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I've been teaching remedial mathematics for the better part of a decade, and I've noticed a big trend in my classes lately. Many of my students are able to grasp the more complex ideas but still struggle with the basic facts (e.g., adding/subtracting/multiplying/dividing negative numbers; properties of 0 and 1; basic multiplication and long division).

My Question: Have other CC teachers noticed a similar trend? For those of you who have--and maybe have more experience dealing with this problem--what have you done when confronted with this issue?

Perhaps an example will help to clarify my question some more and to narrow down the broader questions. In some of my current remedial math courses, I took time out of the regular curriculum to focus specifically on the times table, properties of integer operations, and properties of the identities. Here's an excerpt from one of the worksheets I gave them.

Complete the following sentences by filling in the blanks.

A negative number multiplied or divided by another negative number is...

A positive number multiplied or divided by a negative number is...

...

A number divided by one is ...

A number divided by zero is ...

A number divided by itself is ...

Compute the answers to each of the following.

$1\div 1=$

$-1 \div (-1)=$

$-1\div 1=$

$1\div (-1)=$

$1\times 1=$

$-1\times (-1)=$

...

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    $\begingroup$ I like the question (though it is fairly broad), but I would avoid implicitly suggesting that this is somehow the fault of elementary school teachers. Surely, we can try to do something at all levels to buck this trend. $\endgroup$ – Brendan W. Sullivan Mar 25 '14 at 4:07
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    $\begingroup$ I'm not sure if we want to tackle the question of how to reform math from elementary to an undergraduate level... Would it be better to consider the question of arithmetic review on an undergraduate level? Or how to better prepare high school students for college math? $\endgroup$ – David G Mar 25 '14 at 4:11
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    $\begingroup$ Note, in particular, the excerpt (p. 175): :The bulk of remedial coursework takes place at community colleges. In 1995, the number of students taking a remedial mathematics course at a 4-year college or university was 222k; 10 years later, that number dropped to 201k, to rise again to 334k in 2010. The figures for 2-year colleges were 799k in 1995, 964k in 2005, and 1,150k in 2010 (Blair et al., 2013; Lutzer, Rodi, Kirkman, & Maxwell, 2007). That is, 2-year colleges enroll almost 4 times as many remedial mathematics students as do other institutions of higher education." $\endgroup$ – Benjamin Dickman Mar 25 '14 at 13:44
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    $\begingroup$ (Furthermore, the excerpt above is followed immediately with a foot-note: These figures should give us pause if we think of them as a metric for the success of K–12 mathematics education.) $\endgroup$ – Benjamin Dickman Mar 25 '14 at 13:45
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    $\begingroup$ The edited version seems like a suitable question, possibly still a bit broad, thus I voted to reopen. $\endgroup$ – quid Mar 26 '14 at 11:19
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The question is broad, so my answer will be as well.

Students who have been in school for a sufficient length of time have probably learned highly effective strategies for accomplishing their scholastic goals -- whatever those may be. You (it sounds to me) have identified that the learning strategies many of your students have adopted (and likely adopted for rational reasons) are ineffective for them when it comes to learning what you want them to learn.

Put pithily, this might also be said "what you are trying to teach is simply different from what they are trying to learn."

This cannot be simply corrected by informing them that their learning strategies are broken. Their learning strategies were adopted due to their success in the past, and most students will not re-evaluate these strategies until they witness their failure.

Now, generalities aside, how do you correct this? I use a bit of dishonesty. When I've identified a type of material that I want my students to learn, but that their learning strategies prevent them from learning, I take the following four step procedure in teaching that material:

  1. I teach the material, being explicit about the types of questions I expect them to be able to answer.
  2. I give them a sample exam/quiz question with the correct answer to it as a "study guide."
  3. I ask them in an exam/quiz a question that is almost exactly (or exactly) the same as the one on the study guide. Many students will fail at this point, despite having the study guide. This is (I believe) not mostly due to laziness, as we are quick to assume, but due to the students not recognizing that their learning strategies were ineffective. This failure represents their being confronted by that fact.
  4. Finally, the dishonest bit, I forgive the initial test (which I knew all along I would do, but they did not), and allow a second attempt at a nearly-identical test.

In my experience this procedure accomplishes two things: (a) it ends with a larger percentage of the students knowing the material and (b) they learn a new way to learn. Both of these are valuable.

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Generalizing from your personal experience with a small, local group of students to trends in "mathematics education in the US" is a tremendous leap.

The evidence you have for this "trend" could just as well be evidence that over the last 10 years you have become better at identifying students' weaknesses in arithmetic.

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    $\begingroup$ The belief that students are worse than they used to be seems extremely persistent across different times and places and contexts. In the absence of carefully controlled evidence, I am doubtful that it is a real effect rather than an artifact of selective memory. $\endgroup$ – Neil Strickland Mar 25 '14 at 19:40
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    $\begingroup$ @NeilStrickland, I've been hearing this complaint since I was a freshman (40 years ago). By the trend, we are teaching cabbages today... $\endgroup$ – vonbrand Mar 25 '14 at 21:08
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    $\begingroup$ We could certainly present evidence at our institution of the uniform exams that the university has given getting more and more rudimentary over several decades. $\endgroup$ – Daniel R. Collins May 15 '16 at 5:37
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There is probably not a single answer to this, but there are fads and fashions in the practice of elementary education. Some of the participants here have seen some of them come and go, and the value of practice and drill at basic skills has been questioned by many teachers as well as students. Without slighting the competence or dedication of teachers, there are some philosophies that are better grounded in reality than others.

A systematic failure in a particular educational philosophy may be difficult to detect immediately. It is more likely be detected at a later stage when the student needs the skills that once were taught at an earlier stage. Tutors and teachers in remedial education are more likely to encounter the failures than teachers of advanced mathematics.

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    $\begingroup$ "Teachers of advanced mathematics" just don't get to see those students, ever. $\endgroup$ – vonbrand Mar 25 '14 at 21:09
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My Question: Have other CC teachers noticed a similar trend? For those of you who have--and maybe have more experience dealing with this problem--what have you done when confronted with this issue?

I have definitely seen exactly this trend, in exactly those skills, in my two decades off-and-on of teaching at this same level. My response in the last year or so has been to develop a website designed to drill students on these basic skills with timed quizzes. (esp.: Times tables, negative numbers, and order of operations.)

For many it's the first time that an instructor has clarified that these skills are meant to be automatic; that is, not done by counting on their fingers (which is quite common). Giving exercises and personal feedback every half-hour in class gives me a chance to diagnose and direct students to the website for practice when I see these problems pop up. It also helps if I introduce the site in class on an overhead projector and run through a quiz collectively with the whole class calling out answers. See:

Automatic-Algebra.org

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Just what does dividing by a negative number mean in real life?.....

Some people are just better at things with real life meaning but are being forced to get a degree rather than train for a trade.

So are you seeing a trend in more people going to collage that would have done “on the job” training in past decades?

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    $\begingroup$ That may be the point. A well prepared algebra teacher ought to be able to explain a real-life application of positive and negative numbers. $\endgroup$ – Confutus Mar 27 '14 at 4:35
  • $\begingroup$ @Confutus, how would you explain dividing by a negative number, or multiplying two negative number together. (The other cases all come up in basic book keeping for example.) $\endgroup$ – Ian Mar 27 '14 at 9:25
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    $\begingroup$ Positive and negative numbers are used to indicate direction and distance forward and backward, above or below a reference point chosen as zero. Multiplication or division by a negative is change of direction. $\endgroup$ – Confutus Mar 27 '14 at 18:50
  • $\begingroup$ I think there are some (warranted) downvotes here because this answer addresses OP with another question, rather than really answering the original query. $\endgroup$ – Brendan W. Sullivan Apr 1 '14 at 20:52
  • $\begingroup$ I downvoted this answer because I don't think it answers the question. The question was about teaching basic concepts like how to perform operations on fractional notation or decimal notation, and I think so many jobs rely on that knowledge that it might be better to learn it in school, whereas I probably would have agreed that other information that's getting taught in university courses could instead be taught in a job that uses it. $\endgroup$ – Timothy Jan 24 at 6:59

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