May 12, 2022·edited May 12, 2022Liked by Dan Meyer

I think the unacknowledged trouble with the concern about "tricks" is that the sample state test questions given here are not worth anything more than knowing a trick to get a correct answer easily and go on with your day.

I teach International Baccalaureate curriculum to grades 11 & 12. The course I teach, focuses on Interpretation of results with heavy reliance on use of calculators. For example if you want to know at what time an exponential equation equals a specific value, students are taught to put the equation into Y1 and the value into Y2 to find the intersection. The focus is on understanding what the intersection represents and how to to interpret the coordinates. Students do not have to learn how to solve for an exponent of an exponential equation. Who actually does that by hand in real life anyway!?!?! Instead the need to understand that an intersection of (6.21, 5000) would mean that during the 7th year the function would be greater/less than $5000. Isn't that the same thing that they would find by solving algebraically? Why is one way considered the RIGHT way and the other just a trick?

Using a GDC or Desmos is not "cheating" unless the goal is just for the students just to find an answer without including HOT skills. Which IMO does more disservice to the students than teaching them to solve using technology.

This is pretty interesting . . . I absolutely agree that the idea of graphing and seeing the relationship between the question and the graphs is a big and powerful idea. And that understanding this idea, and how the technology allows you to use that idea to answer the questions relatively easily, is great mathematics.

I also think that the devil is in the details. And the details help me understand who is doing the thinking and making the connections. So if the STUDENTS are figuring out that they can graph and use graphs to answer these questions, and if the STUDENTS are saying that the solution to a system of equations is the intersection of the graphs, then that is a powerful idea. If the teacher is presenting graphing as a shortcut that they can use to get the answer, without having students work to make sense of the reason WHY graphing might help them answer this particular question, then it simply becomes another rule to remember. The key is not the technique, per se, but what knowledge one has to have to know when and how to use the technique.

So I agree with Dan that knowing what graphs mean, and how they relate to solutions and to other representations of functions is a big idea, and that technology then allows us to use this idea to solve a wide range of problems - and even that it allows us to do an end run around less big procedures that people want us to memorize. I also think that the person writing in may have not been talking about big ideas, but about technology being used as a shortcut that forestalls thinking. So it is back to that old idea . . . is the shortcut you are using getting in the way of you thinking about the math, or is it facilitating you thinking about big ideas and connections. I can see how a calculator, just like a mental math strategy, could do either, and it is very much in how they are taught and experienced by students.

A wise educator once said, "You can always add. You can't subtract.”

In Algebra 1 when we teach, say, factoring quadratic expressions initially, there are no connections to graphs yet, but rather, multiplying polynomials. We use an area model for both. Initially, these problems are a “paper / pencil" exercise in equivalence... with symbols.

Later, we realize that quadratic functions exist and have a shape, and that the solutions to equations that we diligently found using our factoring skill can MEAN something in scenarios and show up as x-intercepts on a graph. Students can’t “unknow” this once they know this… and we WANT them to know this…riiiiiiight? Asking test questions that reflect the time of not knowing, and expecting students to default to “paper / pencil” rather than apply their newest knowledge is kinda silly, yes? AND… Desmos is in no way unique here. I’m preaching to the choir when I say one could just as simply press the age-old [ y = ] button on a handheld graphing calculator with a tiny pixelated screen, and *see* equivalence too.

May 14, 2022·edited May 14, 2022Liked by Dan Meyer

Two thoughts: 1.) This method is also possible with a TI-84, it’s just way more of a pain in the a** so fewer students performed the tedious labor of graphing each potential solution. 2.) If the test makers / public don’t view this as a valid demonstration of content knowledge, they simply need to design better questions. See the AP Calculus exam, forced online due to COVID two years ago, which was open to Desmos, notes, and Google and still produced pass rates around 50%.

I must confess, when I first saw Desmos years ago, my gut feeling was that it was "cheating". However, I began to LOVE Desmos as soon as I started using it. My kids are experts at it. A picture is worth a thousand words! It helps them make SENSE of what I am teaching them! Having that visual of a graph transforms math from the abstract to the concrete. Dan, don't you dare be discouraged. What your team is doing is revolutionary! It's the tests that need to change. Keep up the great work! : )

I think that this is an oversimplification of a broader issue. One that has to do with the ineffective nature of state standardized testing in the US. Sure, we can teach kids to take a graphical approach to solving these problems, but I'd implore ya'll in the comments section to really think about how much students are taking away from the experience when they type an equation into desmos and produce a graph that gives them the answer. Dan, you picked a couple of examples where a graphical approach could be an understandable and valid approach. Granted, are they really graphing anything? I see a lot of students that just blindly type a function into the space because that's what they've been told to do. Are the majority of low performing maths students across this country really taking a lesson away from these hacks? I'd argue no, they are not.

I love Desmos. Always have. Have been a big fan since the beginning. There is a huge value in the activities and lessons that are created using the platform. But I've seen a shift in administration and teachers alike that says "Desmos is the way!" and leaves little to no room for actual instruction in mathematics. I'm glad this conversation is being brought up because I'll be the first to acknolwedge that maybe I'm wrong here. I just see a much bigger problem at hand and can't really personally get excited about the fact that Desmos is now available on tests written pre-Desmos that were/are faulty to begin with.

Here's an article that I think should be standard reading when it comes to standardized tests and should really make teachers consider the impact that we're having on our students when we push towards an end goal that is hardly about learning and what we do in the classroom as educators.

Because these *look* like they are questions from my state's end of course exam, I feel like I am in a position to comment... Often when I am teaching using Desmos, students will say things like "isn't this cheating". My response is that we are using a tool, just like any other discipline would, in order to demonstrate knowledge. I have a lot of problem's with my state's standards (we are not a CCSS state) not because they allow for questions like these to be answered using "tricks" but more because they encourage and allow for teaching that is absent of context and relies on more traditional "drill and kill" practice routines rather that developing reasoning and purpose behind the mathematics. Here's hoping the upcoming revision to the standards are better!

When I introduced solving systems to my 8th graders this year, due to my state standards, I had to cover elimination and substitution methods for solving. 10+ years ago, we taught these because it's more efficient than solving by graphing (by hand or with a stone aged TI-83), but with a Chromebook in every hand, my students can graph and explore systems faster and more efficiently than ever. And their understanding of what we are doing is better too. Thanks for addressing this Dan.

This is so interesting. I completely see where teachers are coming from when they talk about "tricks" to do well on state finals. As a teacher who is constantly preparing students to take these exams, I'd like to offer my point of view. I think the issue isn't so much that teachers are teaching "tricks" but rather that these tricks run the risk of delivering the message, "look, just forget about everything that we learned and do the problem just like this!" If powerful teaching and learning are happening constantly, then these "tricks" are not surprising to students at all. Students who are learning at high levels are constantly seeking multiple representations of problems. Expressions or equations can be represented graphically, just like they can also be understood tabularly. So, yeah, it's great to think "equivalent expressions produce equivalent graphs," but let's do this daily as a way to support the algorithms that we learn! (Conic sections come to mind. Converting these equations from general form to standard form is an intricate process. One little, almost meaningless, calculation error can ruin a whole problem. I encourage my students to write out their work on desmos, constantly producing matching graphs to REINFORCE their confidence in their execution of completing the square.)

I absolutely love Desmos. Technology is where we are as a society. Desmos allows for so many options to students and ways to peel apart a problem. When faced with a problem, we automatically "google it" or ask Siri or go on YouTube. Tricks? No. Resources that if you know how to use correctly can help you find solutions to problems. The piece that teachers must be consistent with is helping students work through sticking points when using Desmos, graphing calculator, applets, etc. That's where the true teaching comes in.

GRAPHS ARE GREAT!!! I use that as my "hook" and then teach the "old school" and discuss how the "old school" didn't have tech and isn't this amazing that they were able to use algebra to make the problems simpler. In addition, many students struggle with understanding that the graph is the collection of (x,y) that make the equation true. And we get to go over that again and again every time we use graphs to understand solutions. If the state really wants to assess an algebra technique, then they should create questions to glean that information.

Funny that this should come up just as my Dual Credit College Algebra kids are preparing for their Final. Most "solving equations" questions can be answered by looking at the graph and if they go that route I applaud them.

The main issue long predates Desmos. First article in math ed by Alan Schoenfeld, if I recall correctly, concerns teaching to the test (NY Regents, geometry in this case). I wish the rational function question was rewritten to say "what denominator in this rational fraction makes it equivalent to the quadratic ..." Students need only look at the highest powers in the example and find only one option. And a Desmos user would still need to some intelligent searching.

On top of that, we misuse the tests. A one number summary of a complicated thing like math understanding is never all that good. Summary for students and also summary of schools or programs or curricula. All multidimensional.

## Are We Teaching Tricks or Just Different Math?

edited May 12, 2022I think the unacknowledged trouble with the concern about "tricks" is that the sample state test questions given here are not worth anything more than knowing a trick to get a correct answer easily and go on with your day.

GOAT: Graphs Organize Analytical Thinking

I teach International Baccalaureate curriculum to grades 11 & 12. The course I teach, focuses on Interpretation of results with heavy reliance on use of calculators. For example if you want to know at what time an exponential equation equals a specific value, students are taught to put the equation into Y1 and the value into Y2 to find the intersection. The focus is on understanding what the intersection represents and how to to interpret the coordinates. Students do not have to learn how to solve for an exponent of an exponential equation. Who actually does that by hand in real life anyway!?!?! Instead the need to understand that an intersection of (6.21, 5000) would mean that during the 7th year the function would be greater/less than $5000. Isn't that the same thing that they would find by solving algebraically? Why is one way considered the RIGHT way and the other just a trick?

Using a GDC or Desmos is not "cheating" unless the goal is just for the students just to find an answer without including HOT skills. Which IMO does more disservice to the students than teaching them to solve using technology.

This is pretty interesting . . . I absolutely agree that the idea of graphing and seeing the relationship between the question and the graphs is a big and powerful idea. And that understanding this idea, and how the technology allows you to use that idea to answer the questions relatively easily, is great mathematics.

I also think that the devil is in the details. And the details help me understand who is doing the thinking and making the connections. So if the STUDENTS are figuring out that they can graph and use graphs to answer these questions, and if the STUDENTS are saying that the solution to a system of equations is the intersection of the graphs, then that is a powerful idea. If the teacher is presenting graphing as a shortcut that they can use to get the answer, without having students work to make sense of the reason WHY graphing might help them answer this particular question, then it simply becomes another rule to remember. The key is not the technique, per se, but what knowledge one has to have to know when and how to use the technique.

So I agree with Dan that knowing what graphs mean, and how they relate to solutions and to other representations of functions is a big idea, and that technology then allows us to use this idea to solve a wide range of problems - and even that it allows us to do an end run around less big procedures that people want us to memorize. I also think that the person writing in may have not been talking about big ideas, but about technology being used as a shortcut that forestalls thinking. So it is back to that old idea . . . is the shortcut you are using getting in the way of you thinking about the math, or is it facilitating you thinking about big ideas and connections. I can see how a calculator, just like a mental math strategy, could do either, and it is very much in how they are taught and experienced by students.

A wise educator once said, "You can always add. You can't subtract.”

In Algebra 1 when we teach, say, factoring quadratic expressions initially, there are no connections to graphs yet, but rather, multiplying polynomials. We use an area model for both. Initially, these problems are a “paper / pencil" exercise in equivalence... with symbols.

Later, we realize that quadratic functions exist and have a shape, and that the solutions to equations that we diligently found using our factoring skill can MEAN something in scenarios and show up as x-intercepts on a graph. Students can’t “unknow” this once they know this… and we WANT them to know this…riiiiiiight? Asking test questions that reflect the time of not knowing, and expecting students to default to “paper / pencil” rather than apply their newest knowledge is kinda silly, yes? AND… Desmos is in no way unique here. I’m preaching to the choir when I say one could just as simply press the age-old [ y = ] button on a handheld graphing calculator with a tiny pixelated screen, and *see* equivalence too.

Another wise educator once said, “If a calculator breaks your test, the test was already broken.” Here in Texas, new problem types are on the horizon for our online state tests starting Spring 2023. More preliminary info here: https://tea.texas.gov/sites/default/files/new-question-types-by-grade-and-content.pdf

edited May 14, 2022Two thoughts: 1.) This method is also possible with a TI-84, it’s just way more of a pain in the a** so fewer students performed the tedious labor of graphing each potential solution. 2.) If the test makers / public don’t view this as a valid demonstration of content knowledge, they simply need to design better questions. See the AP Calculus exam, forced online due to COVID two years ago, which was open to Desmos, notes, and Google and still produced pass rates around 50%.

I must confess, when I first saw Desmos years ago, my gut feeling was that it was "cheating". However, I began to LOVE Desmos as soon as I started using it. My kids are experts at it. A picture is worth a thousand words! It helps them make SENSE of what I am teaching them! Having that visual of a graph transforms math from the abstract to the concrete. Dan, don't you dare be discouraged. What your team is doing is revolutionary! It's the tests that need to change. Keep up the great work! : )

😂I hope you and your team at least occasionally cry... with tears of joy for what a service and inspiration Desmos is offering students and teachers.

I think that this is an oversimplification of a broader issue. One that has to do with the ineffective nature of state standardized testing in the US. Sure, we can teach kids to take a graphical approach to solving these problems, but I'd implore ya'll in the comments section to really think about how much students are taking away from the experience when they type an equation into desmos and produce a graph that gives them the answer. Dan, you picked a couple of examples where a graphical approach could be an understandable and valid approach. Granted, are they really graphing anything? I see a lot of students that just blindly type a function into the space because that's what they've been told to do. Are the majority of low performing maths students across this country really taking a lesson away from these hacks? I'd argue no, they are not.

I love Desmos. Always have. Have been a big fan since the beginning. There is a huge value in the activities and lessons that are created using the platform. But I've seen a shift in administration and teachers alike that says "Desmos is the way!" and leaves little to no room for actual instruction in mathematics. I'm glad this conversation is being brought up because I'll be the first to acknolwedge that maybe I'm wrong here. I just see a much bigger problem at hand and can't really personally get excited about the fact that Desmos is now available on tests written pre-Desmos that were/are faulty to begin with.

Here's an article that I think should be standard reading when it comes to standardized tests and should really make teachers consider the impact that we're having on our students when we push towards an end goal that is hardly about learning and what we do in the classroom as educators.

https://www.texasobserver.org/walter-stroup-standardized-testing-pearson/

Because these *look* like they are questions from my state's end of course exam, I feel like I am in a position to comment... Often when I am teaching using Desmos, students will say things like "isn't this cheating". My response is that we are using a tool, just like any other discipline would, in order to demonstrate knowledge. I have a lot of problem's with my state's standards (we are not a CCSS state) not because they allow for questions like these to be answered using "tricks" but more because they encourage and allow for teaching that is absent of context and relies on more traditional "drill and kill" practice routines rather that developing reasoning and purpose behind the mathematics. Here's hoping the upcoming revision to the standards are better!

Thank you. Spot on.

When I introduced solving systems to my 8th graders this year, due to my state standards, I had to cover elimination and substitution methods for solving. 10+ years ago, we taught these because it's more efficient than solving by graphing (by hand or with a stone aged TI-83), but with a Chromebook in every hand, my students can graph and explore systems faster and more efficiently than ever. And their understanding of what we are doing is better too. Thanks for addressing this Dan.

This is so interesting. I completely see where teachers are coming from when they talk about "tricks" to do well on state finals. As a teacher who is constantly preparing students to take these exams, I'd like to offer my point of view. I think the issue isn't so much that teachers are teaching "tricks" but rather that these tricks run the risk of delivering the message, "look, just forget about everything that we learned and do the problem just like this!" If powerful teaching and learning are happening constantly, then these "tricks" are not surprising to students at all. Students who are learning at high levels are constantly seeking multiple representations of problems. Expressions or equations can be represented graphically, just like they can also be understood tabularly. So, yeah, it's great to think "equivalent expressions produce equivalent graphs," but let's do this daily as a way to support the algorithms that we learn! (Conic sections come to mind. Converting these equations from general form to standard form is an intricate process. One little, almost meaningless, calculation error can ruin a whole problem. I encourage my students to write out their work on desmos, constantly producing matching graphs to REINFORCE their confidence in their execution of completing the square.)

I absolutely love Desmos. Technology is where we are as a society. Desmos allows for so many options to students and ways to peel apart a problem. When faced with a problem, we automatically "google it" or ask Siri or go on YouTube. Tricks? No. Resources that if you know how to use correctly can help you find solutions to problems. The piece that teachers must be consistent with is helping students work through sticking points when using Desmos, graphing calculator, applets, etc. That's where the true teaching comes in.

GRAPHS ARE GREAT!!! I use that as my "hook" and then teach the "old school" and discuss how the "old school" didn't have tech and isn't this amazing that they were able to use algebra to make the problems simpler. In addition, many students struggle with understanding that the graph is the collection of (x,y) that make the equation true. And we get to go over that again and again every time we use graphs to understand solutions. If the state really wants to assess an algebra technique, then they should create questions to glean that information.

Funny that this should come up just as my Dual Credit College Algebra kids are preparing for their Final. Most "solving equations" questions can be answered by looking at the graph and if they go that route I applaud them.

The main issue long predates Desmos. First article in math ed by Alan Schoenfeld, if I recall correctly, concerns teaching to the test (NY Regents, geometry in this case). I wish the rational function question was rewritten to say "what denominator in this rational fraction makes it equivalent to the quadratic ..." Students need only look at the highest powers in the example and find only one option. And a Desmos user would still need to some intelligent searching.

On top of that, we misuse the tests. A one number summary of a complicated thing like math understanding is never all that good. Summary for students and also summary of schools or programs or curricula. All multidimensional.