Why do these errors bother me? Most of the errors that I have listed below are of the sort that are also made frequently by politicians, journalists, and various special interest groups. They are the kind of thing that keeps John Allen Paulos employed (he is the author of the book Innumeracy). I had thought that one of the goals of Core Plus was to help the students reason correctly in real-world mathematical settings. Unfortunately, the texts are filled with sloppy reasoning, wrong explanations, faulty physics, and so on. They don’t set a very good example.

I don’t buy the argument that these errors keep students on their toes. It’s true that a good student will be able to find and correct some of the errors. But the average student will either be misled by them without knowing it, or will be quite frustrated if the teacher has to keep explaining what’s wrong. As you will see below, many of these explanations are just too much for your average teenager. Furthermore, in cases where "real-world" applications are involved, the teacher may or may not have the necessary background to give such an explanation.

My examples are divided into two lists: (1) sloppy reasoning or wording (these often occur where the Teacher’s Guide gives a "model response"), and (2) errors associated with "real-world" applications. All references are to the page numbers in the student texts and Teacher’s Guides (the pages in the Teacher’s Guides are preceded by the letter "T", as in T101). For the examples from Core Plus 4, the references are to the prepublication manuscripts that I have.

These are amazingly common. A common pattern is this: the students are asked to discuss some issue that relates to a set of data, or a graph, or some sort of application. They are supposed to give reasons for their conclusions. As an aid to the teacher, the Teacher’s Guide gives model examples of supposedly correct reasoning. Unfortunately, many of these models are faulty. I find this alarming, since it is precisely this kind of thing that a good mathematical education is supposed to help the student avoid. Instead, the message from the Core Plus books seems to be that "anything goes", as long as it sounds good. Some of my examples don't fit this pattern. Instead, they are just faulty or dubious mathematics, plain and simple.

To illustrate my point about the high frequency of this kind of error, I take my first ten examples of sloppiness from the first 2 lessons of Core Plus 1A (about 47 pages).

1. Pages 8-10, questions 2d, 4d, and 6c (invalid conclusions from data, a mathematical error, and also an invalid comparison between two incomparable sets of data): The "Borda count" vote totals given in the tables on pages 8 and 10 are equivalent to average rankings (to see this, divide each vote total by the number of voters). In these questions, the students are asked to draw conclusions about variability in the rankings. While it is possible to draw certain limited conclusions about variability from a list of averages, this is not so easy. In particular, it is a mathematical error to use averages to try to show that there is a large amount of variability in the rankings of any given candidate. Yet this is exactly what is done in the Teacher’s Guide, where "model responses" are given. In the Teacher’s Guide response to 4d, there is no valid justification for the conclusion that "These people seem to agree only that U2 is a hot property", and indeed, such a conclusion cannot be drawn from the given data. The first half of the Teacher’s Guide response for 2d is entirely invalid, since it is easy to construct scenarios consistent with the reasons given such that the conclusion is false. Similarly, the last sentence in the Teacher’s Guide response to 6c is not a valid conclusion. Furthermore, the last three sentences in this response seem to imply that certain properties (including variability) of the two sets of data can be compared to each other. But the sets of "candidates" for the two data sets are quite different: for the first set, all popular music artists and groups were being voted on, while in the second set, only the top 20 artists or groups from the first set were included.
2. Page 12, question 8b (drawing a conclusion about one population when the data concerns a different population): The student is asked about the earnings of "recent high school graduates", but there are no data given for this group of people. The answer in the Teacher’s Guide is based on data about a different group, namely those people whose highest level of education is a "High school diploma only".
3. Page 17, paragraph prior to question 4 (bad definition): In the text, we read that "In a histogram, the horizontal axis is marked with a numerical scale." And then, just to make sure we understand that histograms must have this feature, the text provides us with an example that supposedly would not be appropriate as the horizontal axis of a histogram. This example is the graph on page 12, where the horizontal axis represents the highest attained level of education. But even though this axis is not labeled numerically, it would certainly be an appropriate horizontal axis for a histogram.
4. Page 19, first paragraph in Investigation 2 (weak definition): Here, and elsewhere, the impression is given that "approximately normal" means the same as "unimodal and approximately symmetric, with tapering tails". This characterization is never actually stated as a definition (hardly anything ever is in these texts), but it is reinforced repeatedly. Even in Core Plus 3B, where there is an entire unit about the normal distribution, we find the following question (page 377 of Core Plus 3B): "Is this distribution of heights approximately normal?" The Teacher’s Guide response on page T377 is: "Yes. The distribution is bell-shaped, symmetric, and clustered about the mean and tapers off at the ends of the distribution." To be fair, that unit in the Core Plus 3B text also contains some more sophisticated suggestions for characterizing the normal, but obviously those suggestions are not consistently applied, since they appear before page 377.
5. Page T24, response to question 1c (mathematical error): We read: "Surprisingly, since the values are smaller, the city mpg values are more spread out than the highway mpg." This statement seems to imply (because of the word "Surprisingly") that there is usually some correlation between average value and variability.
6. Page T32, response to question c (conclusions based on pure speculation): In what appears to be an attempt to promote skepticism about data that appear in the news media, a newspaper article from the Washington Post is criticized. We read: "For questions like the number of hours spent on homework, teenagers were probably just asked. That means this information isn’t very reliable." And "The data on the average time interacting with parents seem impossible to collect. How could this be done? If people know they are being watched or if they have been asked to collect this information, their behavior will probably not be typical of their usual behavior." This rather dim view of research in the social sciences borders on downright cynicism.
7. Lesson 2 in general (false impression given about usefulness of the median): There are several examples where the students are cautioned against indiscriminately using the mean, but they are never warned about similar difficulties with the median. For example, on page T37, the students are told that the mean is not useful for the data on page 37. However, the book seems to imply that the median is useful for this case. In fact, this is a good example where the median could also be misleading. If you look at the list of countries in the table on page 37, you see that they hardly belong together. The list of 20 countries in North America contains tiny ones like Barbados, Belize, St. Kitts-Nevis, Trinidad/Tobago, St. Vincent/Grenadines along with large ones like Mexico and the United States. With such lists, the median can be useless. There is no value in a statement like the one in the Teacher’s Guide: "The median of 64.5% means that in half of the countries in North America, more than 64.5% of the households own their own home." In general, if a list contains objects that should not be compared to each other, then the median can be useless. For instance, consider the median weight of the population of all individual living organisms on earth, where such a list contains all bacteria, viruses, plants, fish, humans, etc. There were many opportunities to make this point in Lesson 2. See, for example, the data from question 5b on page 42, where the text seems to accept both the mean and median, even though neither of them is very useful. Other opportunities would have been question 1d on pages 38-39 (see also the next error), and question 3f on page 41.
8. Pages 38 and 39, question 1d (bad answer to a reasonable question): The data on page 38 concerns the start-up costs for the 25 fastest-growing franchises. All types of businesses appear in the table (hotels, fast food restaurants, commercial cleaning, etc.). The question on page 39 is good: "Why might a measure of center of minimum start-up costs be somewhat misleading to a person who wanted to start a franchise?" One reasonable answer to this might be that the data concern a population of objects whose start-up costs are not really comparable to each other. Another answer might be that the start-up costs don’t tell the whole story, since they don’t indicate how fast a new owner might recover these costs. But the response in the Teacher’s Guide misses the point entirely: "These franchises are the fastest growing franchises. Thus we might expect them to cost more than a more typical franchise. So any measure of center of the top 25 franchises would not convey much information to a person interested in starting a franchise." I’m not even sure I agree with the premise of this response, which is that the fastest-growing franchises are more costly to start up. Isn’t it possible that they are on this list partly because their start-up costs are very reasonable? Also, wouldn’t a person starting a franchise be particularly interested in these franchises?
9. Page T41, response to question 4d (non-sequitur): Two lists are shown, giving the five highest countries for annual accidental deaths, and the five highest countries for annual accidental death rates. We read: "France and Poland are the only countries on both lists. Perhaps they have more fatal automobile accidents."
10. Page T41, response to question 3d (bad handling of significant digits): An estimate is made of the population of the US, based on the data in the table on page 40, which shows that 97,100 is 39.5% percent of the population. The Teacher’s Guide gives an answer with 9 significant digits, even though the figures used to make the calculation only have 3 significant digits each. This mistake occurs frequently throughout the Core Plus texts, even though students at this level are already supposed to know about such things as significant digits and rounding off.
11. Lesson 2, Unit 4 of Core Plus 3 (faulty mathematical proof -- circular reasoning): This unit is supposed to teach students about valid mathematical arguments. In Lesson 2, the students are led through a series of "proofs" concerning the minimal conditions that are needed to imply that two triangles are similar or congruent, such as the familiar side-angle-side (SAS) condition. The students are shown how to use either the Law of Sines or the Law of Cosines to derive these results. They are never told, however, that the proofs of the Laws of Sines and Cosines depend on the SAS condition (or on one of the other similarity or congruency conditions). In fact, the very definitions of sine and cosine (found in Core Plus 2) are only valid if you assume a condition like SAS. (In the usual axiomatic treatment of geometry, SAS is one of the axioms.)
12. Page 415, question 3 (silly use of calculator): This one is not really an error, but I still consider it very bad. This question concerns the problem of determining the probability that 6 independent observations taken from the same distribution will happen to come in increasing numerical order. The text suggests using the idea of permutations. Permutations were introduced on page 110, along with factorial notation. I now quote the first three parts of this question: "A. In how many different orders can the digits 1, 2, and 3 be listed? B. In how many different orders can the digits 1, 2, 3, and 4 be listed? C. Compute 4! and 3! using the factorial function on a calculator. Compare the calculator values of 4! and 3! to your answers in Parts A and B." (For the reader who is not familiar with factorial notation, 3! stands for the product 3 x 2 x 1, and 4! stands for the product 4 x 3 x 2 x 1.) Never once does the text suggest actually multiplying the numbers to get the values of 3! or 4!, either here or back on page 110 where the factorial notation is first introduced. I have actually had Core Plus students tell me that they don't know how to do such multiplication without a calculator.

These errors tend to be localized in certain units of the Core Plus texts. In particular, they are most likely to occur where physics is the "real-world context" of choice. Graphs that depict data vs. time are the most likely culprits. The Core Plus authors have real difficulties interpreting these graphs, or even drawing them correctly in the first place. And yet they purport to be teaching such skills to the students. Most of my examples are in Core Plus 1A or Core Plus 4 (Units 1 and 7), since I have not looked much at the rest. . Unlike the examples in my previous list, these errors were found through "spot-checks", rather than from any sort of systematic search. Since I used a pre-publication version of Core Plus 4, it will be interesting to see how many of these are corrected by the time the official version is published

1. Pages 99-100 of Core Plus 1A (bad physics; inappropriate linear model): the students are asked to model a bungee jump apparatus. They are supposed to experiment with a model made from rubber bands and fishing weights, and then plot their results on a graph depicting weight vs. stretch. On page T100, we read that "The graphic pattern that will occur from this experiment should be roughly linear (with exceptions at the extreme limit of the rubber band stretch)." In fact, it is possible to get an explicit expression for the relevant function by using conservation of energy, and this function is not linear. The exceptions mentioned in the book that occur in practice at the extreme limit of the rubber band stretch have nothing to do with this non-linearity. Rather, it is the result of the initial period of free-fall, and it is quite apparent for realistic values of weight and stretch.
2. Page 104 of Core Plus 1A, question 3b (graph depicting motion that is unrealistic in several ways, including infinite velocities): The graph is supposed to depict height vs. time for a person riding a ferris wheel. According to this graph, the ferris wheel makes 9 separate stops for 5 seconds each, and altogether it makes two full revolutions in 90 seconds. Since the ferris wheel has a diameter of 12 meters, the average speed around the outside of the wheel is about 1.67 meters/sec during the 45 seconds of motion, with the 5-second stops occurring every quarter revolution. These features indicate a somewhat unusual ride, but the worst features of the graph are several vertical tangents, which indicate infinite velocities. According to the graph, there are 9 separate places where approximately 2.5 meters of vertical motion occur during time intervals of approximately ½ sec. During these bursts, the wheel is moving about 3 times as fast as it does on the average during its 45 seconds of motion. This is particularly strange since the graph shows these intervals of unusual speed occurring immediately before or after dead stops.
3. Page 109 of Core Plus 1A, question 2 (bad calculation of earnings): A survey is conducted to determine the relationship between price and number of customers. But for this type of enterprise (a dunking booth at a school fair), you shouldn’t calculate the earnings by multiplying price by the number of customers, since repeat customers are quite important. Perhaps it was intended that a person being surveyed could say how many times they would buy a ticket, but no hint about this appears in the text or Teacher’s Guide, and such a survey would be difficult to conduct accurately.
4. Page 215 of Core Plus 1A (inappropriate linear model): The linear model given for this problem assumes that the company will use the same sized box for shipping 1 calculator or 20 calculators.
5. Page 246 of Core Plus 1A, question 3 (inappropriate linear model): Based on two data points, a linear model is found for sales vs. price. As you can see from the Teacher’s Guide, the domain includes the price x=0 (see the response for problem 3a), and extends at least to x=2.5. There is no mention of the absurdity of using a linear model over this domain, or of the inappropriateness of using two data points to determine a linear model. There are several other examples similar to this one.
6. Page 427 of Core Plus 3B, question 6d (bad physics): Finding this example was a little experiment: how long would it take me? I found it in less than 4 minutes after I opened the Core Plus 3B book for the first time. In the Teacher’s Guide response we read: "Since the gravitational pull will be stronger the closer you are to the surface of the earth, the atmospheric gases are more likely to stay close to the surface." This reason is entirely incorrect. In fact, even if the gravitational pull became stronger as you moved away from the earth (imagine each air molecule being attached to the surface by a spring), the atmospheric gases would still stay close to the surface. The change in the strength of the force is irrelevant. It is the change in potential energy that is relevant.

The remaining examples come from Core Plus 4, Units 1 and 7. For these, I don’t mention the Teacher’s Guide as much, but I checked it in each case to see if there was some mitigating explanation for the blunders. No such luck! Once again, I emphasize that these are the results of spot-checking. I didn’t look at other units in Core Plus 4, and I did not even go systematically through Units 1 and 7.

7. Pages 14-15 in Unit 1 of Core Plus 4, question 3 (highly unrealistic graph): On page 15 there is a graph that is supposed to give distance vs. time for a vehicle, as measured by a police radar detector. It is very difficult to imagine a scenario for which this graph would be accurate. Presumably, the x-intercept is supposed to be where the car passes the detector (the Teacher’s Guide says as much). This occurs around t=5 seconds, and between t=5 and 5=6, the car only moves about 10 meters. Taking into account the angle and position of the radar gun, this gives an approximate speed for the car of 10 miles per hour. During the next three seconds, the graph shows the car moving 100 meters, so that it has an average speed of about 75 miles per hour. And during the three seconds after that, the car moves almost 200 meters, with an average speed of about 150 miles per hour, and it is still accelerating! In any realistic scenario that I can imagine, the graph would look very much like a V, rather than like the parabolic figure in the book, even if the motorist slows down somewhat for the policeman (it is easy to derive the formulas explicitly for various cases).
8. Page 27 in Unit 1 of Core Plus 4 (bad modeling choice): This is another bungee jump example. Because of the periods of free-fall, the cosine function is not appropriate. But the more serious error is the damping factor 1/(t+1). This should be an exponential factor. After the first few seconds, this makes quite a difference.
9. Page 54 in Unit 1 of Core Plus 4 (graph showing highly unrealistic motion): This graph depicts the results of an acceleration test for an automobile. Note how the car goes from 0 to approximately 10 mph in a little over a tenth of a second. This gives an initial acceleration of about 4 G’s. Similarly, each time it changes gears, there is an incredibly quick increase in speed. When the vehicle changes into third gear, all of the acceleration occurs within the first second, during which time it goes from about 48 mph to 60 mph. Remarkable!
10. Page 61 in Unit 1 of Core Plus 4 (bad physics): We are back to bungee jumps. Note how the graph consists completely of straight line segments, with two alternating slopes. Since this graph depicts velocity vs. time, we have no choice but to conclude that the forces on the bungee jumper alternate between two constant values. Since the force exerted by the bungee cord is dependent on its stretch, this is clearly wrong.
11. Page 44 of Unit 7 of Core Plus 4, question 2 (bad electrical engineering): We read: "A two-phase alternating current circuit is supplied power by two generators." According to electrical engineers that I have consulted, you will not see a two-phase circuit, although one could be constructed. In practice, circuits are one-phase or three-phase. Also, it is somewhat strange to talk about a circuit being powered by multiple generators, although again it is possible (you find this, for example, when people with windmills sell energy to the power company; but it is quite tricky). In multi-phase circuits, it is customary to have a single generator that simultaneously produces separate currents for each of the phases. But the biggest error in this problem is the question itself, which asks the student to find the average voltage of the two generators. This is an entirely meaningless quantity for an alternating current circuit. In a multi-phase alternating current circuit, the voltages are applied to separate loops. So it makes no sense to either add them or average them. Note, for example, that if we take p=2 in the problem, we get 0 for the average value.
12. Page 56 of Unit 7 of Core Plus 4, problem 10 (ridiculous answer to a reasonable question): In this question, we find the function 37sin(2pi (x-101)/365) + 25. Since it is supposed to model temperature and since x is measured in days, the factor 2pi/365 is appropriate, so that the function has a period of 1 year. In problem 10a, the following (reasonable) question is asked: "How will you decide whether to set your calculator in degree or radian mode?" Here is the entirely absurd and incorrect answer in the Teacher’s Guide: "The calculator should be set in radian mode because it is clear that days are not measured in degrees." (see page 65 of Teacher’s Guide.