Teaching for understanding and teaching procedures

Many critics of the current state of mathematics education take issue with asking students to explain their reasoning. They’d rather students just apply an algorithm and get the answer.

The following is quoted from QED: The Strange Theory of Light and Matter, where Richard Feynman describes how he’s going to explain for a lay audience the techniques behind quantum mechanics that earned him a Nobel Prize. (By the way, I highly recommend this book.)

How am I going to explain to you the things I don’t explain to my students until they are third-year graduate students? Let me explain it by analogy.

The Maya Indians were interested in the rising and setting of Venus as a morning “star” and as an evening “star” – they were very interested in when it would appear. After some years of observation, they noted that five cycles of Venus were very nearly equal to eight of their “nominal years” of 365 days (they were aware that the true year of seasons was different and they made calculations of that also). To make calculations, the Maya had invented a system of bars and dots to represent numbers (including zero), and had rules by which to calculate and predict not only the risings and settings of Venus, but other celestial phenomena, such as lunar eclipses.

In those days, only a few Maya priests could do such elaborate calculations. Now, suppose we were to ask one of them how to do just one step in the process of predicting when Venus will next rise as a morning star – subtracting two numbers. And let’s assume that, unlike today, we had not gone to school and did not know how to subtract. How would the priest explain to us what subtraction is?

He could either teach us the numbers represented by the bars and dots and the rules for “subtracting” them, or he could tell us what he was really doing: “Suppose we want to subtract 236 from 584. First, count out 584 beans and put them in a pot. Then take out 236 beans and put them to one side. Finally, count the beans left in the pot. That number is the result of subtracting 236 from 584.”

You might say, “My Quetzalcoatl! What tedium, counting beans, putting them in, taking them out – what a job!”

To which the priest would reply, “That’s why we have the rules for the bars and dots. The rules are tricky, but they are a much more efficient way of getting the answer than by counting beans. The important thing is, it makes no difference as far as the answer is concerned: we can predict the appearance of Venus by counting beans (which is slow, but easy to understand) or by using the tricky rules (which is much faster, but you must spend years in school to learn them).”

To understand how subtraction works – as long as you don’t have to actually carry it out – is really not so difficult.

That’s my position: I’m going to explain to you what the physicists are doing when they are predicting how Nature will behave, but I’m not going to teach you any tricks so you can do it efficiently. You will discover that in order to make any reasonable predictions with this new scheme of quantum electrodynamics, you would have to make an awful lot of little arrows on a piece of paper. It takes seven years – four undergraduate and three graduate to train our physics students to do that in a tricky, efficient way. That’s where we are going to skip seven years of education in physics: By explaining quantum electrodynamics to you in terms of what we are really doing, I hope you will be able to understand it better than do some of the students!

In the same way, I want students in 2nd and 3rd grades to understand what they are really doing when they subtract, and not just mindlessly follow a procedure to get an answer that they do not really understand.

Where I tend to agree with most critics of the Common Core is that students are asked to write miniature essays to explain their reasoning, and that’s probably a bad idea. Even though I want students to understand why subtraction works, 2nd and 3rd graders are still learning how to write complete sentences and can get easily frustrated with explaining their reasoning in paragraph form. I think there are better ways (like drawing pictures) of assessing whether young children really understand subtraction that is more developmentally appropriate.

Reflections by a teacher on the Common Core

The implementation of the Common Core has left a lot to be desired, but it’s heartening to see that some teachers have embraced what the Common Core attempts to accomplish. I saw the following first-person person referenced in the Washington Post; the original post can be found at http://www.youngedprofessionals.org/1/post/2014/03/is-the-common-core-working-in-the-classroom.html.

The Common Core State Standards are a reality now for teachers in Maryland and DC, while Virginia is one of six states to omit the standards from their state education approach. YEP-DC asked local educators how the Common Core is playing out in their classroom. Are the standards increasing student understanding or presenting obstacles? What’s changed in pedagogical approach, and how are students are reacting to the shift?

Meredith Rosenberg, fourth-grade teacher

Compare 1/4 and 5/6. This seemingly simple problem is a no-brainer for adults. We know right away that 5/6 is greater than 1/4. But where do you begin with a student who has no conceptual understanding of what a fraction is?

One of the most defining features of the Common Core is how it introduces concepts to students through different modes of comprehension. By the end of a six-week Common Core unit on fractions, my students were talking about, writing about, drawing, and playing with fractions. When they encountered the above problem on a quiz, some students drew a picture, while others found common denominators. A few used a strategy called common numerators, which requires a deep understanding of the denominator of a fraction. One student drew the fractions on a number line. The takeaway: The students in my class were able to compare these fractions in no fewer than five different ways.

The Common Core implementation is not without its challenges. Many standards are vague, and there are only small bits of information coming from the Partnership for the Assessment of Readiness in College and Career (PARCC) on how they are to be tested. The inconsistency with which the standards have been implemented result in the need for highly differentiated classrooms. For example, some of my students came into fourth grade with a solid conceptual understanding of fractions, while others from other schools had no idea what a fraction meant.

However, my school has prioritized Common Core implementation and tackled its challenges with consistent professional development, regular refinement of unit plans, daily lessons and assessments, and an intense focus on the Standards for Mathematical Practice. As a result, my students are thinking critically about numbers every day, and they are becoming accustomed to attacking problems with multiple strategies and assessing the validity of those strategies. The Common Core standards choose depth over breadth, and with appropriate teacher development and support, this leads to much more critical thinking and analysis in the classroom.

The Failure of Test-Based Accountability

From Marc Tucker’s blog on Education Week:

In my last blog, I pointed to the data that shows that, after 10 years of federal education policies based on test-based accountability, there has been no perceptible improvement in student performance among high school students (which, when you get right down to it, is what really matters) as a whole, or when the data are broken down by different groupings of disadvantaged students. There is little doubt—whether test-based accountability is being used to hold schools accountable or individual teachers—that it has failed to improve student performance.

That should be reason enough to abandon it. But it is not. The damage that test-based accountability has done goes far deeper than a missed opportunity to improve student achievement. It is doing untold damage to the profession of teaching…

Test-based accountability and teacher evaluation systems are not neutral in their effect. It is not simply that they fail to improve student performance. Their pernicious effect is to create an environment that could not be better calculated to drive the best practitioners out of teaching and to prevent the most promising young people from entering it. If we want broad improvement in student performance and we want to close the gap between disadvantaged students and the majority of our students, then we will abandon test-based accountability and teacher evaluation as key drivers of our education reform program.

But no one, certainly not me, would argue that we should not hold our professional educators accountable for their performance. The question is, what would accountability look like if we actually regarded our teachers as professionals doing professional work, instead of interchangeable blue-collar workers doing blue-collar work? That is the question I will deal with in my next blog.

I encourage you to read the whole thing: http://blogs.edweek.org/edweek/top_performers/2014/02/the_failure_of_test-based_accountability.html

The following video made the rounds a few months ago and ties in with the above point. It is less about the shortcomings of the Common Core than our leaders’ fixation with quantifying educational output. As the speaker says well, “If everything I learned in high school is a measurable objective, then I have not learned anything.”

Thoughts on the Common Core and its implementation

The following picture appeared on the Facebook page of Daniel Bongino, who is running for Congress in Maryland.

Source: https://scontent-a-dfw.xx.fbcdn.net/hphotos-prn1/t1/1184774_620433314716100_343011500_n.jpg

Here was his commentary on this picture:

Like many of you, I am a parent who is passionate about my child’s education in an increasingly competitive and unforgiving global economy.

Having stated that, I cannot condemn the Common Core in strong enough terms. Look at the picture I have attached to this post. I gave my daughter a relatively easy long-division problem to do today, in an attempt to gauge her progress, and this is what she gave back to me.

This is completely unacceptable. How is it that we are replacing a time-tested, efficient method of long-division with an absurd, multi-step process that not only confuses the students, but the parents too?

Compounding the Common Core disaster is the fact that in my daughter’s last school year she was taught the older, more effective method of long-division and is now completely confused.

Friends, all politics are local and it gets no more local than your kitchen table. Fight back against the Common Core, and do it quickly, by calling and emailing your local, state, and federal elected officials.

This is not a partisan issue. Your child’s education is suffering whether you are a Democrat or a Republican. Every second we lose is another second our kids are being exposed to a third-rate curriculum in a first-world economy. Count on me as an ally in this fight.
-Dan

Source: https://www.facebook.com/dan.bongino/photos/a.517057181720381.1073741827.101043269988443/620433314716100/?type=1&theater

Yesterday, I discussed the mathematical logic behind this unorthodox approach to subtraction. Today, I want to briefly talk about the Common Core standards for mathematics and their implementation, as this is a topic that I’ve been following for several years.

To the mindless critics who think that America is headed to communism because of the Common Core: there’s no point having a rational discussion about this. Michael Gerson is one of many conservative commentators who is not ideologically opposed to the Common Core; see http://www.washingtonpost.com/opinions/michael-gerson-gop-fear-of-common-core-education-standards-unfounded/2013/05/20/9db19a94-c177-11e2-8bd8-2788030e6b44_story.html.
Also to the mindless critics: while Texas (where I live) is not a Common Core state, the standards for mathematics that we’ve had for the past 10 years or so align fairly well with the Common Core. And Texas is about as far away from a blue state as any of the 50.
To the thoughtful critics who are worried about the appropriateness of the Common Core standards: as I said, while not in perfect alignment, for the last few years Texas has had content and process standards for mathematics education that are decently close to those stipulated by the Common Core. I’m more than happy to declare that the implementation of the Common Core has been thoroughly botched from sea to shining sea. Still, I believe that a good implementation is possible, and I hope that you don’t throw out the baby with the bath water when critiquing the potential of the Common Core standards.
To the supporters of the Common Core standards: you better read Diane Ravitch’s thoughtful critique of how the standards have been rolled out: http://dianeravitch.net/2013/02/26/why-i-cannot-support-the-common-core-standards/. It seems to me that textbook publishers are driving the rollout of the Common Core, and educators are desperately trying to shift from the previous standards to the new standards while also trying to figure how they are being required to teach because of the textbook… and not because of the standards themselves.
Also to the supporters of the Common Core standards: voters — and, more importantly, parents — will not tolerate these standards if a rationale for these standards are not carefully explained. I do think that most parents do care about the mathematical education of their children and will rationally discuss cutting-edge ways of teaching mathematics, but they have to be convinced that these cutting edge methods actually make sense. The rollout of the Common Core will be studied in public-relation circles for years to come for how *not* to make drastic changes.
And though they are not specifically required by the Common Core, don’t get me started on the hours we’re wasting high-stakes testing, an intellectually lazy and ineffective way of measuring teacher quality.

Thoughts on unorthodox ways of teaching long division

The following picture appeared on the Facebook page of Daniel Bongino, who is running for Congress in Maryland.

Source: https://scontent-a-dfw.xx.fbcdn.net/hphotos-prn1/t1/1184774_620433314716100_343011500_n.jpg

Here was his commentary on this picture:

Like many of you, I am a parent who is passionate about my child’s education in an increasingly competitive and unforgiving global economy.

Having stated that, I cannot condemn the Common Core in strong enough terms. Look at the picture I have attached to this post. I gave my daughter a relatively easy long-division problem to do today, in an attempt to gauge her progress, and this is what she gave back to me.

This is completely unacceptable. How is it that we are replacing a time-tested, efficient method of long-division with an absurd, multi-step process that not only confuses the students, but the parents too?

Compounding the Common Core disaster is the fact that in my daughter’s last school year she was taught the older, more effective method of long-division and is now completely confused.

Friends, all politics are local and it gets no more local than your kitchen table. Fight back against the Common Core, and do it quickly, by calling and emailing your local, state, and federal elected officials.

This is not a partisan issue. Your child’s education is suffering whether you are a Democrat or a Republican. Every second we lose is another second our kids are being exposed to a third-rate curriculum in a first-world economy. Count on me as an ally in this fight.
-Dan

Source: https://www.facebook.com/dan.bongino/photos/a.517057181720381.1073741827.101043269988443/620433314716100/?type=1&theater

This picture was shared by a friend on Facebook; the resulting discussion follows. I’m sharing this because I think the following reactions are typical of parents when their children are taught mathematics using non-traditional methods.

While I don’t think that any of the commentators said anything personally embarrassing, I’m withholding the actual names of the correspondents for the sake of anonymity.

Anonymous #1: What in the world is this?

Me: In the worst case scenario, it’s a waste of time for children who already know how to divide.

In the best case scenario, it’s an effective and pedagogically reasonable first step — for children who don’t yet know how to divide. (FYI, this technique has been used long before the advent of the Common Core.

Here’s the justification: Young children often have a hard time coming up with the “best” first step that 43 divided by 8 is 5 with remainder 3. However, they often can come up with a reasonable first step, whether it’s subtracting off 10 groups of 8 or 40 groups of 8. The important thing is that they’re reducing 432 by a multiple of 80, not necessarily the “best” or “optimal” multiple of 80. With practice, children hopefully get better at guessing the optimal multiple of 80, thus leading to the traditional method of long division.

The idea is that the children can, with time, figure out the reason why long division works, rather than mindlessly following an algorithm that leads them to an answer that they don’t understand.

Anonymous #2: It’s the longest division problem ever. Lol

Anonymous #3: OMG John, that answer was more confusing than the picture!! LOL just kidding! What I want to see from that picture is, did she eventually get the answer right? Did she give up? If the kids learn how to get a right answer, I’m hard pressed to find a valid argument against any teaching method. If it frustrates them to the point that they give up, well then that is a problem. That picture he posted doesn’t give us any real information. It just makes us old farts think “what the hell??” Because it’s so different from what we learned.

Maybe it isn’t pulling up right, but I don’t see an answer in that picture. Is because she couldn’t do it or because he just wanted to post the weird method to promote fear of something new?

My daughter was taught the “lattice” way to do 3 digit multiplication. I wanted to cry trying to figure that out. But it made sense to her and she got the answers right.

But, I will admit that it looks crazy to me, too!

Me: I agree that the person who posted the picture did not (deliberately?) show if the student ultimately got the right answer. I can say that the partial steps that are shown are correct.

I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

I’ll also say this: unorthodox teaching practices usually go over better when both the practices and the rationale for the practices are clearly explained to parents. Sadly, while a lot of thought has gone into improving mathematics education, not much thought has gone into justifying these new practices to parents, and that’s a shame.

Anonymous #2: The problem isn’t teaching the method. I’m all for showing kids multiple ways to do things. The problem is forcing all kids to use this method. We are all different and therefore we all think differently. If it makes sense this way to you great however if it doesn’t make sense then why not let kids use the way that works for them. Yes teaching different methods is great but forcing kids to use methods they don’t understand is foolish.

Me: No argument from me.

Anonymous #4: I am troubled by this and other styles of math that no longer require children to learn and memorize simple mathematical tables of simple addition, subtraction, multiplication, and division. It disappoints me to no end that people allow children to avoid learning thoroughly these tables, as though they are not necessary in life. I am appalled here that kids are encouraged as early as 3rd grade to start using a calculator for basic math!

I appreciate different styles of doing math here, Subtraction and Division are quite different in (European Country) than in America. But sometimes it just seems that so many new methods are obscure attempts to help an overly super small subset of kids which are then exposed to them, and at times, forced on them; much to the chagrin of parents.

Me: (Anonymous #4), I agree about the importance of children memorizing mathematical tables at a young age. I disagree that this particular algorithm — unorthodox long division — necessarily tells children that such memorization isn’t particular useful.

My own daughter struggled with long division when she first learned it. She already knew that 36 divided by 4 was 9 and hence knew the “right” step when computing 368 divided by 4. However, when the problem changed to something like dividing 384 by 4, she had difficult with the first step, as she didn’t have anything memorized for “38 divided by 4.”

My friends who are elementary teachers tell me that this particular conceptual barrier is fairly common when children first learn long division.

For 384 divided by 4, the best first step is subtracting 90 groups of 4 from 384, but she was having trouble immediately coming up with the largest multiple of 10 that would work. However, subtracting *any* multiple makes progress toward the solution, even if it isn’t necessarily the “best” step for solving the problem as fast as possible.

In those early stages of her learning, she computed 384 divided by 4 using suboptimal steps. I can’t remember exactly how she did it, but a reconstruction from memory is shown in the attached picture. She knew that 50 times 4 was less than 384, so it was “safe” to subtract 200. When she did this, I didn’t correct her by telling her that she should have subtracted 90 groups of 4. Instead, I let her make this step (emphasis, step — and not mistake) and let her proceed.

The step that always surprised me was when she’d occasionally subtract 12 groups of 4… she had memorized her multiplication table up to 12 and instinctively knew that subtracting 12 groups of 4 brought her closer to the correct answer than subtracting 10 groups of 4.

Obviously, as she got better at long division, she made fewer and fewer suboptimal steps when dividing. That’s the beauty of this unorthodox method… children don’t have to stress so much about making the best next move, as any next move will bring them closer to the answer. Hopefully, with practice, children get better at making the best moves quicker, but that’s a skill that they develop as they get used to long-division algorithm.

Me: One more thought: (Anonymous #1), I’m sorry if I’ve completely commandeered your original post! 🙂

Anonymous #1: John you crack me up! I have never had such lengthy discussion about anything I have ever posted! I still have NO idea how to do all these extra steps-but I know who I will be asking for help when the time comes for me to deviate from my old school method of math!

The hot hand, or Bayesian updating in basketball

In time for the NCAA basketball tournament, here’s a nice article about the myth of the “hot hand” in basketball: http://www.waterfromtheeyes.com/2012/07/the-hot-hand-or-bayesian-updating-in.html

For more details, see http://www.readcube.com/articles/10.1038%2Fncomms1580.

Is there an easy function without an easy Taylor series expansion?

After class one day, a student approached me with an interesting question:

Is there an easy function without an easy Taylor expansion?

This question really struck me for several reasons.

Most functions do not have an easy Taylor (or Maclaurin) expansion. After all, the formula for a Taylor expansion involves the $n$ th derivative of the original function, and higher-order derivatives usually get progressively messier with each successive differentiation.
Most of the series expansions that are taught in Calculus II arise from functions that somehow violate the above rule, like $f(x) = \sin x$ , $f(x) = \cos x$ , $f(x) = e^x$ , and $f(x) = 1/(1-x)$ .
Therefore, this student was under the misconception that most easy functions have easy Taylor expansions, while in reality most functions do not.

It took me a moment to answer his question, but I answered with $f(x) = tan x$ . Successively using the Quotient Rule makes the derivatives of $tan x$ messier and messier, but $tan x$ definitely qualifies as an easy function that most students have seen since high school. It turns out that the Taylor expansion of $f(x) = \sin x$ can be written as an infinite series using the Bernoulli numbers, but that’s a concept that most calculus students haven’t seen yet.

Earlier posts on Taylor series:

https://meangreenmath.com/2013/07/01/reminding-students-about-taylor-series-part-1/

https://meangreenmath.com/2013/07/02/reminding-students-about-taylor-series-part-2/

https://meangreenmath.com/2013/07/03/giving-students-a-refresher-about-taylor-series-part-3/

https://meangreenmath.com/2013/07/04/giving-students-a-refresher-about-taylor-series-part-4/

https://meangreenmath.com/2013/07/05/reminding-students-about-taylor-series-part-5/

https://meangreenmath.com/2013/07/06/reminding-students-about-taylor-series-part-6/

https://meangreenmath.com/2013/07/24/taylor-series-without-calculus-2/

Engaging students: Inverse Functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Brittney McCash. Her topic, from Algebra II: multiplying binomials.

C3. How has this topic appeared in the news.

For the engagement on this aspect of my topic, I would bring a binomial cube with me. I would pose the question, “What do we do when we multiply two binomials together?” The students, of course would not know the answer. I would then say, “Well let’s what one man did that they even did a news article about him!” This in itself catches the students attention because they are piqued about what exactly I am talking about. I would then pass out a copy of this news article so that the students could read. After popcorn reading out loud, we would discuss the article and about how we could use the binomial cube. I would then take out my cube (If possible, put students in groups and give each group a binomial cube to work with) and ask the students, “How in the world did he use this cube to multiply those binomials (points to equation on board)?” I would give them the hint that they have to add up the sides of the square and solve for the perimeter, and see what they can come up with. This is a great engagement for the kids because not only is it hands on, but the article brings in outside aspects of what they’re learning so that they realize they are not the only ones having to learn the material. It’s also a great way to introduce multiplying binomials because it starts at the beginning of adding variables (which they already know how to do), and it’s a visual representation of concept that is sometimes hard to grasp. It’s also a great way to lead into the FOIL, Box, etc…methods to take it into a deeper explanation. For those that have not heard of the binomial cube, here are some pictures of what the students will be working out.

ARTICLE: News Article about Binomial Cube

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B2. How does this topic extend what your students should have learned in previous courses?

A great way to start off with this engagement would be to take the students back to sixth grade. Start off with asking students, “Who remembers when we had to learn how to add and subtract fractions?” Most, if not all, of the students should raise their hands. You can then ask, “Okay, good. So does anyone remember what the next step was after we learned how to add and subtract fractions? What did we learn how to do next?” The answer I am looking for here is multiplying and dividing. After that is established, you can lead in with, “Okay, so who can tell me what the next step would be with what we have previously been learning (adding and subtracting binomials)?” The answer is multiplication and division. Make sure to let them know that you will only be focusing on the multiplication aspect for now. Then you can pose some questions like, “What does multiplying binomials look like? How do we do it? Is there more than one way?” You can then go into a deeper exploration of multiplying binomials and the different ways you can do so. This is a good way to introduce multiplying binomials because not only did I bring in one concept students were already familiar with, I brought in two. I utilized something they already knew (even if subconsciously) back in middle school, and applied that same order to something more complex. It showed them that there was a purpose for learning what they did, and why there is a reason we go in the order that we do. Then you have the aspect of taking something they had been previously working on this semester and extending it further. This helps the students connect with what they are learning and realizing there is a purpose. Because multiplication is repeated addition, we are taking something they have previously learned, and extending it further. Another reason this is a good plan is because you start off with such a basic question, that every student knows the answer. This allows for immediate attention because all the students know what you are talking about, the more they understand, the more likely they are to participate in classroom discussion.

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E1. How can technology be used to effectively engage students with this topic?

In multiplying binomials, technology is a wonderful thing. It can allow students the opportunity to learn in new and interesting ways. When thinking of an engage for this topic, I thought of the 9^th grade Algebra 1 class I am currently teaching. High School students are sometimes the hardest to keep entertained, and I think I found the perfect video to help keep there attention. This video is a group of students who did a rap about the FOIL method. What better way to relate to students then students themselves! I would start class off by telling the class, “Today we are going to start of by watching a fun video over something we will be learning today.” Proceed to play the video, and observe how every student is watching. The video is fun while also informing. It describes the method, though not thoroughly, but it gives the students an idea of what will be coming. This video helps show that other students all over the state/world are learning the same thing, and are bringing a fun new aspect to the learning of the material. After the video is played, you might ask the class to try and guess at what exactly you will be covering today. It’s always good to see their minds work and try to figure it out. This question also allows them to connect the video back to the classroom environment and settle down. You can then begin your lesson on multiplying binomials. At the end of the lesson, I would bring up the video again, and ask the class if they can recall what FOIL stands for and to give me an example. I would probably make this their exit ticket for the day and have them write it down on a piece of paper. (This video runs a little long, and I would recommend editing some parts out for time sake. )

Resources:

http://www.youtube.com/watch?v=MG-c7NWFS8U

http://www.noozhawk.com/article/santa_barbara_montessori_school_open_house_binomial_cube_20140118

http://montessorimuddle.org/2012/02/02/using-the-binomial-cube-in-algebra/

Large Hadron Rap

I thought something on the lighter side would be appropriate today… like rapping about the Large Hadron Collider at CERN, where the Higgs boson was (tentatively) discovered.

Engaging students: Inverse Functions

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Allison Myers. Her topic, from Algebra II: inverse functions.

CURRICULUM

How can this topic be used in your students’ future courses in mathematics or science?

Functions are a composition of one or more actions that maps one object onto another (each input maps to one output). Inverse functions are a composition of reverse actions that “undo” the actions of the original function.

Inverse functions have real-world applications, but also students will use this concept in future math classes such as Pre-Calculus, where students will find inverse trigonometric functions. Inverse trigonometric functions have a whole new set of real-world applications, such as finding the angle of elevation of the sun, or anything which models harmonic motion.

Students will also see this concept again in Calculus, where they will differentiate inverse trigonometric functions to solve real-world applications involving rate of angular rotation or the rate of change of angular size.

TECHNOLOGY

How can technology be used to effectively engage students with this topic?

In the past, I taught a lesson where the Explore portion of the lesson utilized dry erase markers and transparency sheets to allow students to discover what happens graphically when computing an inverse (trigonometric) function. My goal was for my students to understand why we compute inverses the way we do. To my horror, my theoretical 15-minute, super insightful Explore became messy, full of problems, and confusing to my students.

While reflecting after the lesson, I began to consider how using technology would have better served my students (in their understanding) and myself (in my goals for the lesson). I found Glencoe’s directions for using the TI-Nspire to compute inverse functions (see image below). Using the TI-Nspire, I would start the lesson with a real-world example and data and have my students complete Step 1. Next, I would explain our need to “undo/reverse” the data, and allow the students to come up with different ways to do so. After that, I would ask the students to make conjectures about possible formulas. Using the TI-Nspire would be less messy and time-consuming (as compared to my experience with markers and transparencies), and would also allow the teacher to be within the context of a real-world problem. I believe if we used this (or similar) technology, combined with the constructivist-style teaching, students would come away with not only a better understanding for computing inverse functions but also their real-world applications.

Source: http://glencoe.com/sites/common_assets/mathematics/alg2_2010/other_cal_keystrokes/TI-Nspire/Nspire_423_424_C07L2B_888482.pdf

CULTURE

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Inverse functions are used every day in real life. For example, when a computer reads a number you type in, it converts the number to binary for internal storage, then it prints the number out again onto the screen that you see – it’s utilizing an inverse function. A basic example involves converting temperature from Fahrenheit to Celsius.

Another example, if one considers music notes on paper to be a function of the sound produced, then the software Sibelius can be considered the inverse function, as it takes a musician’s music and converts it back to music notes.