Coffee to Theorems

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

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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.

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

Dan has an quite successful open problem going on over at his blog.

If I give you some cubical candies, what is the least amount of packaging needed for them?

Lots of great problem solving happening in the comments of his thread.  I took a few stabs at it myself. Beginning with making a list of the first few entries and trying to find solutions manually.  1 candy, (1,1,1) cube: surface area 6.  3 candies can be done with (3,1,1).  Surface area 14.  But something like 20 has a few options.  (10,2,1) has a surface area of 64 but (5,2,2) has a surface area of 48.

From that paper-work I was able to generate the beginning of this sequence of minimal Surface Areas: 6, 10, 14, 16, 22, 22, 30, 24… which I then searched on OEIS, resulting with https://oeis.org/A075777 .

I then decided it would be a good exercise in rudimentary python to try to encode that algorithm, so here is my script: 

(caution: this script generates inaccurate results, it is a script of the inaccurate OEIS algorithm.  My improved script is further down in this post)

This algorithm is very similar to some that others were using in Dan’s comment thread.  But here’s where it gets interesting.  Unless I have an error in my code (entirely possible!) then I think we have broken this algorithm.  Dan gives a few frequent algorithm-breaking-numbers here.  And indeed, a few of these break the algorithm on OEIS:

Take n = 1332 using the algorithm described on OEIS:

Cube root is ~11.002

Floor is 11, but neither 11 nor 10 divide n.

9 divides n.  s1 = 9

n / 9 = 148

Square root (148) = ~12.166

Floor is 12, but we need to subtract away 1 at a time until we find a divisor of 148: 4.

s2 = 4

Then s3 = 37

And that gives a surface area of 1034.

However, the minimal surface area is given by a solid of 6*6*37.  Surface Area is 960 in that case.

And here’s one up to 30000 with columns n, s1, s2, s3, minSA: min surfacearea SF upto 30000

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.

Prove that two triangles are congruent.  Sometimes seen as the “first real proofs” of a Geometry course (but they should probably not be the first proofs done, and the types usually done aren’t much of proofs– but let that be for now).

How do you get students to feel the need to prove?  How do we give them a reasonable headache alleviated by the learning target’s aspirin?

The headache-aspirin idea can be illustrated like this:  add the first 100 integers.  1+2+3+…+100. The headache is that it seems like a lot of tedious work, but using Gauss’ arithmetic series folding idea, we fold the long sum onto itself and add pairs inward: 1+100, 2+99, … getting 50 sums each equal to 101, thus finding the sum of integers from 1 to 100 is 5050.  This legend is oft repeated as Gauss’ clever way to get around a teacher’s tedious punishment: notice how even in legend we appreciate the origin of a clever idea as a way to reduce tedium.

Back to triangle proofs.  We want students to see SSS, SAS, ASA, HL (SS-rightA) as useful tools to show that two triangles are congruent.  Dan Meyer made an astute comment: If proofs are the aspirin, the doubt is the headache. The congruency shortcuts are very abstract, but furthermore, a student may not have reason to believe or disbelieve whatever you are going to say on the subject of congruecy.  Two threads emerge here:

(1) why do we want to show triangles congruent?  How can we make a student doubt that triangles may be congruent?

Triangle congruence is our tool of two dimensional congruence.  So we need to have students care about congruence in general.  If your students are philosophical (and many are) then studying the idea of sameness/congruency has some grounding there. But all students are going to want something concrete as well.

(2) why do we use these little 3 part shortcuts?

The accepted shortcuts are an efficient result.  Nothing wrong with showing triangles are congruent by matching all six parts: SASASA — nothing wrong except efficiency. Arriving at the result of the triangle congruency shortcuts can be done by students themselves.

So here’s my scratchwork for a lesson idea: work from the opposite case, can we make a student doubt that different triangles exist? Building Different Triangles We infuse doubt by assigning students to find non-congruent triangles.  When they run up against their sandbox’s boundary– the conditions that cause some triangles to be automatically congruent– they can all of a sudden doubt that triangles can always be made differently.

Students in groups are given the task to build different triangles than their groupmates, given a set of matching triangle parts.  The given parts are either physical manipulatives or digital (advantages with each one)

0. Do we agree that if three sides and three angles are given to us, we can make only one triangle?  Why or why not?

0.5 Think ahead, what if we are given only 5 parts, can we make two different triangles?  What about given other numbers of parts, like 2?  Can we make different triangles then? How many different triangles?

1. Given A=30, AB=3, BC=2.  Make as many different triangles as possible. Related concept: SAS, Law of Cosines eventually.  Geogebra interactive.

2. Given AB=3, BC=2.  Make as many different triangles as possible.  Related concept: triangle inequality.  Geogebra interactive.

3. Given AB=5, BC = 4, CA = 2.  Make as many different triangles as possible.  Related concept: converse of Pythagorean theorem.  Geogebra Interactive.

4. Given A=40, B=30, AB=5.  Make as many different triangles as possible.

5. Given A=40, B=30, BC=5.  Make as many different triangles as possible.

6. Given A=53, B=57, BC=5.35, AC=5.1.  Make as many different triangles as possible.

Be careful with the chosen values.  Sometimes (especially the first one) you want the students to arrive at different triangles.  Save the “we can’t!”/”they’re all the same” moment (SSS (3) / AASS (6) ) for later in the progression.

We also don’t need to restrict ourselves to tech-based manipulatives.  Straws and string work great for the SSS case.

What kind of conjectures would students make when exposed to this kind of task?  Many will apply the triangle angle sum, or have it solidified in their mind: meaning they can conjecture we at most need to be given two angles.  Hey, that’s a proposition we can prove!  That’s one students can prove, or at least justify.  And its not arbitrary or needless: its something that is a significant step towards reducing tedium of triangle differentiation.  If we know two angles, we know the third.

And how about conjecturing and proving statements like: to show two triangles are congruent, at least three parts must be known. (note, hypotenuse-leg gives three parts: two sides and the right angle) Or, at least one side must be known. These are within the grasp of proof by the students because they will have had the experience of seeing the counter examples and of building intuition about what forces the known congruency shortcuts to be what they are.

Thoughts?  extensions?

I’m starting a new movement.  Maybe its not new– and in that case I’m grabbing a banner and rallying the movement!

All too often we talk only about special cases in the math classroom.  All of our teaching revolves around pointing out landmarks and marvels.  But how can our students appreciate such phenomena if their experience never includes the cases that are not marvelous?  If you live your whole life in Yosemite, how would you know how special it is?

That is my thought behind #ContinuumMath.  We need to shake up the special cases by spreading student experiences to include the countless non-special cases.

• Going to talk about SSS or SAS triangle congruency?  Why not talk about “SS” ?  Why are we always using 3 parts of a triangle?  I myself prefer “SASASA” congruency.
• The natural exponential function can be explored in the context of all of the other exponentials whose derivatives are merely proportional instead of equal to their output values.
  

  (click for animated GIF)

• I’ve previously discussed that lines are not always parallel.  Corresponding and Alternate Interior angles are not very interesting if our “desire for congruence” is always fulfilled.
• Algebra: .  “2 step equation”?  Heck, I can solve that equation in 7 steps!  How about this: Students… try using 5 properties of equality (“steps”) for this equation to get to x=6 — I bet you’ll use at least one additive inverse and at least one multiplicative inverse.  The inverse properties come out naturally as the most efficient moves.
• We have lots of new technology to assist us in exploring whats around the special case. Constructing equidistant points with a compass and straightedge is what can be done after we experience [GGB] how distances change.

Students who experience all of these non-special cases then see the special cases as actually special.

Every time your textbook or curriculum offers up a wonderful case– they’re just begging to be shaken up and explored in the context of the structures they depend upon.  This is the Gift of Sometimes True.  Whenever we are giving a theorem or an if-then statement, we can parameterize the condition.  Here’s what I mean:

The diagonals of a parallelogram bisect each other.

We can take that and say, the diagonals of a quadrilateral bisect each other if the quadrilateral is a parallelogram.  And now we want to consider the entire continuum of quadrilaterals– I want students to see quadrilaterals that are almost parallelograms, quadrilaterals that are nowhere near parallelograms, and everything in between!  We further get to talk, as a bonus, about all of the properties of diagonals in other quadrilaterals.  Try the geogebra applet for yourself.

In this applet I parameterized the quadrilatral, but not fully: its impossible to get a rectangle or rhombus here– a tradeoff I made for simplicity of interaction: you can only drag point C.  But as students drag, they observe the diagonals and their dissected lengths.  They can make observations and comparisons in real-time, and they can conjecture and posit as they experience and interact the structure of the geometry.  Students will find the parallelogram.  Students will find the trapezoid.  Students will also find locations where three segments are equal.  Students will find quads that contain isosceles triangles formed by their diagonals… Students will discover so much more because they are seeing so much more than the narrowness of the special case.  Embrace the Continuum.

Check out #ContinuumMath on twitter and/or leave replies here.

Lets look at Triangle Centers: a fun unit for many Geometry classes.  Its a topic with some good “real-world” applications, certainly, but we need not always justify our lessons with application.  And it may hold us back from important mathematical practices.  Pure math is underrated!  Lets compare some traditional textbook style application problems with an interactive style problem.  First, Holt Geometry:

Those two problems are somewhat typical of some application motivators.  But consider a correct response from a student:

18.  Main street should be the angle bisector of the angle between Elm Street and Grove Street.
37.  Bisect the angle between the streets.  Draw the perpendicular bisector between the museum and the library.  The visitors center should be where those two lines cross.

How does this capture the imagination of a student?  If they know what an angle bisector is, they can supply an answer.  But if they do not know it (or haven’t made the connection about equidistance from sides) what kind of feedback can you give them that guides them to the answer without providing the answer?

But the book already provided the direct answer, earlier in this section: Theorem: a point is on the bisector of an angle if and only if it is equidistant from the sides.  #18 is almost explicitly asking for this theorem.  #37 asks a little more: it wants this theorem combined with the perpendicular bisector theorem, which led off this section– no prelude.  These questions are merely dressed up versions of those on Level 1 and Level 2 of Bloom’s Taxonomy.

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With GeoGebra, we can provide loads and loads of information that (1) helps guide the student around the topic we want and (2) does not provide progress towards “the answer” in discrete unassailable steps.  We can be more helpful while being #lesshelpful.

Here is a picture of an applet I created to give some real-time information to students about point D and triangle ABC.  And here is the interactive applet itself, try it!

In this first static picture, what information is given to students?  Most students will be keen enough to see the comparison of the distances DA, DB, DC.  And all students will catch on after they start dragging point D.

What do we need to ask here to get students to think about perpendicular bisectors?  I say: “not much!”  My introduction to the applet is “Drag point D.  Find special locations.”  That’s not to say I’m not communicating with the students.  I’m communicating a great deal more!  I’m communicating through the boundaries and feedback programmed into the Geogebra manipulative.  Manipulatives are powerful in any setting, but computerized manipulatives enable modes of lessons not possible before.

I love doing triangle center lessons via oragami (or patty) paper.  I love doing triangle center lessons via compass and straightedge.  But the boundaries of those manipulatives do not guide the students.  If you make an incorrect fold or line, the paper doesn’t tell you so.  If you want to test distances with your compass or ruler, you must do them one at a time.  If you want to pursue a question other than equidistance (e.g.) then there may be other complicated procedures to follow.

Consider with GeoGebra:  by restricting the interaction to only dragging D, the students may no longer make unproductive moves.  Every move they make produces feedback via the distance “bar graph” and the color shading of D itself.  That feedback isn’t telling the student “incorrect”.  It is telling the student “here are the distances your input asked for.”  The student is then left to parse that information, and respond with more input.  This type of interaction between student and computer occurs within a fraction of a second.  And this type of interaction is repeated tens, hundreds, or thousands of times.  The student can then use the information to start digging into the prompt: what are the special locations for D?  What makes a location special anyway?  These are questions that can be asked and answered by students without teacher interruption.

But wait, there’s more!  Lets say a student has decided upon finding the place where DA=DB=DC.  Computerized Manipulatives now allow a task between this decision and the answer.  They must spend time actually getting point D to the correct spot.  In this Geogebra applet, they have the feedback from color and the bar graph to help them, but here is where the teacher can wait for a student to ask, “is there a better way?”  Boom!  We have arrived at the motivation for the circumcenter construction.  This motivation was driven by “pure” math ideas like equidistance… only we didn’t have to say it.  Proof in geometry should be introduced as a way to perfect our conjectures and hypotheses.  If the students haven’t made a conjecture, how are they going to care about its truth?

In this applet I decided to include an extension: another triangle center.  But this is not the usual 2nd center introduced.  Your students will find the blue point and notice what is special. Try it yourself!  Think about what is also done with this wordless separation of the two points in contrast to a lecture-based introduction: the students will have their own vocabulary to parse the difference that they have played with already, instead of the students having to parse vocabulary that describes a difference that they may not have been aware of.

In this applet I included two “answer” checkboxes.  Depending on where you want to take your class next, it they may steal some thunder from the lesson, but I figured I’d include them to help illustrate the point… and points. (hah)

GeoGebra and other computerized manipulatives enable us to think about Geometry and math in a way unfamiliar from static text or lecture.  Similar to The Lines Are Not Always Parallel, I’ve created a way for students to observe the neighborhood around the special case, and constructed silent helpful barriers and footholds that students can grab onto as they discover what is special for themselves.

Let me know what you think.

 

Recently, Michael Fenton (@mjfenton) posed the following question:

Q4 What advice would you give to a younger version of your teaching self regarding grading? #slowmathchat pic.twitter.com/E9yw2SFemT

— Michael Fenton (@mjfenton) February 27, 2015

I thought back to the types of grading I was doing in my first few years of teaching.  I would go through problems with a fine-toothed comb, taking points off the maximum… minus one here, minus two there… minus 1.5 point here, because “I can tell you know it, but you forgot a negative sign a couple steps ago, and as we all know, negatives signs are worth half a point”.  I would make sure all my tests had appropriate weights on each question, “number 2 is worth 5 points, and number 3 is worth 7 points” (because it’s 140% harder!) and I made sure to write very encouraging comments or constructive criticism that I’m sure contained very little mistakenly copied dialogue from the CSI episode I was watching.  “Jimmy, you have a good idea here when you divide the perimeter by two, but did you gather the DNA sample from the suspect?”   And of course, my gradebook followed precise weights and scales: “yes, 30% tests, 15% quizzes… because if I say the word test I want to make sure you stress exactly twice as much over each problem– also you have less time.”

It was exhausting.  And with more experience I was beginning to see where the extra work came from, such as evaluating the worth of assignments, problems, and categories of problems– a task that required extreme time and effort to nudge the system way from seemingly arbitrary.  Finally, I thought about what students were gaining from all of my extra work– I concluded, “not much.”

If this sounds like you, I offer these steps, in increasing levels of difficulty, to make your grading more efficient.  These are things I gradually adopted over a couple years.  The steps are simple, but the effects transform your teaching itself.

1. All integer points.
2. All problems on the same scale.
3. Skip comments. *
4. (we’ll get to #4 a little later)

Lets talk them through. We’ll see how they tie together and how they alter the culture of the classroom.

1.  Stop giving fractional points.  And further, reduce the total possible point outputs.  Your teacher brain is expending calculation energy dealing with fractional points or a wide range of possible scores.  Why?  Because you want to summarize that the student has the concept but has missed some details.  What message does this send?  “I am somewhat ok with you being not attending to precision” ?  (CCSS.MP6)  Because you want to be completely sure about the value of this student’s work?  Grading is a loss of information, we boil down a student’s work into a numerical value, an irreversible process, and an imperfect science anyway.  Do you know what an 8 point score looks like in comparison to an 8.5 point score?  Or even 8 points vs. 9 points?  (out of 10, lets say) This is like the coastline paradox: you can spend a great deal more energy to find a measure that may be only slightly more useful– or may be less useful even.  Does it make sense to measure the coastline of England on a scale that is affected by the tides?  And are they missing half points a lot?  If so– maybe its not so minor a problem.  If not– then what difference will one half point make in the end?  Now I’m not saying to round it all down.  Where you go from what you might previously have given an 8.5 will depend on other factors.  This step can be adopted by itself, you just have to decide how important those tiny errors are to the overall measure of the student’s work.  You’ll save a lot of energy on your end that can be applied to other areas, and you’ll lose only a little more information than what you previously had.

2. Put all problems on the same scale.  Now that we have no fractional points, lets simplify our life even more: no more deciding what a problem is worth.  No 3 point problems, no 12 point problems, no 11 point problems.  Let us have all problems worth an equal amount with a small number of divisions.  Now, let us also define a “problem” as something that requires work and reasoning to be shown. This can vary depending on the course, but in general, a problem is not multiple choice, or fact recall, or calculation-only.

What do we want to communicate to the student?  We shouldn’t detail what is wrong or incomplete, that should be a skill practiced by the students themselves.  So what do the students need in order to practice that skill?  They only need to know if their work is insufficient.  So to that end, I grade every problem on a very simple rubric: 4, 3, 2, 1, 0.  These numbers communicate to the students the following ideas:

4: Full detailed solution
3: minor revisions needed
2: major revisions needed
1: a mathematical effort in the right direction
0: blank or no mathematical effort

The main idea here is that we want to communicate that the student’s work can be improved.  And that if they were to make revisions, it could be re-evaluated and their grade could improve.  I based my structure on ideas I heard from Dr. Kysh at SFSU during my credentialing program, Riley Lark (now of ActivGrade), and Richard and Rebecca DuFour.  See more comments from Riley here: http://larkolicio.us/blog/?p=800.  The DuFours argument was specifically about late work– why not accept it?  “Hey you didn’t pay your electric bill on time last month– so just forget about it!”  This idea of revising work means that the assignments that you give are automatically more meaningful.  The student remains accountable for their work after they turn it in and they are tasked to figure out how to improve it.  That’s a valuable skill!  And it relates to another Mathematical Practice standard: CCSS.MP3 critique the reasoning of others.  Peers can help each other figure out what is lacking in their work, and what can be improved.

A quick note about scaling: the familiar 90-80-70 scale has a built in pressure towards “completionist” grading.  It assumes that you will be averaging.  It assumes that students will “complete” “50%” of their “work” (feel free to do air quotes along with me) before its even close to being acceptable.  What does it mean to know 70% about solving equations?  What is the difference between knowing 25% and 50% about triangle congruency?  I thought carefully about what is being measured and what was being communicated by those numbers.  I decided to jump to another arbitrary scale– the GPA scale– that corresponds better to the rubric messages I was sending.  (A single 1/4 no longer drags an entire average down like a single 25% does.)

3.  Skip comments. (* 2016-03-19 For additional context on this, and perhaps differing opinions / additional options, see Dylan Wiliam on Feedback)

If your problems are all on the same scale, and that scale has a versatile rubric… what comments do you need to write?  Here are typical comments I used to write:

“Good idea!  What if you apply that here?”
“I like this, but you can’t always do that step because …”

Sure, ideally we could devote unlimited attention towards a single student and use various pedagogy to improve their work.  But we have 150 students.  Time you spend on one student is time you are taking from the others.  What did my comments achieve above?  Was it worth the time I spent on them?  Did the students internalize it?  I do think there is a benefit for personally coaching or cheerleading and encouraging students, but is it a good use of time to handwrite?

Here’s what I started doing instead:

• I would score using the holistic 43210 rubric.  “How good is their solution?”  But that wouldn’t be the only marks on the page.  I might circle things to guide their attention, or put happy faces or stars as shorthand approvals/encouragements.
• I would make notes of “interesting mistakes” ala My Favorite No, and use them as information to structure future lessons.  Because, if one kid has a misconception… how many others share it?
• I carved out class time for students to work on revisions.  This could be structured individually, or peer-review, or office-hours-style, or whole-class error analysis… or some mixture of those styles.

A student who reads comments may gain a lot from them.  But what are they supposed to do?  Wait until the mythical “next time” to address the concerns you wrote? Comments written on a “dead” assignment are asking to be thrown away and forgotten.  But if you give students the expectation of revisions, then all you must do is indicate what must be revised.  The assignment lives on!

This cut my grading time down by at least half.  But also it served as a formative assessment of the class progress, allowing me to still make comments– but this time it was in person, to multiple students at once, and with the structured intent of them figuring out what was wrong.

Revision days sound like they will eat into your classtime, but instead they remove the need for “review/reteach days”.  Plus, the students get the extra practice with critiquing and analyzing work.  PLUS the grading is much less time consuming on the teacher side. PLUUUSSS it encourages culture in your class centered around improvement, growth mindset, and increases the value of every assignment you give.

4.  Ok what’s step 4?  Standards Based Grading.  This one is a doozy.  To switch completely towards assessing student’s understanding, rather than compliance, you can switch to SBG.  This is an entirely new way to grade, as it requires identifying what standards are being assessed when, and how you are going to decide what kind of scores mean what kind of mastery.  It also requires holding the line against some external pressure: the principal and the parents and the students themselves may resist the new style.  But with the previous steps 1,2,3 above, you can be halfway there and then feel out how to make the final leap.  There are various blogs about how to adjust, (I like Sam Shah) and various software to help support you (check out JupiterGrades and/or ActivGrade)

What you’ll find if you head down this road is that you begin by re-thinking what you actually want to assess… and you end by re-thinking how your assignments even work!  First you think, why even grade a drill worksheet of linear equations?  Then you think, what type of assignment will give the students opportunities to vary strategies and produce interesting responses to analyze in revision?

Let me know what you think… Let me know if you have similar or contributing ideas… And definitely let me know if you try it!

Jason Dyer posted an excellent critique of some of the exam items and interface choices made by the American Institute for Research and their development of new Common Core exams.  The one he reviewed is called “SAGE”.

Designing an User Interface is difficult work.  Especially when your UI will serve thousands upon thousands of people.  But the most important part of the UI for education testing purposes is that all efforts should be made to minimize the effect of the UI on the measurement of the student’s test performance/score.

I believe the SAGE test has huge flaws that will affect the validity of the test results.  If students, teachers, schools, districts and even the Common Core standards themselves are to be judged on these results, I believe a lot more work needs to go into the UI Design.

Here is an example flaw, as described by Dyer:

The percents in the problem imply the answer will also be delivered as x%, but there is absolutely no way to type a percent symbol in the line (just typing % with the keyboard is unrecognized). So something like 51% would need to be typed as .51. Fractions are also unrecognized.

(Also the problem is not precisely worded, if I were nitpicky– and I am– then I would restate it to read, “If Ms. Jones chooses a student at random from her class, what is the probability…“)

We may intuit that the authors+designers of this question and its interface have either (a) not given thought to the difference between .51 and 51% or (b) did give though, but decided that the input box’s restrictions may be enough to guide towards the decimal response.  Note that in either case, the authors+designers do not consider the format change from the question to the answer to be significant.  What would Strunk and White say?

To a person with reasonable mathematical maturity, it is clear that .51 and 51% are interchangeable in this context, while not formally stylistically equivalent.  But these tests are supposed to measure a population whose mathematical maturity is in question.

The other more subtle element to this type of UI design is that the input box guides the user to do certain actions.  In a way, it is a part of the question.  A student may not type in 51/100.  A student attempting to type “%” is silently blocked from doing so.  These unspoken rules of communication and interaction should be paid attention to.  Do they affect student/user behavior?  Is the effect “regressive” in the sense that a low-performing student is more likely to have trouble deciphering the interface? (and therefore is more likely to not perform well on this test item)

The other example is rather indicative of the low level of attention the UI has recieved:

 

“Enter each number on a separate line” is kind of the opposite of the issue from the percents problem above.  In this case, the authors+designers consider their interface to need direct instructions on how to respond.  On top of being incongruous in style, it is just plain wordy.

The designers of these tests seem eager to move away from the multiple choice style of test questioning. Multiple Choice has the big drawback in that the answer is given, hidden only by the distractors, which as a set may give clues as to which response is not like the others.  Its something that must be considered carefully when creating a MC test.  So, exploring alternative response systems enabled by the technology is reasonable.

Perhaps the authors did not want to label the input boxes? First Number _____  Second Number_____  I can certainly see the reason to avoid “first” and “second” along with “number 1” or “one number” and “another”.

Perhaps the authors did not want to use variables?  Two numbers, x and y, have a product of 323 and a difference of 2.  x=___ y=___  Ok, maybe variables would be beyond the scope of the test, or would be unfamiliar to the students at that stage.

But there are other options.  The answer input boxes could be embedded inline with a complete sentence.  The two numbers are  ___ and ___.

Finally, Dyer points out that the question implies one solution {17,19} but there is another {-17,-19}.  And it makes me wonder how the test is to be graded.  Wolfram Alpha has demonstrated the capability of software to interpret various mathematical inputs.  Can we grade these tests by computationally checking the answers rather than comparing to an answer key?  If a student inputs -17 and -19, why not compute the product and difference and see if it matches the stem?  Can a computer algebra system or dynamic geometry software check the results of student inputs by calculation?

I believe it is very difficult to design these items well, and I do think it is worthwhile to explore a variety of input methods, however, I believe that more effort needs to be taken to standardize input methods.  I believe that the best interface for these tests is one that the user is minimally aware of, i.e. the user should not have to figure out how to take the test in addition to taking the test.

Earlier today I was in a discussion with @mathhombre on Twitter about what is needed in an College Algebra course (roughly equivalent to Alg2+Precalc for high school).  I came to the (perhaps too radical) idea that lines are simulateously one of the most applicable concepts to a person’s “real life” but also one of the most useful tools for accessing higher math.  There are entire areas of study about linearization as a tool to simplify complicated cases not just in pure math, but economics, physics, engineering, etc.

Any that is just the prelude.  If we are going to talk about lines, what happens to the darling of any high school level algebra course?The quadratic function / The parabola.  Consider f(x) = x^2 – 2x – 15.  Factoring, we obtain f(x) = (x+3)(x-5).  A subtle idea that might be skipped over: f(x) is the product of two lines.  Literally lines y = x + 3 and y = x + 5.  Do our strudents face the idea that a linear factor and a line are the same?  Consider the product of the values of the lines — point by point.  Try it with Geogebra.  The f(3) = (line1 at 3)*(line2 at 3) = (6)(-2) = -12.

We also know the Fundamental Theorem of Algebra will guarantee n roots for a degree n polynomial.  But those n roots may have non-real parts, for example g(x)=x^2 + 4 does not factor over the reals  g(x) = (x+2i)(x-2i)

So can our “line*line” idea survive these new types of models?  And where can we see these complex lines?  The product relationship still holds.  g(3) = (3+2i)(3-2i) because the imaginary terms will cancel since complex roots always come in conjugate pairs.  g(3) = 3*3 + 3*2i + 3*-2i + – 2i*2i = 9 + 4 = 13.  But wait, lets slow down.  3 + 2i is a point.  A point on the line x+2i.  It is a line hidden to our view because we lack the dimensions on our plane to see it.  We have only the 1D real line as our inputs.

So lets consider the imaginary part of our domain.  Click the picture to see a larger view:

That’s g(x) = x^2 + 4.  The red axis is the standard x-axis.  The green axis is the standard y-axis.  But the blue axis represents the imaginary part of our domain.  A normal classroom might be used to plotting 3+2i on a plane, but do we often make note that its a nominally different plane than the one we graph functions upon?  3+2i will be on the plane that passes through the red and blue axes.  I chose to view it this way to keep y outputs as close as possible to our usual view.

Ok so lets see 3+2i and 3-2i and their output product 13.  I included two views here since it is difficult to get a grasp of the 3D situation.

These lines y = x+ 2i and y = x – 2i are kind of abuses of notation.  They should be specified that they are lines in space, namely y=x limited to the level planes Im = 2 and Im = -2.

There’s certainly more to do with exploring this idea, especially making the presentation more robust.

What really appeals to me is the symmetry of a quadratic’s complex conjugate roots is similar to the symmetry of a separate quadratic’s real factors around the axis of symmetry.  Even more, the idea of the imaginary number i as a 90 degree rotation fits puzzle pieces together.  The complex roots are 90 degree rotations around the axis of symmetry of a reflection of the parabola (reflect across a horizontal line passing through the vertex).  This is the part that holds some promise, but that I haven’t quite explained in my own head yet.

I invite you to play around with the Geogebra applet I used to explore this.  It is set up to start with x^2 – 8x +18.  Find the complex roots algebraically first, then see if it meshes with the visuals in the applet. Thoughts?

Students are frequently confused by us harping on the importance of things that seem obvious: because we hardly ever show them the cases where the theorems are false.  One of the areas this comes up is Angles in Parallel Lines.  (The whole course of Geometry may feel like this, actually).  We teachers might feel like, “how do kids get confused here? its so easy and obvious!”  That’s right… so easy and obvious that we teachers have lost sight that we only show them these narrow cases.

We spend time proving that certain pairs of angles on parallel lines are congruent or supplementary.  Over and over and over… but always parallel lines!!  Maybe maybe we make an offhand comment about “if these lines weren’t parallel then the angles wouldn’t be congruent!”.  But those words hardly paint the picture for student.

But think of the infinite number of line that are not parallel!!

So, let us turn to Geogebra.  My philosophy in making this applet (and most of my applets) is to loosen some restrictions in our normal presentation so that more of the “sample space” can be explored.  Lets consider all lines cut by a transversal, not just parallel lines.  Then how many students will not only (1) appreciate parallel’s special nature and (2) make conjectures about congruent angles all by themselves?   This is much in the same vein as Michael Serra‘s Discovering Geometry book: set up a situation so that the students actions will lead them to make those conjectures.

The applet here lets students drag the line to all positions, and illustrates the red and blue angles, giving no measurements.  We want intuition here, not measurement.  Students might already have some idea “yeah they look close enough”.

The check box “Compare Angles” allows them to update their intuition when faced with a little more precision.  This kid who wasn’t able to articulate anything before might now realize in what realm we are trying to explore.

The check box “Compare Lines” allows them to make a connection to the properties of the pairs of lines.  Again, this is some non-verbal feedback and prodding to a student.  The student must assimilate this and then can provide the verbalization themselves.

Finally, I added in a little game at the end.  It will spit out a score based on how close to congruent the angles are.  (The formula is arbitrary, but spikes up very high when you’re close).  Let me know how this plays out with your students!  Do they get invested in beating each other’s high scores?  If so, notice that in trying to beat one-another, they must know that they want to make the lines “more parallel”.  Here we can embrace the difficulty of being precise when doing this with a trackpad or touchscreen.  Will your students say “hey if I could make them parallel, that should be an unbeatable perfect score!”  (I actually think Geogebra will spit out “infinity” score if it gets close enough– a happy consequence of the data structures they’re using)