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Now Playing: Guess My Rule for The Ladder Problem Creates an Algebra Adventure
Guess My Rule for The Ladder Problem Creates an Algebra Adventure
PurposesEffective teaching; Professional development activity; Student engagement; Student model building; Reasoning; Representation
DescriptionThis VMCAnalytic was created as the third “video story” in a set of four. The four VMCAnalytics were used as the central tool of professional development in the Teachers Algebra Workshop (TAW), August, 2016. The TAW was created for a study that examined the use of video stories in the professional development of algebra teachers of minority students with low socio-economic status (Leslie, 2019).
In this study, teacher participants’ responses to this video story were audio-recorded. The teacher participants discussed the mathematical reasoning, student engagement and pedagogy that they noticed in the video. The teacher participants were guided by questions designed to focus the discussion on these three areas. The discussions were audio-recorded and analyzed in the author’s dissertation (Leslie, 2019).
The source video for this analytic was produced by an NSF Study (Informal Mathematics Learning (IML), Award REC-0309062). Clips from this original NSF study were selected for inclusion in this video story/Analytic. The students in the source videos for this analytic were part of a group of 7th graders in the Frank J. Hubbard Middle School in Plainfield who participated in the after school, 3-year NSF study.
This video story begins in the middle of an adventure. Ariel has been working on The Ladder Problem.
This is The Ladder Problem:
****
Ladders are built out of little wooden rods. A one-step ladder requires 5 rods, a two-step ladder
requires 8 rods. (A 1-step ladder in this document it would look like the capital
letter H made of 5 rods. To make a 2-step ladder you would add 3 rods (they look like the letter H
with the bottom “rods” chopped off) to the top of the H, the 1-step ladder.)
Find a rule that will tell us how many rods are needed for a ladder of any size.
****
Ariel first encounters The Ladder Problem when Teacher/Researcher (T/R) John Francisco presents it to him, and asks Ariel to figure out how many rods are needed for a ladder with 8 steps. Ariel starts to build a ladder and, after he builds one with 4 rods he stops and says, “I have an idea.” Then Ariel counts the number of rods in a 4-step ladder; he finds that there are 14 rods and he incorrectly concludes that there must be twice that number of rods in an 8-step ladder. He claims that there will be 28 rods.
As Ariel continues to work on The Ladder Problem, T/R John Francisco asks him to justify his answers – how does he know that his answers are correct? At one point, Ariel claims that an 8-step ladder must use 28 rods. He explains his proportionality argument: 8 = 4 times 2, so the number of rods in a 8-step ladder must be twice the number of rods in a 4-step ladder (see Sigley & Wilkinson VMCAnalytic, 2015, Event 2).
Ariel finally does build an 8-step ladder to show T/R Francisco that he is correct, and he is surprised to discover that there are only 26 rods in his 8-step ladder (not 28 rods). He counts twice to verify that there really are only 26 rods in an 8-step ladder. Then James, a student who is also working on The Ladder Problem, points out that every time you add one step to a ladder, you add 3 more rods. He explains that if you add 4 steps, you add 4 times 3 = 12 rods to the ladder. He says that there are 14 steps in the 4-step ladder and by adding 12 gives you 26. It is unclear whether Ariel, who returns to his own solution, heard James. However, Ariel decides to modify his earlier idea so that it gives him the correct answer. Ariel multiplies the number of rods by 2 (in going from a 4-step ladder to an 8-step ladder) and then he subtracts 2. 28 – 2 = 26 (see, Sigley & Wilkinson VMCAnalytic, 2015, Events 3-4).
This gives Ariel a new path to explore and T/R Francisco challenges him with other problems where he has to test whether his new rule works.
This video story begins as Ariel begins to explore the complexities of The Ladder Problem and three new cases he has to consider:
Event #4: Dividing by Ten and Still Subtracting Two?
Ariel considers whether he has followed his own rule when he is calculating the number of rods required for a 120-step ladder. He begins by following the rule. He divides 120 by 2 and gets 60. Then he departs from his rule and divides the 60-step ladder by 10. Using the fact that there are 20 rods in a 6-step ladder, Ariel multiplies 20 times 10 and then subtracts 2 (and gets 198). T/R Francisco continues to challenge Ariel to explain how he followed his own rule.
(Note that the number of steps in a 60-step ladder is 182. We would have to subtract 18. One could think of this as starting with one 6-step ladder and every time we add another (there are 9 more) we have to subtract 2 of the rods.)
Event #5: If Two Ladders are Combined Do we Just Add the Rods?
Ariel is calculating the number of steps in a 125-step ladder. Using the rule that he created, Ariel subtracts 1 from 125 and works with 124 steps, dividing by 2 and starting with a new problem of finding the number of steps in a 62-step ladder. Ariel pauses and then expresses doubt about whether he will find the number of rods in the 62-step ladder. As he works on this problem, Ariel has produced a new case for his rule: what happens when we add two ladders to make a new ladder? Do we just add the number of steps or do we have to subtract 2 in this case?
Event #6: If Two Ladders are Combined, Add the Rods and Subtract Two
T/R Francisco gives Ariel a simpler case to consider when adding the steps in two ladders to produce a third ladder. He asks Ariel to use a 6-step ladder and add something to create an 8-step ladder. Ariel builds a 6-step ladder and counts the number of rods and gets 20. Then Ariel is prompted by T/R Francisco to think about how to combine the 6-step ladder and the 2-step ladder to make an 8-step ladder. Ariel first adds 20 plus 8 to get 28, but then he counts the number of rods in the 8-step ladder and counts only 26. At this point he remembers his own rule and subtracts 2 to get the correct answer.
This analytic shows T/R Francisco providing opportunities for Ariel to explore his rule in different cases. His explorations lead him to encounter what is true and what is not true about The Ladder Problem. It provides a small window into the complex cognitive algebra trajectory that Ariel worked through before he emerged as a sophisticated algebra student (see, Sigley & Wilkinson VMCAnalytic, 2015, Events 5-8).
REFERENCES
Agnew, G., Mills, C., & Maher, C. M. (2010). VMCAnalytic: Developing a collaborative video analysis tool for education faculty and practicing educators. Proceedings of the 43rd Hawaii International Conference on System Sciences, Honolulu, HI.
Boaler, J., (2016). Designing mathematics classes to promote equity and engagement. The Journal of Mathematical Behavior, 41, 172-178. https://doi.org/10.1016/j.jmathb.2015.01.002
Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM Mathematics Education, 43, 175–187.
Davis, R. B. (1992). Understanding ‘Understanding.’ The Journal of Mathematical Behavior,11, 225- 241.
Leslie, Joyce (2019). Investigating a model using video stories for professional development for algebra teachers of low SES minority students. Dissertation, Rutgers. New Brunswick, NJ.
Maher, C. A. (1998). Constructivism and constructivist teaching, Can they co-exist? In Ole Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 29-42). Proceedings of Topic Group 6 at the International Congress on Mathematical Education [Report on Proceedings] (8th, Seville, Spain, July 14-21, 1996). Faculty of Education Report No. 3.
Sigley, R and Wilkerson L (2015). Tracing Ariel’s Algebraic Problem Solving: A Case Study of Cognitive and Language Growth. Retrieved from https://doi.org/doi:10.7282/T3N0186C
Steele, C. M. (1997). A threat in the air: How stereotypes shape intellectual identity and performance. American Psychologist, 52(6), 613-629.
In this study, teacher participants’ responses to this video story were audio-recorded. The teacher participants discussed the mathematical reasoning, student engagement and pedagogy that they noticed in the video. The teacher participants were guided by questions designed to focus the discussion on these three areas. The discussions were audio-recorded and analyzed in the author’s dissertation (Leslie, 2019).
The source video for this analytic was produced by an NSF Study (Informal Mathematics Learning (IML), Award REC-0309062). Clips from this original NSF study were selected for inclusion in this video story/Analytic. The students in the source videos for this analytic were part of a group of 7th graders in the Frank J. Hubbard Middle School in Plainfield who participated in the after school, 3-year NSF study.
This video story begins in the middle of an adventure. Ariel has been working on The Ladder Problem.
This is The Ladder Problem:
****
Ladders are built out of little wooden rods. A one-step ladder requires 5 rods, a two-step ladder
requires 8 rods. (A 1-step ladder in this document it would look like the capital
letter H made of 5 rods. To make a 2-step ladder you would add 3 rods (they look like the letter H
with the bottom “rods” chopped off) to the top of the H, the 1-step ladder.)
Find a rule that will tell us how many rods are needed for a ladder of any size.
****
Ariel first encounters The Ladder Problem when Teacher/Researcher (T/R) John Francisco presents it to him, and asks Ariel to figure out how many rods are needed for a ladder with 8 steps. Ariel starts to build a ladder and, after he builds one with 4 rods he stops and says, “I have an idea.” Then Ariel counts the number of rods in a 4-step ladder; he finds that there are 14 rods and he incorrectly concludes that there must be twice that number of rods in an 8-step ladder. He claims that there will be 28 rods.
As Ariel continues to work on The Ladder Problem, T/R John Francisco asks him to justify his answers – how does he know that his answers are correct? At one point, Ariel claims that an 8-step ladder must use 28 rods. He explains his proportionality argument: 8 = 4 times 2, so the number of rods in a 8-step ladder must be twice the number of rods in a 4-step ladder (see Sigley & Wilkinson VMCAnalytic, 2015, Event 2).
Ariel finally does build an 8-step ladder to show T/R Francisco that he is correct, and he is surprised to discover that there are only 26 rods in his 8-step ladder (not 28 rods). He counts twice to verify that there really are only 26 rods in an 8-step ladder. Then James, a student who is also working on The Ladder Problem, points out that every time you add one step to a ladder, you add 3 more rods. He explains that if you add 4 steps, you add 4 times 3 = 12 rods to the ladder. He says that there are 14 steps in the 4-step ladder and by adding 12 gives you 26. It is unclear whether Ariel, who returns to his own solution, heard James. However, Ariel decides to modify his earlier idea so that it gives him the correct answer. Ariel multiplies the number of rods by 2 (in going from a 4-step ladder to an 8-step ladder) and then he subtracts 2. 28 – 2 = 26 (see, Sigley & Wilkinson VMCAnalytic, 2015, Events 3-4).
This gives Ariel a new path to explore and T/R Francisco challenges him with other problems where he has to test whether his new rule works.
This video story begins as Ariel begins to explore the complexities of The Ladder Problem and three new cases he has to consider:
Event #4: Dividing by Ten and Still Subtracting Two?
Ariel considers whether he has followed his own rule when he is calculating the number of rods required for a 120-step ladder. He begins by following the rule. He divides 120 by 2 and gets 60. Then he departs from his rule and divides the 60-step ladder by 10. Using the fact that there are 20 rods in a 6-step ladder, Ariel multiplies 20 times 10 and then subtracts 2 (and gets 198). T/R Francisco continues to challenge Ariel to explain how he followed his own rule.
(Note that the number of steps in a 60-step ladder is 182. We would have to subtract 18. One could think of this as starting with one 6-step ladder and every time we add another (there are 9 more) we have to subtract 2 of the rods.)
Event #5: If Two Ladders are Combined Do we Just Add the Rods?
Ariel is calculating the number of steps in a 125-step ladder. Using the rule that he created, Ariel subtracts 1 from 125 and works with 124 steps, dividing by 2 and starting with a new problem of finding the number of steps in a 62-step ladder. Ariel pauses and then expresses doubt about whether he will find the number of rods in the 62-step ladder. As he works on this problem, Ariel has produced a new case for his rule: what happens when we add two ladders to make a new ladder? Do we just add the number of steps or do we have to subtract 2 in this case?
Event #6: If Two Ladders are Combined, Add the Rods and Subtract Two
T/R Francisco gives Ariel a simpler case to consider when adding the steps in two ladders to produce a third ladder. He asks Ariel to use a 6-step ladder and add something to create an 8-step ladder. Ariel builds a 6-step ladder and counts the number of rods and gets 20. Then Ariel is prompted by T/R Francisco to think about how to combine the 6-step ladder and the 2-step ladder to make an 8-step ladder. Ariel first adds 20 plus 8 to get 28, but then he counts the number of rods in the 8-step ladder and counts only 26. At this point he remembers his own rule and subtracts 2 to get the correct answer.
This analytic shows T/R Francisco providing opportunities for Ariel to explore his rule in different cases. His explorations lead him to encounter what is true and what is not true about The Ladder Problem. It provides a small window into the complex cognitive algebra trajectory that Ariel worked through before he emerged as a sophisticated algebra student (see, Sigley & Wilkinson VMCAnalytic, 2015, Events 5-8).
REFERENCES
Agnew, G., Mills, C., & Maher, C. M. (2010). VMCAnalytic: Developing a collaborative video analysis tool for education faculty and practicing educators. Proceedings of the 43rd Hawaii International Conference on System Sciences, Honolulu, HI.
Boaler, J., (2016). Designing mathematics classes to promote equity and engagement. The Journal of Mathematical Behavior, 41, 172-178. https://doi.org/10.1016/j.jmathb.2015.01.002
Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM Mathematics Education, 43, 175–187.
Davis, R. B. (1992). Understanding ‘Understanding.’ The Journal of Mathematical Behavior,11, 225- 241.
Leslie, Joyce (2019). Investigating a model using video stories for professional development for algebra teachers of low SES minority students. Dissertation, Rutgers. New Brunswick, NJ.
Maher, C. A. (1998). Constructivism and constructivist teaching, Can they co-exist? In Ole Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 29-42). Proceedings of Topic Group 6 at the International Congress on Mathematical Education [Report on Proceedings] (8th, Seville, Spain, July 14-21, 1996). Faculty of Education Report No. 3.
Sigley, R and Wilkerson L (2015). Tracing Ariel’s Algebraic Problem Solving: A Case Study of Cognitive and Language Growth. Retrieved from https://doi.org/doi:10.7282/T3N0186C
Steele, C. M. (1997). A threat in the air: How stereotypes shape intellectual identity and performance. American Psychologist, 52(6), 613-629.
Created on2016-07-17T22:09:46-0400
Published on2021-09-15T11:21:28-0400
Persistent URLhttps://doi.org/doi:10.7282/t3-0k8e-ng27
Statistical ProfileVersion 8.5.5
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