# Tommy's "Didaktik1" paper

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This is the final version (1.0) of my paper that is an assignment for the "Didaktik1" course, part of The Programme for Teacher Training at Stockholm University. To fully enjoy this paper, the readers are recommended to speak Swedish, since some minor parts are written in Swedish. Most of it is in English though. Anything can of course be translated if there is enough interest. The paper scored among the top three percent from about 350 students and got graded as an A.

Paper title: "Understanding vs. knowing elementary mathematics"

Subtitle: "An observational report, and experiment in collaboration"

Author: Tomislav Dugandzic

Name of department: Department of Didactic Science and Early Childhood Education

Programme name: The Programme for Teacher Training (270 ECTS-credits)

Course name: Curriculum studies 1 - The School's Knowledge Assignment, 7.5 hp (UDG05L) (7.5 ECTS-credits)

Course name in Swedish: Didaktik 1 - Skolans kunskapsuppdrag, 7.5 hp (UDG05L). Also nicknamed "Didaktik1".

Semester: Spring term 2010

## Contents |

# Background and purpose

## Background

This paper is exploring the difference between "understanding" and "knowing" mathematics. It is not scientifically significant due to it's extremely small observational sample size (the observation of only one teacher, one student and one chosen teaching-learning situation is going to be chosen for analysis, from a total of three consecutive days of observation) but could prove interesting as raw data to researchers who wish to analyze students' first attempts at writing an academic paper as part of their college education.

The observation took place at an elementary school (grade level 7-9) in Sollentuna, Sweden.

## Purpose

The primary purpose of this paper has been to 1. argue for the premise that the author of the paper has indeed read the mandatory course material for this course, and 2. to demonstrate with a real-life example of constructing an actual report (this report) from an actual classroom observation, that the same author, with the help of said course material, has obtained the ability to observe a classroom learning situation and then write a relevant report based on said observation.

A secondary purpose has been to practice writing natural as well as professional academic English.

## Questions this paper will study

The questions that this paper will study are the following:

Suppose I (the observer) observe a student (S1) asking her teacher (T1) a question about mathematics. Suppose T1 answers the question, S1 accepts it, and the interaction between T1 and S1 for the purposes of this report, ends.
Did S1 really *understand* how it is possible to solve that particular mathematical problem, or did S1 *merely learn* how to solve it without mathematical insight, or understanding?

The, for the scope of this paper, proposed difference between "understanding" and "knowing" a mathematical concept is described in the Central concepts section.

The two observations will be analyzed and discussed using the Design-theoretical perspective.

The reason I chose to explore the differences between "understanding" and "knowing" mathematics is that according to the Swedish government's most important steering document, called "läroplan" (approximately equivalent to the American *national curriculum*), only three goals are describing a specific school-subject (Linde, G. 2000). The rest of the document describes important goals that encompass all of the school-subjects. One of these main goals is that the student "masters basic mathematical thinking and can apply it in daily life". Considering the importance I've argued for, of this particular goal, it is worth thinking a few more minutes about this. What is the meaning of teaching even basic math if the student doesn't understand it but only memorize algorithms? Think of all the adult citizens who still struggle with basic percentage calculations just because they never understood the underlying principles and instead memorized the algorithms. And as we've all experienced, insight is never forgotten, but memorization is. Things like housing-price bubbles could be avoided if citizens would master basic math (inflation, interest rates etc). Have I convinced you yet that this is an important question? I hope so.

The second related question: "Is the design of the classroom and the learning environment set up in a way optimal for learning, in this case understanding?", which will be analyzed and discussed using the Design-theoretical perspective.

## Acknowledgments

Many people have helped me with this paper. You are listed in alphabetical order.

Thank you Maria Lindsjö Bjernevik for letting me learn from your doing. Your experience and skill at teaching has been most valuable for me to aspire to imitate, and eventually make it a part of my own.

Thank you Lena Geijer for your support and encouragement both during and after your presentations, and for insisting on lending me your personal copy of a book you thought would give me joy as well as important knowledge.

Thank you Malin Graaf for your brainstorming, feedback and help in general. Without you I would have made many embarrassing mistakes.

Thank you Marita Gustafsson-Jenadri for answering my endless questions and arguments, and your wisdom in designing your feedback and responses in such a way that it would lead me to maximal insight and not just a quick answer.

Thank you Fredrik Lindstrand for not giving up on me when I repeatedly did not understand the difference between a media and a mode. I might have given up on the Multimodal Design theory altogether if it weren't for your persistence.

Thank you Michael Salemsson for helping me with practical wisdom when I was a fresh student and had nothing to offer in return. You're a born teacher.

Thank you Anders Sigrell for taking your time to thoroughly explain and argue for the *good* side of rhetorics and how useful it is to teachers.

Thank you all of my students for letting me practice to eventually become a good teacher. Thank you for letting me use a video camera even though I know it can feel a little private to be taped while asking your teacher math questions. And thank you for being so nice to me even though I'm merely a "n00b".

Thank you everyone who have helped me in any way or form. You know who you are.

Peace.

# Theoretical perspectives

The observations will be analyzed and discussed from the Design-theoretical perspective.

## Central concepts

### Metacognition

John Dewey's "metacognition" (Phillips, D.C. & F. Soltis, Jonas 2009, p. 40).

### Understanding and knowing

"Understanding" and "knowing". There are at least as many definitions of "understanding" and "knowing" as there are cognitive scientists. See the chapter "The structure of disciplines" for one of the examples (Phillips, D.C. & F. Soltis, Jonas 2009, pp. 70-71). Every definition is applicable to different degrees in any situation. The classroom scenarios that this report will be studying would benefit from this definition, best described by an example:

Imagine a student learning how to solve this equation:

*x* + 1 = 3

*x* = 3 − 1

Answer: *x* = 2

The way a student who "knows" mathematics would solve it, would be by thinking: "I move the left *one* to the right side of the equal sign, and toggle its sign. Then I subtract the *one* from *three*, and am left with *two*, which is the correct answer.".

The way a student who "understands" mathematics would solve it, would be by thinking: "I add *one* to both sides of the equal sign. Then both sides would still be equal to each other. Then I subtract the *one* from both sides, and am left with *two*, which is the correct answer.". It would look like this:

*x* + 1 = 3

*x* + 1 − 1 = 3 − 1

Answer: *x* = 2

This definition is functional enough for the scope of this paper, even though many other definitions would likely work just as well. The benefit of this one is that it is as simple as possible, works for the scope of this paper, and therefore is arguably most suitable for this particular report (according to the Occam's Razor principle. But remember that the principle is only a *recommendation*, not a universal truth.).

### Mode and media

The definition of mode (Rostvall, A-L. & Selander, S. 2008 p. 116) and media I've chosen to use is the following:

A bird that uses its song to communicate, does so by using the song as a *media*. The bird doesn't have any other forms of communication, like the voice humans have for example, available so when we talk about the bird's song, we talk about it as a media that communicates some semantic content.

A human who wants to communicate a forbidden political thought, has the choice of designing a song that will be the *mode* for communication. Not the *media* for communication. When said human uses the song as a mode, or "tool" if you wish, she does so *with a conscious purpose*. It is practically useful to distinguish choices of transportation devices for semantic content, where there is a pedagogical purpose to choose a certain transportation device (or "mode") over another, and where we have other reasons for choosing a certain transportation device (or "media") over another.

A song can be both a media and a mode, depending on the purpose you had when you chose the transportation device for semantic content (the song).

Engineers talk about medias, whereas pedagogues talk about modes. It's all about *the purpose* of the discussion, or about *perspective*.

# Method

This report could prove interesting as raw data to researchers who wish to analyze students' first attempts at writing an academic paper as part of their college education.

For that purpose, I've chosen to use a wiki in the creation of this paper so any interested researchers can view many intermediary versions as the work progresses. It is also interesting to experiment using a wiki as a *mode* for the transfer of thoughts, instead of using the stereotypical Microsoft Word document *mode*. Click here to view the wiki version of this paper, as it was intended.

My classmates and teachers have been invited to help me improve this paper while it was being written, to add a form of collaborative metacognition (Phillips, D.C. & F. Soltis, Jonas 2009, p. 40) dimension to the paper.

You may view the discussion page for thoughts and comments I, my classmates and teachers had, while I wrote this paper.

A Microsoft Word version of this paper has been generated to satisfy the mandatory course criteria. It is linked to at the end of this wikiversion of the paper. The Microsoft Word version doesn't contain version history.

The books I've been referencing are written mostly in Swedish.

## Sampling and execution

The observation consists of only one teacher, one student and one chosen teaching-learning situation, from a total of three consecutive days of observation. The observation has taken place in an elementary school (grade level 7-9) in Sollentuna, Sweden.

I made two versions. One primary and one backup version.

In the primary version I used a video camera to capture every detail of what was going on, to have solid data to analyze afterwards. I also discussed the observational scenario with the teacher immediately after, while everyone's memory was still fresh about it.

In the second (backup) version I transcribed by pen and paper, just in case the student and/or parent(s) in the primary version would not give (or later withdraw) their approval for me to use the captured video as source material for my observation. I again discussed the observational scenario with the teacher immediately after, while everyone's memory was still fresh about it. The second version was also intended as a backup in case the primary version would turn out to be not as interesting to analyze, as it would first have seemed.

Both versions were similar enough to warrant analyzing both, for the purpose of having more data to analyze.

## Data processing

The author of this paper is the sole coder (Stigler J, W. & Hiebert, J. 1999, pp. 21-23) of both the video, and "pen and paper" transcriptions. Since there are many ways to design a coding system, we can assume that the arbitrarily chosen coding system is weak in its objectiveness.

This study is however much more narrow in its scope than the TIMSS study, which gives reason to believe that it is easier to intuitively choose a more objective coding system even though there has been only one coder for this report, whereas they had six.

A more detailed analysis of the objectiveness for the chosen coding system is beyond the scope of this course assignment.

## Ethical considerations

The names of the students that were observed and the class name of those students have been kept anonymous. The teacher being observed has been given knowledge at least two weeks prior my request for observation, of the purpose of my observation, and been given the opportunity to decline partaking in my study. The teacher did not decline.

I've asked the student that i videotaped for permission to use the video as data for this paper. I've also asked the student's father by phone as well as in writing. Both the student and father gave me their permission to use the video for transcription with the condition that I delete the video after I've transcribed from it. We agreed to those terms.

# Conclusion

## Description of observation

Please see Appendix A for a description of the two observations that have been made for this report, before you read the analysis and discussion sections.

## Analysis

Summarily I may say that I drew the conclusion that the odds where > 70 % that student S1 from O1 really *understood* the math task described in the observation. Regarding O2 it is very likely that the student truly understood, because it can be argued that S2's prerequisite knowledge and understanding was of a, significantly enough higher level, than the level of her question. Student S2 understood that the line must have become a straight line, but couldn't figure out why it didn't, and therefore raised her hand for help.

The classroom was designed in the classical *form* (also called "*mode*" from the Design-theoretical perspective, explained in the Central concepts section of this report.), i.e. see Figure 5.

Teacher T1 crouched down in order to be situated lower than the sitting student. Teacher T1 situated herself opposite the student, even though that had the consequence of the teacher having to read the student's book upside down.

The classroom had a projector permanently mounted in the roof. The projector could be used as a mode, for powerpoint presentations. The choice made for both observations were a textbook containing text and pictures balanced in an equal amount (as you can see in Figure 1).

The classroom scenarios were multimodal (simultaneously consisting of several modes) in the sense that the textbook varied the use of text as a mode and pictures as a mode. It was not multimodal in the sense that the projector was not used.

## Discussion

### Design of a book/classroom

There is another analysis tool we can use called *resource* (Rostvall, A-L. & Selander, S. 2008 p. 116) that I'll be coming back to in the next paragraph. It is interesting to view the classroom as a three-dimensional information-containing space with objects of information distributed inside this space. Why? Well, if you think of the design of a book, it feels more natural to think about the book as an entity containing knowledge and insights. Why not think of the whole classroom as a huge 3d-book?

Example: It is natural to use your finger to point at the current line of a text you're reading in a book. I would argue that it is practically useful to view the teacher as a kind of finger. The teacher helps the student choose what text to read next, like a finger helps you what row to read next. The friend to the left of the student is good at explaining basic math but not advanced math, whereas the friend to the right is good at explaining advanced but has forgotten how to explain basic. Do you see what I mean? The friends next to the student are a form of *resources*, just like different sections of a book can be viewed as resources. A modern math textbook usually begins a chapter with a section describing the theory behind a new mathematical concept. Then it moves on with a section containing problems the reader is expected to solve.

Just as a student sometimes need to go back to the theory section in a textbook, a student may instead turn to his friend she knows can help her. In this view, there is no difference between a friend, a teacher, a textbook with its chapters and sections. Therefore we can argue that the layout of a classroom is just as important as the layout of a textbook. And haven't we all experienced bad textbooks. It would pay to spend some time thinking about the layout of the classroom we teachers are constantly creating.

### Teacher crouching, situating on opposite side

An advantage this teacher behavior gives, is that the student feels the situation to be less intimate and therefore can focus more on the math problem. One disadvantage can be that the personal space expected by individuals vary from culture to culture. Individuals from the Swedish culture prefer a bigger sphere of personal space to feel comfortable, compared to many other cultures.

A way to adapt to the personal space preferences of a certain student could be to try different distances and see how they react. If a teacher is too far away, the student feels the contact to be too impersonal and stops asking questions (e.g. the teacher waits by her desk in front of the classroom instead of walking around, being more accessible to ask for help) and if the teacher gets too close, the student may simply loose concentration on the math problem being discussed (indicated for example by the student moving her head/body slightly in the the teacher's opposite direction).

After discovering the maximum and minimum distances, one could just calculate the average distance and remember that distance for that particular student. This may sound obvious when written like this, but many things sound obvious when explained in an ordered and structured fashion (Phillips, D.C. & F. Soltis, Jonas 2009, pp. 5-6) even when they are not as obvious as they seem. A critique is perhaps that this behavior comes naturally. There is no need to analyze the situation scientifically. True. But why wait for it to eventually come naturally when you can make it a conscious behavior? And don't see it as a failure to not have this behavior naturally perfectly balanced in the beginning of your teaching career.

You know what? Even great speakers like presidents make every aspect of their speech a conscious activity, even if they are born naturals. Winston Churchill (Prime Minister of the United Kingdom 1951-55) refused to give a speech if given less than two weeks of preparation for a "short but witty" informal speech (Sigrell, Anders 2009, p. 76).

We may not have the president's luxury of preparing each lesson for two weeks, but we can at least do our best in the allotted time.

### Reason for using the Design-theoretical perspective

Now you may ask "what does all of the analysis and discussion about the Design-theoretical perspective have to do with whether the student understood the observed math problems?". Well. The Design-theoretical perspective doesn't really answer the question if one particular student understands or just learns. It answers the question: "Is the design of the classroom and the learning environment set up in a way optimal for learning?" And by "learning" I mean in this case, "understanding", since it's mathematics we're observing.

An attempt to answer the latter question, is that the environment for my observations has been setup in a way that is very common in Swedish schools from grade level 7-9, if I compare to personal experiences and memories I've had about schools of that grade level. It is in this regard also somewhere between American and German schools, with Japanese being the most unlike according to the TIMMS study discussed in The Teaching Gap (Stigler J, W. & Hiebert, J. 1999).

Furthermore the students of these observations each have at least one bench-friend that they can ask questions. Two thirds of them have two bench-friends. Ideally they would all have two bench-friends sitting next to them. But then they would be more distracted. It would be interesting to compare students' results' after one year sitting two-and-two, to students' sitting three-and-three. Do the increased number of modes accessible, weigh more than the disadvantage of the increased distractions?

Furthermore, maybe a textbook is *still* better in some circumstances than using a projector. During my observations the projector was not used as a mode. How could that mode have been used in my observations? One way would've been to ask everyone's attention and project the question S1 and S2 asked, onto the screen so everyone could've learned simultaneously. But the advantage of the good-old book-mode is that everyone can understand at their own pace. Probably some combination of the two modes would be best. Is it possible to answer this question definitely or is the answer too situation-dependent? That is a meta-question interesting in itself.

**I consider the second paper question** "Is the design of the classroom and the learning environment set up in a way optimal for learning?" for my observations **not fully answered**, and beyond my ability to ever answer, fully.

Furthermore **I couldn't answer the first paper question at all** because of space constraints (< 2500 words). I had to choose only one, and I chose the Design-theoretical question because my teacher Marita tried to encourage us to include it in our report. They are all interesting in their own way, so why not.

### Reflection about the research process itself

Next time I'll ask every student I plan to videotape for their and their parents' permission in advance. This time I couldn't keep the video after I transcribed from it, since the father gave that condition to his permission. The data in a study like this is valuable to keep so that independent researchers can have the opportunity to verify the analysis and conclusions. That is not possible this time, but will be next time if I know for which students I have permission to keep the video.

### New questions and further research

The interactions between a teacher and a student changes direction many more times during a learning session than I originally anticipated. Originally I thought that a student asks a question, 1. the teacher answers it, 2. the student asks for clarification, 3. the teacher clarifies, and the session ends. All in all the interactions would've changed direction three times.

In a real life situation, like in my observation, the interaction is much more lively. What implications does this have for other forms of interactions than the traditional "one on one" session between a student and her teacher? The real time communication is almost necessary here, which probably has dire consequences for more lagging forms of communication like for example distance learning by email.

This raises the question; Do these negative consequences weigh more or less, than the benefit lagging forms have over real-time forms, in terms of less stress and more time for contemplation before any change of communication direction?

# Appendixes

## Appendix A

### Digital transcriptions of observations

#### Observation one (O1)

The transcription for this observation was made by pen and paper.

The student (S1) asked the teacher (T1) for help. See Figure 1 (problem number 4).

The translation is: "Draw a graph with the x axis: -5 to 5, and y axis: -5 to 5. Mark the points A: (1,3), B: (2,-2), C: (-1,4), D: (-3,2), E: (0,5), F: (-3,0).".

Coding key:

`Action number`

Person: "Quote of statement", or description of action.

`01`

T1: "What are the x and y values for point A?"`02`

S1: "I don't know."`03`

T1 asked the same question again but this time used her pen to drag it along the y axis as she asked for the coordinates.`04`

S1: "Aha!", "3!".

`05`

T1 asked what the coordinates were for point E.`06`

S1: "I don't know."`07`

T1 asked the same question again but this time used her pen to drag it along the x axis as she asked for the coordinates.`08`

S1: "Aha!", "0!".

`09`

T1 made the conclusion that S1 didn't understand even though S1 answered correctly. Therefore T1 asked what the coordinates were for three more points, this time not helping by dragging the pen along the axises. S1 could neither answer nor tried to guess.`10`

T1 asked what the coordinates were for point B.`11`

S1 suddenly answered correctly, for the x value. S1 didn't mention the y value for the coordinate.

`12`

T1: "What number should you give first?"`13`

S1 looked puzzled, and repeated to give the correct x value but omitted to mention the y value.`14`

T1: "Every coordinate consists of two values. One x value and one y value."`15`

T1 created a new point that didn't previously exist in the book, and then asked S1 to tell her the coordinates for the new point. S1 answered correctly, mentioning both coordinates. T1 reiterated the process two more times and S1 answered correctly, mentioning both coordinates.

`16`

T1 leaves and the student seems satisfied with the help that she got.

I asked T1 ~30 minutes after, in the teachers' room so no students would hear the discussion, if she thought that the student had really *understood* how to give the coordinates for a point or plot a point from given coordinates.

`17`

T1: "Sometimes the [ T1 used: 'poletten trillar ned', an idiomatic Swedish expression' ] student suddenly understands for no apparent reason, immediately after an explanation, and sometimes after several years."

S1 and T1 used the already drawn graph from question 2 in Figure 1, instead of drawing their own, as per instruction from question 4 in Figure 1.

#### Observation two (O2)

The transcription for this observation was made from video footage and complementary pen and paper notes.

The student (S2) asked the teacher (T1) for help.

The task was to plot three points and draw a line through the points on a graph. The function was *f*(*x*) = *x* + 2. Student S2 successfully plotted point A and B, but plotted point C incorrectly (Figure 2).

Coding key:

`Action number`

Person: "Quote of statement", or description of action.

`01`

S2 raises her hand, T1 approaches, and crouches down opposite to S2 with the bench in between them.`02`

S2 takes the initiative to read the problem out loud to T1: "A cab fare costs $USD 2, plus USD$ 1 for each km of transportation. Create a table with some values, and then use the table to plot the points on a graph." [The task was in Swedish currency (SEK), but I translated it to USD for the purposes of this report.]`03`

S2: "Well... I drew one of those [implied a graph], but..."`04`

T1 interrupts.`05`

S2+T1 simultaneousely: S2:"I don't know if it was [this or there, inaudible] [inaudible]" + T1:"Did you start with this table? [points at the table that S2 drew]".`06`

S2: "Yes, it is there [points with her pen at the data table]." [does not seem annoyed to have been interrupted.]`07`

T1: "And then you have chosen three different points in time."`08`

S2: "Mhm [slang for "yes" in Swedish]."`09`

T2: "And calculated the respective price for them."`10`

S2: "Mm [slang for "yes" in Swedish]."`11`

T1: "Mm."`12`

T1: "Then you did one...", pauses a few seconds while thinking, then verifies that the points on the graph matches the values in the table using her pencil to point as she is thinking out loud, while S2 is watching.`13`

T1: "Yes, here it is a little 'crazy' [In Swedish slang "tokigt"; A mild and nonconfrontative way to express that there is an error.]`14`

S2: "Mmm..." [The intonation indicated uncertainty, even though the words were "yes".]`15`

S2 starts to point with her pen at a different point, as if she intends to ask a question.`16`

T1 interrupts: "Because here [points with her pen on point C]... "What is the price here?"`17`

S2: "Yes of course... There, it is*one*[sounds very certain, and emotionally neutral, contemplative of the next line of thought]..."`18`

T1: "*One*, yes..."`19`

S2: "Mmm..."`20`

T1: "*That*is the reason it became 'crazy'." [Quickly but consciously glancing (for the first time in this session) at the body language of S2 while still holding her pen at the point they are currently discussing. It looks like T1 trusts the body language of S2 significantly more than her words. It looks like T1 wants to be absolutely sure S2 understands each step in her explanation before she moves on to the next step of explanation. It looks like T1 anticipates that this is the first time in the session that it is significant to with certainty verify that S2 is really following T1's line of thought. It looks like T1 has been in similar situations many times before.]`21`

S2: "Mmm."`22`

T1: "So you have to find, to be able to plot [pauses while thinking of what the correct coordinates should be]..."`23`

S2 interrupts as if to give a suggestion: "Well, it should become, like..."`24`

T1 interrupts "*Three*... So you'll end up*there*instead [referring to the y coordinate for point C].`25`

S2 "Mmm." [Sound like she simultaneously understood both that the y coordinate must be*three*and that*that*was the reason, the line she was drawing through the points didn't become a straight line.]`26`

T1: "And that will alter your line..."`27`

S2 interrupts: "Yes exactly, that was*that*that was the problem".`28`

T1: "So if you move that [referring to point C, pointing at it with her pen.]... to there" [consciously waiting for S2 to catch up.]`29`

S2: "Mm."`30`

T1 continues to move her pen along the, now straight, line on the graph while saying: "Then it [referring to point C] will lay on the same line as those ones [referring to point A and B.].`31`

S2: "Yes of course." [This time, in addition to her intonation indicating that she truly understands why she got an error and how to fix it, she indicates the same by immediately starting to look for an eraser in her pen purse to correct her plot.]`32`

T1: "So it was just a little 'wrong-pointing' there [In Swedish slang "felprickning"; A mild and nonconfrontative way to reduce the feeling of the magnitude of the error.].`33`

S2: "Aaa [slang for "yes" in Swedish]..."

S2 continues to alter her plot. T1 waits a few moments to see if S2 is going to ask more questions. S2 ignores T1 and T1 walks away. End of session.

## Appendix B

### Mathematica source code to generate Figure 2-4

Mathematica source code to generate Figure 2 graph picture:

apoints = { {2, 4}, {3, 5} } bpoints = { {1, 1} } aplot = ListPlot[apoints, PlotRange -> {{-2, 10}, {-2, 10}}, PlotJoined -> True, AxesLabel -> {x, y}] bplot = ListPlot[bpoints, PlotRange -> {{0, 10}, {0, 10}}] Show[aplot, bplot, PlotRange -> {{-2, 10}, {-2, 10}}]

Mathematica source code to generate Figure 3 graph picture:

apoints = {{-2, 0}, {2, 4}, {3, 5}} bpoints = {{1, 1}} aplot = ListPlot[apoints, PlotRange -> {{-2, 10}, {-2, 10}}, PlotJoined -> True, AxesLabel -> {x, y}] bplot = ListPlot[bpoints, PlotRange -> {{0, 10}, {0, 10}}] Show[aplot, bplot, PlotRange -> {{-2, 10}, {-2, 10}}]

Mathematica source code to generate Figure 4 graph picture:

apoints = {{-2, 0}, {1, 1}, {2, 4}, {3, 5}} bpoints = {{1, 1}} aplot = ListPlot[apoints, PlotRange -> {{-2, 10}, {-2, 10}}, PlotJoined -> True, AxesLabel -> {x, y}] bplot = ListPlot[bpoints, PlotRange -> {{0, 10}, {0, 10}}] Show[aplot, bplot, PlotRange -> {{-2, 10}, {-2, 10}}]

I didn't know how to add labels to the points using Mathematica. I added them with Gimp.

# References

Phillips, D.C. & F. Soltis, Jonas (2009). *Perspectives on learning - Fifth edition.* New York: Teachers College Press.

Sigrell, Anders (2009). *Retorik för lärare - Second edition.* Åstorp: Retorikförlaget AB.

Stigler J, W. & Hiebert, J. (1999). *The Teaching Gap, Best Ideas from the World's teachers for Improving Education in the Classroom.* New York: Free Press.

Linde, G. (2000). *Det ska ni veta! En introduktion till läroplansteori.* Lund: Studentlitteratur.

Rostvall, A-L. & Selander, S. (2008). *Design för lärande.* Stockholm: Norstedts Akademiska förlag.

# MS Word version of this paper

You can download a static MS Word version dated before 2010-04-29 16:00 (deadline for the final version) of this paper at the following link:

http://neo101.org/su/didaktik1-vt10/

You can also view this paper as *any* static version, if you click on the history tab and choose one specific version.

To view the dynamic (the most current) version you can press the article tab.

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