This activity was adapted from one that the really wonderful Jim Noble created and shared with me at a workshop, based on interesting research done by Ipsos-Mori on the Perils of Perception.

This activity fits within any unit on percentage change, for example, as percentage error uses the same concept. Students work in groups to first estimate the percentage of the population of their country who are Muslim, and then to identify the percentage error of countries’ estimations. Along the way, great discussions pop up. Here is the activity:

Again, the non-math part — the part that connects to geography, history, current events — is for me the key element. For example, when discussing how students decided which numbers should math with France versus Australia, students offered explanations like:

- A lot of northern African countries (like Morocco) are former French colonies/protectorates, so there are a lot of Muslims who emigrate to France, because they can already speak the language.
*Woohoo, connections to History!* - France is close to a lot of Muslim-majority nations so it is logical that more Muslims live in France than in Australia, which is far away. A counter argument to that is that Australia is close to Indonesia, the world’s most populous Muslim-majority nation. That fact alone was really interesting for the students, who don’t realize this (though they should know more about Indonesia as it is a former Dutch colony and famous Indo-Dutch people include Mark-Paul Gosselaar [Zack Morris!] and Geert Wilders [ironic?]).
*Woohoo, connections to Geography!* - We hear a lot about Muslims in France, with things like the “burqa ban,” so since that’s in the news a lot, that must mean there are a lot of Muslims in France.
*Woohoo, connections to Current Events!*

The great (sad?) thing is that you can connect this activity to current events no matter when you do it. I did this even earlier in the year, around the time of the Dutch elections, when Geert Wilders was decrying the Islamisation of the Netherlands and saying that he wants to ban the Quran.

I did it with a different class just a few weeks ago right around the time that controversy erupted because the head of the Amsterdam police force was considering reversing the ban on outwardly religious clothing for police agents in an effort to recruit Muslims (specifically hijab-wearing women) for a city that is 52% non-ethnically Dutch (and in which Islam is the second most popular religion; the most common being “none”) but boasts a police force that is only 18% non-ethnically Dutch.

There are so many rich conversations to be had here. You could avoid the Muslim issue altogether and pick a different issue highlighted in the Ispos-Mori survey (I always show the slideshow from Ipsos-Mori to wrap up the lesson — the kids LOVE making predictions and seeing how different countries react to all different things). You could discuss fake news. You could discuss the connection between percentage error and percentage change (so that rather than simply memorizing a formula, there is understanding of how these things work). The important thing is you can *discuss*.

So, what things are you discussing in math class?

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I showed the students the three options for buying pine nuts at the main Dutch supermarket, Albert Heijn, and simultaneously highlighted and obscured important information for them:

Then I asked them the following three questions for them to work out in groups:

- Which of the three packages would you advise me to buy. Justify your answer (and buy justify I mean give a valid reason with some calculations to back up that reason).
- I believe your advice is good, but I choose to buy a different package anyway. Why might I do that?
- Why do you think that Albert Heijn has different pricing for these three packages?

I gave them ten minutes to work on it. Half of the groups had a good sense of where they should start and quickly came to the correct answer. Other groups didn’t have a good sense of where to start, so I asked scaffolding questions like “what makes something a better deal? How can you quickly tell if the medium is a better deal than the small?” and “what important information is given on the price tag? What do you think I covered up?”

You end up with the sort of surprising answer that the medium package is the best deal, even though one assumes (or maybe 7th graders don’t yet have that assumption) that buying in bulk results in a better deal.

So the 3rd, non-math question became the most interesting. The students suggested that Albert Heijn had done market research on the most common amount of pine nuts people want to buy and priced accordingly. They also suggested that the plastic of the medium package, which looked flimsy, cost less than the nice, sturdy looking plastic box that the large package came in. They also suggested that maybe people like the resealable box, so that’s why it’s more expensive.

If I’d had longer, we could’ve delved into this idea more, that the price of something is not just the raw price of the goods itself but the materials used, the convenience of it, etc. I think these sort of discussions are important to have in today’s society and that in math class we should be grappling with the sort of questions of the *real *cost of goods and services. I could imagine a much larger unit about, among other things, environmental justice. Do you have any other good non-math questions that you’ve asked in math class?

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Last school year I started teaching a new-to-me course called Theory of Knowledge, which is basically an epistemology course that asks how we know the things we claim to know. This course has been very rewarding to teach but has also taken up a lot of time and thought-space for me, which is why I haven’t blogged at all.

The current area of knowledge we’re covering is mathematics and one of the questions that arose was what is the point of learning things beyond basic arithmetic. If the majority of the students will never again solve a quadratic equation, why do we spend so much time on factorization, completing the square, the quadratic formula, forms of a quadratic function, etc. In essence, what’s the point of schoolmath?

A student (bless her) said that we study it because it teaches us to think and to problem solve. And my response to her was, “But does it?” I think that is the idealized claim (and certainly that is the impetus behind the MTBoS’ Nix the Tricks), but I’m not 100% convinced that we’re achieving that. I think that despite our best efforts many students simply follow rules or procedures that they only half understand. I saw that play out recently.

This year in my advanced 8th grade classes, we covered the rules of exponents (in an algebra-only context) and scientific notation. In a discovery-style lesson on scientific notation, in one class the students came up with the idea that if 10 raised to the *n*th power was a 1 with *n* zeros behind it, then 10 to the 0th power was just a 1. They also managed to get to the idea that 10 to the -1 should be one-tenth. I felt like in that class that I had taught for sense-making for small numbers in scientific notation, whereas in the other class I had merely taught a trick for it (in the interest of time; they had also had the discovery-style lesson on scientific notation for large numbers).

On the test I put two sense-making questions. The first asked them to find (-1)^{97} (no calculator) and explain their answer. A majority were able to give a completely correct justification. But I also had a fair amount tell me that the answer was -97, two tell me it was -1 x 10 ^{97}, and one tell me it was (-1)^{100} – (-1)^{3}. What mistakes of thinking are here?

The last question asked the students to do a mini-investigation. The first part asked them to calculate 2^{5}, 2^{4}, 2^{3}, 2^{2}, 2^{1}, then give the pattern as you go down in exponents, then use the pattern to find 2^{0} and 2^{-1}. The number of students who could actually *use* their pattern was limited, across both the trick-learning and the sense-making class. From the sense-making class, despite having seen that 10^{0}=1, many students still thought that 2^{0}=0. Among students who did get that part, lots of them then thought that 2^{-1}=0.2! In one particularly egregious case, a student told me that 2^{-1}=0.2, 2^{0}=2, 2^{1}=20, 2^{2}=200, 2^{3}=2000, etc. In other words, there was no sense-making at all! In fact, our class-wide explanation of how the powers of 10 worked had primed some students to *not *think about how powers of 2 might work, or how any power actually works.

I’m still excited about this question and can’t wait to go over the test with them to talk about how we should deal with pattern finding and its application, how to critically think and get over our biases of what we *think* the answer should be, and above all to prove to them that yes, shockingly, any number raised to the power of 0 is 1, by asking them to recall the rules for multiplication and division of exponents.

Which brings me back to Theory of Knowledge, because knowing in mathematics starts with a conjecture (I recognize a pattern that it seems that any number to the power of 0 is 1), followed by a rigorous proof using deductive logic to verify that it’s always the case. Using tricks robs students of what my student says the actual purpose of math is — to learn to think logically!

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I thought a lot about it and this is what I came up with:

I think a good colleague is someone who:

1. Listens and sympathizes with you when you’re struggling so that you feel you’re not alone in your feelings or problems (since teaching can be a very lonely profession in some ways, as most of the day you’re the only adult in the room). While they critically listen, they don’t simply commiserate with you (because then you wallow in your troubles and things can start to fester). Rather, they offer understanding and friendly suggestions (without making you feel like you don’t measure up) that help improve your teaching or your mood.

2. Brainstorms with you the best ways to present material, the best questions to ask (and what your goal is with each question you ask), the richest tasks you can set, the best ways to motivate students, etc. They think deeply about their subject material and how to connect it to the real world, other subjects, and students’ lives. They help make you enthusiastic for the subject you teach and excited by the possibilities of what can be done by a certain class.

3. Is reliable, trustworthy, and always does what they say they will do. They pull their weight and expect you to pull yours, but will step in to lend a helping hand when you’re faltering. They expect the same from you, which means that they trust you in return.

Luckily I think I have many colleagues who meet at least one of my descriptors above.

What’s on your list of attributes of great colleagues?

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```The word "discovery" can mean literally anything. // via @misterpatterson pic.twitter.com/uIGLJYFR0J

— Michael Pershan (@mpershan) May 5, 2015

He was pretty critical of this worksheet, which is definitely not as strong as Kate Nowak’s trigonometry investigation (or my own if saying that doesn’t sound too conceited). But what interested me was that he was also critical of the idea that anything that is worksheet-driven or teacher-led could be classified as “discovery learning.” He seems to suggest that it’s only discovery if the student comes up with it completely on their own.

I’ve already looked at discovery learning on this blog once before, and I think I can say that I’m really not for this narrow idea of discovery learning and think that it can actually be detrimental to a student’s mathematical development. I am, though, very much a fan of **investigation**.

Which brings me to the real reason for this post:

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```#MTBoS – I'm Looking for ideas for introducing rational exponents to Int. Algebra 2 students. Anyone blogging on that? #mathchat

— Jim Pardun (@JimPa23) May 4, 2015

First, this is an investigation I did into Zero & Negative Exponents with MYP3 (US 8th grade age). The students did the first part (worksheet included on third page) under test conditions in class. Upon receiving their graded work back they wrote the essay (fourth page) where they not only used their understanding of a pattern to write a rule but verified that their rule was correct by using the rules of algebra.

A year later, in MYP4 (US 9th grade), I do the following investigation on Rational Exponents with the students, though it is more informal. They work in groups and are not graded on their work for this.

*Scribd is doing some weird things with my document…there are no red question marks in the original. So here’s a picture of what it really looks like for me:*

These investigations are absolutely not “discovery learning,” because there are specific steps (suggested by me) that the students follow to ultimately “discover” the rule. But the steps ensure that the students have a focused approach, and it models for them what I think is an essential chain of mathematical inquiry:

Finally, not an investigation per se on exponents, but more of a hat-tip on the many wonderful ideas I steal from this community, here’s something we spent a class discussing and delving into, which I culled from various blogs and twitter feeds:

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I’m grateful to the MTBoS for giving me an online community, but I also realize that we each only share a small snippet of what we do. And what we rarely share (for probably good reason) are our tests. But this has got to be one of the most difficult (and considering the weight placed on assessment, most important) things we do, actually, and it would probably be better if we weren’t so alone in that endeavor.

I really had no idea what the heck I was doing when I started making tests–I’m embarrassed now to think of how rag-tag and unprofessional they looked, all cut and pasted and mismatched. And as a new teacher (or a veteran) I don’t recall anyone ever sitting with me and saying, “this is how you make a good test,” or asking me why I choose a particular question (or the particular numbers in the question), or suggesting a different question (or way to ask something) instead.

These days I’m trying to look at my own tests critically — looking for the things I can turn into an open-ended or an open-middle question, for example. And I’m constantly bookmarking problems on Twitter or in other people’s blogs as “GREAT QUESTION ON ___.” I’m also trying to find questions that expose misconceptions, but I don’t know how good I am at that yet.

But the point is, although I’m trying it, I’m doing it mostly alone, with the occasional assist from my wonderful coworker (who never blogs, for shame). And sometimes another coworker will hand me their test, like FYI, this is what I’m giving to the class, but I rarely say, “Hey, I’m not sure about this question,” or “Have you thought about asking this instead?” or even “Do you want to work on developing test questions together?” It’s sort of difficult to initiate that conversation.

But I think maybe I’ll propose it to the department, because teaching–and testing–should be less *eenzaam*, I think. Do any of you do test study and have any tips?

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For the last few years, I’ve been introducing it differently and I feel like it sort of addresses the too abstractness and trick-oriented way I used to do it. Now I always first begin with an investigation into special right triangles. I do this with MYP4 (US 9th grade):

In it, students use their knowledge of Pythagoras’ Theorem and simplifying radicals to discover patterns in the relationships between the sides of a 45-45-90 triangle and the more complicated 30-60-90 triangle, write and check algebraic rules, and apply those rules in contexts of varying difficulty.

I do this as a summative assessment under test conditions, though if a student has gone totally off track (for example, by not correctly simplifying radicals), I sit with them to discuss their error and then give them a second go at it.

The idea is that students will see that if we know the angles of a special right triangle and we know one side, we automatically know all the sides and vice versa. My hope is that this primes them to see where trigonometry comes from. My follow-up lesson involves another investigation, but this one done informally in groups.

I ask the students to draw a few lines of varying steepness and investigate the gradient (slope) of the line and the size of the angle of elevation and find a pattern or rule. I’m doing this on Tuesday with my current group, but last year they were really clearly able to say that a gradient of less than 1 gave an angle of less than 45 (and the smaller the gradient, the smaller the angle) and a gradient of more than 1 gave an angle of more than 45 (and the bigger the gradient, the bigger the angle). I then tie it back to their discovery of the sides of a 45-45-90 triangle.

It is only then that I introduce tangent as the rule relating the angle of elevation to the gradient. I have them use the calculator to find the tangent of the angles of elevation that they measured and check them against the gradients that they had. Only after they feel comfortable with using tangent do I introduce the other ratios, again referring to the findings from the investigation on special right triangles.

I hope in this way that trigonometry is less of a black box and a procedure that the students do, but don’t know why they’re doing it. How do you introduce trigonometry? How do you mitigate its abstractness?

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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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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:

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:

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.

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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:

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```A textbook is not a daily lesson plan.

— David Wees (@davidwees) September 5, 2014

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

beuros 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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