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How much detail should you include when providing example solutions for students that involve algebra? I realize that this will of course depend on the level that the students are at. So I guess my question is, how do you know when to show something like this:

\begin{align*} 4x - 3 &= 2x + 5\\ +3 & \hspace{2em} +3\\ 4x &= 2x + 8\\ -2x & \hspace{1em} -2x\\ 2x &= 8\\ x &= 4\\ \end{align*} versus showing a solution like this: \begin{align*} 4x-3 &= 2x+5\\ 2x &= 8\\ x &= 4 \end{align*}

I don't want to lose my slower students but also don't want to bore my students with too much unnecessary detail. I should add that this question is targeted at Algebra I level specifically, but a similar idea would apply across most levels of math.

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Quite closely related matheducators.stackexchange.com/questions/348/… –  quid Mar 29 at 0:11
    
Which kind of course are we talkning about? What do the students know already? –  Roland Mar 29 at 0:11
    
@Roland it is mentioned it's Algebra 1 (presumably referring to US high-school course, which is the second math high schoolcourse, I think) –  quid Mar 29 at 0:12
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6 Answers 6

The most effective manner is to ask the students to provide the steps as you carry them out at the board and use the resulting feedback to determine how much detail to provide going forward. That is, as you are writing at the board, you do not just carry on with the next step yourself, but rather, you say, "and then...?" until the students say "add three to both sides", when you do that and say, "and then...?" and a student shouts out "subtract $2x$ from both sides!" and so on.

The point is that you should make no steps yourself that are not directly provided by the students, and sometimes you have to have the patience to wait for a suggestion, or to prompt suggestions by making things easier. This serves several purposes:

  • Students become more actively engaged in the class, and learn the material more effectively. You can call on different students to explain their ideas, and everyone becomes involved. This is also a way of teaching at several levels at once, since the weaker students might understand the specific step, while advanced students appreciate the more general remarks you might explain surrounding it.

  • You get vital feedback about what the students find easy or difficult. The importance of this information cannot be overstated. If all the students are shouting out "add three to both sides!", then you know immediately that they know that part well and you needn't spend a lot of time on it; but if nobody can say "subtract $2x$" or the equivalent, then you know what needs more attention. And so when students have difficulty providing the step, you give more detail, and when it is easy for them, you may proceed more quickly.

This method works with instruction at virtually any level, whether it is calculus or linear algebra or advanced graduate-level topics. In my experience, many instructors can improve their teaching simply by adopting this one instructional technique, which gives them the important feedback on student understanding that they did not previously even realize they lacked.

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I do this too, but always in conjunction with anonymous feedback. When you are only listening for the students who speak-up in class, you (at least run the risk of) cater(ing) towards the stronger students. I think the most effective method is a combination of yours and mine. –  David Steinberg Mar 29 at 4:15
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@DavidSteinberg Could you explain what you mean by "in conjunction with anonymous feedback"? What I'm talking about is dozens (or hundreds) of verbal interactions in the classroom, and I'm not sure how that could easily be done anonymously. But I do agree that one must take care to ensure that a small group of students does not dominate the interaction. For example, I usually call on students who haven't contributed much yet in order to address this. –  JDH Mar 29 at 11:31
    
Ah, I see that you mention an anonymous email account in your answer. –  JDH Mar 29 at 13:00
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@DavidSteinberg, ask a random student if none comes forward. –  vonbrand Mar 29 at 22:38
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Answer 1: err on the side of too much detail. In my case at least, the students always want more detail.

Answer 2: I think the only way to know for sure is to ask your class what they are comfortable with. Students are often hesitant about volunteering that information, so I created an anonymous email account for them, so that they could provide feedback about this sort of thing (among others).

(p.s. in years of teaching, I have not yet received a single nasty email, either.)

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Of course, the students want more detail. However, a problem is if that is even good for their learning of they have a lot of formula instead of having a short example explaining how things work and what has to be done without having too much details? I have the feeling that some students do not to see the wood for the trees. –  Markus Klein Mar 29 at 7:50
    
Could you give us an idea of how much the anonymous account is used? How many messages do you get this way in a semester for a class of 25? –  JDH Mar 29 at 21:29
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200 students had access to the account last term, and I had 30 anonymous conversations (and many not anonymous conversations). –  David Steinberg Mar 30 at 2:26
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You just have to learn to "read" your audience. Getting them to speak up is usually hard, and most of the time too late. Do one step by little step, and start omitting steps until you see they aren't following anymore. Or start the other way, adding steps if you see lost faces.

Ask them to do (a part of) an example. The level of detail shown is a clue.

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The amount you write in the calculation of things should be inversely proportional to how many times the students have seen these calculations. The first example or two should have everything. Later examples should omit the easiest steps. I think a rule of thumb should be to ask yourself "Can the students show the missing steps in example N by looking at examples N-1 and below?". There are a couple of things I'd like to mention though.

  1. Are the examples shown in different lectures? Will the students still remember the previous lectures (perhaps, but not fully)? If there was homework assigned in between did the students do it (probably not)? In this case the decay of steps shown should probably be on the lower side. You can repeat an example from the last lecture with the same amount of detail so as to get students up to speed and make them more comfortable to go to the next step.

  2. Are the steps you want to omit touching on knowledge learned in the previous term/year (This is just like 1. but on a longer time scale)? If so, perhaps it is safe to assume that students don't remember perfectly and it's better to err on the side of repeating yourself.

I know you are targeting Algebra I specifically but I'd like to highlight that there are a lot of first year university students who are not comfortable with something like your second example (albeit the examples faced may involve a more complicated expression with fractions).

In general, when I tutor students the usual complaints are that teachers either don't show enough properly worked examples that they can then mimic and learn the material from or the examples are too easy and the exercises they need to work on are too complex comparatively.

  • don't show enough properly worked examples that they can then mimic and learn the material from
  • or the examples shown are too easy and the exercises they need to work on are too complex comparatively
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You can test, how detailed the students' understanding is, by using alternative exercises. Examples:

  • Give them a rather long solution without the transformations or with some steps missing. Ask them to provide the missing transformations or steps.
  • Give them a solution with mistakes in several places. Ask them to find and correct the solution. You should also give one solution with unnecessary or wrong leading transformations, which are nonetheless carried out correctly.
  • Let them talk each other trough a rather long solution and listen to what transformations and steps they name or leave out and which ones the student talked to asks for.
  • One students presents a solution uninterrupted and the others give feedback after a small break.
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By default, show the level of detail that you want to see from the average student in the class. Be willing to show more detail if students ask (at least, up to a point), but don't ever show any less detail than you'd want to see from them.

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"don't ever show any less detail than you'd want to see from them" Why? –  Benjamin Dickman Mar 29 at 4:14
    
I disagree on "don't ever show any less detail than you'd want to see from them.". For example, if you have a more complicated task which you trace back on, e.g., solving a linear equation, then I usally say: "Okay, you have been solving linear equations for years: You know how to do it - or ask me after we finished. Here, I will only give a the result." –  Markus Klein Mar 29 at 7:55
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The question said "when providing example solutions for students" so I read that as saying that the point is to model what a solution should look like. If you show less detail than you'd want to see from the students, you're providing a poor model and leaning toward telling the students to do as you say and not as you do regarding how much work they should show. –  Isaac Mar 29 at 11:11
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