**I am Desperate**

I am on a desperate search to find out who or what broke my students. In fact I am so desperate that I stopped class today to ask them who broke them. Was it their parents, a former teacher, society, our education system or me that took away their inquisitive nature and made math only about getting a right answer? I have known this was a problem for a while but today was the last straw.

**A Probability Lesson Gone Wrong**

It started out innocently enough working on the seventh grade Common Core standard 7.SP.C.5 about understanding that all probabilities occur between zero and one and differentiating between likely and unlikely events which I thought would be simple enough. After the introduction and class discussion we began partner work on this activity from the Georgia Common Core Resource Document (see page 9). The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability. I thought it would be an interactive way to gauge student understanding. Instead it turned into a ten minute nightmare where I was asked no less than 52 times if their answers were “right”. I took it well until I was asked for the 53rd time and then I lost it. We stopped class right there and proceeded to have a ten minute discussion on who broke them.

**If You Can Type the Problem into Wolfram Alpha and Get an Answer You Aren’t Doing Math**

When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one “answer” and then call it a day. They rarely think about what they are doing as long as at the end of the day their answer is “correct”. Today they were given a task with no real correct answer and they lost it. It did however lead us to have a very productive discussion about that fact that they are lucky, after all they live in the 21 century where they can solve any computation problem with technology with no issue. The problem I told them lies in the fact that they have no idea how to interpret that answer. We talked about the need for them to stop worrying about if I think their answer is right and to start worrying about whether or not they thought their answer was right. I told them I was sorry someone (maybe me) broke their desire to think about math and instead taught them that math was a means to an end where there was always one right and one wrong answer and then I told them to try their assignment again.

**Probability Revisited**

Things went so much better the second time around with not one student asking me if their answer was “right” (perhaps out of fear of another Powers meltdown). For the first time I heard some really rich discussions that were sometimes correct and sometimes were not but the important thing was the kids were talking about the math. There was a great discussion about the circle that was shaded in vs. the circle that was not shaded in. The obvious answer was that the shaded in circle represented one and the unshaded represented zero but another group of students thought maybe the non-shaded circle was actually shaded in with white and would therefore represent one as well. Fabulous. Another group focused on the statement “it will rain tomorrow”. One student had seen the weather and knew there was a 90% chance of rain the other had not seen the weather and though the probability was 50% since it would either rain or not. They compromised and picked the middle but that’s not the part I cared about, I cared that they had a reasonable discussion about their thoughts.

**Why Constructing a Viable Argument and Making Sense of the Reasoning of Others is Crucial**

This may seriously be the most vital mathematical practice. If students can’t share their ideas and understand the ideas of others is there any real point in them “doing the math”. After our lesson redo I paired each group with another group to share their number line. Their goal was to look for similarities and differences and explain their rationale about why they placed controversial cards where they did. I heard some of most logical and articulate arguments we have had all year. I think I heard, “I like what you did there, but…” repeatedly along with “I hear what you are saying”. We brought it back together as a whole class to follow it up and each group shared the most interesting conversation that they had.

In the end our meltdown and redo took more time than anticipated by me but it was time well worth it. If we are to truly make progress in getting our students to understand the concepts presented in the Common Core to the depth intended we must help them learn to stop looking for a right answer and start looking for a right reason. I still don’t know who broke my kids but I know it is up to me to fix them one argument at a time.

Ah, this breaks my heart. It happened in my classroom, and resulted in an cruel parent calling me the day spring break began, saying truly vile things to me about how unfair it was to demand that her child communicate in class. The girl got a perfect score on her end of year state test and I left that school.

Ugh I hate to hear that! It is so frustrating and hard to flip that culture. I try my best but it is only 177 days out of a lifetime of being programmed to not talk about the math. I am hopeful we start to see it improve as more states roll out Common Core!

NOTHING broke your kids.

at the seventh grade level, they are just making the transition from arithmetic to math. Those are two completely different subjects. Arithmetic is all about getting the correct answer, all about 2+2=4. Math is about understanding WHY it equals four.

The problem comes in that you, as a teacher, do not comprehend that. Were this an art class, it would be like you asking why children who have had only finger painting in kindergarten have no understanding of brush stroke technique.

I am 55. I was one of the first classes that got what was called “New Math” in the 1960’s. It is what is taught exclusively today, and basically attempts to teach the kids math philosophy first and facts later. The problem is, that is not how kids learn. They learn facts first, then the philosophy that binds them to make them true second. Schools have put the cart before the horse, and wonder why they are failing. How many times have we heard how other countries are surpassing us in STEM scores? Its because they are still teaching “old math”. Why did *I* succeed in math? My teachers were still skilled at Old Math techniques, and taught both. Kids today fail because most of their teachers learned only the New Math philosophy, and can only teach what they learned.

Kids are “broken”, they just aren’t adults. We need to stop trying to treat them like they are.

Actually, you’re pretty much entirely wrong about how children learn about the world. Maybe it’s because you haven’t kept up with pedagogy since your time in primary school. In any event, your condescending tone seems more hubris than wisdom.

Students actually do end up retaining more and having a better grasp of underlying concepts if mathematics are taught as applying modes that they use every day to abstract problems; while abstraction is a developmental change, the modes generally aren’t. Things like counting, pairing and sorting are already available in kids; using those algorithms as algorithms gives better outcomes than the drill-and-repeat nonsense of hidebound antediluvians.

The “show-tell-do” method is inadequate for teaching math, especially to students with diverse learning styles. It may have worked for you; it’s not a generalizable method.

As for this teacher, this quote is worthwhile from the International Bureau of Education, run through Unesco: “Tasks should not have a single-minded focus on right answers; they should provide opportunities for students to struggle with ideas and to develop and use an increasingly sophisticated range of mathematical processes (for example, justification, abstraction, and generalization).”

So maybe back off on the jerkish pronouncements and recognize that this teacher’s actually using the best tools at her disposal. Tools that are, actually, more supported by math (empirical research) than yours are.

Thanks, glad you got to post this. I will post my own thought but I felt the same way. Kids may be broken but not because of not learning math

I’m a British maths teacher (that’s right, maths, plural). In primary (elementary) school, I learned traditional pen and paper methods from teachers who would have been horrified at the idea that children might use calculators. Even then, all agreed that doing the calculation was the easy bit. The hard, important and useful bit was to know which to do and why.

This is even more the case in a world where calculators cost almost nothing. I find children who see why almost always do better in exams and in real-life problem solving than those who don’t see why but know how to perform techniques.

And yes, the UK did a bit better than the US in the latest PISA rankings

Amen. I don’t understand why the teacher was unhappy about students using “whatever method was about right” and then later expressed satisfaction when they were making up methods (to combine probabilities of weather). The message is unclear. In any case, it would have helped the kids a lot of they had been taught the concept of a probability prior. Without that, the discussion of probability is, in fact, meaningless. It looks like the Common Core curriculum doesn’t cover that. It raises the question (some rocks float, after all – what is the chance you’re tossing one of those into water?) but doesn’t teach the required concepts to answer it. No wonder the kids were confused.

Thank you so much for your comments. I wrote a follow-up post here: https://powersfulmath.wordpress.com/2014/06/03/a-reflection-on-who-or-what-broke-my-kids/

that I think better explains the lesson. Again I do appreciate hearing your thoughts on the matter too!

All through my academic career (junior high all the way through college) I’ve come across so many math teachers & professors who take the same attitude as you: “Why don’t any of you care about math?” “It isn’t about the correct answer!” etc etc. You yourself ask what “took away their inquisitive nature and made math only about getting a right answer”.

The answer is simple math:

Why should I put the effort into learning the material when I can bullshit my way to a 100 in 1/10th the time? I could use Wolfram and be done with my homework in 20 minutes with a perfect 100. Or I could do the work by hand, spend 1.5 hours on it, and get a lower grade.

Now, if those kind of assignments were only handed out rarely, it might be reasonable to expect students to spend the 1.5 hours and do the learning. But they aren’t rare, and it doesn’t matter how much of an academic white knight a student is: after a few years of those kind of assignments, they just stop caring. And when they stop caring, they’re rewarded with a higher grade and more free time. I’m a prime example — I’m 19, and graduated two weeks ago. I have both a high school diploma and an Associate’s degree, which I earned by taking classes at my local community college while in high school. I can say with great certainty that, on average, I haven’t spent more than 1 hour total on homework per week for the past 8 years.

tl;dr Your students aren’t broken — they’re perfectly adapted to succeed in the educational environment they’re in.

I think you are directing your student anger at the wrong target.

I’m not angry at anyone.

I hate to hear that’s what your education experience was like. I think that is my point. We need to shift our focus from the doing problems that can be typed into wolfram alpha to problems that require a deeper level of thinking and understanding.

Au contraire! I loved my educational experience! The system is structured in such a way that you can learn the material you’re interested in, halfass everything else and get solid grades all across the board.

As for transitioning to problems that require deep though vs problems that can be solved by a computer algebra system:

The dichotomy between computer-solvable problems and problems that require deep thought is essentially non-existent at this point. If you want problems that a CAS can’t solve, you have to either make problems that aren’t mathematically solvable, that are expressed in natural language that is sufficiently dense to obfuscate the symbolic / numeric representation of the problem (at which point the real exercise is to transform the problem from words to mathematics to make it CAS-friendly — word problems much), or that use mathematics that are just too advanced for the system (which I’m certain isn’t in middle school — or high school, for that matter — common core).

Fethawit, I feel as though you’re speaking to the issue that I observed in my own scholastic journey… I found performance was valued over intrinsic knowledge at practically all times.

Brooke, what a great day you had! I am lucky that I had a similarly passionate set of teachers about the core of mathematics. Honestly I have always been thankful for the teachers I had throughout school, as for the subjects I was interested in they all took great care to emphasize the theory and understanding. Unfortunately, this did not solve the problem though. While as students we were educated in a manner that emphasized the proper core of the subject, the curriculum and the school culture itself, cultivated the very performance metric you described.

At the end of the day, even with tests that were themselves a mixture of demonstrating that you could solve the problem and also that you could explain the why and how of those solutions… the true importance was driven by standardized testing that emphasized just being able to get the right answer. To make matters worse, entire “sections” were devoted simply to the tests rather than the course we were taking. I never investigated whether this was state or district driven, but ultimately that was what I faced.

We were also driven to achievement and performance rather than preparation… get the best grades, get all of your credits. Emphasis was put on completion rather than interest or even just simply taking your time. Even though we ultimately had 4 years of high school regardless of anything, it was always a rush to make sure you’d be ready at those 4 years.

We need a change and bad. That “ready” status they supposedly prepared us for, was the exact opposite – I was only prepared because of my personal desire to learn and grow – thanks to that and the teachers (I’m sure like you) that put up with my questions and pestering after school, feeding my appetite for the juicy details around the answer. I had a great time at college.

My favorite after school discussion was perfect numbers and all of the crazy things that perfect numbers ALSO happen to be… because that singular discussion lead to about a month of new subjects and interesting things about history in mathematics.

I agree with you completely. I was fortunate enough to have a great teacher back in 10th grade that pulled me out of this grading based mess that education has become and I am grateful to him for that. However, most students aren’t so fortunate.

It’s as you say – students adapt. Straight and simple.

fethawit, you are exactly the kind of person I will reject for a job when you come looking for one. I want people who are able to independently analyze and solve problems they encounter in the workplace, because in the real world the correct answer is rarely, if ever, available via Wolfram or Google.

You may have adapted perfectly to your educational environment. Unfortunately you have failed to learn that those adaptations are ill suited outside of it.

That’s precisely the point! Problems assigned in schools rarely, if ever, prepare a student for ‘real world’ problems, and thus a student who succeeds in school is often still unprepared for the ‘real world’. I can’t speak for all U.S. students, but most of my fellow students who cared about succeeding in the ‘real world’ knew this, and most of us took steps on our own to prepare themselves, self-educating in line with our interests. Sometimes that even meant paying attention in class 😮

We left the boring stuff to Wolfram.

As for my desirability to an employer: if you assume that adaptations are mutually exclusive — that if I’m a good bullshitter, then I must have no real problem analysis / solving skills — then I’m not really interested in working for you anyway 😉

And what exactly is your plan for rejecting the world’s Fethawits, Mr. Employer? Give them some kind of quiz? With right and wrong answers? Most of them do pretty good on those.

Oh, wait, I know this one, it’s “I can tell from the way they communicate in the interview.” Ha ha ha. No you can’t. Did you notice how intelligent and articulate Fethawit was there? If she had been slightly dishonest, you’d be out of luck.

These students have just spent 12 years of their life, 9 months a year, 5 days a week, preparing themselves for your employment screening rather than your job. You’re pretty much screwed. And so the system continues.

fethawit, I think you have very good points. However, there is a flipside: I pretty much shared your math experience in school. Then, in a linguistics class in my final year at university we revisited some of the stuff. Only then, I understood what I had been doing all those years as a teenager.

Sadly, I now enjoyed it more than most other things I was doing. I say sadly because the fact that I was able to half bullshit, half charm my way through school-maths meant I missed out greatly – I firmly believe that had I known earlier what maths could be about, I would have studied it instead of one of my two minor subjects.

tl;dr: if one doesn’t understand something, one can’t make a proper judgement about it.

My 19 year old friend. If you took a math, physics, engineering, computer science, or chemistry degree you would get the shit kicked out of you so hard it would circle the earth and re-enter you.

While you’re correct in middle school and high school and non-math intensive programs you can just punch it into wolfram and be on your way. When you start taking anything above that level math becomes very different.

I am going to assume you have not taken a course on integral calculus? If not I suggest you look into it and try to do some of the problems. Sure sometimes wolfram will do them for you, but it’s rare. You normally run out of computation time. Alright so you go torrent Maple, Mathematica, and MATLAB and learn to write scripts to do the work for you. It’s a time investment but you finally understand the scripting languages to do those problems and you can go back to doing the work in 20 minutes, right? Nope.

You’re going to start running into err(x), si(x), and all these different things you never knew about. It turns out math stops actually HAVING an answer for some questions. And math stops having a sure-fire way to get to the answer. We find out about this in your very first math class at university. Introduction to calculus 1.

Now okay, you have to do maybe 10 minutes a problem and you have 10 problems per assignment… if you’re lucky or smart. That’s 1 assignment a week. You memorize the “traditional” way of doing integrals and you figure out the pattern and get lucky and bullshit your way through integral calculus. I had a friend who did this who is now in 4th year physics with me. His mark was the same as mine coming out, about mid-80s. He spent maybe 20 hours of work on the assignments and 10 hours of cramming for the midterm, and another 10 on the final. I spent maybe 140 hours on the assignments trying to understand them because I fell a bit behind at the start of the year and about 10 on the midterm and final. Sweet you saved a ton of time right? WRONG

Now comes 2nd year math and physics, the math itself gets way harder and the concepts get simply insane, you have to understand what an integral is doing because they throw differential equations at you in physics before you even know what they are, they throw Eulers formula at you relating trig functions to exponentials and imaginary numbers and you start working with phasors and vectors simultaneously. You start having to do proofs and you have no idea what to use because you have 9 other classes that were all prerequisites for the class you’re doing and all that material is fair game. You’ve forgotten everything you’ve memorized and you completely BOMB everything and now you’re putting in 30 hours of work on an assignment, wolfram isn’t working because it’s taking more time, you don’t understand the math well enough for you to know what to even enter into Mathematica or Maple or MATLAB, and you’re getting a 0% on your assignment. My friend barley passed his 2nd year math and was saved on his physics because some of the questions were on concepts, which he worked on in physics but he ignored in math. Turns out that you NEED to understand the math to be able to do it at the 2nd year level. I understood the material from first year perfectly and when I went into second year I had no catching up to do like I did in first year, and with the SAME amount of work my friend did I got an award for the highest average across my year in my department along with five scholarships.

Now 3rd year. My friend had to back down and take less classes and go back and re-learn from SCRATCH all 4 integral calculus courses he took because he was risking flunking out of university. He DID catch up and did well on some of the courses he was required to take… but it hit him like a train. I had some health problems and butchered first semester. I took incompletes, went into 2nd semester with 7 courses and still challenged for highest marks in all of them.

And I hate to tell you, you can have 1 question in physics or math, and you can spend 15 pages doing it and it can take you and a small group of people working on it over a day before you get it.

So yeah, if you are NEVER going to go into 5 of the biggest and most well paid disciplines not understanding a math and using a calculator isn’t THAT big of a deal… but if teachers don’t try to get THIS message across to students, that they are completely screwed if they don’t understand the math, the students will just end up failing out and having to rethink their life, or go back to high-school level classes and relearn the material and try to understand it if they are lucky enough to fail out in 1st year and haven’t wasted too much of their time.

As part of earning my Associate of Sciences I took three integral calculus courses, two calc-based physics classes, and two computer science courses (although they were entry-level). How coincidental that you bring up the various computer algebra systems — my final exam for my final integral calculus course was a Sage worksheet.

I think what you and other commenters are assuming, based on two paragraphs of opinion and personal anecdote, is that just because I said I am a good bullshitter and well adapted to succeed in a test-driven environment that I therefore have no ability to learn and possess no analytical intelligence. The two are not mutually exclusive.

Clearly you are having difficulty with math. Your own post tells me this. You say that your homework can be done in 20 minutes by using WolframAlpha to get it done 10x faster. 20*10=200. 1.5 hours, which you admit to being the full time by working by hand, is 90 minutes. 90=/=200.

So you’ve done some homework quickly. But what happens on the test? If WolframAlpha solved the polynomial of x^2-5x-25 and you have no idea how to solve for X, you’re in for some trouble.

I never had math homework take longer than an hour. And prior to calculus, the homework took 30 minutes and I had it done with in class. “I can say with great certainty that, on average, I haven’t spent more than 1 hour total on homework per week for the past 8 years.” So it seems we’re on the same page. But your outcomes are reflected in my first paragraph.

I might have had a similar attitude to yours, except 1) I had a really great trig teacher in high school, and 2) I had to do college math with those skills developed in high school and middle school. Half-assing it leaves you either unaware of or unable to do a lot of the math that really does make life richer.

fethawit

Everything I’ve read about you tells me that you’re not an average student. Most students can’t get through one integral calculus by the age of 19 much less three.

So while your comments are lovely anecdotes, don’t assume that your experiences would in any way be representative of the general population.

I didn’t get anger out of that at all. I’m 30 and completely agree. I can anecdotally say, “ditto”.

I vehemently believe that debating skills should be taught to all children (without exception, be it race, religion or disablity) from nursery. Start simple with – “Why is your favourite toy better than all the other toys?” – and work form there. At least an hour a day of reasoning, arguing and understanding. By the time they’re 18 they’ll confidently be discussing complex and difficult topics in a constructive way and there will be some hope for the future, as opposed to todays culture where difficult conversations must be avoided at all costs because it might offend (or worse, “bore”) someone. They’ll also “know” more, instead of just “remembering” more.

I couldn’t agree more!

That being said, this could lead to a lot of bullying. It would have to be done very carefully.

How is healthy debate in any way bullying?

I agree. My husband has said the same thing for years.

I couldn’t agree more.

This story gives me hope. My outlook on US education is bleak, for a number of reasons. We are falling behind internationally, we have a culture that seems to promote anti intellectualism (especially for math), and our curriculums are focused on teaching for the test. I was bearish on common core (specifically, the methods of teaching arithmetic to younger students look obtuse and confusing), but this probability exercise is fantastic – better than anything I got in 7th grade – but only when properly taught.

There is still room for pessimism – the first attempt to teach this exercise makes this a cautionary tale as much as a hopeful one. I just have to hope there are more teachers like you out there.

Thank you so much for your comments. I am optimistic that we will start to do more critical thinking and problem solving and less skill and drill with the implementation of the Core. It is desperately needed.

That’s the crucial problem, though, isn’t it? Teaching, especially in public schools, seems to be the refuge of the talentless and the dogmatic. The tenure system, so preserved by the teacher unions, rewards obtuseness and punishes enthusiasm.

In my own recent tutoring experience, I’ve been working with a high schooler on algebra. It’s a modern book with a bunch of approaches and classroom discussions. He says the teacher always skips the discussions. According to the NCLB accountability report, only like a third of the students in his school actually learn the math that they’re supposed to learn.

You don’t have any idea what you’re talking about, do you? “Tenure” in primary schools doesn’t mean “job for life with freedom to research whatever you like”, the way it does at the university level. It means that one can only be fired for cause, and all disciplinary action is subject to due process requirements.

That’s all it means. Seriously.

There’s a lot of reasons that the average talented person doesn’t go into the teaching profession, but job security is not one of them.

I think one issue is that the problem you gave them is one which admits to guessing. One where guessing makes it seem like some reasonable amount of work has been done (because it’s followed by some glueing or computing some arithmetic), rather than the truth that it was a random guess followed by gibberish.

I’m surprised you never use the word proof, because this is what “making sense” and “reasoning” and “discussing” mathematics is. Though I am not a full time teacher, I have found the best way to get students in the right motivation is to give them an open ended problem. A problem whose answer is either yes or no, but the answer by itself is completely unsatisfactory. Moreover, the explanation should be elegant, but not obvious, so that once they spend a long time thinking about it and getting nowhere (or do!), the lecturer can show the power of reasoning with a truly beautiful proof.

I can give you an example of such a problem, and an entire lecture I give around the problem, which doubles as an introduction to graph theory: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-theory/ I usually give this lecture to high school students, but I have given a simplified version to eighth graders. But I hope it’s the ideas, not the content, that make it worth reading.

I use graph theory for two reasons: the students are almost guaranteed to have never heard of it, and there are no preconceptions about how best to teach it to high school students. Moreover, it’s extremely easy to counter nearly every claim a student makes with “why?” and when they respond there’s a noticeable transition for the student. It becomes obvious to them and everyone exactly how much gibberish they’re actually spouting when they guess randomly.

Thank you so much to for your graphing theory lecture. I love it and will make sure and share with my department.

In his “Intuitive Explanation of Bayes’ Theorem”, Eliezer Yudkovsky wrote:

It’s like the experiment in which you ask a second-grader: “If eighteen people get on a bus, and then seven more people get on the bus, how old is the bus driver?” Many second-graders will respond: “Twenty-five.” They understand when they’re being prompted to carry out a particular mental procedure, but they haven’t quite connected the procedure to reality.

I was awe-struck, so I asked a friend who sometimes teaches second-graders to try this. 11/18 wrote “25”, 5/18 wrote “25 passengers on the bus” and 2/18 returned a blank note.

I think this is a big part of the explanation. If you’re taught addition as a process that happens in a notebook, not in

reality, then you have no way to separate answers that make sense from those that don’t. You also have no way to connect math to things you experience in your life, and I think the most common way to develop an interest in something is to find out it’s related to something else that you’re already interested in.In the last Super Bowl, the Seattle Seahawks scored 8 points in the first quarter, and 14 in the second quarter. Who won the match?

I couldn’t agree more!

(Hoping you see this and I’m not too late)

Slightly related request: Do you teach primary school at least some of the time? Can you try this experiment yourself? Right now I know the result for one class of 18 second graders in Israel, I wonder how it is in different cultures and age groups.

We are already out of school for the summer otherwise I would have been glad to. I would be happy to do it when school starts back again and let you know!

My son is in middle school (sixth grade). He has so much assigned homework that he has little time to contemplate or think. Last Tuesday night, he had written homework in Spanish, Math, Language Arts (English), and Science. He had to finish up a long term science project due on Thursday. He also had to work on a long term project (English) due on Friday. Also, he had to study for upcoming quizzes on Thursday (Math), and Friday (Science and Spanish). He was also supposed to have read two chapters of a book they are reading, which he didn’t get to (surprise quiz on that on Wednesday– oh well). Also, on Wednesday, surprise quiz in Spanish. The homework is all graded, with the students losing points for incorrect answers. The Math homework was a 37 problem online multiple choice test, automatically graded, and electronically submitted to the teacher. He gets home from school at 3:30, and goes to bed at 9:00. We eat dinner at 6:00-6:30. Not much time for contemplation. Just get it done.

I hate that this has been your son’s experience. One of the things I push for is less quantity in terms of homework and more quality. I typically give three problems a night that really require a deep understanding of the math.

This doesn’t sound like a traditional public school. Or at least not one in north america. I remember complaining about homework and it seemed like a lot and I felt overworked until I hit university. Now I’m lucky if I sleep more than 5 times a week.

Georgia. Are these religiously-raised kids from authoritarian families?

No. I am at a public school in an urban area.

What is sad is I feel this article _just_ missed the point.

It;’s not _math_ but “thinking”. The same goes on in history where you memorise dates but little emphasis is place on leveraging what you know about history to think about history. Or in English memorising quotes, but missing the author’s message. Etc. The trivial and mechanical is easier to test and mark, but understanding is truly truly hard to measure so we don’t; but it’s the only thing that matters.Anyone who thinks that there is a problem with math specifically is broken too.

I agree that it is a part of a larger problem that is due to a lack of focus on thinking and reasoning. My teaching focus is math so that is why I shared this experience but I think it is a problem across content.

You broke them. Every state/county/local school board that institutionalized standards testings. Every teacher who followed the core curriculum to get the scores that get the money and recognition for the schools.

Although I agree there is a problem with too much standardized testing I don’t think I am the reason they are broken. I work myself to death to shift their focus from short term assessment knowledge to long term thinking and understanding.

One way I solve the “Is this right?” problem is by giving my middle school writing students a bead at the beginning of their first draft. Since I’ve already pre-taught brainstorming and instructions, they need to finish their first draft without constant help. I tell them that their bead is their one question: they can choose to waste it on “Is this right?” or “Can I get another pencil?” or save it for a question that will truly help their progress. It’s the only way I’ve found to get them to trust themselves to write a rough draft which inherently will never be “right” the first time.

Thanks so much for your thoughts. I will most definitely use some of those ideas!

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I truly wish mathematics education was like this. I think learning how to approach problems and reason about them is absolutely vital. I took every single math course I could through high school and many through university (earning me a minor in mathematics). The problem at least in the schools I attended were the teachers sung high praise about understanding and process, but their grading reflected the importance of the exact correctness of answers. What they graded and what they preached conflicted so frequently I just started ignoring their preaching and went straight for optimizing grades. Why bother going through the effort of showing some creativity and elaborating on solutions when if the last line is even slightly wrong I get a 0 on the question? I hope math education can be reformed because it’s an amazingly fun subject that can applied to so many places in life.

I agree with so much of what you are saying! Trust me when I say they are teachers out there that do recognize and reward our critical thinking students. You are correct when you say that most grading practices reflect exact correctness. Reform is most definitely needed in terms of grading in a lot of schools.

I don’t think this is unusual or even new. Many students find a refuge in math because there are right answers and you can always get them if you understand the concepts and have the skills down. Students often hate word problems because no one really teaches them how to go from the words on the page to the math problems they’ve been drilling (and because, in the end, the extra steps mean they take more time).

Most children, teens, and young adults struggle with the idea of how to apply math, and probability and statistics is probably the most important area for them to understand, because people are going to use it against them every day for the rest of their lives. What’s difficult to get across to them is that very few people ever use math in the purely abstract way in which it is drilled in school. Nearly every math problem you see in the real world is a word problem, and sometimes we don’t even recognize a given problem as a math problem.

When you throw this type of lesson into the mix, it’s going to drive some students out of their comfort zones, and they’re going to look for reassurance. While it’s important to get the concepts across, it’s probably just as important to preface a lesson like this with a discussion of what you’re expecting from them.

For the most part, I think our system of education is really breaking the teachers, not the students (it may be failing some of the students, but they’re not broken). You teach these kids every day, you planned the lesson, and YOU had the breakdown. Why didn’t you expect this type of response? More importantly, if you ever use this lesson in the future, I would hope that you will start the lesson with the discussion.

We spend so much time focusing on test scores and it seems like sometimes we all forget that we’re stuffing a room full of kids and expecting one person to keep them under control all day and, on top of that, actually teach them something. I’m thankful that teachers are willing to try new things in educating our children in an environment which seems determined to turn schools into factories, with their high school diploma having all the value of the quality control tag that you sometimes find in a new package of socks.

Problem too is that once you hit algebra and geometry, word problems are more like math puzzles than corresponding to real world needs. Like a young kid will make use of how many apples they have or how to make change, but not many teens are going to determine how long of a ladder they need by calculating the longest side of the triangle. They’ll just ask why they can’t eyeball it instead.

They don’t teach you how to apply math to real world situations, but they teach you that you need to decode a word problem into a equation and then solve the equation. The words 90% of the time are pointless dribble in algebra problems and up, because they don’t correspond to the way any rational person engages reality. Unless they are an engineer, scientist, or someone who might actually need to calculate the doppler shift or something. Most higher-order math is more or less useless in that regard.

I’ve lost count of the number of times that Colin has posted an interesting article on mathematics education on Hacker News over a weekend. This submission here is particularly good. The problem indicated is quite stark, and very commonplace. Most pupils in a mathematics lesson in elementary school swiftly learn that “getting the right answer” is the point, and some check out and soon begin to doubt their own ability to REASON to the right answer.

The author of the submitted article writes, about a seventh grade class, “The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability.” As the author makes clear, the particular lesson arises from the new Common Core State Standards in mathematics,[1] which are only recently being implemented in most (not all) states of the United States, following a period of more than a decade of “reform math” curricula that ended up not working very well. I am favorably impressed that the lesson asked students to put their numerical estimates of probability on a number line–the real number line is a fundamental model of the real number system and its ordering that historically has been much too unfamiliar for American pupils.

The author continues by elaborating on his main point: “When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one ‘answer’ and then call it a day.” I like the author’s discussion of that issue, but I think she misses one contributing causal factor–TEACHER education in the United States in elementary mathematics is so poor[2] that most teacher editions of mathematics textbooks at all levels differ from the student editions mostly just in having the answers included[3] and don’t do anything to develop teacher readiness to respond to a different approach in a student’s reasoning.

What I LOVE about the Singapore Primary Mathematics series,[4] which I have used for homeschooling all four of my children, is that the textbooks encourage children to come up with alternative ways to solve problems and to be able to explain their reasoning to other children. The teacher support materials for those textbooks are much richer in alternative representations of problems and discussions of possible student misconceptions than typical United States mathematical instruction materials before the Common Core. Similarly, the Miquon Math materials[5], which I have always used to start out my children in their mathematics instruction before starting the Singapore materials, take care to encourage children to play around with different approaches to a problem and to THINK why an answer might or might not be correct. (Those materials, both of them, are very powerful for introducing the number line model of the real number system to young learners, as well as introducing rationales as well as rote procedures for common computational algorithms. I highly recommend them to all my parent friends.)

I try to counteract the “what’s the correct answer” habit in my own local mathematics classes (self-selected courses in prealgebra mathematics for elementary-age learners, using the Art of Problem Solving prealgebra textbook[6]). I happily encourage class discussion along the lines of “Here is a problem. [point to problem written on whiteboard] Does anyone have a solution? Can you show us on the whiteboard how you would solve this?” Sometimes I have two or three volunteer pupils working different solutions–which sometimes come out to different answers [smile]–at the same time. We DISCUSS what steps make mathematical sense according to the field properties of the real numbers and other rules we learn as axioms or theorems in the course, and we discuss ways to reality-check our answers for plausibility. We don’t do any arithmetic with calculators in my math classes.

[1] http://www.corestandards.org/Math/

[2] http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic…

http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma…

[3] When I last lived overseas, I had access to the textbook storage room of an expatriate school that used English-language textbooks from the United States, and I could borrow for long-term use surplus teacher editions of United States mathematics textbooks. They were mostly terrible, including no thoughtful discussion at all of possible student misconceptions about the lesson topics or of alternative lesson approaches–but they were all careful to show the teachers all the answers for the day’s lesson in the margins next to the exercise questions.

[4] http://www.singaporemath.com/category_s/252.htm

[5] http://miquonmath.com/

[6] http://www.artofproblemsolving.com/Store/viewitem.php?item=p…

It’s a depressing answer but as long as the tests won’t or can’t measure critical thinking and as long as you keep getting told that your score on the test is going to affect your future for the rest of your life, this is just logical. School math doesn’t reward the right kind of work so in a way the students are actually being kind of smart in that they’ll optimise for the amount of work they need to do to get what they want. It’s broken incentives that are breaking the students.

FWIW I’m not a teacher, just an engineer with vivid memories of why I hated math in school and of what I missed out on by thinking I wasn’t ‘good at math’. Luckily I discovered I actually really like math, even if I find manual arithmetic extremely laborious and need context to really integrate ideas well 🙂

I appreciate your feedback as an engineer that thought they weren’t good at math. That is the one thing that is hardest about teaching in the middle grades. A lot of my kids have always decided that they aren’t good at math when I get them. I do the best I can to provide them with rich and in context activities and lessons to try and reverse that thought process.

I think you would love Jo Boaler’s course called How to Learn Math for teachers, there is also one for students. She is a hero of mine. I did her course last August and took ideas like you have and took them into the stratosphere. http://joboaler.com/how-to-learn-math-2014/

Also if you aren’t aware of these three, spend some time with them…

Dan Meyer – http://blog.mrmeyer.com/ (I want to be him in the classroom)

Fawn Nguyen – http://fawnnguyen.com/ (I also want to be her in the classroom)

Andrew Stadel – check out his http://www.estimation180.com/ for getting kids to reason

I have little and messy blog where I try to post cool things about mathematics to show it’s more than right and wrong answers. It has a tiny following but includes kids, adults including some parents

Thanks so much for the links. I had no previously known of Jo Boaler but am excited to delve into her info today. I am a huge fan of Dan, Fawn, and Andrew. I am definitely going to start following your blog it sounds like we are pretty similar in our methodology!

I took How to Learn Math last summer and loved it! It’s free online.

Oh my poor blog – it’s been neglected while I’ve been trying to maintain a blog for kids.

Enjoy Jo Boaler’s course, it’s just so so good.

I started the student one yesterday to see what my kids will experience – great stuff. My school is planning to get the community on board with the free student course – all teachers (I hope), parent community and middle and high school kids when we come back in August. Exciting!

Hi, Brooke–

As a fellow conceptually-focused math teacher, I just wanted to thank you for sharing, and to say keep it up! I’m both haunted and fascinated by the same questions of how we re-define math for our students, and teach them an entirely different way of understanding what it *means* to “do the math” or to be “good at math”.

In addition to lissgriffin’s resources, I’d recommend Martin Haberman’s short but classic paper, “The Pedagogy of Poverty versus Good Teaching” https://www.ithaca.edu/compass/pdf/pedagogy.pdf and Paul Lockhart’s “A Mathematician’s Lament” http://www.maa.org/external_archive/devlin/LockhartsLament.pdf as ways of understanding how, systemically, school math has come to be defined and enacted in ways you’ve [quite rightly] described as broken…

Thank you so much for resources you listed. I am especially interested in the poverty vs.

good teaching and will spend some them today digging into that more. I appreciate your thoughts it is nice to talk to other teachers who believe in a conceptual math education process.

What a great learning experience you’ve described! I also celebrate great discussions of mathematics when I can induce them in my students.

But you didn’t answer the question of what “broke” your kids. There are many answers–the culture, the parents, all or most of their former teachers. Even the discussion raised by fethwit’s wonderful comments has focused on getting right answers. And I’m afraid I can’t be so optimistic about you that Common Core standards will bring about change in this regard; almost all of them can easily be interpreted as requiring right answers. Even the “understanding” parts assume that there are single best ways to understand. Whereas our understanding of an idea in mathematics changes over time, just as our understanding of a piece of literature changes each time we revisit it.

Moreover, like most teachers, the Common Core standards–even those of mathematical practice–ignore an essential characteristic of mathematics since about 1850: that answers depend on context, just as in every other field. The number of lines through a point and parallel to a given line depends on context; are we talking about lines as shortest paths on a sphere, or the surface of a mountain? If a student claims that 8 + 5 = 1, what teacher would inquire enough to find out that the student is thinking in the context of a 12-hour clock? Before I flip a fair coin, the probability that it will come up heads 1/2; what is that probability after I flip it? What if I’ve seen the result but you haven’t? Is the probability different for the two of us? Answers to all these questions depend on context, on the assumptions we’re making? Questions like these are at the heart of mathematics, but students rarely encounter them. Why? Because they might get “confused.” That is, they may not be able to produce “the right answer.”

I agree completely on your thoughts about the “context”. I enjoy hearing Dan Meyer speak and write on this topic. I too hate that we seem to withhold topics and conversations with kids to avoid “confusing” them. Thanks so much for all your thoughts, they were really eye opening to me.

“Moreover, like most teachers, the Common Core standards–even those of mathematical practice–ignore an essential characteristic of mathematics since about 1850: that answers depend on context, just as in every other field”

I don’t see how you can reconcile that statement with, for example, the actual text of the practice standards. I’m thinking of things like “Mathematically proficient students start by explaining to themselves the meaning of a problem” and “the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents… creating a coherent representation of

the problem at hand; considering the units involved; attending to the meaning of quantities” or even “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace… They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.”

Thanks for your response, James. I agree that applying mathematics to solve real-world problems is an art, not a science, and requires attention to context–just as doing pure mathematics does. But your first two quotes indicate that perhaps the writers still had right answers in mind. “Mathematically proficient students start by explaining to themselves the meaning of a problem”: Which meaning? Does a problem have only one meaning? To everyone? Is that real life?

I have the same question about “the meaning of quantities.” An old problem is “There are 3 feet in a yard. If f represents feet and y represents yards, write an equation relating the two quantities.” What’s the meaning of the quantities in this problem?

What concerns me most, however, is not so much the writers’ concepts of mathematics but the perspectives of those who are working with the standards. Besides teaching mathematics, I work closely with teachers. As if I weren’t busy enough, I’m a professional technical and academic editor, with an emphasis in mathematics. These days my clients are publishers churning out online materials tied to Common Core standards. Unfortunately, the teachers I know ignore the mathematical practice standards altogether, and any reference these publishers make to these process standards is superficial. They certainly don’t deviate from “right-answer” practice problems, because they’re preparing students to take standardized tests, most of which require single right answers, and none of which require explaining context.

Before I read the prior responses I wanted to add my own, Who broke our kids?

WE did. As much as anything without intention and without wisdom. An amazing proportion of our ‘learning” and knowledge base is not the result of intentional academic or didactic endeavor but more often than not inadvertent encounter and misadventure in the urban environment. Consider what the following demonstration of a mentalist’s skill reveals about how the human mind soaks up information from the ambient environment. Obviously no one has to put elements in place for this to work. The senses poll the environment and whatever repeated elements become more familiar are more likely to stick. Would that we could arrange our home and school environments as mindfully as Derren Brown has arranged his drive across town for the purposes of his demonstration.

Andy Warhol observed that “It’s the movies that have really been running things in America ever since they were invented. They show you what to do, how to do it, when to do it, how to feel about it, and how to look how you feel about it.” Which also seems on point.

Plato was something of an extremist in so far as his social prescriptions run toward “command and control” But the tutor of Aristotle was quite possibly wise in the role of observer. If our minds are as open to influence, as Brown’s demonstration might suggest, them the responsibilities of ‘free expression’ could be very very heavy, Artistic license is not necessarily freedom ofrom responsibility. No free expression really can be. We have been ignoring and denying the consequences of indulgence for a very long time and our students in every generation have borne the consequences further forward.

On top of all this has been a growing indulgence and entitlement of youth to the point where neotenous characteristics are becoming plain, and young people increasingly comprehend ageing and maturity as undesirable conditions to be avoided and eschewed at almost any cost. A loss of value for age and maturity is to a great degree a loss of respect. The indulgence of our youth is a very profitable consumer enterprise much approved of by “children of all ages”. You deserve a break to day, have it your way, indulge yourself, Do it!

As students require increasing management, the processes of management favored by the institutions of education increasingly undermine student autonomy. The more students are carefully stepped through a lesson without an opportunity to engage in self-managed play the more they learn to feel comfortable only under extremely structured conditions. Asked to direct themselves they fall quickly apart..

Academics having embraced industrial protocols of manufacturing feel it necessary to establish the primacy of a convergent praxis wjhere only one correct answer is possible.

While the Arts represent a divergent praxis of individuation and personal preference for several generations now young scholars are expected to pick this up on their own time, it being essentially trivial development.

I would argue otherwise, but tomorrow i finish emptying my classroom due to a reduction in force. Maybe there will be another classroom maybe not. Time will tell.

I am so sorry to hear that you are emptying your classroom tomorrow due to a reduction in force. I was there once and luckily did find another classroom but I remember well the feelings. It is obvious that you are a passionate and articulate educator and I hate to hear that the profession is losing you due to cuts.

I agree with you about the “learned helplessness” part. I work hard to engage my kids in the “productive struggle” that is such a buzz word right now but fight that helplessness the entire way. That was really the point of the post. I hate that the my students do not feel confident enough in their abilities to explore on their own without me guiding their every thought.

I think the thing your should take from this is that you told a classroom of students that they are broken and can’t think. A bunch of children who have no experience with what you (a well-educated adult who created the lesson) are trying to teach suddenly pulls the proverbial rug right out from under them. These students are there to learn what you are teaching (that is, math), not to be tricked and then have a sociology lesson.

Your students look up to you and you tricked them. One of my strongest memories of elementary school was in a math class, actually – it was a sheet of paper with a number of silly little commands like ‘calculate this’ or ‘shout “Almost done!”‘. Of course, before the exam we were told to read the entire sheet throughly before beginning – and of course the last question said something like “Don’t do any of these commands except #1”. Question #1 was “write your name at the top of this paper, turn it in to the teacher and sit quietly until everyone else is finished”. I felt SO STUPID at the end. Of course I was the first one done with the whole thing. What did I learn from that? Not “read the directions carefully”, but rather that this person I looked up to as a benign parental figure who otherwise was teaching me things I loved to learn had betrayed me for no apparent reason than for their own amusement. Luckily, everyone else in the class made the same mistake so I wasn’t the only one shouting out stupid things. So, the strongest memory I have of that class is that I felt stupid, not what I learned. Like it or not, you are a surrogate parent to many of these children and so making them feel ‘broken’ is not really conducive to their learning.

Let’s face it, right or wrong, pre-university math classes ARE about getting the right answer because all that matters at the end of the class/year/twelve years of education is passing that test and getting a good grade. Standardized tests only want the correct bubble be filled in on a sheet, not if the studen can think. So, again – right or wrong – these kids aren’t broken. This is what our society (and our lawmakers) want them to be. Certainly there is a valid debate about if this is what we as parents should want our children to learn or how to learn, but most parents are (like their children) just trying to get through the day.

Man, what a weird takeaway. I didn’t get the sense she directly told her students they were broken. And the chip on your shoulder about being tricked is weird too.

As far as right/wrong Manichean thinking, one of the more powerful math lessons I remember from high school was learning the ability to translate modes of problem solving from one to another, e.g. straight algebra to matrix. That ability to think laterally about questions really has helped me in the long run. Oh, and my test scores were pretty fantastic too.

Could you please post a more detailed explanation of the probability activity with the number line? If sounds fascinating and useful. I teach English to international students, and I’d like to try something like this with them the next time we talk about hedging and expressions of probability.

There is a pdf link in the post that will take you to the original version of the activity. I also get a lot of really great card sorting activities from map.mathshell.org. I hope that helps!

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You’ll probably be interested in reading A Mathematician’s Lament by Paul Lockhart.

http://www.maa.org/external_archive/devlin/LockhartsLament.pdf

Thanks for this, Brooke. I am a relatively new teacher educated in Canada, but I’ve spent the past two years teaching in England, where standardized testing is the norm. I am constantly fighting this battle, both with my classes and myself—I want to create activties that are open and promote more serious thinking and communication, but sometimes it just feels easier to surrender to the idea of “teaching to the test” and just covering the basics they need for their exams. Of course, if I don’t somehow manage to do the latter as well as teach more meaningful skills, then they won’t be prepared for those tests either.

There is something more broken with the system than just the issue around math. Our industrialized system of education through classrooms and standardized tests has not caught up to the realities of today’s workplace. And anyone else who tells you that you are lying to these students that they are broken, that you are wrong—well, they are wrong. The students are broken. The system is broken, because it’s still trying to compensate for the past half-century of cultural and technological change that seems to have taken it by surprise.

I’ve learned a lot in my two years of teaching so far. Some of it is about teaching and managing a classroom. Most is about my own naive faith in the education system as it was versus how it needs to change for the better. I don’t have solutions. But I’m becoming better at seeing the problems, and maybe if enough of us can see them and work on them together, we can find a better way.

Thank you so much for your comments You sound like a fabulous teacher and the system is lucky to have you. I am with you in the fact that I may not have the solution yet but I can definitely see the problems which is the first step for change. I am looking forward to see what the future holds for education in the U.S.

Hi, Ben! I’m in my 10th year teaching in Chicago public schools, and my advice is to *ignore* the standardized tests, as often as you can. Talk to your students about what *they* think of the tests, and help them to recognize and understand the problems associated with testing. In at least 80% of your class time, avoid any mention of the tests (even if you happen to be working with material that’s relevant to the test). If, at some point, your students say, “But how come we’re not learning about the test?” then you can show them sample questions or content guides, and point out that they have learned those things, just not in a crappy multiple-choice format…

Make the classroom experience real, and trust that once they have ways of thinking about the content, they’ll figure some stuff out, even if they’ve never seen it before, when asked on the test…

I love what you have said here James. I agree completely. If you provide kids with authentic learning experiences they will figure it out when it comes test time regardless.

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I don´t understand it, Maths are cool! http://trtrcamp.tumblr.com/post/85140925457/199

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Thumbs up! Here in Austria we are starting with common core standards next year in our job-schools. Thanks for sharing your experience with us. Your lesson and your blog-post seems to be a good example, how teacher should think, act and collaborate.

Thanks so much for your comments. Good luck implementing your standards next year. The implementation of common core standards here has in my opinion made my classroom a much ore productive place.

Some historical context in this wonderful article from 1935 entitled “The Teaching of Arithmetic I: The Story of an experiment” in which a superintendent of schools in New Hampshire describes the results of “abandoning all formal instruction in arithmetic below the seventh grade”. There’s quite a bit to read but it’s fascinating. http://www.inference.phy.cam.ac.uk/sanjoy/benezet/1.html.

dan

Thank you for sharing this. This one of the most interesting articles on education that I have ever read. He is very convincing. If I were home schooling, I think I would try this approach.

Thanks so much for sharing! I added this to my weekend reading list. Thanks again!

I read this and felt I had to chip in: good on you, Brooke! This reminded me of a similar experience when I was teaching undergraduates: in a computing lab session, where students were learning how to code in C++, and a few of us were on hand to answer questions, I encountered one student who would call one of us over and point to his code, only ever asking one question: “Is this right?”

I was a bit taken aback – no matter how many times I explained that there was more than one way to code that particular task, and that perhaps he should try running it to see what result it produced, he simply persisted, “no, but is it right?”

I wish I’d had your presence of mind when confronted with this particular brick wall!

I appreciate Dan’s reference. Historical perspective can be useful. The fact that math (maths?) has (have?) been taught virtually the same way for hundreds of years might be cited, however, as a reason not to change, despite the fact that the way it’s been taught is successful for maybe only 5 to 10 percent of today’s students.

Another perspective I find even more relevant is research in the intellectual/ethical/identity development of college students, spawned by William G. Perry’s longitudinal study in the 1960s. To oversimplify, at that time, many students entered college thinking of knowledge in right/wrong (intellectual) and parallel good/bad (ethical) terms. Condemning their peers for being evil because they had different values didn’t go so well, so they moved to the opposite extreme of “everyone has a right to their own opinion.” Eventually, confronted with the occasional failure of bs-ing their way through courses, they encountered the ideas that opinions need to be consistent with data, but even so conflicting opinions may be equally well supportable. This “contextual relativism,” and the consequent need to choose their own path, was such a challenge to their identity that a few retreated to the more comfortable good/bad worldview with which they had started. Most plowed on, committing to a set of values even while realizing that alternative perspectives can be intellectually and ethically legitimate.

Knowing this developmental scheme–and realizing that a student’s position with respect to mathematics may differ from that with respect to other fields–makes me tolerant of mathematics students at all levels. I’m less sanguine about adults who still hold the good/bad, right/wrong view of knowledge.

Nevertheless, I love the fact that this discussion has displayed several of these perspectives on the tangible and mathematical universes.

Thanks so much for the comments. We deal with the same thing when we do robotics. The kids would generally rather us tell them the right series of events needed for their robot to complete their challenge rather than testing and adjusting piece by piece.

As a parent of a 6th grader, I can tell you we were completely frustrated with the probability lessons as well. Same as with your students, our daughter kept asking us if she had the right answer and we couldn’t tell her yes or no! We had no idea what the expectation was; why this lessons was being taught. From our point of view – a math class teaches you 2+2=4. There is one right answer for that problem. Communication with the teacher should have been our first route, but I think most of us are reluctant to ask a teacher “why are you teaching this” or “why is this important.” This is where explaining CC can be difficult to us parents. We expect math to be about getting the right number. Again, 2+2=4. That was the math we were taught; it always started out by making sure we came up with the right answer. The same answer as everyone else. Most of us still did the rote memorization of times tables, etc. That was our math. Why is it different for our kids? CC explains why – but getting that message out is hard.

Thank you so much for the comment! It is always nice to hear a parents perspective. I currently have a son in kindergarten and he has already started changing my view points on math education. I look forward to learning right along with him about primary math education and look forward to that experience helping me to become a better teacher.

Thanks for your hard work. The point of public education is learning to participate in our democracy with all the tools available. Math is becoming more and more central to proper negotiation of our problems—not just arithmetic or calculation, but working with the concepts to see what they mean. (For instance, Feynmann saying to the head of the NASA Challenger team, “1 in 300 chance of failure? You really believe you can launch this rocket every day for a year and have one failure?”) Thanks again. More Powers to you.

Thank you for your comments. I agree that students need more work understanding the meaning of concepts and “answers”. I think we are getting there but it is a long road.

It should occur to everyone that their assessment of what/how should be taught at school is based on their experience of being in the classroom as a student. (If you’re not a trained teacher)

I would like to offer an analogy: a person walks up to the scene of a car accident and says to the cops investigating, “Hey, I was in an accident once….I can help”

If you’ve never faced the challenges of teaching in a ridiculously underfunded and logistically challenged environment like public school, your anecdotal input is just that, input. Thank you for your story and your courage in sharing your feelings.

Everyone offers some prescription that they are woefully under-qualified to make. It would be refreshing, from time to time, to hear folks asking the professionals what help they could offer.

I love your example about walking up on the accident. That is the perfect analogy for some people’s view points on sharing their opinions on teaching. I appreciate you taking the time to comment.

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I was a private tutor in Maths, from elementary until Master/MBAs, during 17 years. I don’t like teach in class, I like small groups, preferably individuals. And in EVERY person that I had taught, I could noticeably perceive a bad and ill-trained math teacher that this person had the bad fate of having.

Deficiencies in logical thinking, lots of deficiencies in transpose the word problem to a math problem (since text interpretation until being able to find the real proposed problem), you name it. And as a tutor, a crafting job of make their brains talk, not their Pavlov’s conditioning.

My last deceive was my oldest daughter’s AP Statistics teacher. Well, I’m a statistician, scientific initiation in Stochastic Processes. He was wrong, passing the concepts in a wrong way. As a control group, I did one of her homework, and he graded it wrong, and when he was argued about the grading, he refused to comment on it. Her final grade has 69, and she got a 5 in the final AP exam (those applied by the AP company, I don’t know the name). If she didn’t had me (and her father, obviously, he is a mathematician and a Ph.D. in economics), how could she face this kind of bad pressure?

To be a Maths teacher you need first TO KNOW Maths. College level, at least, well studied. In Brazil (our original country) there is this wrong concept that you need to know a lot of Pedagogy, but the content is secondary. This is completely wrong. I have no training in Pedagogy (if you think that having 3 daughters don’t qualify me to this), but all the content I know, and I developed and used and taught and discussed in home and discussed in TAs session at USP and ENCE/IBGE, it made the lives of my students easier, and the usual answers from them were, “Are you sure? Is that so easy? But the teacher makes it seems so complicated! But your way to solve the problem is not in the answers from the teacher, but seems so logical and so reasonable, that I really can understand!”

And sorry by my English, I still think in Portuguese to write….:)

My (brief) two cents based on my experience growing up:

I believe there’s something to be said about the notion of “teaching for the test” being prevalent in schools. It wasn’t until I was in grad school that a professor told me I could get “failing” grades on all the class exams but still get an A for the course. That was a radically different concept to me.

Growing up, it was all about having the “right answer” in an effort to pass a test to move on to the next course / level and (very generally speaking) “succeed” in life. If you don’t pass the regents exam, you don’t move on to the next level; if you don’t pass the SAT you don’t get into college; etc. It was just hammered into our heads that we HAD to have the right answers – not know how to THINK.

I didn’t learn until much later that knowing what / how / why to apply to a given challenge or problem is necessary in this world….because it just wasn’t something that was rewarded in any way at the time.

Again, just my two cents – I’m sure my experience 20 years ago in a NY school doesn’t apply to everyone of every generation in every part of the country.

Sounds like you’re a great teacher!

I understand what you are saying. I’m so thrown when kids are allowed to use calculators on tests. If that’s not cutting out reasoning, I don’t know what is.

I read your article and many of the comments thereafter. I’m fairly sure no one has ‘broken’ anyone. And for my part, you seem to have demonstrated both the right passions and efforts to teach well. I suspect though, looking for a culprit is somewhat of a snipe hunt. As well, if I may, I believe that part of the issue may be found in the pervasive need of Western society to find an absolute solution, a “this is definitely it.” Lately, it seems to be all we can think about. Yet, seldom in any endeavor is this the case. The world operates in grey; yet, we measure the black and white.

My youngest is just getting ready to go off to Uni. She’s finished a grueling two years of IB, seventeen exams, and all the stress and fuss a 17 year old can manage. She has, and will likely be, quite successful. Yet, and even at her age, she recognizes the disparity between right thinking and the right answer; and, she sees how life wants the former and ‘systems’ demand the latter.

Kind regards,

Why? Two people (among others) have delved into the why and have done some really good work on the subject:

John Taylor Ghatto

John Holt

My son uses Math U See by Demme Learning. It is definitely a good curriculum as it is intended to provide enjoyable practice of lesson concepts. It stimulates thinking by presenting concepts in different formats and includes activities suitable for a wide range of learning styles.

As far as the enthusiasm for Common Core, I’ve learned that there is really not much to be excited about. Looks like a catastrophe is on the horizon. I saw this well documented video that describes this weird odyssey into education:

This begs the question, as usual: whose children are these anyway? You can see, in the related videos to this one that a lot of teachers are not buying it and thus, are leaving their occupations by conviction.

Something to ponder and reflect on, given the times we are in. And where we are heading as a country…

Reblogged this on keithm77's Blog.

Boy I need a life long teacher for my math skills. I skipped 6th and 7th grade and I believe that “broke” my brain when I got to high school level mathematics. I missed a lot of the foundation points.

I applaud your determination. Yet, it is not the only quality that is necessary to mend a broken class moment. First, the children are not broken, only the learning moment. Second, the quality of mercy should not strained, paraphrasing the Immoral Bard (Shakespeare). If you see there is a basic failure of understanding, not just rebelling, think of your own difficulties in understanding and try to help one little step at a time. Service, like teaching, requires at least a modicum of humility; that understanding of others and that your way may not have been the only way to get through to them.

I am not berating you, but pointing out the unfortunate snag of the computerized education plan. Not every child thinks in the a+b=c mode of reasoning; many think in concentric circular mode of reasoning. It is your job to help them all to grasp that THEY ARE NOT BROKEN; only they need to learn the lesson.

God bless you and keep trying.

Reblogged this on marietales and commented:

Now THIS is interesting. Math has NEVER been a strong point…Who or What broke our brains? haha

I doubt my response will even get approved because it isn’t what this teacher wants to hear. But i’m a mature woman and have seen the times change, school has changed, concepts are the same however getting to the answer you want is not the same as how a student today works to get the answer.

This is a society of simple and fastest way to get to the answer society. Computers and technology. Your methods for teaching are out of date and if I was in your class, I would want a transfer. Today’s teachers are also broken in my opinion, if they expect kids to read their mind, and not ask questions at all. Chastised for asking is this correct is entirely wrong!

I am currently taking classes in college at my age yes, because technology has changed so much, I want to know more my need to learn has not diminished. But if you ask me did I need to know about the type of Math your teaching then or now I would tell you no. I do not see a need for algebra and have never used it ever in life. I think that there is a place for it among those who want to learn it not those who are forced at 7th grade to grasp a concept of adding letters and not having an answer what the heck is that?

I hate the fact that in college I will need to take a course that will force me to learn something I won’t ever use in my years in future or ever used in my past.

Basic math is all you need for Accounting.

The way this teacher has lashed out and stomped her feet because her students don’t get it does not deserve the good response to such a harsh blog post.

The trouble with kids is not that they are broken as not all are, some are just less enthusiastic about being forced to grasp a concept that they will never use in life more than likely.

A+b=c does not compute. Even a normal calculator wont accept a letter you punch in. There are no letters on a calculator keyboard.

Bashing these kids was uncalled for. Not everyone needs to KNOW what you know, far as I remember we were free to choose what we want to learn. If you go into science field then you can expect to be forced to know a plus b = c. But if these students are in 7th grade or high school they should have choices.

NO one needs to brainwash kids into thinking math is to get answers, because that is what equations of numbers are. Its logic. And calling it a day should have been precisely what it was. They want to be correct because that is what all classes require to have correct response and answers. You saying that its not true is incorrect in my view. You chastise them for giving you feedback or asking questions? That is incorrect. I give you a Fail on teaching and feel sorry for these kids.

And if you don’t approve my response ill just blog it myself, if you post something in anger you should realize there are other sides to the subject.

Talk about power happy Teachers you wonder why kids behave as they do.

Thank you for your comment. I would never not approve a comment due to differing opinions. I would encourage you to read my follow up post where I reflect on the lessons mentioned in this post. I answer a lot of the misconceptions you mention here in that post. I actually think you and I may agree on more than you realize. My goal is for kids to not exit my class feeling like they will never use the math. I try to bring real world and context to ever lesson so that students can find success. I am sorry your educational experience wasn’t the same.

Thank you for the response but why would you count 52 questions and chastise them for asking. Your 2nd attempt should have been your first attempt to work with them. Not to discourage them. The only positive thing is that I feel that you do want to teach just your methods should take into account attitude toward the kids. I know kids are frustrating and teaching 7th grade I would not wish on any teacher. But you do what you have to but I would say take into account that if the kids are asking questions then they are eager to learn. Hope that makes sense.

Yes I totally understand what is you are saying. That is why I wrote the follow up post which is much more detailed about how the lesson went. It is much less about throwing a tantrum and much more about getting the kids to think. I hope you will take a look.

Nice post! I think fear of mistakes may be one factor, and also lack of confidence in math due to the new subject.

Great to see that the students are now focusing more on the thought processes instead of just the answer.