# Infusing Mindset through Mathematical Problem Solving and Collaboration: Studying the Impact of a Short College Intervention

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Problem-Based Approaches in Mathematics

#### 1.2. Infusing Mindset and Shifting Students’ Beliefs

#### 1.3. Collaboration

#### 1.4. A Mathematical Mindset Approach to Problem-Based Learning

#### 1.5. Course Design

## 2. Materials and Methods

#### 2.1. Data Sources

#### 2.2. Data Analysis

#### 2.2.1. Student Written Reflections

#### 2.2.2. Pre and Post Mathematical Problem-Solving Assessment

#### 2.2.3. Survey Analysis

## 3. Results

#### 3.1. A Shift in Students’ Mindset and Beliefs about Mathematics

Upon coming to this class, I began to see math differently, as something creative, something without a right or wrong answer, but rather a means of positive struggle, where I embraced all my missteps as part of a growing experience. I took the growth mindset to heart--not only did I apply it academically, I began going to the gym regularly, taking on a vegan diet, and doing even the little things that used to scare me, such as karaoke night (which, might I add, was extremely fun.) (Michelle, Final reflection)

My relationship with math going in was awful, trash, garbage. I hated it and, to be frank, it didn’t make much sense. The way it had been taught to me hadn’t clicked and it didn’t look like it was going to click anytime soon. I was scared for math in college, and honestly considering going from Chemistry to Linguistics to avoid having to take calculus and physics. I am no longer considering switching. (Ricardo, Final Reflection)

#### 3.2. A Shift in Students’ Problem Solving and Collaborative Skills

I was made sure of this today when we approached a problem nearly identical to the one we attempted to solve on the first day. Instead of jumping straight into the concepts we learned in statistics and calculus, my partner and I made visual representations of our thoughts and identified the patterns within the problem. Even as we were doing it, my partner and I realized the differences in our thought processes and how that led to us finding an equation that could represent what we were trying to solve. (Kim, Final Reflection)

This class taught me how to think. It taught me how to make connections within the realm of calculus and with the people around me. Because of this class, I am looking forward to working collaboratively on work, especially math, because I have found that some of the best learning comes from learning with others. (Tasha, Final Reflections)

#### 3.3. Centering Mindset through Mathematics Problems: An Example

When we began to engage in more of the hands-on activities, I started to shift my mindset and continue to try things out. The moment that I believe changed me the most was the cube task. In that task, we worked so well together, and the work was distributed pretty evenly. I felt accomplished, as well as did my group surrounding me. (Chantelle, Final Reflection)

The first problem that really helped open my eyes was the lemon problem. My group thought really creatively about the three methods that we tried, and actually physically manipulating the lemon helped me see why the different methods worked well. But it was at the end of it, when we discussed the problem as a class, that I saw all of my groups’ solutions were basically just different ways to perform summation/integration. It was the first time that I saw the integration formula/graph, and it actually made sense to me. Since that problem I have been riding a kind of high in the class. I now feel like if I try hard enough, and if I think creatively enough, then I can genuinely figure anything out. (Esther, Final Reflection)

#### 3.4. Students Who Resisted the Messages of the Course

Being forced to learn math at a fundamental level was very frustrating at first because beneath the memorization of formulas and functions, it made me feel like I didn’t actually understand it at all. (Marina, Final Reflection)

## 4. Discussion and Conclusions

Taking this class has been my best decision at college so far. I feel like I will go into my first-year maths courses knowing a secret that no one else knows: Maths does not have to be intimidating. It is not maths’ fault that it has been portrayed to be an evil subject only conquered by "geniuses" and/or white males. With a little love and understanding, maths can be kind, compassionate, and even fun to be around. (Briaunna, Final Reflection)

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Mathematics | Calculus | |
---|---|---|

Day 1 | Maths Pre-Task Opening Algebra | |

Day 2 | Rod Trains | |

Day 3 | Negative Space | |

Days4 & 5 | Lemon Task Volume of a round object and integrals as Riemann sums | |

Day 6 | Koch Snowflake Task Geometric Series | |

Day 7 | Bicycle Path Task Derivatives as tangent lines | |

Days8 & 9 | Galileo’s Investigation Derivatives as movement | |

Day 10 | Collatz Conjecture/Four 4’s | |

Day 11 | Walking the graph Derivatives as velocity and acceleration | |

Day 12 | Donut to Coffee Mug Topology | |

Day 13 | Curved Shapes Task Integrals as Riemann Sums | |

Day 14 | Maths Post-Task |

## Appendix B

**Table A2.**One pair of students’ pre- and post-assessments, coded non-collaborative at pre, and collaborative at post.

Non-Collaborative Pre-Assessment | Collaborative Post-Assessment |
---|---|

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**Figure 1.**Each circle represents the same pair of students that completed both pre- and posts-assessments at the beginning and end of the course.

Negative | Neutral | Positive | Extremely Positive | |
---|---|---|---|---|

Criteria | Majority of response is about a bad experience in this class (does not need to include critique) | Mixed experience = some positive, some negative (about the same for each) Or Mostly neutral sentiments | Majority of response is about a good experience, without going deeply into personal transformation | Includes clear evidence (multiple indicators) for personal transformation and plans to apply the ideas of the course to their future |

Task | |
---|---|

Pre-task | Leo the Rabbit is climbing up a flight of 10 steps. Leo can only hop up 1 or 2 steps each time he hops. He never hops down, only up. How many different ways can Leo hop up the flight of 10 steps? Provide evidence to justify your thinking. |

Post-task | You are tiling a straight path that is 2 units wide and 10 units long. The titles you have available are 1 unit by 2 units. How many different ways can you tile the path? Provide evidence to justify your thinking. |

Principal Component | Increased Agreement | t-Value | Decreased Agreement | t-Value | McDonald’s omega ^{+} | Cronbach’s alpha ^{+} |
---|---|---|---|---|---|---|

Math is complex | I like to solve complex problems | 0.17 | N/A ^{~} | 0.616 | ||

I like math | 0.9 | |||||

Nature of Math | Math is creative | 2.68 *** | In math, answers are either right or wrong | −5.42 *** | 0.596 | 0.588 |

Math is a subject with lots of connections between ideas | 1.97 ** | Mathematics involves mostly facts and procedures that have to be memorized | −3.34 *** | |||

Mindset | When I make a mistake in math, I feel bad | −2.35 *** | 0.429 | 0.427 | ||

People can learn more math, but they can’t really change their basic math intelligence | −4.12 *** | |||||

People who really understand math will get an answer quickly | −3.88 *** | |||||

Sometimes math makes me feel afraid | −2.21 ** | |||||

Personal Orientation | If I put in enough effort, I can succeed in mathematics | 1.34 * | People can learn more math, but they can’t really change their basic math intelligence | −4.12 *** | 0.380 | 0.238 |

It is really helpful to talk about math with others | 1.83 ** | Sometimes math makes me feel afraid | −2.21 ** |

^{+}Scale reliability coefficient;

^{~}Minimum of three continuous variables required for calculation; * p < 0.1, ** p < 0.05, *** p < 0.01.

Strategy | Image | Description |
---|---|---|

Water displacement | Students filled the vase to its capacity with water, submerged the lemon, and captured the water that spilled out. The measure of the spilled water approximated the lemon’s volume. | |

Approximating to other solids | Students approximated the shape of the lemon as a sphere, ellipsoid, or two cone shapes. They cut into the lemon to take measurements. | |

Cylinder Slices | Students sliced the lemon into multiple cylinder shapes, using digital calipers to collect measurements and the volume of a cylinder formula. The sum of the cylinders’ volumes approximated the lemon’s volume. | |

Play-Doh Mold | Students molded play-doh around the lemon to make a hollow shell that they filled with water. Next, they measured the amount of water the mold held. | |

Negative Space | Students sliced the lemon in half, inverted the pieces to represent the base and top of a cylinder, filled negative space with play-doh, and calculated the volume. Next, they removed the play-doh, reshaped it into a rectangular prism, and calculated its volume and subtracted it from the cylinder volume. | |

Sectors to rectangles | Students sliced the lemon into multiple thin disk-shaped slices. Each disk was sliced into multiple sectors which were then rearranged in an alternating pattern to create a rectangular prism shape. This process was repeated for each disk, and the sum of the volumes of the resulting rectangular prisms approximated the lemon’s volume. | |

Disk method | Students approximated the side of the lemon as a curve using the equation of a parabola. Next, the students rotated this parabola about the x-axis and found the volume of the resulting figure using integration. |

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**MDPI and ACS Style**

Boaler, J.; Brown, K.; LaMar, T.; Leshin, M.; Selbach-Allen, M.
Infusing Mindset through Mathematical Problem Solving and Collaboration: Studying the Impact of a Short College Intervention. *Educ. Sci.* **2022**, *12*, 694.
https://doi.org/10.3390/educsci12100694

**AMA Style**

Boaler J, Brown K, LaMar T, Leshin M, Selbach-Allen M.
Infusing Mindset through Mathematical Problem Solving and Collaboration: Studying the Impact of a Short College Intervention. *Education Sciences*. 2022; 12(10):694.
https://doi.org/10.3390/educsci12100694

**Chicago/Turabian Style**

Boaler, Jo, Kyalamboka Brown, Tanya LaMar, Miriam Leshin, and Megan Selbach-Allen.
2022. "Infusing Mindset through Mathematical Problem Solving and Collaboration: Studying the Impact of a Short College Intervention" *Education Sciences* 12, no. 10: 694.
https://doi.org/10.3390/educsci12100694