Why Questioning?

Found at From-Student-to-Teacher Tumblr
From Joy of Literacy

Last week became questioning week with student teachers. It came up in two action plans and we were able to have some really interesting discussions about it.

A lot of what I'm writing about here is from work with David Coffey, inspired by Kathy Coffey, and processed from Mosaic of Thought (link includes Chap.1 as a sample) among other books. Neither Dave nor I can remember the actual origin... which is sometimes symptomatic of having done it yourself.

In my own growth as a teacher questioning is definitely one of the places where effort and reflection have helped me improve. The first level was just asking better math problems. More open-ended, that required more problem solving. When the problems are better, there's more interesting things to ask the students about later. The next was to ask more appropriate questions. Some of those excellent problems I gave to students were too much for students. This is a zone of proximal development idea. When it was too much, I needed to scaffold. Later I learned about rephrasing the question instead of narrowing it. Later still I learned about demonstrations of how I think about a problem, sometimes more appropriate than guiding students through it. Then you ask 'what did you notice?', which is - of course - one of the all time great math questions.

One of the best things I ever learned was to stop being the authority. I don't say what is right or wrong. I ask the students if they 'agree or disagree?' That might start an actual conversation. 

One of the first things I really noticed about teaching was that students would tell me that they couldn't do it (whatever it was at the time) but when I asked them questions they could get from beginning to end with no difficulty. So, obviously, I needed to teach them to ask those questions of themselves.   It was difficult. Really difficult. Shifting from asking them 'how to do (next step)?' to 'now what?' and 'how do you know?'

Literacy learning experts are good about this idea of questioning as a process, with the idea that questions are how we move ourselves forward. I like this framework to help me think about the kinds of questions that I'm asking.


Obviously, we have had a tendency to ask too many literal and application questions in math class.  I think about inference questions being predictions, reading between the lines, hypothetical questions and the like. Analysis questions are reflections, synthesis, connections, recommendations and so on.

What I shared that seemed to tie it together for the student teachers is a simple idea: ask to find out what I want to know about. I don't need to ask for answers - I know those. I don't need to ask right or wrong. I need to ask about what they are thinking. Students know when a question is genuine, and this simple idea has improved my assessment more than anything else.  I'm more persistent in getting answers when I really want to know them, also.

From A Softer World

Some good student teacher writing on questioning:

• Caitlin - Open Ended Questions
• Stephanie - Questioning Techniques

Math Memoirs

There's one of my favorite math teaching videos up in Keith Devlin's most recent blogpost.  (Thrilled that it's now on YouTube with several other of her interviews.) It's from the wonderful Marilyn Burns, who has really done more for math education than anyone in recent decades. Her stuff was good enough to get word of mouth sharing before there was an internet. (Dan Meyer is quite reminiscent of her in several ways.) Devlin uses the video to make the point that apparent understanding is not proof. A better assessment can uncover the truth, and Burns is excellent at questioning to get an accurate picture.




As a novice teacher I quickly found that the students giving an answer did not tell me whether they understood or not, and I moved to asking for how they got it. It took me years, however, to get to understand that how is better, but also does not get at understanding. It's their thinking that I want to hear. Then I should ask for it! Unfortunately, even spoiled university teachers don't have time to sit down and interview students over all of our major objectives. I can, however, ask questions to which I really

Dave Coffey found this excellent solution by Carl Barnard, a high school student, to the painted cube problem that crossed the line to be a memoir. We often introduce the idea of a memoir by having our students read it and comment on it. (Here's my most recent version of the workshop; Dave was almost certainly the original writer of it.) It's a start to learning how to communicate your thinking. While it's important and beneficial for any math student to do this, it is unbelievably crucial for math teachers, and a big part of pedagogical content knowledge to me.  Most students on end of term evaluations comment that this is an area where they grew significantly.

I wanted to experiment with what this looked like in elementary school, so I asked my daughter to give it a go. These are from the summer between her 4th and 5th grade. She was a bit precocious with her writing, but had had pretty traditional math instruction. (And not interested in talking with me about to a large extent.)

Just writing the method. A first attempt at memoir. I supported with a kind of questioning framework.




She was game enough to try twice more.






This is the sort of information that I want from an assessment. I get so much more an image of what ideas she has about number and operation. And it is far more interesting reading than traditional assignments where I'm looking for what I expect or not.

Two Final Problems

Trig Problem 2

For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.



The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.




Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

Have to love easy-to-draw memes.

Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.

SBG Resources

From Rainbowcatz @ Flickr

I was gathering Standards Based Grading (SBG) resources for a colleague and thought that would be worth sharing.  Maybe this should be a LiveBinder? There's a definite math focus to my selections below, those it's not strict. Many people refer to SBG as Standards Based Assessment and Reporting (SBAR), which gets the whole 'grade' idea right out.

People: (Name links to Twitter)

• Jason Buell - SBG resources that includes the SBG galas.
• Shawn Cornally - collected resources at his blog
• Riley Lark - SBG blogposts
• Frank Noschese - SBG blogposts

Fundamentals and Further:

• Ed Leadership article: Seven Reasons for Standards-Based Grading by Patricia Scriffiny
• SBG Beginners Wiki, which also has math skills lists , started by Elissa Miller
• SBG page at the MSmathwiki.
• Blue Harvest - Shawn's SBG software
• ActiveGrade - Riley's SBG software

Twitter discussion

• #sbar - find more SBG folk, or people trying it in your discipline, by a Twitter search.
• #sbarbook - book group that 'meets' weekly for discussion about a particular book on assessment. Doesn't look like this semester's book is very engaging, though.

For completeness sake, here's my 2 (so far) SBG posts. Hey, this makes 3! If there are more examples of SBG in college, especially college math, please help me find them.

Patterning

I gave an algebra assessment this week (in SBG style) and a student really surprised me. So I thought I'd share.

The assessment:
Possibly relevant standards.
A. Algebra: representation, operations and modeling

1. Linear equations and functions 
2. Quadratic equations and functions 
3. Exponential and logarithmic equations and functions 
4. Higher polynomial and rational equations and functions 
5. Functional representation and operations. 

The problems: take 10 to 15 minutes to consider the following problems. You may do 1 a couple or all. Your SBG score will not depend on a correct answer, but rather on showing your understanding of the ideas involved. (Of course, good understanding helps find solutions, so it’s not totally unrelated; but it’s easy to give an answer and show no thinking.) Since the answers are not the central issue, be sure to communicate your thinking, process and understanding. If you read these instructions, clap your hands once.


1. Using the parabolas at right, estimate the equations of the curves.
2. Discuss: how do you recognize a quadratic pattern from a table, and what does that have to do with the familiar parabola shape of the graph?
3. Use the idea of parallel and perpendicular slopes to give the coordinates for vertices of a square that has no sides parallel to an axis or to y=x. What is the area of your square?
4. Find a symbolic rule for one of the patterns below and look for connections between the symbolic and the visual. Extend the pattern one step to check your rule.







Analysis:
Problem 1 gives lots of nice information. Whether the student uses standard (harder) or vertex form, what the coefficients mean to them, and if they are consistent in applying that understanding amongst the different parabolas. Almost no one uses the roots, and I've yet to see someone apply regression to points.

Problem 2 pointed out that students over-identify parabolas with quadratics, as most students who struggled talked about increasing slope and symmetry as opposed to first or second differences. (Also that there was a homework that they didn't get to!)

Problem 3 was interesting in that no one started with the points. Everyone who tried it gave linear equations, established parallel and perpendicular, and struggled with how to get the sides to be equal length.

Problem 4 was the surprise. Most people chose the pattern on the left, and used visual identification of pieces to generate a direct rule. Only one student tried the pattern on the right. But instead of finding the quadratic pattern I intended, she noticed that each element was as long as the three previous elements summed. Definitely true for the picture... and got me to wondering if it would be quadratic. Nope. but definitely non-quadratic. I made a Google spreadsheet to compare, and this was unusual that the quadratic matched the sum of the first three pattern so well. I got interested in the ratios as they seemed to be converging. (Here's the spreadsheet if you want to play around for yourself.)


Finally I gave in, and looked up the sequence in the On-line Encyclopedia of Integer Sequences, where it is known as the tribonacci series. (Cute, eh?) This limit ratio is the solution of x^3 - x^2 - x - 1 = 0. I think I'll call it the Golden Ratio.

May your quizzes ever surprise you!

Grading: SBG and U

Math Monster
by Mister Awesome @ Flickr

Standards Based Grading to me is the idea that the teacher lays out what students are responsible for demonstrating ahead of teaching, and students have a long period during which to demonstrate them, possibly up until grades are finalized.  And students have multiple opportunities to demonstrate.  (This is a Part II to the previous grading post.)

Other people describe it better and more thoroughly.  Especially Sam Shah and Shawn Cornally.  Also please check out the beginner's wiki started Elissa Miller and the SBG gala hosted by Matt Townsley. (Note that you could be interacting with these outstanding professionals on Twitter: @samjshah, @thinkthankthunk, @misscalcul8, @mctownsley) Frank Noschese is thinking about it powerfully, too, in physics, but I haven't had the chance to interact with him about it.

I will say that I've only used it with preservice teachers so far, but they were mostly an appreciative audience for it, and would like to see it in their content classes.  I will be doing it in my content classes, starting with a graduate calculus class in the fall, but we're so pinched for math educators right now that I don't get to teach any straight content courses.

The preservice teachers have been helpful for improving my practice of it with their feedback.  If you're making the change, I'd encourage you to discuss it with your students, give your reasons, and involve them in the process.  I was only going to do it through in class assessments and similar things in office hours, but I added an SBG option to portfolio submissions and added an interview option for office hours.  The biggest remaining thing is how to communicate it better at the outset, with which the resources in the second paragraph will help.

Another Speedbump Classic

The most powerful concept to the shift has been giving the students a clearer purpose on the assessments: to demonstrate what you understand by communicating your thinking.  Much of the emphasis on the right answer is gone, as is the expectation that test questions will be trivial repeats of tasks already done.  Not that my tests were like that lately (have to go back over 20 years for one of those), but it was a bone of contention with students.  Now it makes (more) sense to them that they couldn't show understanding on a question like that.  I've had a few students reject a problem because they knew how to do it already.  (That's not the majority, but some day...)

It's different from K-12 use because in the university we see the students so much less. We give up class time for independent work outside of class, which minimizes time for summative assessment.  I struggled to provide multiple assessment points.  Put lots of former standards on assessments as choice, and polled students as to what previous standards they wanted on.  My standards were much broader than they would be in a content course, as math ed classes wind up covering things like "all of high school mathematics."  So I made my standards pretty broad, but we looked at examples of more focused grade level standards.  In the future as I reuse, I'll try to add some of those specifics as ways to demonstrate the broad standards.  I also let them know that the final grade would take into account which standards we had covered and assessed in class.  Some of the content I don't set until the preassessment is in, so it's hard to know ahead of the semester.

Here's the policy on my middle school math syllabus.

Standards Based Grading: SBG is a relatively new way to assess students that seeks to get a higher correlation between grade and understanding. On each of the objectives below, you will have opportunities to demonstrate your understanding. These objectives are a bit broader than you would expect in a secondary classroom, since we are seeing content from three years of schooling. In a secondary classroom, the teacher identifies the standard demonstrated, but in this preservice teacher preparation course you will also be trying to identify which of your work is evidence of which standard.

Scores do not mean an answer is right/wrong, but are meant to reflect how much understanding was demonstrated. It is possible to demonstrate good understanding of a concept without even finishing a particular problem. The score for each category is the average of the 2 highest scores. If there is only one score it is discounted by 1; a single A becomes a B, etc. You can reassess on specific objectives during office hours or at arranged times.

A+ complete understanding and can extend on your own
A complete understanding, can apply when appropriate
B some small difficulty applying or missing a small point of understanding
C significant difficulty in application or missing a major point of understanding
D mechanical application of ideas without understanding
F little to no understanding or evidence of understanding

Mathematical Content Objectives

A. Number: representations and operational concepts

1. Integers
2. Operations on integers
3. Rational numbers: fractions
4. Operations on fractions
5. Rational numbers: decimals
6. Operations on decimals

B. Algebra: representation, operations and modeling

1. Patterns: recognizing and generalizing
2. Variable: as unknown and changing quantities
3. Linear and exponential relationships

C. Geometry

1. Similar figures and proportional reasoning
2. 2-D figures: characteristics and sorting
3. 3-D figures: characteristics and sorting
4. 3-D representation

After a messy fall semester of trying to run parallel SBG and traditional, and a messy winter semester of struggling with full implementation, I'm very happy I came down this road.  I have four basic goals for my grading:

From Comically Vintage
Don't be a Dodo!

• fair - reassessment helps this.
• measures real understanding - move away from non-problems helps this.
• not fear or anxiety inducing - students said this was a big improvement.
• measures where the student is at the end of the course - clear improvement.

While I didn't get a lot of out of classroom reassessment until the end of the semester, I did get people using the in class assessments to reassess.  Students were more responsible for their own marks than ever before, and rather than tracking grades, they were attending to objectives. Broad over-generalized objectives, but I had to start someplace!

I strongly recommend you consider SBG, whether you be K-12 or 13-19.  If you do, let's talk!

Practice Problems

I'm trying to shift towards standards based grading (SBG) this semester for the content grade in my geometry class, and so I'm paying a lot of attention to the kinds of problems I write. My tendency is to write big, open, sprawling problems that cross many standards, but I haven't yet worked out how to do that and SBG.  I know that I want problems where students can show understanding without necessarily getting to a correct answer.  Mathematicians are wrong a lot. What distinguishes us, I think, is that we often know whether we are wrong or right.

My students asked to have the time before the test to practice, instead of the book group I wanted to do. (110 min class and a 1 hour exam.) Very reasonable.  But that meant that I had to come up with practice problems! I'll post those now, then the test when all students have taken it.

Photo by scrappy annie @ Flickr

Do you give students practice?  What's the relationship between practice and the assessment problems?  This was a big topic in my student teacher observation this morning also.  

 322 Midterm Practice

The Standards

A. Analysis of characteristics and properties of
2-D geometric objects: number of edges; side length; angle size; parallel; perpendicular; convexity, etc.

1. concept: definition and recognition
2. application: use to sort and characterize, build or draw
3. combination: consider multiple properties in a single object
4. familiarity with examples: triangles, quadrilaterals, polygons

D. Distance, area, angle

1. concept: definition and key properties
2. formulas: use and derivation
3. connections among formulas

Try your choice of the following problems. Look for problems that allow you to problem solve and share your thinking.

Which standards could you demonstrate on which problems?


1) Connect each side property to an angle property and draw a different polygon to match each pair.

(at least) 3 congruent sides		no adjacent congruent angles
pair of perpendicular sides		(at least) 1 angle > 180 degrees
3 parallel sides		pair of adjacent congruent angles

Draw 3 different connections and try again!

2) Sometimes quadrilaterals are defined by their diagonals rather than by sides and angles. Determine which quadrilateral goes with which definition below, and make your argument. If no quadrilateral goes with a definition, state why.
a. Diagonals both bisect each other.
b. At least one diagonal bisects the other.
c. Diagonals are equal length.
d. Diagonals are perpendicular.
e. Diagonals do not intersect.
f. Diagonals are perpendicular bisectors.

3) On our Area on a Grid class workshop, find the areas of the shapes using formulas.

Area on a Grid


4) On graph paper, divide a square up into exactly 7 triangles with as many different triangle types as possible. Can you get all 7 types?

5) Area
a) Make an area formula for a trapezoid, or prove the one you know, using the formulas for rectangles and triangles.
b) Make an area formula for a kite.
c) Make an area formula for a chevron, using the diagonal lengths.

6) On graph paper, find squares with areas listed or argue why you can’t: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
a. Note: 5 is possible.
b. Find the edge length for each square you found using the Pythagorean Theorem. What do you notice?

7) Draw a 2 circle Venn diagram with the labels only in your mind (eg. a line of symmetry and at least one pair of parallel sides). Write in the quadrilateral types where they go, and give to a tablemate to figure out the labels.

Growth Model

Dweck.

Last summer, my colleague Dave Coffey's summer reading revolved around the idea of a growth mindset and its relationahip with learning.  Carol Dweck is the most often cited researcher in this field.  There's a pretty good piece on her and the related research in a recent Chronicle of Higher Education article.  Some of her ideas have been popularized by Daniel Pink.  (Jamie Feild Baker has a post about that connection.)  There's a commercialization of her work by Brainology, that seems to be based on proving to students that this brain research is true by showing them it's how their brain works.  (Brainology is also on Twitter, it turns out. Not bad linkage.)

The oversimplified summary of the research is that people's growth is limited by their own assessment of their possibility for growth.  People who think ability is fixed find it hard to grow.  People who think it's possible to grow find it hard not to.

My response was to try asking Dr. Dweck's questions of my future teachers and see what happened.  It was definitely interesting, and the discussion of the growth mindset idea was definitely a great discussion.  But there's always so much to do, that I just let it slide.

Recently Sue Van Hattum (aka Math Mama Writes) posted that she was thinking about it also and put up a sample questionaire.  I dug up the one I had given last fall to send her, and then started thinking about making it more math-centered.  And that's what I wanted to share today.  I merged them with some mathematical process questions I had used before, and a few other math attitude chestnuts.  I am more and more convinced that assessment and evaluation is where I need to concentrate so that I can teach my actual students and not figments of my imagination.

Dweck's original questions (2 forms)

Implicit Theories of Intelligence Self-Eval.v2

Modified for math

Implicit Theory of Mathematics Learning

A time problem

Wrote this problem for my spring final and fell a little too in love with it.  Has some fun algebra behind it.

Jane glanced up at the clock and noticed that when the second hand was on the 12, the three clock hands divided up the clock into a right, acute and obtuse angle.  What time was it:

• 3:30, 
• 5: 43, 
• 1:22, 
• 9:15, 
• more than one possibility or 
• none of those  

Remember the hour hand moves during the hour.

When grading, many students ignored the idea of the hour hand moving in between, so I evaluated based on their assumptions.  In general on the test, I was trying to create the possibility of seeing some problem solving, where they could demonstrate understanding of ideas without necessarily having to get a right answer.  It was partially successful.

As I was trying to write the problem, I posed myself this question: 

If it is y o'clock and x minutes past the hour, what is the angle formed by the clock hands?

If you're considering either, I'd love to hear what you think in the comments.  How do you evaluate the first?  In the second, would you expect the equation to be linear?  Why?

Cartoon from xkcd, of course.



Some of the exam problems were pretty open-ended, like:

1.    Find an L-shaped figure with an area of 84 sq.cm and a perimeter of 44 cm.  Is there more than one?
2.    What kind of triangles can be made by connecting vertices on a regular octagon?  Specify the side-angle type.  Did you find all of the types? 

And some were more closed, but hopefully with multiple ways to do them.

5.    Sort the quadrilaterals into two overlapping Venn diagram circles: one for rotational symmetry, one for reflectional symmetry.  Quadrilaterals that don’t fit either should go outside.
6.    A Hershey’s chocolate bar is 43 g.  A kiss is 4.56g.  You remember that 1 pound is 454g and 1 pound is 16 oz.  How many ounces is a Hershey bar?  A Hershey kiss?  How many kisses in a bar?  (Make a joke if you want.)

Nobody made a joke.  How many kisses in a bar?  Come on!