Compilation: Linearizing data
Compilation: Linearizing data
Date: Sun, 16 Jul 2006
From: Laura Sloma
Subject: Linearization
I e-mentor a newer physics teacher and he recently asked me this question: " I'm debating the usefulness of teaching students how to linearize data during data analysis for the labs. I know that it is useful in determining units for the derived coefficients, but I'm wondering if it is worth all of the effort when so many students are familiar with how to use a calculator or computer to get and equation." What are other people's thoughts on this?
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Date: Mon, 17 Jul 2006
From: "Malone, John"
Keep linearizing during labs and for data analysis, but do try to give them a variety of relationship types to keep them alert. Linearizing can be repetitive, but if it gets boring, the solution is not to stop linearizing, but create activities where there is need for matching data patterns to a model.
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Date: Tue, 18 Jul 2006
From: Paul Bianchi
I think it's very important, because it makes explicit a very important step in data analysis. One of the most important reasons for doing labs and taking data (for me) is for the students to derive an important formula in a concrete and meaningful way. To let a calculator do the linearizing is to throw in a "magic step" that suddenly makes the resulting formula somewhat arbitrary in the students' minds.
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Date: Wed, 19 Jul 2006
From: Aaron Titus
I do not disagree that there may be strong pedagogical reasons for linearizing data. However, this strikes me as a similar issue to so many other issues regarding technology. Do we use the tool effectively and focus on application of the results, or do we not use the tool so that we can understand some "idea" (in this case linear relationships) more clearly? A question is whether students should use a computer to do the linear regression. Students can understand more clearly what the linear regression is all about if they do the calculation by hand and/or use a program that calculates the "squares of the differences" for various lines that the student draws. Then, the student can find the "least squares" by manually adjusting the line. While I think that students calculating least squares by hand and by "trial and error" of drawing lines, I do not do it in my class because it simply takes too much time. It's not as important as the other topics that I wish to teach. As a result, I have students use the linear regression tool in Logger Pro. For similar reasons, I do not ask students to linearize the data. But rather, I ask them to write a function and fit it to the data. That is, in Logger Pro, they have to define the function (such as y=Asqrt (x) for example) that they will fit to the data. Logger Pro then calculates the constants.
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Date: Wed, 19 Jul 2006
From: Frank Noschese
This year I am going to try a new approach, which seems to combine the pros of both the "automatic" and "hand-drawn" method of finding best fit lines... Inspired by what I read in the book A Den of Inquiry http://www.denofinquiry.com students will use Logger Pro 3.4 to get best-fit equations as follows:
1. Enter data as usual to plot the data points.
2. From the "Analyze" menu, choose "MODEL" (how appropriate!)
3. Pick the general form of the function they think best applies and click "OK".
4. In the Manual Fit box that appears on the graph, they click each coefficient and guess their values. They can then tweak the coefficients using the up/down arrows (this is cool b/c the shape of the function changes instantly on the screen...great links to precalculus).
5. Stop when they get a graph that fits "good enough".
The equation does not have to be perfect, in my opinion. Science isn't perfect. (The automatic fit isn't perfect, either). The process of seeing how the coefficients control the function is important. Making a good guess as to what they might be (and HOW they reasoned the guess) is important. You can even have students produce THREE curves of "best fit", ala Goldilocks and the Three Bears: one that is acceptable, but too low, another too high, and the one they feel is "just right". That way, they can get a "just right" value for the slope and an upper and lower limit; much cooler than using esoteric error propagation formulas. Another suggestion made in the Den of Inquiry book is to NOT AVERAGE data for repeated trials, but to have students plot ALL of their data on the same graph. This gives the appearance of error-bars and students can easily see which trials are outliers or don't fit in well with the others. I hope all that made sense. Please let me know if I need to clarify or if you have any questions...or if you disagree! And check out the book! There are sample chapters on the website...especially the "software training" section...it gives detailed student directions with screen shots for the process I just described. All the activities are very modeling-friendly!!
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Date: Sun, 23 Jul 2006
From: John Clement
The issue of understanding the linear regression is actually a very important issue. There is a strong correlation between the Lawson Classroom Test of Scientific Reasoning and the scores on the FCI. Several items account for most of this difference. The ability to do proportional reasoning and two-variable reasoning account for most of the strong correlation. As a result, it is vital to increase both of these. Linearization gives the students more experience with proportional reasoning, so it may have some effect on this vital ability. But apparently it is not enough, because this ability does not seem to rise much even when linearization is used. The other pedagogical reason for using linearization, especially by hand, is the building of the perception of the need for accuracy and precision. This particular thinking skill appears to be low in students who score low on the FCI. But I will admit that just doing hand calculations may not be the answer to improving the need for accuracy. Again, specific targeting of this skill ala Feuerstein's Instrumental Enrichment may be necessary. Unfortunately it may not be possible to improve FCI scores without improving the general problem of low proportional reasoning of many students. Now if your students score high in proportional reasoning, then the linearization method may be completely irrelevant to improving physics understanding. But the 2-variable reasoning is also vital, especially when you have multiple forces.
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Date: Mon, 24 Jul 2006
From: Bernadine Samson
In the first post asking about linearizing data, I thought the issue was whether to make the students re-graph parabolas to straighten out the graph before getting the best linear fit. Other posts seem to question whether getting the best linear fit should be done with or without technology. In my class the students sometimes question the re-graphing to change a parabola to a line instead of using their graphing calculators or the computer software to do a fit to y=a x2 or y2 =b x. I tell my students that we are working with real data, not an ideal set of numbers as they do in their math classes. By straightening out the graph before doing the linear fit it’s easier to use the 5% rule to rule out y-intercepts that should be zero.
Regarding using technology, I think it is the job of the math teacher to teach the least square process and have the students find the line of best fit without the use of technology other than a four-function calculator at least once. I taught Algebra I for 9 years and that is the way I taught the concept. After they had worked at least one line of best fit problem on their own, I taught them how to let their scientific or graphing calculators do the work. In physics class since I have been using modeling, I teach students how to use Graphical Analysis and they analyze all their data using GA.
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Date: Fri, 28 Jul 2006
From: Bob.Allison
Letting the calculator (or computer) get the equation would be great if all we wanted was to get the students to solve a problem: the problem being “what is the equation for this data?” I agree wholeheartedly with Paul Bianchi. I believe that letting a calculator get the equation would teach zero physics and zero understanding. A slightly better way is to have the students calculate the square of the data and plot the square, either on a computer or by hand. This is an old (traditional) method for linearization. It has an advantage in being "useful in determining units for the derived coefficients", but I do not believe it teaches the physics or leads to true understanding. For motion the graph becomes position vs time2. Time squared has no physical meaning, is confusing for students, and brings no understanding.
What I require my students to do is linearize the data by doing a graphical derivation. Yes, it is very time consuming, but I believe essential to do for the students to understand the concept. I require this during one lab in the first study of acceleration. The first time we do this, the students take their graph of position vs time and draw a best fit line with a 'bendy ruler'. This helps get across the idea that a line does not need to be straight. (If you question your students I think you will be surprised how many don't know this.) I then have them 'slice it up' into time segments of 1/4 or 1/2 second, depending on the data, and calculate the slope of each segment. From the prior study of velocity they know this slope is the average velocity. The final result is a velocity vs time graph. The units are X/t vs t and they see that we are plotting the change in velocity, not some strange x vs t2 thing.
The second time we do this in lab, usually for a falling object, I have them put their data in a table and calculate the t2. I do a unit analysis and show them how this is equivalent to what we did before. After this I let the students do their labs and calculations in anyway that they understand.
I find I always have a few students in each class who continue to solve problems by filling a table like the one they did in the graphical derivation. When I ask why they use this harder, more time-consuming method even on quizzes, their answer is that they understand it, and it is easier for them than manipulating algebra equations.