## Posts Tagged ‘Elementary Education’

### Review of _The Schools We Need_ by E.D. Hirsch Jr.

November 21, 2013

I read The Schools We Need by E.D. Hirsch last month, and I wanted to get my ideas down here. This is a post that I had hoped to spend more time on, but I have had a tough time finding time to blog about it. I have 30 minutes now, so I am going to see what I can do.

I wanted to read Hirsch’s book for a couple of reasons. I have heard about Hirsch since I was an undergraduate. I have always viewed him as a “bad guy” in education, in that Hirsch and I would probably disagree about a lot of things. But I was never really informed about his views, and I was hoping this book would help (spoiler: it only sort of did); if I am going to disagree with someone, I figured I should know what they actually are saying. Additionally, we have friends with kids in a Hirsch-inspired charter school, and I wanted to be able to speak knowledgeably to them about the school.

I had a tough time figuring out what to think about the book. I vacillated between thinking that his ideas were completely uncontroversial and thinking that his ideas were bad. In the end, my opinion is that he has some reasonable ideas, although he has more bad ideas (again, my opinion). Most of all, it seems to me like he mostly likes attacking straw men.

Here is my summary of his ideas (again, I read this book a month ago, so take this with a grain of salt): a big problem with education is that different students learn different things in grade $n$, which makes it difficult to teach grade $n+1$. Compounding this is that the U.S. has a pretty transient student population, so it can be impossible to know what a transfer student knows. His solution is to have a set of national standards.

But more than this, he thinks that the standards should be a set of facts that students know. For instance, it is very important to know what the capital of Egypt is after the first grade.

He emphasizes that these facts need not be learned through rote memorization. On the other hand, all of his recommendations about what to do seem to suggest that he thinks that rote memorization is the way to do.

My biggest question is whether he correctly describes the attitudes of K-12 teachers. He repeatedly talks about K–12 teachers’ disdain for facts. Listening to Hirsch, one would think that K–12 teachers go out of their way to make sure that students don’t learn any facts; that is how much he thinks that the teachers hate facts.

I have spent some time around teachers—enough to see how one could possibly get this impression. I have heard teachers say things to the effect of, “They just want us to teach the kids a bunch of facts.” But my interpretation of this is that the teachers were complaining that they were being told to only teach facts, and nothing else.

(Coincidentally, I also recently learned about classical homeschooling. This seems to be the sort of fact-based education that Hirsch might like).

I also found it interesting that he complains that progressive educators say that progressive education has never been tried and should be given a chance, when (Hirsch says) we have actually had a progressive education system for almost 100 years. He then goes on to complain that a traditional education has never been tried (recently, anyway), and should be given a chance. So Hirsch makes exactly the same complaint that the progressives do, yet provides little evidence that he is more correct than they are.

Here are my main takeaways:

1. Hirsch is helping to convince me that some sort of national standards is probably a good idea. I was leaning this way already, although I still could change my mind on this.
2. I would like to find out if the culture of K–12 education is as hostile toward facts as he says it is. I suspect that he is wrong about this, but I would like to hear from people who know more about this than I do.
3. Even though I understood much of what he wrote as being very reasonable (I am a big fan of facts), I think that I am not correctly understanding the severity of his stance. He makes several statements that suggest that he is much more extreme than I would like (e.g. he seems to implicitly endorse doing a lot of rote memorization).

Any sort of background on Hirsch’s ideas would be welcome in the comments.

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### The Many Ways of IBL Conference

June 26, 2013

I attended the University of Chicago’s “Many Ways of IBL” conference last week. Here is a brief list of my thoughts for the week, in no particular order.

1. It was utterly great to see a couple old friends. I have been blessed to have had good colleagues everywhere I have been, and I wish that I could have taken many of them with me to my current position.
2. It was great to meet a bunch of new friends. I hope to stay in touch with many of them.
3. Part of the conference was to watch John Boller teach an IBL class on real analysis to a bunch of super-motivated high school students. Both John and the students did a fantastic job. I told John that it was so enjoyable that he could charge admission.
4. One big thing I was failing at with IBL last year: I did not discuss the statements and meanings of the theorems before students presented. Boller did this, and it must help students understand everything about the course better.
5. Paul Sally continues to be amazing. He is also hilarious.
6. In many classes, I have students read the textbook rather than lecture. I have no idea how to mesh this with IBL, but it is something I value. I realized from the conference that the reason why I value this is that it helps students learn how to learn on their own.
7. Even though I have been calling my recent hybrid classes “a mix of Peer Instruction (PI) and IBL,” I no longer think that I have been doing IBL. At best, it is IBL-Lite, although it is probably just “students presenting problems.”
8. This will lead me to alter a paper that I recently wrote on a PI/”IBL” calculus class; I will now qualify that my IBL is pretty weak.
9. I am now fairly certain that my courses for pre-service elementary education majors are IBL.
10. I might do IBL in my abstract algebra course this spring. If so, I might interweave IBL and PI differently: I might mainly do IBL, but then have some PI days to make sure students understand the ideas that have already been presented.
11. In abstract algebra, I might also create a class journal, where students can submit homework problems to an editorial board (of students) for peer review.
12. In IBL classes, have students take pictures of the board work. They can then upload the pictures to the course website as a record of what happened.
13. Matthew Leingang gave me a nice way of communicating course rules. He has “The Vegas Rule” for his class: “What happens in Vegas, stays in Vegas” where “Vegas” is defined as “the world outside of this classroom.” This is a nice concise way of reminding students to not use previous knowledge and outside sources.
14. Leingang also got me excited about paperless grading. Now I just need to find \$1200 for an iPad and scanner.
15. Ken Gross uses an “adjective-noun” metaphor for fractions, where the adjective is the number and the noun is the whole. That is, you can explain common denominators by doing something like: $2/3$ “units” $+ 1/2$ “units” equals $4/6$ “units” $+ 3/6$ “units,” which is equivalent to $4$ “sixths of a unit” $+ 3$ “sixths of a units” $= 7$ “sixths of a unit” $= 7/6$ “units.” Most of the work then is just changing the “noun” and finding the appropriate “adjective” for each of the new nouns.

### What oral exams taught me

June 8, 2012

In my course for elementary education students, I once again gave oral exams—this time for the final exam. Here are two take-aways from the oral exams.

First, I need to do some peer instruction next time. In particular, students had a difficult time understanding the difference between the “whole” of a fraction and the “denominator” of a fraction (Consider “$\frac{1}{2}$ of a mouse” and “$\frac{1}{2}$ of an elephant.” Both have a denominator of “2,” but the whole of the first is “mouse” and the whole of the second is “elephant.” This leads to different meanings. I think that three clicker questions would eliminate this.

Second, I was shocked at how ineffective my lectures were. The oral exam questions (which they also had to create screencasts for) were ones that were previous done in class (for example: why does inverting and multiplying give the correct answer to a division problem?). The process was this: students would figure out why the algorithm works, and then present at the end of a class period. I begin the next class period by giving the same argument. Other class periods begin with students presenting on similar questions, the class evaluating the presentations, and—if needed—me presenting the correct explanation.

Furthermore, I gave the answers to each of the oral exam questions on the last day of class. Test test So students saw the answer to each oral exam question at least three times, and probably more (especially since I had students view other students’ video solutions).

I was concerned that students would simply memorize these explanations. This simply did not happen. Either students understood the algorithm (I can tell from the oral exams—these students could answer any question that I had on the algorithm) or students did not understand any portion of the algorithm.

Most puzzling is that, in my student evaluations, some of my students complained that they were never shown how to do the algorithms correctly. This is in spite of seeing a completely correct solution to every problem between 3 and 10 times. I can only explain this in two ways:

1. Somehow students did not understand that the solutions they saw were solutions to the problems from the oral exams and screencasts. This would mean that I did not clearly communicate the intent of presenting the solutions.
2. Lecture was monumentally ineffective in helping them learn—so much so that students did not even remember that they occurred.

Do you have any other ideas?

### Jigsawing

April 19, 2012

My elementary education students are creating vlogs that explain why different algorithms work for different operations. They have been creating roughly one video per week, posting them, and then getting feedback from the course grader. The only graded part of this is at the end of the semester after many drafts.

This week, we did a jigsawing-type activity to improve the videos (like most everything else, this idea was inspired by Andy Rundquist. On Tuesday, I split the students into four groups: one for addition, one for subtraction, one for multiplication, and one for division. The students came to class having watched all of the videos on their particular operation, and the class period was spent deciding what makes for a good explanation for that operation. At the end of the class, we split into new groups where one member of the group had just studied addition, one subtraction, one multiplication, and one division.

Today, we spent the entire class period reviewing videos in these teams. One team member was an “expert” on each operation from Tuesday, and they made suggestions on how to improve the explanations.

I asked everyone if this was useful enough to repeat on our fractions algorithms, and every student said that it was (most were emphatic). This appears to be a success.

My one reservation: although I am not sure, it appears that some students are trying to memorize a good explanation rather than understand. I know that I will be able to tell which students really understand from the oral exams, but I am wondering if it will be clear from the videos. Does anyone have any experience with memorizers?

### Assessing with Student-Generated Videos

January 17, 2012

I regularly teach a course for future elementary education majors. The point of the class is for the students to be able to do things like explain why you “invert and multiply” when you want to divide fractions. This involves defining division (which, itself, requires two definitions—measurement division and partitive division are conceptually different), determining the answer using the definition, and justifying why the “invert and multiply” algorithm is guaranteed to give the same answer. At this stage, I simply tweak the course from semester to semester. This semester, though, I am making a major change in how I will assess the students.

Since this class is for future teachers, it makes sense to assess them teaching ideas. So there are three main ways of assessing the students this semester:

1. The students will have two examinations. Part of each examination will be standard (a take-home portion and an in-class portion), but there will also be an oral part of the examination. The oral portion will require students to explain why portions of the standard arithmetic algorithms work the way they do.

I only have 31 students in this class (I have two sections), so hopefully this will be doable. Moreover, I am going to distribute the in-class portion of the exams over a period of weeks: many classes will have a 5 minute quiz that will actually be a portion of the midterm.

2. The students will regularly be presenting on the standard algorithms in class. This is only for feedback, and not for a grade. I am hoping that the audience will listen more skeptically to another student than they listen to me.
3. The students will be creating short screencasts explaining each of the standard algorithms (Thanks to Andy Rundquist for this idea). Students will be given feedback throughout the semester on how to improve their screencasts, but they will create a final portfolio blog that contains all of their (hopefully improved) screencasts for the semester. This portfolio blog will be graded.

I will keep you posted. I welcome any ideas on how to improve this.

### Scholarship and Creativity Day 2011

May 5, 2011

As I did last semester, I had my students (all elementary education majors) do mini-research projects and present at a small poster session.

As before, these posters were optional, although a student cannot get an A for the semester without doing one. I have 37 students, and 24 choose to do a poster. Unlike last semester, there was no paper that accompanied the poster.

Also unlike last semester, I did not hold the poster session during class time. Instead, I integrated it into the campus-wide “Scholarship and Creativity Day.” There were no classes this day—it is a day completely devoted to showing off students’ creative projects.

Here were my suggested projects:

1. Note that $\frac{1}{2}=0.5$ and $\frac{3}{4}=0.75$ do not have repeating decimals; we say that they “terminate.” How can you tell which fractions in Martian arithmetic will terminate?
2. Consider extensions of our Last Cookie game (basically, a Nim game). What is you could remove either 2 or 3 cookies per round, but not 1? What if you could do 1,2, or 4 What about other combinations?
3. There is a division algorithm called “Egyptian division.” Explain (as we have been doing) why this gives the correct answer to a division problem.
4. Learn about “casting out nines,” a method that helps you determine if you did an arithmetic question correctly. Explain why this method works.
5. There is a fast and easy way to determine if a number is divisible by 3 in base ten. Explain why this method works.
6. There is are not-so-fast and not-so-easy ways to determine if a number is divisible by 7 in base ten. Explain why one of these methods work.
7. Explain divisibilty results for other bases (can you easily tell when a number is even/divisible by 3/5/7/etc in base six? Base eight?)
8. Research one algorithm from the Trachtenberg System, and explain why it is guaranteed to give the correct answer.
9. Teach Mayan students how to use our number system.
10. Come up with your own topic (talk to me about it first).

By far, most students choose the “divisibility by 3” or “casting out nines” problems, a reasonable amount choose “teach Mayan students about base ten” “the Last Cookie” problem. Three others did a Trachtenberg problem, one student chose to explain “Egyptian Division,” and two explained why a finger trick works for multiplication by nine.

Many of the presentations were excellent, and many still had trouble understanding what the question is. This was expected. What was not expected was the number of students who participated: I expected about half the number I had.

Finally, many professors from other departments approached me to compliment the poster session. In fact, the dean of the college referenced one of my students’ posters in an address later that evening.

I must remember to try to do this again in most of my classes.

### Course Projects

April 11, 2011

I have decided to start incorporating projects in most of my courses. This semester, I am teaching a content course for elementary education majors, and I am again doing projects. See here for a list of potential projects and a brief summary of the format.

In short, the projects are required if you want to get an A in the course, but optional otherwise. However, doing a project might help a student’s grade if it is lower than an A (it might be the difference between, say, a BC and a B).

I am looking forward to the poster session, which will probably be in a couple of weeks.

### Assessment FOR Learning, Take 1

April 1, 2011

I love working with pre-service elementary education majors. I frequently teach their content courses, and I am teaching them this semester. I usually spend a decent amount of time in my elementary education courses having the students explain why the standard algorithms for the operations on integers and fractions give correct answers. That is, I work to get the students to understand how the algorithm relates to the definition of the operation. This is something that they always have trouble with (I have taught the course 5-6 times).

But I think that this semester may be significantly better. The reason why is that I am applying techniques I learned in Black’s Assessment for Learning: Putting it into Practice. Here is what we did in class yesterday:

1. I had the students determine qualities that make an explanation “good.” I prodded them on a couple of these, but we came up with:
• The explanation is relevant; that is, the explanation answers the question at hand.
• The explanation is appropriate for the audience (i.e. the explanation uses knowledge common to both the explainer and the explainee).
• The answer is correct.
• The answer is complete; there are no gaps that the audience would need to understand the explanation.
• The answer is concise; it is long enough, but no longer.
2. I gave them three explanations for why the standard addition algorithm is really the same as definition of addition (roughly, “combining and counting”). Here are the explanation: one was decent, another was solely an explanation of how (not why) the algorithm works, and a third was somewhere in between.
3. I asked them how well each of the explanations did in each of our categories from 1.
4. Initially, the students all loved the “how but not why” explanation (the second one). But when we delved into relevance, several students started saying that it did not answer the question. I could almost literally see light bulbs going off over several of the students’ heads. I think that this will greatly help their justification of several multiplication and division algorithms; I will keep you posted.

In some sense, I am kicking myself for not doing this before. I have (in theory, at least) been a proponent of helping students develop their metacognitive skills. It seems like that is what I was doing yesterday: giving them tools to think about how they are thinking about explanations.

### APOS and Computer Programming in Mathematics Classes

March 4, 2011

My students had their first programming experience this week, and it went reasonably well. I perhaps gave too much background on programming, resulting in only a small amount of class time being devoted to have the students write code. In fact, here is an outline of the day:

1. I showed them how to use codepad.org.
2. I showed them if/elif/else statements in Python.
3. I showed them for loops in Python.
4. We created a program that would add two single-digit numbers in base six.

This took the entire class period, leaving the students almost no time to start writing a program that would add two four-digit numbers in base six.

Finishing that program was one of several ways students could get credit in the course. As I expected, some students decided to do the coding, although most did not.

I had been wondering about the worth of doing this in a class for elementary education courses. I am attempting to focus on why these algorithms work, whereas the programming seems to only help with understanding how to do the algorithms. I am still undecided for this class, although Ed Dubinsky and Robert Moses wrote an interesting article for this month’s AMS Notices (I also recommend reading Beckmann’s article and Wu’s article).

In short, Dubinsky and Moses write about APOS theory, which roughly (I am not an expert) says that students progress through several stages of understanding any sort of mathematical idea:

1. Action: a student can do a particular mathematical “move,” but cannot really reflect on that move. For instance, a student could cube a given number.
2. Procedure: a student can reflect on the action, and consider it abstractly. For instance, a student could imagine cubing a number without actually doing it. Now, for instance, a student might be able to think of how you one might attempt to find an inverse to cubing a number.
3. Object: a student things of the former action/procedure as a “thing,” and realizes that he/she can act on it. For instance, the student could imagine taking $f(x)=x^3$, and transforming it by taking the derivative. The former action/process becomes more “tangible.”
4. Schema: I am particularly hazy on this one, but it seems to me like a student is able to abstract the object and consider other actions and procedures that are related to it. My best guess is that the student might realize that $f(x)=x^3$ fits into a larger category called functions, and one can add/compose/differentiate this type of object.

To be sure, students do not progress through these four stages in a step-like manner; it is messier than that (it is sometimes useful to think of something as a Process, and other times it is useful to think of it as an Object). I would also welcome people to correct my inevitable misconceptions on APOS theory in the comments.

Anyway, Dubinsky and Moses wrote that computer programming can help students rapidly ascend through these four steps. Once the student has already internalized a mathematical idea into an Action, having the student write a computer program is a great help to turn that idea into a Process. Having the student code a different program that calls the first program turns that idea into an Object. “Going back and forth between object and process conceptualizations of a mathematical idea, so necessary in doing mathematics, resulted from this pedagogy almost effortlessly (Weller et al., 2003).”

This makes me think that requiring my elementary education students to code might be a good idea, since it would seem that they would definitely need to view addition, say, as a Process (rather than solely as an Action).

Comments on the programming project’s worth in my class an on APOS Theory in general are welcome.

### Library of Virtual Manipulatives

February 5, 2011

I am teaching mathematics for elementary education majors this semester, and we are currently discussing number systems. In discussing our base ten number system, I found the Library of Virtual Manipulative’s section on base blocks to be extremely useful. It turns out that it takes much less time to click on a button to create a virtual 10-by-10-by-10 cube than it does to build an actual 10-by-10-by-10 cube. This allowed me to run through many examples in class.