The Importance of Not Following Plans

Us Math teachers have been talking about lesson plans. “If something is worth learning, then it should be in the lesson plan” is one viewpoint. Substantially changing things on the fly is a sign you haven’t thought hard enough about the learning. Small adjustments to deal with student misconceptions are fine, but don’t diverge for a significant chunk of a lesson.

I disagree. I think that every chance to respond to student’s instinctive thoughts should be grabbed with both hands, as a way of mimicking the process of discovering Math and to pivot around student interest. No matter how hard you think before a lesson you cannot predict what will happen in the heat of the moment. I find that lessons that follow the plan lack zest and joy. When totally unexpected things are happening the students get excited, and I get excited.

 

Here is what I think is a great example of when not following a plan is great.

  • Mr J: Does anybody have any questions about the Pythagorean theorem, before we move on?”
  • D: “I just noticed… 5 squared is 12 + 13. 3 squared is 4 + 5”

Note that D did not answer my question at all. I was looking for people who found Pythagoras hard. He just wanted to share an observation.

At this stage, I had two options. Ignore and continue on with the plan, or explore this pattern further. In the end we explored it a tiny bit, then the class voted on which path to go down. They voted unanimously to explore D’s path, which ended up taking the rest of the lesson (30 minutes)

  • Mr J: “But 8 squared is 64, and that is not 15 + 17”
  • D: “Yes, but 64 is double 15 + 17”
  • Mr J: “Why do you suddenly double?!”
  • L: “Maybe because the difference between 17 and 15 is 2?”

And now, the game was on. I scribbled a plan on the board:

  1. Google a list of Pythagorean triples
  2. Test the conjecture with some more triples
  3. Prove the conjecture

Students frantically finding the largest triples they could find. Navigating the mathematical internet is a good skill to encourage. Testing, finding counterexamples, realizing their mistakes (“I thought the difference was 25, not 15!”), finding more conjectures  (“every triple with smallest number odd contains two consecutive numbers”). G moved to an algebraic statement of the theorem, and noticed it reminded her of something from her Algebra days – the difference of two squares.

 

1

A final pattern at the end of the lesson, for triples where the smallest number is even…

  • 2 lots of 8 is the average of 15 and 17
  • 3 lots of 12 is the average of 25 and 37
  • 4 lots of 16 is the average of 63 and 65

What is going on here?

unnamed.jpg
A student proudly showed me their conjectures that they had worked on over the long weekend…

This conjecture is approaching one algorithm for generating triples:

Untitled picture.png

 

Thoughts:

  • Was everyone involved? How do I monitor engagement while also thinking flat-out as a mathematician?
  • How to persuade my teaching colleagues that this is worthwhile? I am starting from the assumption that the purpose of school Math is to give students a chance to act like Professional Mathematicians. Is that true?
  • Payton is currently trying to move towards a model where students in different classes get the same experience. How can that mesh with the idea that lessons should be substantially changed by student conjectures?

 

By the way, we started this lesson with students reflecting on why they study Math:

 

Why do I study Math? If I had to guess, what would I say are Mr Judge’s reasons?
  • For a grade
  • For interest
  • Focus on things not in outside world
  • To understand the world around me
  • Improve decision making
  • To help financial math in real world
  • For a career in future
  • Encourage deep thinking
  • To understand things e.g. why a circle is a circle
  • To verbalize thoughts
  • Foundation of other subjects that we learn
  • We will always need to use math
  • To have a different outlook on the world
  • For fun
  • To challenge yourself
  • Its fundamental
  • To create new things
  • To become professional mathematician
  • Curious thinker and trust nothing
  • Has endless possibilities to explore
  • To discover different topics in math
  • To get critical thinking and problem solving skills
  • To encourage exploration and discovery
  • To prove an argument

 

Advertisements

Love of Lockhart

I am re-reading Measurement, by Paul Lockhart. His mathematical values precisely match with mine. I want to be him. Read the book read the book read the book.

I wonder what he would say to the idea that the ability to view Math as an imaginary land where you explore and play is a privilege that only some students are able to access? That, due to systemic oppression, we would be doing students a disservice by skipping past mastery of skills straight to open-ended tasks?

Here is me attempting to recreate Lockhart’s ethos in my geometry class, full of curiosity at wandering in the mathematical jungle.

 

img-0848
Circles only exist in our minds… But let’s still go and assume that measuring this circle is useful? Ha!

Other People’s Children

Lent by Ginna.  Some disorganized thoughts:

 

The dichotomy between traditionalist and progressive education is a false one, created by academics for ease of analysis. Good teachers know that both have their strengths, and a balance is the only way forwards.

 

Dry slow-paced practice is terrible. Every task has to be meaningful, taught in context, aimed at a real audience. What does this mean in Math?

 

If you only focus on open-ended rich tasks, thinking about fluency rather than skills, then you widen the achievement gap. Students who come to you with basic skills in place will fly. Students who don’t have this foundation will never be given an opportunity to build them, and will fall further behind.

 

Don’t be shy about exhibiting the fact that I have knowledge-based power. By pretending the power doesn’t exist I am not helping my students. Instead I should be explicit about what the students need to do to gain power themselves. Simultaneously, we should also work to destroy the power imbalances of society.  “Pretending that gatekeeping points don’t exist is to ensure that many students will not pass through them”

 

Students already have a voice. My job is to not help them to find their voice. My job is to help them to hone their voice, and make it work in the dominant discourse of American society.

 

The teacher should act primarily as an ethnographer – actively seeking to  understand the diverse cultures of my students.

 

Teachers are experts in their fields. Students are experts in their lives and communities. Relish and use both.

 

 

Questions:

  • How can I be more active as an ethnographer? Without overstepping the teacher/student professional boundary, and without embarking on poverty porn?
  • I agree that there are many skills in English that are essential to modern life – the ability to write a letter/email or speak in a “sophisticated” way is essential to have a successful professional life. I disagree that this idea transfers well to Math. If my students are unable to write a written argument well, they will be limited when communicating with colleagues. If they are unable to apply the cosine rule, there is no inherent limitation (apart from in the arbitrary world of assessments). Surely the role of Math is to encourage deeper thinking? It’s not that I naively assume that through rich problem-solving my students will magically pick up an understanding of the cosine rule (that thinking would be putting diverse students at a disadvantage and is something that I thought last year), I just don’t care at all about the cosine rule.

Viviani’s Theorem

In Geometry, we proved Viviani’s Theorem. Individual groups’ lightbulb moments diffused across the room, ensuring that everybody had a chance to experience that giddy moment of discovery. Collaborative work to share knowledge, individual write-up to assess internal understanding.

I have been giving the students problems that I would find interesting. “Treat them like experts and they will become experts” is an unwritten mantra. Is there any justification for this? Am I prioritizing my intellectual enjoyment over the students’ needs? In any case, there are some outstanding proofs here:

IMG_0587IMG_0593

A great idea to use circles, but is it justified?

IMG_0590

Excellent attempt at dynamic proof without words:

IMG_0588

This student is thinking very precisely, but I have no idea what is going on:

IMG_0589

Equilateral triangles within equilateral triangles…

IMG_0591

As a starting point, this is ripe for extension. What about other properties of equilateral triangles? What about other triangles? What about other polygons? what about other dimensions?

A South African high-school student stumbled across a different invariant. His name is now immortalized in the Clough Conjecture. Students began this journey the following lesson…

IMG_0585IMG_0584

Acceleration or Enrichment – The Tower Built on Sand

At Payton, we seem to be moving rapidly through difficult content, in order to prepare our students for AP Calc. The aim, within the next few years, is for every student to study Calculus by the end of their time with us. Students are now even able to take Algebra 1 and Geometry combined into a one year course (where traditionally it would take 2), in order to get them ready.

I have been mulling over this, reading this article and this report about the phenomenon of increasing numbers of high school students taking college-level Calculus courses, in America.

Some factoids:

  • In the 1980’s, 5% of high school students took Calc. Only those students who were planning to do intensive STEM degrees at college. Now it is more like 20%.
  • 80% of AP Calc students take it because it looks good on their college application.
  • 90% of AP Calc students re-take calculus in college anyway.
  • 47% of Asian American students take AP Calc. Only 8% of black students do.

Overall message: in order to learn Calculus, you have to want to learn it. Too many students are now taking AP courses, and for the wrong reasons. Simultaneously, not enough students from disadvantaged areas are taking AP courses, since their teachers do not have the required subject expertise.

Some ideas:

  • There is a difference between acceleration and enrichment. Rushing through the foundations of Math, in order to get to Calculus before the end of high school, often creates “a tower built on sand” – students might be able to differentiate a polynomial, but they don’t really understand what a polynomial is or why calculus is beautiful and important.
  • When these students re-learn Calculus at college they often lose confidence – unable to understand the content when presented at a greater pace and more formally. They are lacking the fundamental prerequisite understanding.
  • Seeing the idea of Calculus in high school is beneficial for students, but there is no need to do a formal exam in it.
  • How would you differentiate (x^3 -1)/x? If you are an expert, you would simplify first. If you are a student, you would just use the quotient rule.
  • In order to get students ready for Calculus, often teachers in middle schools who lack subjects expertise are teaching Algebra. Is this what’s best for the students?

What can we do at Payton?

  • Encourage colleges to no longer value AP courses. Unsure of how feasible a task this is for one school.
  • Only encourage students to study Calculus if it is suitable for them. (What does this mean? How can we do this with equity?)
  • Offer a wider variety of Math courses for older students. Probability? Geometry? Coding? Number theory? Graph theory? Interdisciplinary projects? Problem-solving? Financial Math? History of Math? Calculus is great, but so are many other ideas. (We would have to work hard to ensure these are not seen merely as the cop-out options for students who are not “clever enough” for Calculus).
  • Can we survey the current students, both those who are doing AP Calc, those who are about to do it, and those who are doing Pre-Calc in their senior year? What are their thoughts and motivations?

The overall picture of the High School math journey.

Group A – stronger mathematicians. Group C – weaker.

What should a pre-calc course include?

Students Shaping New Definitions of Success

The Equity Principle that I am focusing on this semester is:

wyIB2gv720VvHEFLCERjbJAoVpg4AW_lAFxNdjhLK1n-lNNhJX_x6WY-Qo7kGplj3SaRLlzpIPST0ECK8x4hJo_2HTgr9vfCaTg0nJUXeNqD5WkG7U0j1BuoggBLdpHdx17DhDPz.png

What this currently means in my classroom:

  1. Students create norms
  2. Students create Math

Firstly, students are given space to define norms in the classroom. I had started the year with a strong vision of what I wanted the collaboration dynamics to be – minimal teacher help and maximal peer help. Students perceived this to mean that I don’t care about their learning. We spent a sequence of 3 lessons building a contract. Now, all I have to do is remember to stick to these norms…

3
Planning the session
1
Difficult to hear, but very worthwhile
2
Thank you for listening to us
1
Contract
2
What does each of those slogans actually mean?

 

Interestingly, I only felt a need to re-build class norms with older students in HPC – the fresh-faced Geometry students were much more willing to go with my flow. Is this due to their age, their confidence in their mathematical ability, their newness to the school?

Space is now given in most lessons for a quick oracy task, focussing on ethical dilemmas in the news. Should this teacher have been fired? How should Britain deal with the issue of grooming gangs in North England, where the men are mostly of Pakistsani heritage? What does the equality sign mean in Math? Should you all be treated equally in this class? Should you all get equal grades? Do you treat each other equally?

Secondly, students define what mathematical success looks like. Whether that is inventing new notation, discovering Professor Miller’s Quadsector Theorem (named after a Freshman student who found it), or celebrating the multiple routes to take on a rich problem. There is no longer such a thing as “the right answer”. This provides space for people to use and celebrate their diverse skills.

 

What this could mean in the future:

  • Students, rather than just creating norms and small chunks of Math, help to curate their entire Mathematical journey through high school.
  • Parents and communities having an active role in what their children learn, and how they learn it.
  • Adults, who use Math in their lives in some way, help to refine the learning.

What are some challenges?

  • If each class and each student has a different definition of success, how can everybody receive a consistent education?
  • How do we reconcile celebrating multiple methods and student voice with the closed ability to master specific methods?
  • When the course is moving at a fast pace, how can you provide space for students to own and direct their learning?

Notes from convo with Tamara and Latoya

  • Do not be afraid to assert my views within my teams. Luckily, everybody in my teams are friendly and open to change, but if I don’t tell them, they won’t know. (Simple fix: spend more time working in the department office).
  • Which parts of being new are inevitable, and which can be avoided? Who knows.
  • Too much flexibility –> last-minute planning, done on a whim without proper consultation with communities. Too much structure –> lack of teacher autonomy and ability to respond to student needs. How can you have structure and flexibility?
  • Why do we not have a teacher-mentoring structure at Payton?

Notation is Human-made

As a precursor to:

3.png

Equality is a weird thing…

  • Should everyone in this class get an equal grade?
  • Should everyone in this class be graded equally?
  • Should the teacher treat everyone in this class equally?
  • Should you treat everyone in this class equally?
  • Should everyone in this class do an equal amount of work in class? At home?

 

Notation is not given to us by God or the universe. We, at some stage, invented it. Robert Recorde, from Tenby (Wales), invented the equals sign after getting sick of having to write out “is equal to” every time… Took 200 years to be fully accepted – slow-moving meme.

What other equals signs should exist? Think of a gap in the market that needs filling…

Possible ideas:

  • Is a cat equal to a dog?
  • Are two cats equal?
  • Are two triangles with same side lengths equal?
  • Are two triangles with the same angles equal?
  • Is Mr Judge equal to your Math teacher?
  • Is your age the same as your neighbors?
  • Is 4 + 1 equal to 3 + 2?
  • Are women equal to men?
  • Are two meter-sticks equal?
  • Is a + a equal to 2a?
  • Is 3x equal to 9?

 

Student notation:

12

We have to use the Rafael and the Olivia signs as much as possible. Maybe in 200 years’ time they will be standard…

Geometry –> Algebra. A progression

Step 1: Handshake problem

If 10 people all shake hands with each other, how many handshakes in total?

Purpose: to relate every object in an algebraic expression (in this case 1/2n(n-1) ) to a verbal/geometric reason.

Step 2: Moser Circle problem

If I have 10 dots on the circumference of a circle, and join them all up, how many regions are there?

Purpose: to realise that not all patterns have easily identifiable algebraic rules. to TRUST NOTHING.

Step 3: Maximum Right-Angles?

If I have a 100-sided polygon, what is the maximum number of right-angles there can be?

Purpose: to combine the motivation for algebra from Step 1 and the skepticism from Step 2, to go on a deep and joyous journey.

Possible prompts:

  • Can you prove that a 100-sided shape definitely cannot have 99 right-angles? Can you generalize?
  • You think you know the pattern… Can you find an 11-sided shape with 9 right-angles? And look – here is a 12 sided shape with 9 right-angles!

Students gave the following answers:

  • There is on answer since there are infinite different polygons
  • Between 50 and 100
  • 52
  • 66
  • 67

Possible Extensions:

  • What if you replaced right-angles with acute angles? Obtuse angles?
  • What if you allowed the polygon to have lines that overlap?
  • What if you allowed both interior and exterior right-angles?
  • What if you only allowed exterior right-angles?
  • What if the polygon has to be convex?
  • How many different 9 sided shapes have 7 right angles?
  • What if you…

 

Student work

1
Great whiteboard

IMG_0380

2
Not the best use of a table…
IMG_0378
When your theory breaks, you either give up or just get more excited!
IMG_0381.jpg
Great to see slogans being repeated back to you by students…
3
Teacher approximation
IMG_0388.jpg
… which a student independently came up with. 

Miller’s Quadsector Theorem

We spent a happy half-hour exploring the difference between proof and illustration on Geogebra.

3.png

  • Angle bisectors of a triangle meet at a single point. A single point?! Madness.
  • Angle trisectors meet to form an equilateral triangle. Morley proved this.
  • I asked students to construct this on Geogebra. Professor Miller found trisectors too tricky, so went for quadsectors instead – bisect and bisect again. (fascinating and unknowing nod to the futile attempts to trisect an angle using straight-edge and compass). Excitement fizzed around the room at the new conjecture. Professor Miller is very good at Geogebra, after only one lesson.
1
Seems legit?
2
Hmm. What’s going on?
  • Another group managed to quint-cept the angles. No idea how. They also found that the central triangle seems to be equilateral.
  • A final group suggested that the pattern could continue for any n-cepted angles?

Excellent playing with initial circumstances, and students pumped up at the possibility of creating new knowledge. Good times.

Update: Professors Stitch and Madhava explored Morley’s Thoerem, but trisecting sides rather than angles…

Exploring Chicago 1: Pilsen

C expertly led us through Pilsen, a Mexican neighborhood of Chicago. We travelled on the pink line to get there – a surprising proportion of students had never been on it before. Is this just because the city is sprawling, or signifying a segregated city?

We started our tour by a brand new mural. Fence in the background – the ever-present symbol of the border.

Local artist Hector Duarte’s house, covered entirely by a huge and beautiful mural:

Next, the Mexican Art Museum. The students in the group with Mexican heritage led the tour, showing greater knowledge of their folklore than I know of British stories. We learnt of the grape boycott, the merging of the Virgin Mary with preexisting beliefs, the way Mexicans honor their dead.

In allyship with local residents affected by gentrification, C was wearing a badge saying something like “Forced out of Pilsen”. One of the museum staff asked her about it, and seemed offended that she was wearing a badge without actually being forced out herself. Gentrification is a very sensitive topic here.

Another incredible mural, on 18th Street:

Lunch at 5 Rabanitos – cilantro and avocado make everything amazing. Live music from passing buskers. Continued to walk along 18th street, picking up Conchas from one of the multitude of bakeries.

Thank you C for an excellent visit to Pilsen! Can’t wait for the next adventure.