Author: Sabine Harvey

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CAMP 2025 Day 5

We have officially reached the end of Bard Math CAMP 2025! This week has really flown by, occupied by great fun as well as great math. Students enter the lobby and make their way to morning activities to pass time as they wait for their last math class of the week.

The Sine group starts by exploring a grid, noticing how to get from one intersection point to another, recognizing a pattern and making it easy to traverse from “Point A” to “Point B”. They discover Pascal’s Triangle within the pattern!

CAMPers then use exactly three Cuisenaire Rods to see how many different trains could be made for a set of sequential numbers. The CAMPers collaboratively fill out a data table on the board, taking turns each adding a component. They discuss patterns between length and number of trains. CAMPers see how “choosing” played a role in finding the numbers and the “magical” appearance of the triangular numbers.

Class ends with CAMPers having the opportunity to create even more trains, with each group getting a different criteria. Unfortunately, time runs out to finish this activity- but with enough information to continue on their own!

The Cosine group starts class going over the methods of solving the train problem they were given yesterday, counting the number of trains of a certain length, composed of a certain number of cars. CAMPers are then tasked with explaining what those lengths mean in the context of a formula that one CAMPer conjectured: The number of trains of length n made from k cars is just (n-1) choose (k-1)!

They use blocks to create a visual representation of the trains in the problem and are challenged to use these visuals to investigate correlations between the numbers in the problem and triangular numbers. Class ends with CAMPers visiting each others tables, taking turns explaining patterns and thought processes to solve the problem.

CAMPers then move on to their second class of the day with the Cotangent group heading to art and the Tangent group going to computer science.

In art, using the vanishing point and perspective concepts that they learned earlier this week combined with their shading practice yesterday, CAMPers make optical illusion-like doodles! They create squiggly oceans, descending rabbit holes, colorful patterns, and so much more.

In computer science, the Tangent group starts by collaboratively completing the Towers of Hanoi. Each CAMPer is assigned a disk and is responsible for said disk and the one above it. The goal is to discover a pattern of behavior that will be translated into a recursive algorithm!

The CAMPers then use the driver-navigator partner system to write a code that efficiently solves the Towers of Hanoi using the patterns they explored and their newfound understanding of recursive functions.

CAMPers break for lunch, enjoying the beautiful weather! Skipping our elective period, once CAMPers are done with lunch, they head straight to their third class of the day. The Cotangent group goes to computer science while the Tangent group heads to art to have their turn in creating and designing optical illusions.

In computer science the Cotangent group gets a new challenge to work on! Some are continuing work on logic problems like determining whether inputs are positive or negative. Others use the driver-navigator partner system to create a function that doesn’t use loops, but finds the factorial of a number and test their thought processes in this.

When CAMPers finish this challenge, they are given the same Tower of Hanoi problem that was given to the Tangent group. They work to figure out a pattern so they can create a function that will demonstrate it, and solve this puzzle.

After their last class at Bard Math CAMP, CAMPers take a survey asking their opinions on the time they spent here. They trade their survey in for their CAMP t-shirt! All CAMPers walk to the RKC building to take a photo together. Once back, everyone sets up and gets ready to show their parents everything they’ve worked on this week during open house.

We have officially reached the end of Bard Math CAMP 2025! We hope that all CAMPers were able to learn something new and will continue exploring any curiosities outside of CAMP. It is such a joy to see so many CAMPers grow intellectually while having so much fun doing math!

This week went by so fast! Although we hate to see this week go, saying goodbye means enthusiasm for next year just grows, as more ideas for what to do next continues to form!

Photo credits: Featured image- Bard student on campus, 1-39- Sabine Harvey

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CAMP 2025 Day 4

Happy Thursday! We are slowly nearing the end of the week, but that just means that CAMPers are beginning to dive even further into what they have been working on. Despite some gloomy weather, the lobby still fills with smiles as CAMPers sign up for an elective of their choice and make their way to the art room to play with various morning activities while they wait for math class.

The Sine group looks at Pascal’s Triangle to see what patterns they can find and particularly ones we studied. Many realize a connection between Pascal’s Triangle and their strategy in solving problems from earlier in the week.

The CAMPers are next given Cuisenaire Rods of lengths one and two to build “trains”, recording lengths, number of trains that could be made for each length, and recording their findings on square grid paper. They are excited to discover that the number that could be made was the Fibonacci’s Sequence. They share their findings on the board and see how the diagram shows the additive relationships.

The Cosine group starts off with Pascal’s Triangle as well, learning four new counting techniques and using them to fill out their pages. They work to find paths and patterns that will allow them to solve various scenarios using triangular numbers and Pascal’s Triangle.

CAMPers then use a worksheet to practice using and identifying counting techniques before receiving a main question that allows them to apply what they’ve learned to a real world scenario involving train cars.

In three small groups, CAMPers work through the problem, discussing formulas, ideas, and strategies. They use paper to write out thinking and blocks to create a visual for solving a train problem that has been presented. Many CAMPers are enthusiastic when they begin finding relationships between the problem and their previous work with Pascal’s Triangle.

CAMPers then move on to their second class of the day, with the Tangent group heading to computer science and the Cotangent group going to art.

In art, CAMPers get to create their own mazes! Using graph paper, they shade squares- some being made lighter and others darker. They then form various pathways, creating one to be the correct way and others to trick someone trying to complete their maze. Once they have figured out their separate routes, they are tasked with erasing their lighter shading to make the path clearer. CAMPers try to find the easiest and most efficient way to complete their mazes, challenged to come up with as many possibilities as they can.

The Tangent group begins with a review of what they’ve gone over throughout the week and are given the chance to ask questions. They then collaboratively go over challenges from yesterday and take turns writing lines to solve them. CAMPers learn about compound boolean statements and test using them.

The CAMPers are then given more challenges to work on in their respective driver-navigator pairings, they experiment with creating statements that receive true and false outputs. Class ends with CAMPers being thrown a “curveball” where they are asked to write a function to find the factorial of a number without using loops.

CAMP breaks for a well deserved lunch. Many CAMPers finish eating quickly and play various games with each other inside and outside while waiting for electives to begin. Today for electives CAMPers experiment with different tic-tac-toe variations, go for a walk on campus-stopping at the cemetery, library, and on the “secret path”, or complete the second of the two-day paper bridge engineering challenge elective.

Yesterday in the engineering challenge, CAMPers worked to stack as many textbooks as possible on only one index card, eventually learning that cutting the index card and distributing the weight among the pieces is the most efficient strategy.

Today, CAMPers got to make paper bridges! In groups of two and three they begin constructing with paper and tape taking as many trials as they need to build the strongest bridge possible.

Time runs out during elective time to finish testing so it will be finished during afternoon activities.

Once electives are done, CAMPers head to their third and final class of the day. With the Cotangent group heading to computer science and the Tangent group going to art to have their turn creating mazes and being challenged to do so in the simplest way possible.

In computer science, the Cotangent group starts by recapping what they have learned and asking questions. They then explore boolean operators and incorporate them while going over a challenge from yesterday- collaborating to write a code that solves it.

CAMPers are then given more challenges to do on their own and use the driver-navigator partner system to complete them. They experiment with different inputs as they continue to explore the Python language.

Today while waiting for parents, paper bridge testing continued. Many teams’ bridges max out- their strings breaking! However, Daphne, Ariana, and Camille’s bridge is able to hold the weight without the string breaking, deeming them the winners!

What a fun day at CAMP! We are saddened at the idea of CAMP coming to an end but so excited for parents to come tomorrow for open house to see everything the CAMPers have worked on this week.

Photo credits: Images 10, 17-19, 33, and 36-37 – Japheth Wood, Featured image, 1-9, 11-16, 20-32, 34-35, and 38-55 – Sabine Harvey

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CAMP 2025 Day 3

Today marks the midpoint of our favorite week! As CAMPers arrive, they sign up for an elective and make their way to the art classroom to enjoy some games and puzzles, waking up their brains before heading to the first class of the day.

The Sine group continues their work on combinatorics. Previously, CAMPers had been put in pairs to work on different combinatoric problems. Today, they share their problems, explaining their thought processes.

The CAMPers realize that their problems have related to a common answer- 10- allowing them to work together to search for connections and patterns in their work that would generate this conclusion.

The CAMPers use what they notice to begin looking for strategies to solve more problems. They end class introducing Pascal’s Triangle, encouraged to look for patterns within the triangle that relate with what they did.

At the same time, the Cosine group continues their work on the triangular numbers and various representations. The past few days, CAMPers had been paired up and given different scenarios on the triangular numbers. Today, they get the opportunity to present those scenarios in front of the class, explaining thinking and strategies.

Their work shows a connection in what they have been working on, allowing CAMPers to collectively work through the formulas and figure out the problems. They discuss patterns and observations.

Once math classes is wrapped up, CAMPers head to their second class of the day with the Tangent group going to the computer lab and the Cotangent group to art class.

In art class, CAMPers learn the history of perspective, and explore paintings from before this concept was used in artwork. They talk about how art has changed geometrically now that there is greater knowledge of math in a creative context.

CAMPers use this discussion to create pictures using vanishing point and horizon line, making their own geometrical images with perspective.

In computer science, the Tangent group recaps what they’ve learned so far before getting the opportunity to work on challenges. They do one together, experimenting with inputs and collectively calculating their outputs. They write a function to find the absolute value between two numbers to calculate differences between numbers before working with each other to write code as a class.

CAMPers are then given three additional challenges. Using the driver-navigator partner system, they continue exploring Python and finding answers to questions.

After a relaxing and plentiful lunch, CAMPers go to the elective of their choice.

In today’s electives, CAMPers can either test their hand-eye coordination and learn juggling, decorate a dilapidated dodecahedron sculpture on campus, or begin Shiven’s two-day paper bridge engineering challenge.

Today in Shiven’s elective CAMPers were tasked with stacking as many textbooks as they could on only one index card. The winner stacked 39 total textbooks! They finished with a discussion of tactics and observations.

What fun electives! CAMPers then move to their third and final class of the day. The Tangent group goes to art where they have their own discussion about the history of perspective in art, adding new observations to the subject. They use this new knowledge to create drawings using the same aspects they just saw.

In afternoon computer science, the Cotangent group first goes over what they’ve already done before being given the opportunity to ask any questions they may have about what they’ve learned.

They collaboratively run through a line of code, working together to fix any mishaps and make corrections. CAMPers learn the difference between arithmetic and boolean operators before being challenged with using the driver-navigator system to write their own functions in Python.

Thanks for another great day at Bard Math CAMP! Today was filled with a lot of excitement and even more learning. We cannot wait to pick up where we left off tomorrow!

Photo credits: Image 34- Shoshi Cohen, Image 35- Japheth Wood, Images 1-33, 36-50, and headliner image- Sabine Harvey

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CAMP 2025 Day 2

CAMP Day 2 has commenced! As CAMPers return to the lobby, their smiles even bigger this morning, they grab their name tags, choose from one of the offered electives for the day, and head to the art room to wait for their first class to begin.

The art classroom is much louder this morning as CAMPers walk in today to see familiar faces rather than new. With names now learned and game preferences found, they quickly begin to engage in the various activities set out before heading to math class- the Cosine group with Japheth and the Sine group with Frances.

The Sine group starts class with a new visual of the handshake problem they were given yesterday. This visual allows CAMPers to see a geometric connection to the formula and to internalize it to better understand the problem.

They return to their pairs from yesterday and continue with their combinatorics problems to find patterns and formulas. They are encouraged to leave all work on their paper to display the entire thought process and steps that go into solving.

As CAMPers better understand their problems together, they are excited with what they are able to find. They continue to ask for more combinatorics challenges to work on and further their learning.

Meanwhile, the Cosine group is split into pairs with each pair being given a different scenario to think about in the context of what they learned with numbers of chords yesterday. These pairs work separately before combining into groups of four to discuss findings and look for correspondences in their problems. They are set off to work with the big question, “Can you create a mathematical argument that states what you should always find? Is there something to always find? A definite outcome?” Triangular numbers seem to come up quite a bit as students discuss their findings.

CAMPers are then given “Cuisenaire” rods to study, form questions, and compare. They are given a starting question to expand on: How many combinations or lengths can you make with five rods? CAMPers are encouraged to ask questions and share their findings.

We then move on to the second class of the day, with the Tangent group going to computer science and the Cotangent group going to art.

Today in art, CAMPers use the dice they created yesterday to create a character with randomly generated aspects such as clothing length and color, as well as any accessories. They are then tasked with creating a name and backstory for their character. CAMPers are asked to think about how many different types of character possibilities there are while they work.

When they are done with their characters, they create monsters! Starting with a body, they roll their dice until they’ve rolled every number, each one pertaining to a different body part. Creativity is encouraged, adding whatever the dice says- even if it means four eyes! Once they are done drawing, they give their monster a name and backstory as well.

In computer science the Tangent group first recaps what they learned yesterday with Python Syntax and remind each other of what to remember as they continue working.

The CAMPers then work together to create a function that will always be true if the input is even, and false if the input is odd.

After their collective coding, CAMPers pair up and use the driver-navigator partner system to create their own code using this statement and expanding on what outcomes could come with different inputs. They learn the difference between print and return statements and how they affect the behavior of a function.

Finally, CAMPers are challenged to write a code that will tell them if a variable is even or odd without using if statements. They continue asking for more challenges as they figure out the given problems and get the wanted answers.

All CAMPers head to lunch, excited to replenish their minds- and stomachs- before returning to learning.

Once done eating, everyone goes to the elective that they chose earlier. Today, CAMPers either explore the world of Rubix cubes, participate in more mind-blowing math magic, or chill and do paper puzzles.

After electives, we move to our final class of the day. The Tangent group goes to the art classroom where they have their turn in using their die to create their very own characters and monsters. They creatively use the mathematical aspect of art to design funny outfits and creatures.

When some CAMPers finish early, then are given the opportunity to make necklaces. They pair beads up by color, seeing how many pairs of a color they can find before stringing them on a thread to wear.

The Cotangent group continues to explore Python in computer science class. They begin by reviewing what they learned yesterday before using the driver-navigator system to find out what the percentage symbol does as a python operator.

After experimenting in pairs, they discuss as a collective what they found and learn how to advance. The CAMPers take turns providing test cases for discovering how Python handles different operations and inputs.

CAMPers learn about the if statement and experiment with numbers as boolean value commands before analyzing discoveries as a collective. They end class with a challenge: using their investigations to come up with a statement that will determine if a number is even or odd.

All the CAMPers end their day in the art classroom, occupying their time with varied activities and games while waiting to be picked up.

Time flies when you’re having fun! The day is over but the good news is that there’s still three more. We can’t wait to see everyone tomorrow to let the fun continue.

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CAMP 2025 Day 1

A warm welcome to all parents and CAMPers returning and new. This morning is the beginning of the week we all look forward to- Bard Math CAMP! Nervous excitement fills the lobby as CAMPers arrive.

They are first given the choice of which thrilling elective they would like to participate in later on in the day- so many good choices!

They then make their way to the art room where many games are at their discretion to warm up to the environment and meet new people.

Once staff have been introduced, the campers are split up and eagerly dismissed to the first class of the day- math! The Sine group goes with Frances and the Cosine group follows Japheth up to their corresponding classrooms.

The Sine group starts off the week with combinatorics. These CAMPers are faced with the starting question; “If everyone in the class shook hands with everyone else once, how many handshakes would there be in total?”

Everyone gets right to work testing possibilities and calculating outcomes before coming together as a collective to discuss findings. They are given more to think about with this question; Is there a pattern? How many patterns are there? This inquiry is demonstrated geometrically by creating a visual on the board and also by trying to create a formula.

CAMPers are then paired up with each pair getting their own combinatoric problem to solve together. The groups keenly work until the time runs out, excited to continue their findings in class tomorrow.

Simultaneously the Cosine group starts off with a fun math stretch to warm up their bodies and minds before jumping into silent ice breakers: CAMPers are faced with the challenge to line up in various ways (shortest to tallest, alphabetical order by first name, and birthday month and day) without speaking with each other.

Once names have been learned and seats have been found, the class jumps into discussion of circles. The CAMPers split into three groups, each with a large sheet of paper and markers. They review circumference, drawing a circle on their paper, then expand into talk of line segments, chords, and arcs.

They discover that if one line segment is drawn from one point on the circumference of a circle to another point on the circumference of the same circle, it creates a chord and two regions in that circle. If there are three points, then three chords are drawn yielding four regions. If there are four points, then six chords are drawn, yielding eight regions. Various regions include segments of the circle, triangles, and pentagons.

CAMPers are then given the opportunity to explore how many regions they can make in their circles with their groups and what polygons can be found when it comes to five points, six points, and seven points. They explore, observe, and make conjectures as a class, eager to continue their mathematical discussion tomorrow.

As CAMPers head to their second class of the day, questions still linger: Is there a pattern? Is there a formula?- Tune back in later on in the week to find out!

CAMPers split into two new groups; Tangent and Cotangent. The Tangent group explores computer science with Yulia while the Cotangent group goes to art class with Shoshi and Ariella.

In computer science the Tangent group uses an ice breaker to learn names and favorite computer science memories as well as fun new ice cream flavors to try. CAMPers then split up into pairs with one pair at each computer.

They use the “pair-programming” driver-navigator system to explore the coding language Python. They spend time taking turns getting familiar with the language to ensure that they are comfortable getting “errors”, and figuring out how to expand on their learnings later on in the week. For right now they focus on defining variables, comparing numbers, and figuring out how to use exponents in their coding.

CAMPers spend time as a collective discussing what they discovered using the Python command prompt, talking about different behaviors that they found. They discuss syntax, experimenting with code using trial and error to figure out formulas to use in their program. They end class by working to create a statement that will allow them to input a number and find out whether it’s even or odd.

In art class, the Cotangent group starts off with a fun ice breaker where they learn favorite numbers along with names. They then begin creating their very own die, keeping in mind that the formula of a die states that opposite sides must add up to 7. While decorating their die, they explore many questions: “How many different ways are there to number a die?” “What’s the probability you roll the same number twice? Three times?” “How could you roll two six-sided dice so that they have the same probabilities/outcomes as two standard six-sided dice, provided that one of the dice has no number larger than 4?”

They then use their die to create randomly generated drawings using a code that lines up numbers with what they draw and how many of that image they draw. They add a mathematical aspect to this activity by asking questions like “Out of five rolls, what’s the probability you get the same number twice? Twice in a row?” “What’s the probability that your plant has an even number of branches on top?” CAMPers end by cleaning up supplies and setting their work aside to be displayed on Friday for visiting parents.

CAMP then breaks for lunch where CAMPers choose between eating inside or outside to refuel before heading to one of the various electives they chose earlier.

Today’s electives are to experiment with shocking math magic tricks, create nifty shapes with origami, or to go outside and play lawn games for a physical aspect to recharge their brains.

Once electives are done, CAMPers head to their third and final class of the day with the Tangent group going to art class and the Cotangent group exploring computer science class.

The Tangent group get their turn to make and decorate dice and use them to generate random images. They’re faced with the same mathematical questions to ponder while they creatively challenge their brains and share fun riddles with each other.

The Cotangent group split up to create directions for Gemma “Beep Boop” Robot on how to draw a square on the board. With many giggles, they learn the importance of being specific in their coding and giving the correct amount of directions in the correct order so that they are interpreted in they way they want in order to get the intended outcome. The CAMPers then split into pairs using the “driver-navigator” system to explore coding in Python with three questions in mind: “How do you do arithmetic in Python?” “How do you compare values in Python?” “What do the “//” and “%” operators do?” CAMPers end with a full group discussion of what they learned and found using Python.

To end the day, all CAMPers gather in the art classroom to play more fun games while waiting to be picked up.

What an exciting first day at CAMP! We can not wait to see everyone again tomorrow for more fun and even more math!