I am spending a bit of time a couple days a week in kindergarten this year. It was part of the now-changed sabbatical plan, but important to me to follow through on.

Today was my first day. It was awesome.

The young ones are working on patterning. AB patterns, AAB patterns, ABB patterns and ABC patterns. I’ll leave the curricular questions for when I know more. Today I’ll take these activities at face value, which is to say: this is the mathematics these children were working on today.

The children were instructed to use square tiles to make an ABC pattern. If you haven’t spent time studying curricular approaches to patterning in early elementary, this means that they were to use the tiles to make something such as this:

Color is the only variable attribute of the tiles the children were using.

I had several interesting conversations about this task with children today. The one I want to report is the following.

I made this pattern:

I asked the girl sitting next to me whether I had made an ABC pattern.

Girl: Yes.

Me: How do you know?

Girl: [blank look; long pause]

Me: Show me how you know it’s an ABC pattern.

She carefully points to each tile, saying one letter per tile in the following way.

She pauses.

Girl: And you need another C.

Learning is messy—beautifully messy.

I left today with two big questions on my mind; each relating to this exchange, and to others not documented here.

What would these children have done if asked—prior to instruction—to make a pattern with their tiles?

How does this kind of patterning work interact with learning to count?

I hope my readers will see that these are not questions I expect to be answered in the comments. I hope you will see that these are big and important questions worthy of wondering about for days, weeks, and beyond. I hope you’ll join me in wondering about these questions, and the consequences of potential answers to them.

Here are a few questions that have surfaced in the early weeks of the semester. These are all student questions in College Algebra.

(1) Can it still be a variable if it only has one value?

This was asked by a student as we were sorting out whether counts as a function, and whether it counts as a one-to-one function.

(2) How do you solve for ?

This was asked by a student as were considering the relationships among functions, inverses and inverse functions.

(3) Is the inverse of a circle an inside-out circle?

See, we were using a set of equations, considering x as the domain and y as the range. We were asking whether each equation—so viewed—is a function and whether it is one-to-one.

Then we were switching domain and range (i.e. swapping x and y) and asking the same questions about this new equation. Bonus question was to solve each of the new equations for y.

One of our equations was . Swap and and get back the same thing. Thus, a circle (as a relation) is its own inverse. Which fact I had never considered.

But my purpose here is to check in on the progress I am making in fostering and noticing student questions as evidence of learning.

One thing Malke Rosenfeld and I agreed on over breakfast the other day is that the question, What did you learn? makes us uncomfortable. Weird, right? We are teachers and find both answering and asking this question makes us uncomfortable.

I have many reasons for not liking the question: that it implies the process has ended; that when I ask it of my students, they may be inclined to say what they think I want to hear; that it doesn’t invite further questions; on and on.

This coming school year, I will characterize learning—for myself and for students—in the following way.

Learning is having new questions to ask.

If I have learned something, it is because I can ask questions that I previously could not. Some examples…

example 1: Algebra II

Reading Nicholson Baker’s article on Algebra II in Harper’s [behind pay wall; also available at your local library. And seriously, a Harper’s subscription is like $15 a year.] recently, I didn’t learn anything. Much of what he had to say about the course and the way students experience it is pretty familiar and the tone resonated with many of my feelings. But when I read Jose Vilson’s response to it, I had questions. Jose writes,

If someone said, “Let’s end compulsory higher-order math tomorrow,” and the fallout happens across racial, gender, class lines, then I could be convinced that this was a step towards reform.

I wondered whether I would view Algebra II differently if I were a man (or woman) of color. I wondered yet again about the place and effect of developmental math and College Algebra on the economically and culturally diverse population of my community college. I have new questions to ask, so I learned something from my colleague Mr. Vilson that I didn’t learn from Mr. Baker.

And you are reading Jose Vilson’s blog on a regular basis, right? If not, now would be a good time to start.

Example 2: Percussive Dance

At Malke’s workshop this week, I began asking about:

the relationship between variable and attribute,

the importance of decomposing things by their attributes and paying attention to one of these attributes at a time, and whether that is a fundamental characteristic of mathematical activity,

whether a characteristic of a novice is an inability to distinguish noise from pattern,

how children’s experiences with sameness in their non-mathematical lives informs and constrains their ability to work with sameness in mathematics,

whether I was taking seriously my responsibility and opportunity to use physical classroom space for student learning, and

what kinds of equivalence relations we could use in Malke’s percussive dance work, and whether we can form a group from the resulting elements, together with composition (my hunch is yes and that the resulting group is non-Abelian, but I haven’t worked out the details).

Now you should watch Malke in action. I’ll be surprised if this 3-minute video doesn’t give you some new questions to ask.

Conclusion

See, in math classes asking questions is usually a sign that youhave not learned.

“Any questions?” is a signal to students to speak up if they don’t get what has just been explained.

We have it all backwards.

It shouldn’t be, “What questions do you have?” [I hope you have none so that I can tell myself you learnedsomething.]

It should be, “What new questions can you ask?” [I hope you have some because otherwise our work is having no effect on your mind.]

—

*Asked by someone who is not Malke, for the record.

Quick break from prepping and grading final exams.

My future elementary teachers always struggle to name the denominator when they need to find of .

They draw the picture.

They know that the numerator needs to be 3. And then they argue about whether the denominator should be 12 or 16.

I struggle every year to get 8 on the table as an acceptable answer. I usually end up being a voice of authority for 8, and we discuss what the whole is if you use 12 or 16 as the denominator.

My students don’t like 8 because that means the answer is of one square, but the pieces come from different squares.

This year, I had an insight that helped a lot. The question was this:

What are some situations in life when you get two same-sized parts of distinct wholes?

I opened the class session following our usual denominator debate with this question and it helped us to focus on the issue at hand.

After a few false starts (i.e. examples that didn’t really exemplify what we were after), we settled on this scenario.

When you buy a 75¢ pop from a vending machine, by inserting a dollar you get back a quarter. Do it again and now you have two quarters. Each quarter came from a different dollar, but they are still quarters. Each is one-fourth of a dollar and together they are half a dollar (even though collectively, they are one-fourth of the money you started with).

Back to the squares and we had a frame of reference for eighths.

I happened across a review sheet for another instructor’s College Algebra exam today. I know not whose, nor do I wish to know. I just want to use it as an example of what my poor students have to go through.

There were eight tasks on the review sheet. I would like my students to have the skills represented in those tasks for sure. But I wouldn’t happy with just those skills.

So here are the tasks. Each is followed up by the sort of question I would ask my students on an exam. Pity them.

Original: Find the domain and range of . Follow up: Give two more functions: one that has the same range as f but a different domain, and one that has the same domain as f but a different range.

Original: Is even or odd or neither. Follow up: Can a function be both?

Original: Solve the absolute value inequality and graph the solutions. Follow up: How do these solutions relate to the function ?

Original: Graph the function . Then find the intervals on which f is increasing and decreasing. Are there any local maxima or minima? If yes, where are they located? Follow up: Choose two points near a maximum or minimum value (if such a value exists). Find and comment on the average rate of change between these two points.

Original: If , find the net change and the average rate of change between and Follow up: Why are these values different? BONUS: Give a new function for which these values are equal between the same two points.

Original: If , write the transformations that yield . Also graph both and on the same coordinate axes. Follow up: How many solutions are there to this system of equations?

Original: If and , then what are and ? Also list the domain for each case. Follow up: Choose one of the four functions you wrote. Write its inverse (if such a thing exists).

Original: Graph this piecewise function Follow up: There is a gap in the graph. Change the second piece of the function to eliminate this gap.

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