Today we continued to build norms for the classroom by working with figurate numbers. Figurate numbers come from ancient Greece and were an attempt to connect geometric figures to numbers that would somehow unlock secrets of nature and the universe.
The first problems looked at square numbers. The class was asked to look at these without using calculators and to try to make direct connections between the figures and the values. I used the physical connections made between Pascal's triangle and the number of names said in the Circle-Name game as an example.
The square problems are hard for students to think of from a fresh perspective since they are so familiar with square numbers. I push them to look for patterns and make connections to the figures. Typical patterns include that the number of dots in a side match the figure number and the difference in square values are odd numbers that increase by two. The discussion around the first square number shows that it is a logical extension of the patterns but in fact creates degenerate square.
Next came triangular numbers. This forces students to look at patterns and make connections. Most realize that you need to sum up values. You may see other interesting patterns and recursive formulas that crop up. Student presentations are always interesting because you typically can have students present in such a way that the story about triangular numbers builds upon itself. The discussion of what the first triangular number should be is similar to the first square number.
Most students will not be familiar with the Gaussian summation formula and this is a good time to tell an anecdote, introduce the Greek capital letter sigma as a symbol for summation, and to connect the formula to the work that was done.
The presentations of these simpler problems help develop classroom community and establish the norms and expectations as to how the classroom operates. Students are starting to realize that solutions and insights come from them.
Visit the class summary for a student's perspective of the class and to view a copy of the day's lesson slides.
Friday, January 11, 2013
Thursday, January 10, 2013
Discrete Math - Day 1
This blog will discuss my course in Discrete Math. My Discrete Math web site explains the orientation of the course, provides summaries of daily lessons and slides for those lessons. This class is an inquiry-based course. The purpose of this is to have students make sense and connections with and between the mathematical concepts presented. Students are encouraged to explore, question, and investigate the mathematics with an eye toward how mathematicians work. This is much different than learning procedures or problem solving techniques that most students experience in their math courses.
The first day is always interesting. I start with the Circle-Name game. It provides an opportunity to learn student names and build classroom community. I then assign students randomly to groups of three or four.
The first task is for students to determine how many names said during the Circle-Name game. I have students work on their own (individual think time based on brain research) that allows students to engage in the mathematics. I then have students discuss their results at their tables and then have a whole class discussion. I don't push for formulas at this time. I am trying to establish class norms on working through problems, sharing results, and discussing methods.
We then explored Pascal's triangle (see below). I like to start with Pascal's triangle because the patterns are observed in so many situations. Depending on what topics you want to cover, Pascal's triangle will show up in combinatorics, number theory, and graph theory. The NCTM's Navigating Through Discrete Mathematics in Grades 6-12 provides several Pascal triangle explorations and is a good resource for other Discrete Math topics.
I gave students five minutes to work on the problems on their own before working in their groups. Not every group came to resolution on the results. We shared partial solutions and results that differed. This gave an opportunity to show how thinking from different groups can build upon each other and how do to go about determining which of many offered solutions is correct. For patterns in the triangle, students made observations about row content, diagonal patterns, and the general structure of the triangle. One student noted that the triangle had a line of symmetry. I was able to illustrate mathematical thinking as one student pointed out that every other row had a middle value. I suggested that given all the patterns in the triangle I wondered if the middle values followed a pattern and whether that pattern connected to some known entities.
After discussing course policies and course topics covered in the semester, we dove back into looking at Pascal's triangle with an eye toward making connections to the number of names said in the circle. This discussion provided some nice connections. One student said she saw the ones on the outside of the triangle as the people standing in the circle. She said she didn't feel it was all that insightful but it was what she saw. Another student made a connection with the third diagonal, noting that these represented the total names said for any given number of students. Students did not have any other connections. As I was listening to the class discussion I thought about the first diagonal representing the individuals and then thought of the second diagonal as representing the number of names an individual would say. The third diagonal represented the total names said. This made a nice connection between the observations and the reality of the situation. I didn't pursue it further but it would be interesting to see if there are any connections between the fourth diagonal and the Circle-Name game.
The class wrap-up was for students to summarize connections between Pascal's triangle and the number of names stated in their own words. This summary is based upon brain research that says that giving students time to think about their learning at the end of class and writing down their thoughts helps to embed the learning in long-term memory.
The first day is always interesting. I start with the Circle-Name game. It provides an opportunity to learn student names and build classroom community. I then assign students randomly to groups of three or four.
The first task is for students to determine how many names said during the Circle-Name game. I have students work on their own (individual think time based on brain research) that allows students to engage in the mathematics. I then have students discuss their results at their tables and then have a whole class discussion. I don't push for formulas at this time. I am trying to establish class norms on working through problems, sharing results, and discussing methods.
We then explored Pascal's triangle (see below). I like to start with Pascal's triangle because the patterns are observed in so many situations. Depending on what topics you want to cover, Pascal's triangle will show up in combinatorics, number theory, and graph theory. The NCTM's Navigating Through Discrete Mathematics in Grades 6-12 provides several Pascal triangle explorations and is a good resource for other Discrete Math topics.
I gave students five minutes to work on the problems on their own before working in their groups. Not every group came to resolution on the results. We shared partial solutions and results that differed. This gave an opportunity to show how thinking from different groups can build upon each other and how do to go about determining which of many offered solutions is correct. For patterns in the triangle, students made observations about row content, diagonal patterns, and the general structure of the triangle. One student noted that the triangle had a line of symmetry. I was able to illustrate mathematical thinking as one student pointed out that every other row had a middle value. I suggested that given all the patterns in the triangle I wondered if the middle values followed a pattern and whether that pattern connected to some known entities.
After discussing course policies and course topics covered in the semester, we dove back into looking at Pascal's triangle with an eye toward making connections to the number of names said in the circle. This discussion provided some nice connections. One student said she saw the ones on the outside of the triangle as the people standing in the circle. She said she didn't feel it was all that insightful but it was what she saw. Another student made a connection with the third diagonal, noting that these represented the total names said for any given number of students. Students did not have any other connections. As I was listening to the class discussion I thought about the first diagonal representing the individuals and then thought of the second diagonal as representing the number of names an individual would say. The third diagonal represented the total names said. This made a nice connection between the observations and the reality of the situation. I didn't pursue it further but it would be interesting to see if there are any connections between the fourth diagonal and the Circle-Name game.
The class wrap-up was for students to summarize connections between Pascal's triangle and the number of names stated in their own words. This summary is based upon brain research that says that giving students time to think about their learning at the end of class and writing down their thoughts helps to embed the learning in long-term memory.
IPS - Day 1
The posts referencing IPS refer to my Inferential Probability and Statistics (IPS) class. My IPS web site describes the course and provides summaries daily lessons with lesson slides.
In this blog I will be re-capping some of the days activities and my reflections on the lessons.
The first day was devoted to conveying a sense of what statistics is about, what will be covered in class, and how the class will work. The Ted video I show lets students know that seemingly simple probability problems do not result in the obvious answer. The disease example and the trial evidence example get students thinking about the ideas about what happens randomly. I am actually able to relate a personal experience similar to the video's trial example.
The idea of what do random events look like and how does actual data compare to expectations of what randomness looks like will be a recurring theme throughout the course. The first unit covers probability and simulations, building the foundation of what randomness should look like.
In this blog I will be re-capping some of the days activities and my reflections on the lessons.
The first day was devoted to conveying a sense of what statistics is about, what will be covered in class, and how the class will work. The Ted video I show lets students know that seemingly simple probability problems do not result in the obvious answer. The disease example and the trial evidence example get students thinking about the ideas about what happens randomly. I am actually able to relate a personal experience similar to the video's trial example.
The idea of what do random events look like and how does actual data compare to expectations of what randomness looks like will be a recurring theme throughout the course. The first unit covers probability and simulations, building the foundation of what randomness should look like.
Wednesday, December 8, 2010
AP Stat Soup Survey Project: Writing Surveys
Here's the information presented on 12/7 regarding survey writing basics.
Survey Basics
Survey Writing Specifics
Directions to Respondents
1. Include a brief explanation of the purpose of the questionnaire.
2. Include clear explanation of how to complete the questionnaire.
Content of Questions
1. Will the respondent be able to answer your question, i.e., do they know the answer?
2. Will respondents want to answer the question, i.e., is it too private or silly?
Wording of Questions
1. Will the respondent understand the wording?
2. Are any words so strong that they might influence the respondent's answer?
3. Are you asking more than one question at a time by using "and" in your question?
4. Do multiple choice question choices mutually exclusive and encompass the total range of answers?
Order of Questions
1. Be careful not to include so many questions that potential respondents are dissuaded from responding.
2. Attempt to get recruit respondents' motivation to complete the questionnaire. Start with fact-based questions and then go on to opinion-based questions.
3. Attempt to get respondents' commentary in addition to their ratings.
4. Put a date on the form so you can keep track of all future versions.
Source: http://www.managementhelp.org/evaluatn/questnrs.htm
Scales in Surveys
1) Much Worse 2) Worse 3) Slightly Worse 4) Slightly Better 5) Better 6) Much Better
(This example shows the use of a six point scale that forces the respondent to make a decision as to whether or not the soup tastes better or worse.)
1) Much Worse 2) Worse 3) About the Same 4) Better 5) Much Better
(This example shows the use of a five point scale that allows the respondent to be neutral as to the taste.)
Additional information and resources can be found at:
http://4h.uwex.edu/evaluation/documents/Wordingforratingscales.pdf or
http://www.gifted.uconn.edu/siegle/research/Instrument%20Reliability%20and%20Validity/Likert.html
Survey Basics
- Decide what information you want to gather from the survey.
- Ask only those questions that will provide the information you need.
- Keep the survey as short as possible
- Use multiple-choice questions whenever possible.
- Avoid leading and biased questions.
- Use the same rating scale throughout your survey.
- Test the survey before you roll it out.
Survey Writing Specifics
Directions to Respondents
1. Include a brief explanation of the purpose of the questionnaire.
2. Include clear explanation of how to complete the questionnaire.
Content of Questions
1. Will the respondent be able to answer your question, i.e., do they know the answer?
2. Will respondents want to answer the question, i.e., is it too private or silly?
Wording of Questions
1. Will the respondent understand the wording?
2. Are any words so strong that they might influence the respondent's answer?
3. Are you asking more than one question at a time by using "and" in your question?
4. Do multiple choice question choices mutually exclusive and encompass the total range of answers?
Order of Questions
1. Be careful not to include so many questions that potential respondents are dissuaded from responding.
2. Attempt to get recruit respondents' motivation to complete the questionnaire. Start with fact-based questions and then go on to opinion-based questions.
3. Attempt to get respondents' commentary in addition to their ratings.
4. Put a date on the form so you can keep track of all future versions.
Source: http://www.managementhelp.org/evaluatn/questnrs.htm
Scales in Surveys
- When using scales, create a balance between positive and negative options.
- An even number of choices forces the respondent to decide whether they lean to the positive or negative.
- An odd number of choices allows the respondent to select a neutral response.
1) Much Worse 2) Worse 3) Slightly Worse 4) Slightly Better 5) Better 6) Much Better
(This example shows the use of a six point scale that forces the respondent to make a decision as to whether or not the soup tastes better or worse.)
1) Much Worse 2) Worse 3) About the Same 4) Better 5) Much Better
(This example shows the use of a five point scale that allows the respondent to be neutral as to the taste.)
Additional information and resources can be found at:
http://4h.uwex.edu/evaluation/documents/Wordingforratingscales.pdf or
http://www.gifted.uconn.edu/siegle/research/Instrument%20Reliability%20and%20Validity/Likert.html
Tuesday, November 30, 2010
AP Stat: ESP Investigation Guidelines
For the Chapter 11 ESP Investigation, be sure you do the following:
1. Describe how you are conducting your simulation
2. Describe what your response variable is
3. Report the results for each trial
4. Summarize your results
5. State a conclusion that is backed by your analysis
1. Describe how you are conducting your simulation
2. Describe what your response variable is
3. Report the results for each trial
4. Summarize your results
5. State a conclusion that is backed by your analysis
Friday, November 12, 2010
AP Stat: US Census Population Totals
Here's the US Population data that we looked at in class. You are to re-express the data so that it yields an appropriate linear model. Please be sure to check your residuals.
2010 310,232,863
2000 281,421,906
1990 248,709,873
1980 226,542,199
1970 203,302,031
1960 179,323,175
1950 151,325,798
1940 132,164,569
1930 123,202,624
1920 106,021,537
1910 92,228,496
1900 76,212,168
1890 62,979,766
1880 50,189,209
1870 38,588,371
1860 31,443,321
1850 23,191,876
1840 17,069,453
1830 12,866,020
1820 9,638,453
1810 7,239,881
1800 5,308,483
1790 3,929,214
2010 310,232,863
2000 281,421,906
1990 248,709,873
1980 226,542,199
1970 203,302,031
1960 179,323,175
1950 151,325,798
1940 132,164,569
1930 123,202,624
1920 106,021,537
1910 92,228,496
1900 76,212,168
1890 62,979,766
1880 50,189,209
1870 38,588,371
1860 31,443,321
1850 23,191,876
1840 17,069,453
1830 12,866,020
1820 9,638,453
1810 7,239,881
1800 5,308,483
1790 3,929,214
Monday, November 8, 2010
AP Stat: Smoking Investigation Feedback
You are to comment on at least two reports in the discussion section of the AP Stat wiki.
Here are a few things to look for when giving feedback on the Smoking Investigation report:
1. Were the W’s addressed in the paper?
2. How did the person demonstrate that a linear model is appropriate?
3. Are graphs labeled and understandable?
4. Was the model described in context and written using proper notation?
5. Do any predictions made using the model look reasonable?
Here are a few things to look for when giving feedback on the Smoking Investigation report:
1. Were the W’s addressed in the paper?
2. How did the person demonstrate that a linear model is appropriate?
3. Are graphs labeled and understandable?
4. Was the model described in context and written using proper notation?
5. Do any predictions made using the model look reasonable?
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