Comments from other sites: Four animated gifs of the same awesome problem (dy/dan)

CFOS: Comments from other sites: I do a pretty good amount of writing online about math and math ed… just its not always here. In these posts I will link to the other site and copy over my comment with some context if needed. Mostly this is for me, so that I have a journal of what my thoughts were kept in one place.


Today’s CFOS is from Dan Meyer’s blog regarding the wonderful and somewhat famous task posed by Malcom Swan.  Dan writes,

Here is the original Malcolm Swan task, which I love:

Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc? Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.

We could talk about adding a context here, but a change of that magnitude would prevent a precise conversation about pedagogy. It’d be like comparing tigers to penguins. We’d learn some high-level differences between quadripeds and bipeds, but comparing tigers to lions, jaguars, and cheetahs gets us down into the details. That’s where I’d like to be with this discussion.

So look at these four representations of the task. What features of the math do they reveal and conceal? What are their advantages and disadvantages?

paper

processing (code from Dan Anderson)

 

Geogebra by Scott Farrar

Desmos activity by Dan’s twitter followers

 


And my comments:

I’d love a combination of all of the above.

It appears that the Desmos activity had a drawback of no error checking. (Or, to call back your talk at CMC Asilomar, if this were a video game it would ask you to try again if your perim/area were impossible) In a customized Desmos applet, could you ask for width and height during the brainstorm rectangle phase, and then the second screen asks for perim and area? (couple more options here… you could still accept perim and area that do not match the given width, height: just flag them separately and then you can filter them out of the overlay or include them depending on where you want to take the lesson next)

It also appears that someone was investigating rectangles of a certain type, thus creating a rather straight line. Nice conversation starter there…

I think I am partial to the ability of my geogebra one to have the non-numerical entry point. On the other hand, they aren’t directly improving their skills in that arena either. A student is not hindered by calculation, but neither are they supported in it. But… I have liked giving a task to students where their interaction is manual and analog– meaning they are digging into the constraints and boundaries of a mathematical concept by physical motion and visual feedback. Does a student become immediately aware of and curious about this region that they cannot seem to get the dots into?

The processing one I like as a contrast to the hand-implemented geogebra. How would we systematically experiment around this idea? And when we devise a systematic approach, computers can be helpful in executing it. We can see the density of the dots is not uniform — what is the interaction between this perim v. area relationship and the way we have implemented our experiment? (I would guess the rectangle dimensions are chosen at random, thus creating a would-be-uniform display if not for the fundamental perimeter-area idea.)

The hand graph is limited in iteration, prone to calculation error, and slow. However, its requirements are all internal (save paper and pencil) rather than computer-based. Graphing by hand is perhaps the handle onto all of these other ideas. Is there research around how the concept of cartesian graphing changes if we grow up experiencing it in the computerized sense without the pencil and paper sense? I wonder!

For now I’d love to mix all of these in the classroom. When I spoke about this problem at Asilomar 2013, I used Malcom Swan’s initial prompts and provided grid paper for the attendees to grapple with the problem. Only afterwards did I move to the geogebra version (2013 version here, that has a slightly different scope)
And I would start the same way with students now. Here’s a rough timeline for a 1-2 day lesson:

1. paper rectangle in groups. Make a Perim v. Area graph on paper in your group that has your four rectangles. (this may correct some errors right away, like if kids put width and height instead of perim/area, their group may correct them)
2. Groups share on Desmos, add more rectangles. They are released from having to graph them on paper now. We are letting that go because Desmos will pick it up. Thus allowing students to step up their conceptual / abstracted approach.
3. class discussion, noticings, wonderings, address errors… blank region may become clearly curious here. Ok, now we’ve used the Desmos tool, but if we want to dig in with more rectangles, lets switch tools.
4. Geogebra sketch and/or programmed iteration. Groups come up with things to try and experiment with on the geogebra sketch, or devise a script that would generate perims and areas and plot them like Dan Anderson’s.
5. Return of the single case. Which points are on the edge? Can investigate with Geogebra sketch by hand. Or: can hand the class a mission on Desmos: “come up with rectangles that you think are on the edge”.
6. depending on the level of the class you can lead this towards some kind of proof or justification, entice students to make reasoned conjectures about what is going on.

Now we’ve taken the stepping stones from drawing a single rectangle to an aggregate set of many rectangles and their (p,a), to a set larger than what we could human-ly create. When we’re on step 4 there, we’re getting a handle on the abstraction of perimeters and areas of rectangles, thus allowing us to experiment with the abstraction just as easily as we worked with a single rectangle on paper at the beginning of class. But then we can go back and use each tool for what it helps with.

That, I believe, is the power of education technology: giving concrete handholds to abstract concepts. But tech is at its most powerful when working in concert with non-tech methods. And the possibilities of tech are so wide that no single tool was the magic bullet that enables this lesson– each one exposed a new vector to take.

Finally, remember Swan’s prompts (that did not aim to use technology) originally got at the existence of this concept by suggesting an impossible point and then suggesting that there are many others to find. His prompt defines these two regions (that we might call white and red for our purposes) with two examples: one red and one white. I wonder what the difference is about investigating the existence of white and red by imagining and reasoning on paper vs. bouncing up against their boundary on the geogebra applet.

 

Reddit Selection: Motivating geometry students to do proofs

In these posts I will share some of my reddit comments that I feel are worthy of saving.


You ask a good question, “How do you teach the act of proving without showing proofs?” In short, I don’t think you should not show them– but I don’t encourage memorization of them, and you should show more “proto-arguments” early in the year, becoming more and more formal as the year goes on.

I think my top level reply answers a little bit of my thoughts on the matter:https://www.reddit.com/r/matheducation/comments/3nhn28/motivating_geometry_students_to_do_proofs/cvon7ffbut I’ll say more here…

So first, I’m definitely NOT saying get rid of the theory. Proofs are the most important part of a typical geometry course. The course should be all about justification. But, I take issue with the development of proof in many textbooks– they often leap into it as either chapter 1 or 2 and say, “ok got it? good now the rest of the chapters will just ask for harder or longer proofs”

Consider this analogy. You’re taking a class in how to build a car. It would be silly for the class to have you first build a fully functional toy car, then a fully functional half size car, and finally a fully functional full size car. The size of the car is irrelevant to how difficult it is to create it. If we don’t know how to prove, then doing any proof is difficult.

So we should back up and think: what are the interior parts towards proof? i.e. what are the wheels, the engine, the gears?

I argue some of those internal parts are conjecture, precision of description, representing an idea in a format other than what it started as.

So consider something like the huge timewaster traditional vertical angles proof (see https://www.youtube.com/watch?t=201&v=wRBMmiNHQaE for how to turn a simple idea into 4:51 of boredom).

Instead of showing that proof or anything like it, ask students to draw two lines intersecting at any angle. Have them notice and wonder ( https://www.youtube.com/watch?v=a-Fth6sOaRA ) about what they see. You WILL have a kid claim that there are angles that are always equal– if you wait long enough. And then you can pounce, “hey, Bobby says that the angles he calls ‘across from each other’ are always equal, who agrees?” Use the vocab that the students came up with– or if its unclear, ask for clarification… ask other students if they can help clear up Bobby’s naming… reach a class consensus.

 

 

 

What you’re doing there is building the pre-requisite skills needed for proof. The students first need to be able to talk about their mathematical ideas with some level of precision. Many of them have never been asked to try (and then how are they going to prove anything?) As they get better at making mathematical statements and asking mathematical questions, they get better about linking answers together, and formalizing their informal reasons for belief in certain truths.

As you do these, you can also provide tools to students like examining how to draw conclusions from conditional statements (I wear my boots only if it is snowing. Its snowing… do I need my boots? (not necessarily) I only wear my boots if it is snowing. (oh, aren’t you cold then?) I wear my boots if it is snowing. (sounds reasonable)) and making a chain of logic using things like the law of syllogism or equivalent statements.

But these things should be developed over the course of the year. Students should always be justifying — but its is the formality and rigor that should increase. In the beginning they are making guesses, then conjectures, then argument ideas, then providing possible reasons, then linking reasons together, then referring to larger concepts and tie-ins… and then by the end of the year they can write a formal proof instead of a rough ‘because’.

I said this metaphor in the other post, but I’ll say it again here. Instead of doing ‘easy’/short proofs, then ‘harder’/longer proofs, building the structure of proof skills as if stacking blocks — think of proof skills going from unfocused to focused as if you are adjusting a lens throughout the year, bringing the structure of proof skills into sharper and sharper clarity.


 

https://www.reddit.com/r/matheducation/comments/3nhn28/motivating_geometry_students_to_do_proofs/cvphfrb?context=4

Academic Paper Breakdown: Lampert “Teaching while students work independently”

As I’m working on my M.A., I plan on sharing my thoughts on some of the articles I read either for a class or for my own research. I hope these serve as both an accessible summary of the article for current classroom teachers and a place for me to share some of my thoughts on the article.  So here is our first one:

Lampert, M. (2001). Teaching while students work independently. In Teaching problems and the problems of teaching (pp. 121–142). New Haven: Yale University Press.

Lampert is a 5th grade teacher who is also a researcher.  During the writing of this book she was teaching part time as a math specialist for an elementary school, but she also has 8 years of full time experience.   Her background includes the philosophy of mathematics and what it truly means to know a mathematical concept. (And she believes that memorization does not constitute knowing.) She describes her teaching style as one that uses problems and therefore problem solving. “Teaching mathematics would have to engage students in doing mathematics as they were learning it.” (p5)

Chapter 6: Teaching While Students Work Independently

Lampert’s students are given an activity that is based around problems of the form (  ) groups of 2 = (  ) groups of 4.  Some of the learning targets in this lesson are for students to use multiplication to create a true statement, and to utilize problem solving strategies.

Notice a few more things here: (1) there are many ways a student can answer this.  (2) its possible but not necessary for students to bring up fractions, (3) the equals sign is not treated as a “do” symbol.  Number 1 means its more open than a normal exercise and provides opportunities for students to analyze each other’s work (great Common Core thinking 12 years early, Lampert!).  Number 2 means we can extend the problem naturally but also we have not put unnatural boundaries on the problem.  2*5 = [  ] is pure arithmetic calculation. But “x groups of 2 = [  ] groups of 4” is all of a sudden touching on algebra.  Finally number 3: the equals sign.  In a problem like “2*5 = [   ]” the equals sign is more of a symbol that the multiplication should be performed and the answer written down.  It is common for elementary students, when faced with “9 + 4 = [  ] + 5” to write, “13” in the blank.  Liping Ma, in her book Knowing and Teaching Elementary Mathematics, even notes that elementary teachers will misunderstand the equals sign: a statement like 3 + 3*4 = 12 = 15 was not flagged as incorrect by teachers.  Back to Lampert’s activity, we have the equals sign properly used as a relation, not an instruction.

Lampert describes some of the actions a teacher performs in order to keep students working on a task, while also getting the most learning out of that task.  In other words, this is not “do the worksheet then check answers at the front while I grade at my computer.”  I broke Lampert’s 11 item list (p140) down into a few categories:

  • A – Assessment – finding out what students know
  • C – Content – using content knowledge to produce help or challenges from many angles
  • S – Structure – keeping the task on track by managing student behavior or task instructions
  • P – Problem Solving – providing and modeling tools, suggestions for general problem solving strategies.

Many items had more than one category, such as “providing and maintaining appropriate use of notebooks and seating assignments” which I labelled as P and S.  Or “clarifying, inquiring, probing” as C and A: the teacher must be able to see what the student is thinking and then place themselves in that viewpoint in order to provide the next tool, hint, or question that would best support the student’s thinking.  This mental agility requires deep content knowledge.

Lampert also focused on a few case studies of her interactions with students.  One student, Varouna, had begun a problem like this:  “[ 1 ] groups of 7 = [   ] groups of 21“.  (At this point it is important to note that Varouna had completed parts (a) and (b) which were “[5] groups of 12 = 10 groups of 6” and “30 groups of 2 = [15] groups of 4”.  Note there is only one blank.)  Lampert brings up my favorite takeaway at this moment.  “She had tried an experiment and was now thinking about how to cope with its consequences.” (p123)

Lampert assessed that Varouna was unlikely to use fractions to complete the statement correctly, but to tell Varouna to erase the “1” and choose another number was to devalue Varouna’s efforts and thinking thus far.  It may appear that Varouna hasn’t done much of anything yet, but I believe that Varouna has already accomplished a lot:

  • She has applied a strategy: the other problems had only one blank, and I can’t think about this one until I fill something in
  • She (perhaps) chose a number with intent: “1” may have been a strategic choice since she assumed the problem would be easier with a smaller number.
  • She assessed her own knowledge: she stopped, realizing that she did not know the number for the second part.  But– she knows that she doesn’t know.

Lampert works a little with Varouna around how to make sense of the problem with diagrams (showing that drawing is an acceptable way to do math) and after a short bit, Varouna writes “3”: “[ 1 ] groups of 7 = [3] groups of 21”  Lampert again does not flinch to signal incorrectness:

My work here was to interpret and respond in a way that taught her something about the mathematics of multiplication and that also respected her efforts to make sense.

How often are we tempted to make the quick correction that the student is “almost right” or has the answer “backwards” ?  In doing so, we are signaling that the student can stop thinking about the problem.  Furthermore, perhaps the student has constructed the concepts in their heads in a way that is internally consistent, just not in agreement with the outside prompts. By making a quick correction on the result of their thinking, we risk destroying a correct conceptual construction.  We as teachers must take the time to get at the heart of the student’s misconception so that we can bolster their conceptual ideas.  Varouna has knowledge that 3 sevens are 21.  She has written an incorrect statment, but it is the result of that knowledge.  Telling Varouna that she is incorrect may harm her confidence in the foundational fact.

I do recommend reading the rest of the chapter for the other case studies.  Here is a link to the book, or you can find excerpts perhaps on academic journal site.

Coin Problems — what are they structuring?

Dan Meyer posted a classic example of a “coin problem” the other day on Twitter.  The problem was in Pearson’s Common Core Algebra 2 text.  Lets assume positive intent from Pearson’s authors in their choice of inclusion here, since at first glance, it may not seem very “common corey”.  But actually, regardless of their intent, lets see where this kind of problem has (1) traditionally taken us and (2) what we can do with it to explore non-traditional approaches.

This post is adapted from my comments in Dan’s thread.

Mark has 42 coins consisting of dimes and quarters. The total value of his coins is $6. How many of each type of coin does he have? Show all your work and explain what method you used to solve the problem.

What kind of problem is this?  Usually this comes up during a systems of linear equations unit in an Algebra 1 class.  The text itself is an Algebra 2 book, so we’re probably safe to assume their intent is to use it as an example of a slightly dressed up system of lines exercise.

So my thoughts are to step back a little bit.  (1) its 2 equations and 2 unknowns — what is interesting and important about those?  What kind of mathematical structure is this?  (2) What does the context of coins do to our structure?  (3) Does the coin context add to our ability to uncover the underlying structure?

Intro

Consider its a little strange to know how many coins you have but not how many of each denomination.  An initial sign of contrived context that students pick up on.  Mostly by high school, students have formed some identities in the classroom that oblige them to cooperate with teacher instruction, so if they’re not initially interested, they may hang with you a little bit. But perhaps some loudmouth in your class will point out some of the strangeness of the contrived situation. #embracetheloudmouth

So perhaps at some point during the lesson series you then have the opportunity to say, “you’re totally right, Lou D’mouth, in what situation could we know the number of coins total and their value total, but not their partition?”

This is where you can set up the class to turn this coin problem on its head, as it were.  (rimshot!)

Letting the kids “get meta”

I like Denise Gaskins’s @letsplaymath idea:   Turn it into a 20-questions game: Stu reach into coin jar & grab handful, others ask Qs to find out what coins they have.?

This changes the requirements of the task: not only do we want to find a solution but we care ostensibly about how efficiently we can do it with respect to the number of questions asked.  Why does it take at least two questions?  Denise’s activity might not go on very long but that’s fine because why will the students stop doing it? They’ll figure out how easy it is!  They’ll wrap their heads around it.  Maybe the teacher can explicitly challenge groups to try to ask a single question to answer # of quarters and # of dimes.

(Presumably, we’re going to disallow “how many quarters?” although its still interesting that we still need two questions there…)

Furthering this section could be questions like:
Q.  If you ask “how many quarters?” and they respond “8” then why does that not let us know how many coins are in their hand?

Expand on the Structure

Now we can acknowledge the contrivance because its going to lead us to something special about this mathematical structure. We acknowledge and loosen our contrivance: parameterize the # of coins.  A number of people on twitter responding to Dan had this idea in addition to myself.

(Lets make the numbers smaller) Say we have $2.00 If we only use quarters and pennies… how many coins do we need? N=8 coins (8q0p) is one way… N=104 is another (ask how!).

1. spend some time finding Ns. mathematical practice standard: make use of structure! (How do we know when we’re done?)

2. To add up to $2.00 does every N (coin total) break down into only one quarter and penny partition? why?

3. Why do some Ns (105 e.g.) not have any solutions?

4. ***What is it about the quarters and pennies, Ns, and $ totals that forces us to have unique solutions or not?***

and 5. Can we contrive a coin-situation so that there are $s,Ns with multiple quarter-penny partitions? Is this possible?

Think back to when you were a student

I remember a lot of handwaving and “worship” in my own education towards the fact that 2 equations and 2 unknowns has a solution. (and 3 eq, 3 unknowns, etc…)

But statements like “ok so this is 2 equations and 2 unknowns so we know how to do it” are exactly what contribute to the magic spellbook idea of math. Kids feel that those who are good at math know the right spells to incant, and “2 equations 2 unknowns” must be a pretty good spell if it makes answers appear out of nowhere!

Don’t solve too quickly

I think this would be a rare case in which I would NOT go for a graph quickly. Yeah, we know about intersections of lines. Yeah we can show these coin relationships are linear.

But can we justify to ourselves that this idea of limited solutions based upon # of constraints is generalizable? We start with quarters pennies sure. In the middle maybe we prove something about # of solutions of systems and we talk about other ideas like mixtures and such. But by the end? We need students to grok (http://en.wikipedia.org/wiki/Grok) this idea of systems and solutions.

Descartes before dehorse

Maybe this is something that the teacher holds in his or her head.  But the deeper and richer the teacher is thinking about the mathematical structure, the more links and hooks and connections can be made between student ideas.  Here we might think about the genius of the Cartesian plane a little bit:

The graph informs us of:

– the infinite nature of the linear models
– the monotonic nature of the linear models
– the difference in ratios between coinA:coinB and valueA:valueB (thus different slopes)
– all of the ordered pairs that fit the conditions of the coin count
– all of the ordered pairs that fit the conditions of the value total

And combining our geometric knowledge about lines at different angles, mapping that onto the linear models, mapping THAT onto the coin situations… that’s how we are justifying our singular solution.

How this all might play out in a classroom would be just that extra 15 seconds of wait time, that extra question for group discussion, that incremental food for thought…

Not so much that we want a full answer, but that we want the students to take up the role of justification.

T, to S: “Why DO we know there is one solution here…?”

Let the students bring it out into the discussion

And if S reasons via graph, great!

But S might say something like: “well if we take off a quarter, and add a time, we keep the coin total at 42, but lose value…. we’ll always lose value at this way so there can be no solutions with more quarters!”

That’s (a) proof by contradiction and (b) using concepts that apply for theory of functions: decreasing, monotonic. (in this case its a sequence) But also the S has synthesized a hypothetical that generalizes to a larger case: take off a quarter, add a dime. Then they’ve upped their abstraction by arguing that any case like that will not provide a solution. (And similar arguments can work for adding a quarter, subtracing a dime)

Then T has the opportunity to engage the class in a discussion of bridging the graphical and the deductive reasonings.

It may very well end up becoming a discussion on why Cartesian graphs are so great, and all the better! Lots of students struggle with graphing because they are stuck on procedure without realizing what a graph represents specifically in an instance, or generally.

Bouncing off the Literature

In the beginning of this post I brought up the idea that a student will go along with some ridiculous contrived situations because they feel obligated to conform to a certain identity in a math class.  I’m considering this in the context of Dr. Paul Cobb’s 2009 article regarding how students create, maintain and alter their identities during their education.  This can be a bad thing, as in the case study course:

The frustration and disenchantment that all the students voiced indicated that they were not identifying with mathematical activity as it was realized in this classroom but were instead merely cooperating with the teacher. (Cobb, 2009)

P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms.Journal for Research in Mathematics Education, 40-68.

When thinking about classroom discussions, I enjoyed Walshaw and Anthony’s 2008 paper on classroom discussions.  A favorite quote:

Unless teachers make good sense of the mathematical ideas they hear in class, they will not develop the flexibility they need for spotting the golden opportunities and wise points of entry that they can use for moving students toward more sophisticated and mathematically grounded understandings. Reflecting on the spot and dealing with contested mathematical thinking demand sound teacher knowledge. Importantly, the way in which teachers manage multiple viewpoints is very much dependent on what they know and believe about mathematics and on what they understand about the teaching and learning of mathematics.

Walshaw, M., & Anthony, G. (2008). The Teacher’s Role in Classroom Discourse: A Review of Recent Research Into Mathematics Classrooms. Review of Educational Research, 78(3), 516–551.