teaching the most advanced courses (usually taken in the 11th and 12th grades) are better prepared in mathematics, we need to consider several caveats. It may be even more important to have well prepared teachers in entry-level courses usually taught in 9th and 10th grades. These courses serve as the foundation for more-advanced courses, may be even more difficult to teach, and are just as important in terms of preparing students for further study. But for these foundational courses, teacher content knowledge was not nearly as strong. It is worth noting that on some of the more advanced mathematics topics (number theory, geometric transformations, logarithmic and trigonometric functions, and calculus) up to half of the teachers did not feel very well prepared to teach them. Perhaps these same 50% were those who did not have a major in mathematics.

**Mathematics Knowledge**

One key part of the PROM/SE project was planning and carrying out content-based capacity building for teachers. As a part of this component, we administered a test of mathematics knowledge to a sample of teachers. [The results] strongly suggest that elementary and middle school teachers perceived their weaknesses accurately and reported them honestly. They appeared to be reporting that they were not well prepared academically to teach the mathematics content that they were being asked to teach.

Across grades, the percentage of teachers who did not have a major or minor in mathematics ranged from nearly all of the teachers at 1st grade to around one-half of them at 8th grade. These same teachers were able to answer correctly only about half of the items, as compared with teachers with mathematics majors who were teaching at the corresponding grades. The teachers with mathematics majors were able to correctly answer about 70% of the same items. This gap of almost 20% is sizable and very important. It confirms what the teachers told us when they said that they were not well prepared.

The problem at high school is more a problem of variability. The data indicated that most of the teachers had mathematics majors and that their mathematics knowledge was reasonably good. However, about one-third still did not have strong academic preparation.

**The Effects of Teachers’ Mathematics Knowledge on Opportunities to Learn**

Given these results about teachers’ mathematics knowledge, it is tempting to blame elementary and middle school teachers for not being prepared, but we believe blaming teachers is a mistake. Why? Because teachers prepare themselves according to the standards and guidelines established by the states that certify them and the teacher preparation programs that train them. Our point here is that such variation in academic content knowledge is likely to affect the quality of content coverage. Since the content coverage described previously varied appreciably, these data indicate one possible reason for such variation as it is very likely that this lack of knowledge influences not only the quality of the coverage of particular topics but also the bigger picture as to how the teacher makes choices about which topics to cover, for how long (to what depth), and in what sequence. Such

lack of knowledge further exacerbates the variation in content coverage in mathematics across classrooms, schools, and districts, resulting in further inequalities in opportunities to learn.

Used with permission from the Publisher. From Schmidt & McKnight, Inequality for All: The Challenge of Unequal Opportunity in American Schools, New York: Teachers College Press, © 2012 by Teachers College, Columbia University. All rights reserved.

Rights & Permissions

Add a Comment

You must sign in or register as a ScientificAmerican.com member to submit a comment.

our head teacher requires us teachers to be generalists so she can easily reallocate timetables or replace sick teachers.

The problem with teaching different subjects every semester is we feel unable to achieve any real depth of understanding in each subject – just musical chairs.

Link to thisultimobo,

“we feel unable to achieve any real depth of understanding in each subject”

Are you suggesting that you do not already have a mastery of all of the mathematics subject matter at the elementary and secondary education levels?

Link to thisI think part of the problem is that, to achieve a real understanding of math and science requires a good deal of studying on one’s own: it cannot be gotten by group discussions as in subjects like english or history. So it requires a certain amount of introversion to be able to do this. Teaching, on the other hand, is about dealing with people full time, so requires a lot of extroversion. So we have a conflict in personality type which is hard to find. My own experience bears this out. The most knowledgeable math and sciences teachers I had just were not nearly as interesting or fun to learn from as the best teachers in the social studies subjects.

Link to thisI know how he feels. My ex took my children to the Lansing area from the Washington, DC area. They are far behind where they were in Maryland schools. NCLB may seem like a boondoggle to many, and it has not been well-implemented, but quite apparently, many states and school districts are shorting both our children and our country.

Link to thisQuote: Back home, his 2nd grade daughter would be learning multiplication tables up through the number 5, yet in Charlottesville, multiplication was not even part of his local school’s second grade curriculum.

Unquote.

While teacher prep is not what it should be, it’s hard to lay off the problem in Charlettesville on lack of teacher preparation. OTH, it’s easy to figure out the source of this particular problem, and equally easy to figure out why nothing will be done about it for many years.

Link to thisI hate to challenge a Michigan State colleague, but I work with schools both in Michigan and “around Charlottesville, Virginia,” and I disagree with a number of the assumptions suggested here.

The first, of course, is that I’m with Conrad Wolfram on the concept that “order” is really not important in maths education (you can find Wolfram’s TED talk on YouTube or via my blog – http://speedchange.blogspot.com/2012/01/changing-gears-2012-maths-are-creative.html in January 2012 “Changing Gears: Maths are creative, Maths are not arithmetic” post) and thus I strongly disagree with those who might measure either students or schools according to some pre-determined pacing guide.

The other assumption which I find flawed is the comparative measurement structure suggested here. I see low quality maths instruction in many places, I see high quality maths instruction many places, but mostly I see that high quality maths instruction is suggested when students have a deep grasp of mathematics concepts no matter what their arithmetical skills are… and this I am quite certain we have been building in the schools around Charlottesville (see my blog, http://speedchange.blogspot.com/2011/12/among-schoolchildren-december-2011.html “Among Schoolchildren”).

We are doing that by embracing a different concept of professional development, one based in “teacher entrepreneurship,” social engagement, and grounded theory action research combined with a view that we look at the whole student “result” not any single measure. We do this because we believe that students are humans who develop different skills at different rates.

As for TIMSS and other “international comparatives,” I wish to draw everyone’s attention to the work of Yong Zhao who demonstrates the inverse relationship between international standardized test score results and national economic creativity http://www.slideshare.net/irasocol/ictedu-2012-keynote-lit-tipperary-part-2 (see slide8/8). The nations which score lowest on these exams lead the world in patents per capita, and new product and service introductions per capita. It is the choice between being a nation which invents iPads and a nation which builds iPads at slave labor prices.

“Around Charlottesville” our goal is to raise a generation ready to lead, to invent, to solve problems. It is not our goal to raise test takers. And it is our goal to help teachers develop into educators who can support that mission.

Link to thisA striking omission in the article is any mention of the quality of the instructional materials. As a college teacher I am extremely grateful that I have the privilege of choosing my own textbooks. A good textbook allows me to simply, “follow the book”: the explanations are good, the examples and assignments are appropriate, and I don’t have to spend much time figuring out supplemental or remedial material for my students.

When my children were in elementary school, they were issued the new math textbooks purchased by the district. The each page was profusely illustrated, had multi-color callouts and resembled nothing more than People magazine. Unlike People, the illustrations typically had nothing to do with the text and were all taken from a stock photo houses. The supplemental materials such as homework assignments were provided by the publisher, not the author, and were often wrong and of poor quality.

When my son qualified for the Advance Match class he was issued a different, 20 year old textbook. This textbook had no illustrations and was all in black and white. The other, however, had clearly diagrammed the concepts he was teaching. Of course, this textbook was much easier to understand.

Individual teachers at the elementary school level exercise little influence on which textbooks are assigned. Acting collectively, through teacher’s unions, they exercise a great deal of power. I often wonder why they don’t use this power to influence textbook quality – the tools they use every day.

Link to thisIt shouldn’t be surprising that teachers feel unprepared when the curriculum changes from that which they were trained to teach. But a professional teacher should be able to learn new curriculum. I hope that they are able to teach themselves – that is what a true education does.

But there is a bigger problem with how math is taught and how math curriculum is developed — is is done by mathematicians who have only been taught mathematics.

Mathematics for the 99% is an applied science, not a theoretical one. Math has been developed to DO things. But math is often taught extracted and isolated from the reasons it exists. True math education ends up getting taught when it is applied in science or business.

I took an advanced math course in graduate school that is normally taken as an undergrad. As I went through the course I’d recognize many equations from my physics and chemistry work. But there was no context in the book, and the professor had no idea what contexts were there.

At the same time, students don’t seem to make the connection between the theory of math and the use of it. As a teaching assistant for an introductory chemistry class, I had a visit by one student for some help with a lab. I started to explain how to extract the slope and intercept from the data. He looked at me, almost with anger, and said (I’m not kidding here) “what is slope and intercept doing in chemistry” !

Link to this@mihondo said it! “…students don’t seem to make the connection between the theory of math and the use of it.”

I struggled mightily with math up through calculus, and algebra and calculus are still like Greek to me. However, when I took physics, I easily grasped the necessary mathematics, to the point where I actually tested out of 8 hours of university-level laboratory physics.

‘Nuff said.

Link to thisNorm Augustine, former CEO/Chairmen of Lockheed Martin, chaired a committee (The Gathering Storm) to renew America’s economic competiveness. It addressed the need for better math and science achievement in the public schools. It reflected the teacher problem you have described. His group attributed it to the very low percentage of teachers who had degrees in math and the hard sciences and a general lack of proficiency in them.

He said if teachers are uncomfortable teaching math, the students won’t generate the enthusiasm to master it.

Recognizing the problem is one thing, fixing it is another. Augustine’s committee recommended developing a subsantial cadre of math and hard science teachers who had majored in those subjects. To accomplish it, they recommended providing full scholarships for math and science majors if they agreed to commit to 5 years of public school teaching. Even with this he told our group he expected it would be difficult to recruit unless a supplement were paid for teaching. That he said would be difficult to sell to the unions.

I would hope Scientific American could offer some workable solutions to the problem. Most agree we have one. Staying on the same course won’t solve it.

Link to thisI have an idea. Why don’t we give teachers 1/4 of the year off so they can work on improving their skills?

I’m an engineer. Very little of what I learned in college thirty years ago is useful due to changing technology and practices. Everything I use at work now is from things that I taught myself over the years.

Link to thisI feel the need to review the comments here, and consider the realities:

Link to thisIn #10 Corky Boyd writes, “Augustine’s committee recommended developing a substantial cadre of math and hard science teachers who had majored in those subjects. To accomplish it, they recommended providing full scholarships for math and science majors if they agreed to commit to 5 years of public school teaching.”

This is the “Teach for America” idea – that content knowledge trumps pedagogy. Yet we know that this specifically does not work in mathematics. If it did, then I could ask any 25-year-old who had attended university to tell me about “College Algebra” and they would prove highly knowledgable. After all, they were taught by exactly the cadre Augustine’s committee was proposing. But the fact is, if I was to ask 25-year-old to converse about their university mathematics courses they will stare blankly at me, because they absorbed absolutely nothing.

On the other hand in Comment #9 “Arynix” explains the issue. Like “Arynix” I failed high school maths and struggled with basic university maths, but I had absolutely no problem with structural engineering. When the mathematics are taught around interests, and in project-based form (as they often are in physics, chemistry, programming, and engineering), students succeed. When we try to explain things like Algebra – or even multiplication tables – without giving anyone any reason why they should be learned, students shut down – for very good reasons.

“Mihondo” in #8 notes this. I believe that maths are both applied and conceptual, and that we can bring both to students, but not through the “training regime” Schmidt suggests nor through the textbooks “Bgulino” (#7) suggests. In one of those Michigan districts I work with mathematics are usually done outside. We can easily prove the value of understanding triangles if we start to build, we can easily measure size and volume if we begin to look around – say (in this instance) farms.

This experience argues against “TTLG’s” (#3) “lone scholar” theory. Certainly there are students who will do this on their own, but it is far more common for humans to benefit strongly from social learning environments. This is as true in mathematics as it is in any other subject, we learn from each other, we push each other, we help each other. I see this not just in elementary schools but in graduate mathematics departments at universities.

What is missing, however, from this conversation is the “why,” as in “why should students learn mathematics?” We rarely explain this beyond the nonsense of typical story problems. Why do we memorize “times tables” when we have easy solutions to not committing them to memory? (I encourage students to know what they can, but also to solve what they can’t recall – What is 5×9? For me it is 5×10 – 5×1.) Why do we teach Algebra, a system of logic used to discover unknowns from knowns, almost exclusively with numbers and arcane mathematical symbols? Wouldn’t an episode of House, MD or Sherlock prove far more effective for most learners? Why don’t we tell students from the start that mathematics – yes even 2+2=4 – is a construct of rules and theories, so they are not put off by the fact that what we are saying is, quite obviously, not always true?

If there is a “maths issue” “around Charlottesville” or across North America it is not an issue of teacher content training, or one of textbooks, or one of “standards,” it is a basic question of the need to shake off the antiquated pedagogies of 19th Century industrialism and to embrace the concept of “education” rather than the concept of “training.”

I have been teaching mathematics at virtually all levels for more than fifty years. In my experience the content of the article and the comments are for the most part right on. I find that the preparation of my community college students to be appalling. When I ask my students if they have ever had a teacher (who was teaching them math) tell them that math was their worst subject, half of my students raise their hands. How do these teachers expect their students to believe they can learn math when their teachers are essentially telling that they themselves had a hard time doing so.

Indeed students need to be able to apply their understanding to the real world, just like their language skills have to relate their experiences. Yet most of them see math as a manipulation of symbols without any understanding of the meaning of those symbols. That is akin to saying that grammar is more important than content. My students complain that I don’t teach them nemonic techniques like the FOIL method when what they need to understand is the distributive property. They also shy away from word problems which is the essential to understanding math applications.

Pedagogy is very important but it can not replace understanding deeply the content. I was one of those math teachers who was put into a math class without any training in how to teach math. I had to learn it on my own. The incredible advances in brain functioning has had a profound influence in how I now teach. (Thanks to Scientific American) When students tell me they can’t learn math I tell them that chickens can count. There is no reason that our students and especially our teachers can’t be totally prepared to teach all math at least up through calculus and statistics.

Statistics is often seen by university math departments as peripheral to mathematics. At the university where I spent most of my career, students were required to take calculus or statistics and the math department pushed calculus and only included statistics at my insistence. Every elementary and secondary teacher needs to have taken statistics. Much of our scientific knowledge in all sciences is statistical in nature. Unfortunately, most teachers are Inadequately prepared and do not understand things like confidence intervals or margin of error.

Finally, learning math does take discipline and motivation. Teaching my students the math of finance often gets their attention, especially compound interest. My experience is that often teachers of math, especially at the elementary level, do not teach the understanding of math, but teach the tricks to passing the standardized exams at that level. So sequence is important given that our students do move.

Link to thisA brilliant article that’s rather disturbing. I get an extra shudder out of it, since I feel that if you replaced the word “mathematics” with “science” the article would read almost identically.

Link to this-Undergraduate in BS-Secondary Education – Biology.

-Husband is a BS Mathematics, graduate student PhD mathematics.