## What’s the best teaching style?

Yesterday Robin Matthews tweeted a link to an article in The Guardian that says there’s apparently no evidence to back “discovery learning.”

I felt like that sort of went against the whole #mtbos ethos and Dan Meyer’s idea of being less helpful, so I was curious to delve in a bit more. Robin linked me to this scholarly article:

Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching

which has a sort of in-your-face title. Failure? Ok then. The article is certainly interesting. It advocates for direct instruction, specifically the use of worked examples, especially with lower-performing students, and suggests that inquiry-based learning is helpful for only top performers and is even harmful for lower performers, with students knowing much less in the end.

One thing that stuck out to me was this phrase “unguided” or “minimally guided” instruction. Is that what we advocate or mean when we talk about investigations and inquiry-based learning?

The way I see it is that our job as teachers designing the tasks is to have the end goal in mind (what we want the students to discover) and also imagine in advance all the things the students will struggle with. Therefore we’re always scaffolding the discovery learning with pointed questions and sometimes hints along the way to point the students in the right direction. But feedback during the task is also critically important. The article is clear that when students are going the wrong way we should step in, otherwise they may codify mistakes that are later hard to shake.

I find myself constantly revising my worksheets and tasks to bring more clarity and help the students focus where I want them to. Is this minimally guided or is this guided instruction? I don’t think it’s direct instruction, per se. So am I doing something research-supported or not? Hard to say.

Further:

Tuovinen and Sweller (1999) showed that exploration practice (a discovery technique) caused a much larger cognitive load and led to poorer learning than worked-examples practice. The more knowledgeable learners did not experience a negative effect and benefited equally from both types of treatments.

I agree that worked examples are really important! I think after investigating something, there should always be structured note-taking (which can clarify any misconceptions and make sure that all students have arrived at the same framework), worked examples, and then practice.

But one thing I think is that the point of investigations is not necessarily the content at hand, but rather the flexible approach of taking knowledge one already has and building upon it in new ways. That is what we are committing to long-term memory; rather than *what* is investigated it is the process of investigation itself. How often have we seen students stymied by a slight change in the wording of a problem/how it’s presented/application?

We constantly lament that they lack the tools for this — isn’t this what discovery learning attempts to remedy? Of course I, as the teacher, can explain the concept best — distilling its pitfalls and connections and intricacies into an outline format with examples — and that needs to happen as well. I can’t abdicate my role as the content expert in the room (both mathematically and the learning of math). But in my mind, the discovery process isn’t (just) about the content/concept, but about the journey that the student takes applying known knowledge to the just-out-of-reach knowledge.

This little bit was the most fascinating to me:

…the worked-example effect first disappears and then reverses as the learners’ expertise increases. Problem solving only becomes relatively effective when learners are sufficiently experienced so that studying a worked example is, for them, a redundant activity that increases working memory load compared to generating a known solution (Kalyuga, Chandler, Tuovinen, & Sweller, 2001). This phenomenon is an example of the expertise reversal effect (Kalyuga, Ayres, Chandler, & Sweller, 2003). It emphasizes the importance of providing novices in an area with extensive guidance because they do not have sufficient knowledge in long-term memory to prevent unproductive problem-solving search. That guidance can be relaxed only with increased expertise as knowledge in long-term memory can take over from external guidance.

Before I worked in The Netherlands, I worked in the South Bronx in District 7, a really struggling area. When I taught there, I had a really regimented approach. Every day began with a warm-up (I couldn’t bear to call it a “Do Now” as they were trying to get us to do–it was too imperative sounding) that would either review the work from yesterday or prep them for the day’s work with some sort of small discovery learning. Just a short example:

1) 3(x-4) —> ________
2) ______ —> 7x + 35

Then we did notes in a really structured outline format (roman numerals and all), a worked example, and then the students got to work. I prided myself on my super clear explanations and step-by-step instructions. And you know what, I had great results (as measured by the NYS tests and Regents tests, of course, so grain of salt)! I felt really good about how I taught math.

But when I came to The Netherlands, a lot of students chafed at this style (though they all said I explained well and had good results). They didn’t want to take notes. They felt like they already got it and I was forcing them to sit through an explanation that they didn’t need and making them write down things they were never going to look at. I also started to feel like maybe I was holding their hands too much, like I was doing all the mental heavy lifting. So I’ve drifted from this model.

So here are my questions:
Does the kind of instruction you provide depend on the level of the class you have (and I don’t just mean differentiating a bit but the whole approach)? It feels wrong somehow to deny a weaker group the experiential learning…but at the same time, providing them with direct instruction is what helps them grow, according to this research. And does all this point to the idea that tracking is better? Because you are able to provide direct instruction to lower performers, who will benefit most from it, and provide discovery learning to higher performers, who will benefit most from that?

I don’t know the answers to these questions, and I wish I had two parallel classes so I could try out the two different styles (but no, this year I have one class each of 7th, 8th, 9th, 10th, 11th, and 12th, UGH) and see how it plays out. What are your experiences? Do you agree with the article? How do you reconcile the kind of teaching advocated by Elizabeth Green’s Building a Better Teacher and this new study?

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## How Many Ancestors Do You Have?

This whole fall has gotten completely away from me. So many great things have been going on and I’ve just been too swamped to write a thing. Forcing myself now, because I loved this project and must share!

Our unit “What’s in a Number?” contained a substantial bit of work on exponents, specifically rational exponents. One of my assessment criterion is Applying Math in Real World Contexts, which specifically asks students to model and critique their models, which is something I’ve been interested in for a while (h/t Frank Nochese).

So I came up with a modeling project based on this YouTube video from the always awesome Vsauce:

In the first part the students had to investigate how many ancestors, A they had g generations ago, with themselves as g = 0 (and the number of ancestors being 1 — themselves). The ultimate goal was to write a rule to find the number of ancestors they had in any given generation.

In part two the students had to decide how long a generation lasts (meaning, how long before a new generation starts) and use it to approximate the year of their great-great-great grandparents’ birth, writing a rule to find the birth year of any given generation. Here’s some interesting things that occurred here:

• Some students misinterpreted this — one student said a generation was a 100 years because that’s how old people live to!
• Most reasoned about 25-30 years per generation. Many of them said 30 years was a good amount of time because it’s how old their parents were when they had them.
• Just under half of them made comment on the fact that this was not a very good/accurate assumption because in “the olden days” women had children much younger.”
• About a third of the students took into their own age (the fact that g = 0 is in its 15th year) into account, but many of them had rules that said things like their parents were born only 25 years ago, oops.
• In part three they were asked to approximate the number of ancestors they had in different time periods, such as the year of the founding of the Netherlands (where we are) and around 1200 (the time period in which 0 was introduced to Europe). They had to put together their two models/rules to calculate this, which most did with no problems.

But then I gave them the estimated population of the world in 1200 and asked them to comment on their model with this in mind. I asked them what was good about the model, what was bad, how accurate was it, what could they conclude? Because if they’d done their calculations correctly, they should come up with the fact that they had more ancestors than people alive at that time period (or that more than half of the world was related to them).

Their answers are the really interesting stuff:

• Some students are just trained to justify themselves constantly so they say things like, “My answers are very accurate because I used a calculator.” As if that is what makes things accurate!
• Some students who calculated that they had more ancestors than people alive did not appear to notice this fact. Do they really critically look at their answers? Or are they just concerned with getting to a “right” answer?
• One student said his model didn’t take into account cultural norms like taking multiple wives (I thought this was interesting) or child brides or that a generational length can vary tremendously around times of war
• About 2/3 of the students said it’s impossible to be accurate without knowing the exact length of a generation, even though they felt they had a good model.
• A few people said that their model didn’t take into account incest — the fact that at some point people must’ve been having offspring with people related to them.
• One student who had about a third of the world related to him said that if this model was true for him and the other 7 billion people alive today, we all had to be related, and that not only that, if you go back far enough you’d be related to every person. He even had a nice diagram to illustrate this point.
• I liked this project because it takes something that’s a clear obvious rule A = 2^g and then muddies the water with estimating generation length. It gives the students a chance to actually create their own model and then hopefully analyze it. It’s clear they still need some work at this, but it’s a good start, I think.

## Why I’m already discouraged for my Dutch math classes

Edited to clarify: This post is not about the IB but about the Dutch national curriculum

This year I teach almost all of my classes within the International Baccalaureate (IB), so I have only two classes within the regular Dutch curriculum, one 7th and one 8th grade. These are students who have decided to take their schooling bilingually (though the majority of them are fully Dutch and have never spoken [much] English before). They begin in 7th grade directly with half of their classes, including math, in English. That’s a pretty daunting feat and these kids are all very bright. Yet I feel discouraged and sometimes even dread these classes.

Why? It starts with this tweet from David Wees:

And this:

That’s an image of the “study planner” that I need to follow for this term. It was made by a colleague, and if I don’t follow it then we end up out of sync and it becomes unfair for the tests, not to mention I don’t get through the material that the students “need to know for the next year.”

We have just two lessons (of 75 minutes each) per week with the students for an 8-week term, but are constantly losing days from our schedule by things like a day off from school, or a test planned during our lesson, etc. So you figure approximately 14 or 15 lessons per term. Times three terms plus one term where we have them only once a week (yes, that is as productive as it sounds), and that means I have to cram 8 chapters worth of material into about 52 lessons.

And honestly sometimes the curriculum careens wildly around. In the first “real” lesson I was meant to do:

• 1.1 Theory A: Using Letters (Writing Expressions)
• 1.1 Theory B: Addition and Multiplication using Letters
• 1.2 Theory A: Multiplying out brackets (Distributive Property)
• Just to give a quick example of what sort of problems the students are expected to do in one lesson, it veers from:

“Tessa earns b euros per month. Her sister Sandra earns three times as much. How much does Sandra earn?”

to:

-2p * 3q – 2p * -5 – 3q * -2p – 4p

and finally to:

2/3x(18a – 12b – 3c) – 1/4a(48x + 4b – 8c) – b(c – 8x – a)

What an absolute SWEAT to get through that. My brain hurts thinking about it.

So in the second lesson, a full quarter of my class had clean-up duty, which means that they have to sweep the school after lunch (YES THIS IS REAL, I AM NOT JOKING!!!) so they came 15 minutes late to my class. So I took that time to have the rest of the class work together on this worksheet I made for them on writing expressions and combining like terms:

I’m not trying to act like this is the most brilliant worksheet ever, but I think it did serve a purpose, because I saw kids write abcd instead of a+b+c+d, kids who wrote a^2 + b^2 + c^2 + d^2 instead of 2a + 2b + 2c + 2d, kids who wrote a + b + c + d * 2 instead of 2(a + b + c + d), kids who couldn’t come up with 9 – (a + b + c + d) at all and more. So what that says to me is that kids cannot just steamroll through everything. They need recap, revision, and time to reflect. It is helpful for them if I bring in extra things for them to do and let them puzzle through it.

But no matter because in lesson two I was supposed to do:

• 1.2 Theory B: Multiplying Binomials
• 1.3 Theory A: The Remarkable Product (a+b)(a-b)
• 1.3 Theory B: The Remarkable Products (a+b)^2 and (a-b)^2
• After “wasting” so much time on the variables on a map thing, I could hardly get through this and ended up assigning the majority of it for homework.

Sure kids, I know you’re only 13 and you’ve just learned how to multiply binomials for the first time ten minutes ago, but do you think you could please now do this problem for me: (2a + 3b)^2 – (3a + 2b)^2? Because I really don’t have a lot of time to make sure you understand it and give you cool projects around this like Babylonian Multiplication, because in the next lesson I have to do:

• 1.4 Theory A: Simplifying Algebraic Fractions
• 1.4 Theory B: Adding Algebraic Fractions
• 1.4 Theory C: Multiplying and Dividing Algebraic Fractions
• 1.5 Theory A: Scientific Notation of large numbers (WOW, how is this related??)
• 1.5 Theory B: Scientific Notation for small numbers
• I felt so damn shitty after this lesson (and also my 7th grade lesson, which is even more intense, especially when you consider that these kids are in high school for the first time ever and don’t even speak the damn language of their textbook) and horrified looking at what I have to do in one lesson next week. Basically, what it comes down to for me in these classes is that if I do anything that’s not in the textbook or that takes any serious amount of time for thinking and discussion, I won’t be able to get through the curriculum. I always like to start Scientific Notation with a reading from Bill Bryson but it’s like, AHHH, that will take at least 10 minutes!

I really don’t know what to do about this. I agree with David Wees. I feel like I have to teach in a manner that’s directly counter to what I think good teaching is. So that’s why I’m already discouraged for my Dutch math classes. Any advice?

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## Establishing a Rigorous Culture from Day One

One of the nice things about the IBO MYP curriculum is that it’s completely untracked and all students have access to it. Except for math…in the last two years of the program math is split into Math Standard and Math Extended. The idea is that Math Extended is supposed to cater to the top students to prepare them for DP Higher Level Math and a future STEM career, but in our school things have worked out differently.

Instead any student that has any inclination at all towards math takes Math Extended in an increasingly packed (and diverse) class, leaving the students who despise math to wallow in what’s almost become slow math instead of the standard thing. And so the rigor of the class has also declined in response…I hate that!!!

This year I have my usual MYP4 Math Extended (9th grade) but I also have MYP5 Math Standard, and I’m so excited to try to really get these kids doing math!

Our first unit is on quadratic equations and functions and their first big investigation is on how the form of the equation affects the graph of the function. I decided to jump right in on the first day with a formative assessment of their knowledge on graphing linear functions as a way to introduce several things. It started off nice and slow (so I thought):

Question One…Should Work, No?

See the entire formative assessment here.

I gave them each two pieces of paper and asked them to keep track of their “noticings” on one and their “wonderings” on the other with these prompts up on the beamer:

• “I notice/see that _____”
• “I realize that _______”
• “I’m confused by _______”
• “I wonder if/how/why ______”
• “How do/can you _______”
• “How/Why does ____ work?”
• “What would happen if ______”
• I had them work individually for 6 minutes — I intended it to be 10 minutes, but I noticed right away that one or two students were unable to complete even that first question. I stopped them and asked them each to jot down at least one noticing or wondering and then begin to discuss in their groups.

I was extremely pleased with how much they started helping each other and working together! In one group there was even a disagreement and a debate over how a line with a gradient of -3 should look.

I had let them pick their own groups and three students who are particularly down on math had sat together and they really struggled to get any of the problems done, but in each case someone had an idea and they were able to be the group leader for that particular problem, which was nice for them (though I don’t know that it’s tenable to have them remain as a group).

They all worked really diligently for about 30 minutes and which point I asked them to consider as a group the following questions:

• What was the point of this activity?
• What do you think I, the teacher, should learn from it?
• What do you think you, the student, should learn from it?
• How do you feel about what you did?

Here’s the list of what they came up with as answers to the first three:

• To refresh their memory and get them ready for the new year
• To check what level they are at so I know how to help them
• To work together in a group
• “Math”

To which I said, “math isn’t specific enough! Be specific.” And then they were — they said graphing, equations and how it relates to the graph (BINGO!), linear equations (YES, so we can next distinguish between linear and quadratic!), gradient. I love that they came up with everything I wanted them to learn for themselves!

Then I asked them to flip to the last problem in the formative assessment (no one had gotten close to it), which was meant as an introductory exploration to quadratics:

We managed to go through the first two parts of the problem together (though it took a lot of prodding they were able to discover the answers for themselves, leading one student to say, “I feel smart!”) and then I told them about their HW, which I said would be different than what they’re used to and difficult but to not give up and if they don’t know one question, to write their thoughts anyway and then go on to the next.

Here’s that HW, which might be way optimistic, but I’m excited with how I hope it helps them develop (we shall see and I will report back!).

As they left, I collected their noticings and wonderings. A handful wondered what the connection between an equation and a graph was (yes!) and a full 11 of them noticed that they’d forgotten a lot over the summer. But my favorite, even though it’s not about the math itself, is this one:

“I notice that I have no clue what I’m doing…I notice that I understand a lot more now that we worked in a group.”

I think I’ve gotten off to the right start with them…I hope!

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## Educating Myself First

I’ve been thinking about Elizabeth Green’s wonderful NY Times article “Why do Americans stink at math?” and wondering how my own math education has affected the way I teach math. So what was my math education?

I really struggled with math as a kid. I couldn’t count money or tell time (and haughtily told my 3rd grade teacher that I’d just get a digital watch) or subtract (to do 12 – 7, I had no strategy other than 7 + ? = 12).

In middle school, I was in the honors class, but the teacher felt there were too many kids in the class and set out to eliminate as many as possible. I failed the first term (which unfortunately included percents). I started going to extra help every week and when I got my first 100 on an exam, I said, “Mr. Epler, aren’t you proud of me?” He replied, “No.” He was not a warm and fuzzy kind of guy.

When I got to high school, numbers disappeared and suddenly everything made sense and I loved math. But please don’t think the tenor of my education changed — one of my high school math teachers was a nun who wore full habit. She is the only teacher I’ve ever had who hit me (when she caught me checking what sin 60 was, which I was supposed to have memorized) and once took off 5 points from my test because I had dropped the edges of my paper on the floor and forgotten to pick them up.

So, I didn’t really have the sort of progressive math education Green posits is best for learning. But since I started teaching 7 years ago, I tried to go in that direction even though I had little experience with it as a learner. Each year I’ve pushed myself more and, especially with my international classes, I feel good about a lot of the activities and investigations my classes do.

But then two nights ago I couldn’t sleep; I got really bothered by something and felt ashamed of myself: circles. When doing circumference with my Dutch 2nd graders (8th grade), I’d always shown the students that the circumference was always a bit more than 3x the diameter.

Ugh how I cringe to write that. Shown. Why didn’t I let them find out themselves? And then I didn’t even explain where the formula for area came from, just gave it to them. Then worst of all, I taught them something that was taught to me by a colleague in my first year of teaching: Cherry Pies — Delicious! Apple Pies Are Too!

How simple! Of course! Now it’d be so easy for them to remember the formulas! Except it wasn’t. They’d constantly forget whether it was A = $\pi D$ or A = $\pi r^2$. This year I finally (after three years! How could I have let this go on for THREE YEARS??) started saying things like, “well, what should the unit be for area?” hoping that if they could remember centimeters squared, they could also remember radius squared.

Tricks just do not work. I couldn’t sleep thinking about it. I’d been doing it so wrong.

How could I get the students to discover the formula for area themselves, just like they should’ve discovered circumference for themselves? I am embarrassed to admit that I had no idea how one could even come up with this formula — after all, I’d never been taught any way but memorizing it!

Finally I hit upon this:

The area of a circle is a little bit more than 3x the area of a square with side length equal to the radius of the circle

Christopher Danielson helped me refine my thinking as well to include another idea:

the area of a circle is a bit less than 4x the area of a square with the side length the same size as the radius

Before today I literally did not truly know this myself.

I think that before we can teach in the progressive way that Green advocates, we need to teach ourselves this way, to pull apart the math and at every turn ask ourselves how a student could come up with this themselves, and then have them do it. Even though I’d been paying lip service to this idea for years and actually implementing it quite a bit, there’s still so many holes in my own practice.

So I’m filling up some holes. This year, in addition to actually having my students measure circumferences and diameters, I’m going to have them draw a circle of a certain radius and ask them to come up with an estimate of the area of it, using what they already know about how to find area. I want them to give me a reasonable guess that’s too low and one that’s too high, articulate how they came up with it and then write the damn formula themselves.

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## A Better Modeling Task Part 2

After yesterday’s flurry of thought I settled down today to actually write up the new modeling task. In my document, I included Geoff Krall‘s screen shot with the introduction that this problem was taken from an American book.

I then gave them the exact conversions (at this point in the year, they’ll have done a whole report on converting temperatures) but for ease’s sake told them to change the problem so that the temperature at sunrise was 18 degrees Celsius and the temperature rose (by the way, hate the repetition of sunrise and rise in the problem) 3 degrees Celsius per hour.

Here’s the new assignment:

1. As instructed in the American problem, find the equation of this scenario.

2. Model this situation on a graph. Make sure you label your axes and pick an appropriate scale. Indicate any important points.

3. Critically look at your model – this can not be correct…why not? What’s wrong with this model?

4. Is it realistic to assume that the temperature rises 3℃ per hour? Why or why not?

5. What other assumptions are you and the model making? What isn’t the model taking into account?

6. Create a new model. You can either first create a table that relates temperature, T, with hours during the day, h, or first create a graph of how the temperature should look. (You can use actual times of day to help you orient yourself). Briefly explain how you picked your temperatures/graph.

7. If you were to continue this over a couple of days, how would your graph look? Give a small sketch.

9. How would the graph compare if you were the daily temperatures in another country (like for example Saudi Arabia or Greenland)? What would be the same, what would be different?

I envision this taking an entire class period for my MYP 4 (9th graders). I do not expect them to come up with an appropriate equation for their new model, but they should still evaluate its validity. What do you think?

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