TORPEDO - Universiteit Utrecht€¦ · TORPEDO In 2018 I received a Comenius scholarship. I designed TORPEDO. TORPEDO is a digital learning environment to support and encourage students

TORPEDO

A digital learning environment todevelop mathematical problem solving

ability in primary teacher training

TORPEDO

Marjolein Kool, senior lecturer mathematics educationPrimary Teacher Training Institute, Hogeschool [email protected]

People need problem solving ability to survive in modern society.

TORPEDOMathematical problem solving ability

Mathematical problem solving ability is the ability to construct suitable problem approaches for unknown, so-called non-routine mathematical problems

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(Polya, 1990; Selden et al., 2000; Drijvers, 2015)

Mathematical problem solving ability

Prospective teachers who want to develop their students' problem-solving ability need themselves sufficient problem-solving ability.

How can we achieve this?

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To develop problem solving abilityreflecting is needed.

Reflecting mathematically

- is distancing oneself from the action of doing mathematics; (Wheatley , 2009)


- is looking back after solving a non-routine mathematical problemand linking new knowledge to existing knowledge.

(Van Streun, 2001; Ambrose et al., 2010)

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Reflecting

Reflecting to develop problem solving ability means reflecting on the:

- problem structure- problem approach- problem solving process

(Van Streun, 2001; Ambrose et al., 2010)

When students work on non-routine problems at home they hardly spendany time to reflect.

(Kool & Keijzer, 2018)

During mathematics lessons the teacher trainer pays much attention to this necessary reflection.


TORPEDO

In 2018 I received a Comenius scholarship.

I designed TORPEDO. TORPEDO is a digital learning environment to support and encourage students to reflect when they work on non-routine math problems, to develop their problem solving ability.

May 16th, 2018


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TORPEDO

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TORPEDO offers the following reflective elements:

- (meta) cognitive reflective support- (meta) cognitive reflective challenge - affective-motivating support


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TORPEDOAn example from

The blue area

Given:The figure shown on the left

Asked:How large is the area of the blue area in square centimeters?

The problem-solving step-by-step path with support:

1. The problem 2. Prior knowledge 3. Hints 4. Your answer 5. Reflection Groupe of problems

1. Understand the problem- Visualization of the problem- Reformulation of the problem

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The blue area






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You can draw a square around it

or divide the figure into pieces


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The blue area






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The blue area






TORPEDOReflection

Watch video 2.0-1 about problem approach 1: Transform the figure6 reflective videos:- Problem approach 1: Transform the figure- Problem approach 2: Find a pattern or regularity- Problem approach 3: Use a formula- What are the advantages and disadvantages of the various problem approaches? - To which other problems does this problem belong? - What is useful to remember for next time?

Het blauwe gebied oplossingsmanier 1 korte versie.mp4

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Is it also possible to transform the blue figure into a square?

No, because the blue triangles are smaller than the white ones.

No, because the blue triangles are bigger than the white ones.

Yes, into a square of 10 cm x 10 cm

Yes, into a square of 5 cm x 5 cm

Question 1

Reflection

Watch video 2.0-1 about problem approach 1: Transform the figure

Het blauwe gebied oplossingsmanier 1 korte versie.mp4

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The blue area






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The blue area





Groupe of problems 2: The blue area (domains: measuring and geometry)

2.0 Main problem: The blue area

2.1 Problem to practise: Optical illusion

2.2 Problem to practise: Dividing cookies

2.3 Variant problem: The church window

2.4 Variant problem with student work: The yellow arrow.

practise, but non-routine problems







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The blue area






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The problem-solving step-by-step path:


The church windowGiven:A church has round windows with a diameter of 2 meters. Gold-colored ornaments have been painted around it.

Asked:What is the surface of such a gold ornament?Give your answer in square meters.







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The blue area



The problem-solving step path with support:



With discussion

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Groups of problems

Choose a group of problems

1. Stacking glasses (domains: whole numbers and patterns2. The blue area (domains: measuring and geometry)3. The fast little dog (domain: measuring)4. The outdoor cinema (domains: whole numbers and patterns5. Jamie Oliver (domain: percentages)6. Square terrace (domain: whole numbers)7. Saudi Arabia (domain: measuring and geometry)8. The Christ Statue (domain: measuring and proportions)9. Long jump (domain: whole numbers)10. Fuel consumption (domain: proportions and fractions)

TORPEDOAffective-motivating elements in TORPEDO

- Important aim

- Appropriate method

- Freedom of action

- Success experiences

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Research question:

How and to what extent are students supported and encouraged to study reflectively when working independently in TORPEDO?

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The TORPEDO trial period took place from December 15, 2018 to January 15, 2019

390 students from 17 different teacher training colleges were interested

271 students actually logged in and spent time in TORPEDO

They showed mutual differences in: - study variant- number of study years - number of times they had already participated in the national

maths test

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213

43

15

0

50

100

150

200

250

full-time part-time compactcurriculum

study variant(N = 271)

Features of the participating students

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7

22

152

59

209

1 10

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 unknown

Number of study years(N = 271)


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138

46

26 28

12 116 2 1 0 1

0

20

40

60

80

100

120

140

160

0x 1x 2x 3x 4x 5x 6x 7x 8x 9x 10x

Number of times participated in the National Mathematics Test

(N = 271)


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During the experiment the following data was collected:- Log data of 271 students who worked in TORPEDO - 181 completed surveys- 41 other forms of feedback on TORPEDO

Quantitative use of TORPEDO

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80

103

54

1420

0

20

40

60

80

100

120

0 - 1 1 - 5 5 - 10 10 - 15 > 15

Number of hours spent in TORPEDO (N = 271)

Mean = 4 hours and 46 minutes sd = 5 hours and 49 minutes


TORPEDO

- mean = 12,4 problems, sd = 13,7.- major differences in the quantitative use of TORPEDO- no correlation between the time spent in TORPEDO and the number of solved problems


46

112

52

2519 17

0

20

40

60

80

100

120

0 1 - 10 11 - 20 21 - 30 31 - 40 41 - 50

Number of solved problems(N = 271)

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Indicators of reflective study behavior

Concerning reflection-supporting elements:1. The students use the worked-out problem-solving step-by-step approach when working on the main problem

Concerning reflection-stimulating elements:The students:2. answer the reflective questions that go with the reflective videos3. work on one or more related follow-up problems after they have worked on the main problem4. if they get stuck with the follow-up problems look for help within TORPEDO5. solve the variant problems with student work6. contribute to the discussion groups, and / or read the contributions in the discussion groups.

General indicator of reflective study behavior:7. The students take notes while working in TORPEDO

Qualitative use of TORPEDO

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Concerning reflection-supporting elements:1. The students use the worked-out problem-solving step-by-step path when working on the main problem


Data from survey: 1. Never used5. Always used

N Mean Standard-deviation

Understand the problem 181 3,39 1,25

Activate prior knowledge 181 3,18 1,45

Use hints 181 3,52 1,15

3 reflective videos on different problem approaches

181 3,40 1,36

3 reflective videos on problem structure, problem approaches and working process

181 3,05 1,38

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Concerning reflection-stimulating elements:2. The students answer the reflective questions that go with the reflective videos


Log-data


The students answer the reflective questions that go with the reflective videos

271 1,77

out of 60 reflectivequestions

2,97

123 students (N = 271) answered at least 1 reflective question.8 students answered all the 60 reflective questions

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Concerning reflection-stimulating elements:3. The students work on one or more related follow-up problems after they have worked on the main problem


Log-data


Number of problem groups in which the students work on one or more related follow-up problems after the main problem

271 3,4

out of 4,3 problemgroups

2,55

100 students always solved at least 1 follow-up problem after they have solved the main problem

27 students only solved main problems

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Concerning reflection-stimulating elements:4. The students if they get stuck with follow-up problems look for help within TORPEDO.


Open question in the survey.

Number of students who get stuck while working on follow-up problems ...

that look for support within TORPEDO 41

that look for support outside TORPEDO 119

No answer on this question 21

Total number of students 181

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Concerning reflection-stimulating elements:5. The students solve the variant problems with student work


Log-data

Problem category Number of solvedproblems

Main problems 927 27,6%Problems to practise 1432 *) 42,7%Variant problems 536 16,0%Variant problems with student work

460 13,7%

TOTAL 3355 100%

*) The number of problems to practise in TORPEDO is double that of any other category, so the number and percentage of solved problems to practise must be halved in order to be able to compare them with the other categories.

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Concerning reflection-stimulating elements:6. The students contribute to the discussion groups, and / or read the contributions in the discussiongroups


Survey1. Never done5. Always done


The students contribute to the discussion groups

271 0,36 times

to 20 discussion groups.

1,40

Log data


The students read the contributions in the discussion groups

181 3,03 1,36

Contribute to the discussion

Read the discussion

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General indicator of reflective study behavior:7. The students take notes while working in TORPEDO



The students take notes to record the results of reflections.

181 3,19 1,53

Survey1. Never done5. Always done

Comparing indicators of reflective study behaviorTORPEDO

Indicators of reflective behavior

1a. Using stap 1 to 3 from the worked-out problem-solving step-by-steppath: 1 understand the problem; 2. activate prior knowledge; 3. Use hints.

175 of 181 = 96,7%

1b. Using 3 reflective videos on different problem approaches 154 of 181 = 85,1%

1c. Using 3 reflective videos on problem structure, problem approaches and working process

145 of 181 = 80,1%

2. Answering the reflective questions that go with the reflective videos 122 of 271 = 45,0%

3. Working on one or more related follow-up problems after working on the main problem

195 of 271 = 72,0%

4. After getting stuck with the follow-up problems looking for help within TORPEDO

41 of 181 = 22,7%

5. Solving the variant problems with student work 153 of 271 = 56,5%

6a. Contributing to the discussion groups 31 of 271 = 11,4%

6b. Reading the contributions in the discussion groups

149 of 181 = 82,3%

7. Taking notes while working in TORPEDO 141 of 181 = 77,9%

Percentage of students for whom these indicators have been observed at least once




175 of 181 = 96,7%



145 of 181 = 80,1%



195 of 271 = 72,0%

4. After getting stuck with the follow-up problems looking for help within TORPEDO:

41 of 181 = 22,7%




149 of 181 = 82,3%



Reflection –supportingelements are usedquite often




175 of 181 = 96,7%



145 of 181 = 80,1%



195 of 271 = 72,0%


41 of 181 = 22,7%




149 of 181 = 82,3%



Reflection – stimulatingelements are less popular. But some of these have been observed relatively often.




175 of 181 = 96,7%



145 of 181 = 80,1%



195 of 271 = 72,0%195 of 225 = 87%


41 of 181 = 22,7%

5. Solving the variant problems with student work 153 of 271 = 56,5%153 of 225 = 68,0%



149 of 181 = 82,3%



Reflection – stimulatingelements are less popular. But some of these have been observed relatively often.

46 students didn’t enter any answer




175 of 181 = 96,7%



145 of 181 = 80,1%



195 of 271 = 72,0%

4. After getting stuck with the follow-up problems looking for help within TORPEDO

41 of 181 = 22,7%




149 of 181 = 82,3%



Some reflection –stimulatingelements have been observedrarely.

TORPEDOEvaluation of affective-motivating elements

The evaluation is based on:

Several closed questions, with a Likert scale from 1 to 5

4 open questions:What do you want to explain or comment on:- motivating and demotivating aspects of TORPEDO?- what studying in TORPEDO has brought you?- the technical quality of TORPEDO?- how we can improve / expand TORPEDO?



The aim of TORPEDO (i.e. working on your mathematical problem-solving ability), that is 1. very demotivating ...5. very motivating

181 3,82 0,87

a. An important aim

TORPEDO focuses primarily on the development of mathematical problem-solving ability. How do students evaluate this aim?

1 student scores a 1,43 students score a 5

Some students seem to have forgotten that the original aim was to develop problem-solving ability. They focus on passing the National Mathematics Test:

"The problems from TORPEDO did not match the problems on the National Mathematics Test.""I saw no similarity at all with the National Mathematics Test"“(Unlike TORPEDO) you recognize the problems in the Test right away and then you know

which strategy to apply."


b. An appropriate method

To what extent do students think the reflection-supporting and reflection-stimulating elements suitable for developing their problem-solving ability?


The presentation of different problem approaches, that is1. very demotivating5. very motivating

181 4,12 0,85

- Evaluation of the reflection-supporting elements

1 student scores a 170 students score a 5


A lot of support is given with the main problem, that is1. very demotivating5. very motivating

181 4,14 0,89

2 students score a 175 students score a 5



- Evaluation of the reflection-stimulating elements


No support is given with the follow-up problems, that is ...1. very demotivating5. very motivating

181 2,16 1,19



Each problem is presented with the problem-solving step-by-step path that is ...1. very demotivating5. very motivating

181 3,51 0,89

4 students score a 123 studenten score a 5


TORPEDO includes problems with student work, that is ...1. very motivating5. very demotivating

181 3,77 0,92




Evaluation of the reflection-supporting and reflection-stimulating elements

- Both reflection supporting and stimulating elements score generally high.- The reflection-stimulating elements score slightly lower than the

reflection- supporting elements.

- Students find the use of reflection-stimulating elements more difficult.

"In working on the extra problems you suddenly had to do the reflection yourself. A very big step for me.”

"It is a pity that you cannot view hints and discussions with all the problems."

- Some students reject reflective study behavior because they find it cumbersome, time-consuming or ineffective.

"I want to practise more problems in less time.""I find it very demotivating that solving just one problem takes more than half an hour."

"I want to solve a huge amount of problems to improve, but before you could enter an answer you had to take many steps."

- The high rating for the problem category ‘variant problems with student work’ is surprising. (mean = 3,77 sd = 0,92)


c. Freedom of action

To what extent do students feel free to use TORPEDO in their own way?

TORPEDO provides advices for reflective study behavior

Students are free to ignore these advices. Nevertheless most of the students follow them.

Although compulsory activities are lacking, some students find TORPEDO too guiding. For instance they find the lack of support for the follow-up problems patronizing.


d. Success experiences

- Problems at the right level


The difficulty of the problems was for you ...1. way too low ……5. way too high

181 3,47 0,71


"My knowledge is apparently not high enough to investigate my own problem approach.""I found it difficult, but it did challenge me."

The average good score of all made problems in TORPEDO is 52,9%


d. Success experiences

- Learning results

Self-confidence N Mean St.dev.

I have got more confidence with regard to solving new mathematical problems.

1. Not at all ...5. Certainly

181 3,02 1,14


Tools for new math problems N Mean St.dev.I know better what to do if I have to solve a new mathematical problem.1. Not at all ...

5. Certainly

181 3,26 1,05

12 students score a 117 studenten score a 5

Dare to leave problemapproaches

N Mean St.dev

I dare to adjust my problem approach or to replace it with another one.1. Not at all ...

5. Certainly

181 3,24 1,02

11 students scores a 114 students score a 5


Open question: "What do you want to explain or comment on what working in TORPEDO has brought you?”

20 students: “Hardly anything”.

22 students feel that they work more structured, systematic, thoughtful:

"By using TORPEDO I was much calmer during working on the test problems.

"I worked in a more structured way."

"(I went) to look at the problems differently. What exactly is being asked here? "

"I can now (...) think which steps I can take if I don't understand a problem right away."

"I was much more aware of the steps that had to be taken."

They developed their problem solving ability, but this was not achieved by everyone.

TORPEDOFinal discussionQuite a few students showed reflective study behavior

The students that solved at least one problem, showed at least once that they:- use the step-by-step-path (96,7%)- solve follow-up problems (87%)- use reflective videos (82,5%)- read discussion groups (82,3%)- take notes (72,9%)- solve problems with student work (68,0%)

Do the students that showed oneor more of the indicators of reflective study behavior, experienced that they developproblem solving ability?

Quite a few students believe thatTORPEDO can develop theirproblem solving ability.But no correlation was found between this effect and any of theindicators of reflective behavior.

12

28

56

65

17

3

0

10

20

30

40

50

60

70

1 2 3 4 5 No answer

1 = Not at all 5 = Certainly

I know better what I have to do to solve a new math problem

(N= 181)Mean: 3,26Sd: 1,05

TORPEDOFinal discussion

Why wasn’t TORPEDO effective for developing problem solving ability according to some of the students? Why didn’t they reflect very much?- Mathematical level: Working/reflecting in TORPEDO is (too) difficult (31 of 181 = 17,1%)

(They try to reflect but it was (too) hard)- Believe: Reflection is useless as preparation for the National Mathematics Test (21 of

181 = 11,6%)(They found reflecting time consuming and cumbersome. They wanted practiseand dril in stead of reflecting.)


Students need to have suitable mathematical knowledge and skills to work in TORPEDO.Working on non-routine problems makes no sense if you can’t solve the routine problems. Reflecting on your math work is difficult. Students need a mathematical base.Perhaps some of the participating students lack this base.

According to the students this problem can be solved:


No support is given with the follow-up problems, that is ...1. very demotivating5. very motivating

181 2,16 1,19


32,6% of the students ask for video’s, hints, support, etc. for the follow-up problems.They find it patronizing that they can’t choose for help after they have solved these problems.

A student wrote: ‘Tell me how I can solve these problems’.

Dilemma: if Torpedo offers this kind of support, how will the students use this if reflection is- too difficult for them- and/or useless according to their opinion?


Students will not reflect if they do not see what it can bring them.

Inviting students to study reflectively in a specially designed digital learning environment is not enough to change their view of problem solving. Students must be regularly encouraged to do this. (Nelissen, 1999)

Maybe TORPEDO will inspire more students to study reflectively if it is used in a blended way.

Considering learning maths as memorizing rules and recipes, and problem solving as remembering the right problem approach at the right time, can hinder the development of problem-solving ability.(Crawford, 1994; Kloosterman, 2002)

The vision/believe of some students about learning mathematics is not in line with the principles of TORPEDO.


TORPEDO doesn’t work for all students, but…students with appropriate mathematical knowledge, skills and believes - like Torpedo- reflect- and feel their problem solving ability is growing


Look at the following example of a problem with student workand the corresponding discussion.

What do you think of these two reflection- stimulating elements?

Do you have questions?

Do you have suggestions to improve TORPEDO?

Students have among other wishes, the following requests:- Give us a test exam at the National Mathematics Test level- TORPEDO must use problems that may occur in the National Mathematics Test.- Provide follow-up problems with hints, discussions and instruction

Which requests would you honor?

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The problem-solving step-by-step path:


Given:The large square is 10 cm x 10 cm.The small square is 8 cm x 8 cm.How large is the area of the yellow figure?A student says:a. I do half of 8 x 8b. And I add half of 10 x 2 to thatc. I subtract the result from half of 10 x 10d. And then I have calculated a quarter of the yellow arrow, so I multiply that by 4.Asked:Which step or steps are wrong?

The yellow arrow

A problem with student work

DiscussionTORPEDO

Y.S. : But is that possible?Because if you do ½ x 2 x 10 then you are not quite right because the yellow arrow does not touch the ground there. And then you can't double because what remains of the yellow arrow must then be ½ x 2 x 8. I can not get out.

K.M .: It is about calculating the area of the beige areas. If you subtract these, the area of the yellow figure remains:- Square of 8 x 8 and two triangles of ½ x 2 x 10.- Area square is 64 m² and the area of two triangles is 20 m². That is 84 m² together.- The area of the entire square is 10 x 10 = 100 m², so 100 - 84 = 16 m².The area of the yellow figure is therefore 16 m².

S.M .: Oh yes, a trap. For a while my focus was only on the yellow figure ... Good practice in reversing your way of thinking;) NB I think it goes wrong with D. He / she has calculated half of the yellow arrow and not a fourth .. It must be times 2. Then the answer is correct....

L.R.R: The area of the triangle is ½ x 2 x 10.You have to double that.Given:

The large square is 10 cm x 10 cm.The small square is 8 cm x 8 cm.How large is the area of the yellow figure?A student says:a. I do half of 8 x 8b. And I add half of 10 x 2 to thatc. I subtract the result from half of 10 x 10d. And then I have calculated a quarter of the yellow arrow, so I multiply that by 4.Asked:Which step or steps are wrong?

The yellow arrow


Suggestions for adapting TORPEDO:

a. An important aimMain aim: the development of problem-solving ability, two indirect aims.

b. An effective methodCreate a gradual transition from reflection-supporting to reflection-stimulating elements.

c. Freedom of actionGive students the opportunity to choose for more support, hints, discussions, instruction.

d. Opportunities for success experiencesBridge the gap between the starting level of the students and the level of the problems.Give the opportunity to solve problems more than once and organize a flexible reward system.

e. Moral supportCreate opportunities for collaboration and personal guidance.

f. Good technical quality and designImprove the navigation on the website.

TORPEDO

Kool, M. & Keijzer, R. (2018). To what extent do student teachers develop their mathematical problem solving ability by self-study? Proceedings, EAPRIL-conference 2017, Hämeenlinna, Finland, 80-90. Zie: https://www.eapril.org/sites/default/files/2018-04/ConfProceedings2017.pdf

Van Streun, A. (2001). Het denken bevorderen. Oratie. Groningen: RUG.

Ambrose, S. A., Bridges, M. W., DiPietro, M., Lovett, M. C., & Norman, M. K. (2010). How learning works: Seven research-based principles for smart teaching. San Francisco: Jossey-Bass.

Drijvers, P. (2015). Denken over Wiskunde, onderwijs en ICT. Oratie, Universiteit Utrecht, Freudenthal Instituut.

Bronnen:

Pólya, G. (1990). How to Solve It. Londen: Penguin Mathematics.

Selden, A., Selden, J., Hauk, S., & Mason, A. (2000). Why can't calculus students access their knowledge to solve non-routine problems? In Dubinsky, E., Schoenfeld, A., & Kaput, J. (Eds.) Research in collegiate mathematics education. IV. Providence, RI: American Mathematical Society.

https://www.eapril.org/sites/default/files/2018-04/ConfProceedings2017.pdf

Documents

TORPEDO - Universiteit Utrecht€¦ · TORPEDO In 2018 I received a Comenius scholarship. I designed TORPEDO. TORPEDO is a digital learning environment to support and encourage students