reflection – Page 3

Summer Updates

One of my favorite parts of summer is reflecting on the past year and brainstorming new ideas for the next. On my mind these days:

Refining my standards based grading system after this past semester and year’s implementation
Building my IB courses for math(s) and physics, which will have both HL and SL in the same class period
Sharing ways that programming has made my teaching life easier and richer
Better making the most of in-class time, as well as maximizing the benefit of time students spend on their own

There are posts brewing in my head on each of these. At the moment, I’m on a road trip headed west and plan to enjoy my time enjoying the views and life, so these will likely live only in my head for now.

Stay tuned for the roll out.

Curated review for finals

I really don’t like reviewing for exams. I don’t think I’m the only one that thinks this, by far.

If I create a the review sheet, I’m the one going through all of the content of the unit and identifying what might be important. It would be much more valuable to have students do this. I’ve also been filling the school server with notes and handouts of what we do each day, so they could be the ones deciding which problems are representative of the unit.

Suppose I do make a new set of review problems available to students. If students have this set of problems to work through during class, I spend my time circulating and answering questions and giving feedback, which is the best use of my time with students. Better yet, students answer each other questions, and give each other feedback. They lose the opportunity to see the scope of the entire semester themselves because, outside of the set of problems I prepare for them, they don’t actually take the time to see that scope on their own. They only see my curated sample and interpret it according to their own understanding of the relationship between review problems I select and problems I select for an exam.

I’ve had students themselves create review sheets, but this always has its own set of issues. Is it on paper or online? If on paper, how does this sheet efficiently get shared with other students? The benefit of an online resource is the ease of sharing. The difficulty comes from (1) the difficulty of communicating mathematics on a computer and (2) compiling that resource in one place. It’s a lot of work to scan student work and paste it into a document. Unless I am meticulous in making sure that all students are using the same program (which is a lot of work for a class of twenty-four students all with their own laptops) this becomes a lot of work (again) for me. I’ll do it if I really believe it is worth the effort for students, but I’m always looking to be efficient in that effort. I also don’t want to put this effort on the shoulders of a student to together. And before someone tells me to use Google Docs and its amazing collaborative tools, I’ll bring up the governmental disruption of Google services and leave it to you to figure out why that isn’t an option for me and my students.

In the end, I have to decide which is the most valuable for students relative to a review. Is it getting feedback on what a student does and does not understand? Is it going back over the entire semester’s material and figuring out what is important relative to a cumulative final?

If I have to pick a theme of my online experiments this year, it has been the search for effective ways to leverage social pressure and student use of technology to improve the quality of the time we spend in the classroom together. In the past, I have been the one collecting student work and putting it in one place when I’ve tried doing things differently for exam review. That organization is precisely something computers do well if we design a scheme for them to use.

Here’s what I have had students do this year:

Each student has a blog where they post their own review sheet for one standard. They submit the URL of their post and their standard number through the same site through which they sign up for SBG reassessments. They see a list of the pages submitted by other students:

This serves as a central portal through which students can access each other’s pages. Each student controls their own page and URL information, which saves me the effort to collect it all.

Why am I really excited about this list?

I curate the list. I decide whether a page has met the requirements of the assignment, and students can see those pages with a checkmark and a WB for my initials. If a student needs to improve something, I can tell them specifically what isn’t meeting the requirements and help them fix it. Everyone doesn’t have to wait for everyone else to be finished for the review process to begin. I don’t decide what goes into each page generally, but I do help students decide what should be there. Beyond that, I don’t have to do any compilation myself.
Students (ideally) vote on a page if they think it meets the requirements. Students can each vote once for each page, and see a checkmark once they have voted. This gets them thinking about the quality of what they see in the work of other students. I have been largely impressed with what students have put together for this project, and students are being fairly generous with this. I’m ok with that at this point because of the next point:
Students have an incentive to actually visit each other’s pages. I have no idea how many students actually use the review sheets we’ve produced together in the past. I doubt it is very many. There’s some aspect of game theory involved here, but if a student sees that others are visiting his or her own pages, that student might feel more compelled to visit the pages of other students. Everyone benefits from seeing what everyone else is doing. If some review happens as a result, that’s a major bonus. They love seeing the numbers adjust real time as votes come in. There is a requirement that each vote include a code that is embedded in the post they are voting for, just so someone isn’t voting for them all without visiting the page.
Students were actually using the pages to review today. Students were answering each other’s questions and getting feedback sometimes from the authors themselves.
I get to have valuable conversations about citing resources online.

Right now, students can vote as much as they want, but I plan to introduce one more voting option before this is entirely done which allows students to vote on their top three favorites in terms of usefulness. I am not sure how I would do this without it turning into a popularity contest, but I might try it and see how their sense of quality relates to mine. I would also love to use this next year as a Reddit style resource where students are posting problems and solutions potentially for specific standards and can vote on what is particularly helpful to them. Again, just an experiment.

I really loved how engaged students were today in either developing their pages or working on each other’s review problems. It was one of the most productive review days I’ve had, particularly in light of the fact that I didn’t have to write a single problem of my own. I did have to write the code, of course, but that was a lot more interesting to me today than thinking of interesting assessment items that I’d rather just put on an exam.

Testing probability theories with students

One of the things that has excited me after building computational tools for my students is using those tools to facilitate play. I really enjoyed, for example, doing Dan Meyer’s money duck lesson with my 10th grade students as the opener for the probability unit. My experiences doing it weren’t substantially different that what others have written about it, so I won’t comment too much on that here.

The big thing that hampered the hook of the lesson (which motivated the need for knowing how to calculate expected value) was that about a third of the class took AP statistics this year, so they already knew how to do this. This knowledge spread quickly as the students taught the rest how to do it. It was a beautiful thing to watch.

I modified the sequel. I’ll explain, but first some back story.

My students have been using a tool I created for them to sign up for reassessments. Since they are all logged in there, I can also use those unique logins to track pretty much anything else I am interested in doing with them.

After learning a bit about crypto currency a couple of months ago, I found myself on this site related to gambling Doge coins. Doge coins is a virtual currency that isn’t in the news as much as Bitcoin and seems to have a more wholesome usage pattern since inception. What is interesting to me is not making money this way through speculation – that’s the unfortunate downside of any attempt to develop virtual currency. What I’ve been amazed by is the multitude of sites dedicated to gambling this virtual currency away. People have fun getting this currency and playing with it. You can get Dogecoins for free from different online faucets that will just give them away, and then gamble them to try to get more.

Long story short, I created my own currency called WeinbergCash. I gave all of my students $100 of WeinbergCash (after making clear written and verbal disclaimers that there is no real world value to this currency). More on this later.

After the Money Duck lesson, I gave my students the following options with which to manage their new fortune in WeinbergCash:

Then I waited.

After more than 3,000 clicks later, I had quite a bit of data to play with. I can see which wagers individual students are making. I can track the rise and fall of a user’s balance over time. More importantly, I can notice the fact that just over 50% of the students are choosing the 4x option, 30% chose 2x, and the remaining 20% chose 3x. Is this related to knowledge about expected value? I haven’t looked into it yet, but it’s there. To foster discussion today, I threw up a sample of WeinbergCash balance graphs like this:

Clearly most people are converging to the same result over time.

My interests in continuing this experiment are buzzing with two separate questions:

To what extent are students actually using expected value to play this game intelligently? If you make the calculations yourself, you might have an answer to this question. I haven’t parsed the data yet to see the relationship between balances and grade level, but I will say that most students are closer to zero than they are their starting balance. How do I best use this to discuss probability, uncertainty, predictions, volatility?
To what extent do students assign value to this currency? I briefly posted a realtime list of WeinbergCash totals in the classroom when I first showed them this activity. Students saw this and scrambled to click their little hearts away hoping to see their ranking rise (though it usually did the opposite). Does one student see value in this number merely because it reflects their performance relative to others? Is it merely having something (even though it is value-less by definition) and wanting more of it, knowing that such a possibility is potentially a click away?

I had a few students ask this afternoon if I could give them more so they could continue to play. One proposed that I give them an allowance every week or every day. Another said there should be a way to trade reassessment credits for WeinbergCash (which I will never do, by the way). Clearly they have fun doing this. The perplexing parts of this for me is first, why, and second, how do I use this to push students toward mastery of learning objectives?

I keep the real-time list open during the day, so if students are doing it during any of their academic classes, I just deactivate them from the gambling system. For me, it was more of an experiment and a way to gather data. I’d like to use this data as a way to teach students some basic database queries for the purposes of calculating experimental probability and statistics about people’s tendencies here. I think the potential for using this to generate conversation starters is pretty high, and definitely underutilized at this point. It might require a summer away from teaching duties to think about using this potential for good.

After a hiatus: circular functions

It has been a busy time in gealgerobophysiculus land. By land, I of course mean school, and by busy, I mean what results when you have multiple exciting projects going on, school functions to organize, and the normal operations of a classroom to sort through and organize.

I haven’t taught the unit circle in three years. Before that, I took the approach of throwing a definition of the radian up on the board and discussing it as this strange thing that mathematicians decided would be a good idea. When I learned this in high school, we did some cool activities involving string and wrapping functions. At that time, it wasn’t clear to me how the string wrapping around a circular object really related to measuring an angle around it. I was always relating the idea of the radian angle back to degrees, because the angle part never made sense.

After some thinking and coding, I put together an activity that I thought would make this concept more concrete for the students in my tenth grade class. You can check it out at http://apps.evanweinberg.org/circlemeetsradius/

It starts with the premise of moving around a circle at distances of integer multiples of the radius. Looking at your own work doesn’t really establish how this relates to measuring angles at all. When you look at what happens when many people do the same thing to differently sized circles, the result makes clear that this could be a fairly natural way to measure out angles:

I didn’t have the networked part of this applet working when I did this with students, so I collected screenshots of students and their different circles together. I asked students what they expected would be different about the locations of these six points for circles of different sizes, and there was pretty solid agreement that they would be in roughly the same point around the circle, but this was still too abstract to establish the idea that these points measure out angles. The students weren’t too surprised by the result, either, but I think the activity in this form still left me as the teacher to connect the dots.

I wish I had an extra day to configure the final screen of this activity. I wouldn’t have had to work so hard.

The rest of the unit walked the line from this concrete idea of moving around the circle up the ladder of abstraction to what we ask students to typically do with these functions. We went from identifying points around the circle for a given angle measured in radians, to using our knowledge of 30-60-90 triangles to find the coordinates of some of these points, to formal definitions of sine, cosine, and tangent functions using these points. Every time I could, I related this idea back to the first activity of moving around the circle, but by the time we got to graphing these functions, I think I was demanding a high level of abstraction without also demanding the deliberate practice necessary to connect the angles and coordinates to each other. Students struggled to evaluate the trigonometric functions at different angles not because they couldn’t piece it together with time, but because they always felt compelled to go all the way back to the circle. I suppose it’s the trigonometric equivalent of going back to counting on your fingers.

I also was a bit disappointed to see that only a third of the class answered this question correctly on the unit exam:

To those that recognized the similarity to our opening activity, it was quite easy. The bulk did not see it this way though.

At this stage, however, I’m not too concerned. Many students admitted immediately after the exam that they did not practice the unit circle as much as they should have. They reported that they understood much of the unit up to the graphing part, where I think I pushed them a bit more quickly to piece together the graphs than would have been ideal for them to get an intuitive sense for them. I’m confident that a second and more rigorous look at these functions next year in IB year one will help solidify some of these concepts for them.

Standards Based Grading: Bridging the Gap

I ended last semester feeling pretty discouraged about my implementation of standards based grading. I still felt pretty good about what it was doing for students. For the students that were actually taking advantage of the system and initiating reassessments, it seemed to be helping. Some examples:

Students were getting frequent assessment over the different standards of the course through quizzes, and got feedback on their learning the same day since the quizzes were short.
Some students understood that they could request reassessment on their own to show progress on standards that previously were not marked as proficient.
On a day that I gave one of these standards quizzes, I was guaranteed to have a conversation on content, albeit sometimes a short one, with every single student. I can’t guarantee that I get to every student, every period, though I’d like to think that I do.
Students seemed to be less frustrated or scared by unit exams, and I received (anecdotally) fewer blank problems since students had an incentive to try anything.

By the end of the semester, I had 248 student-initiated reassessments. I don’t know how that compares to others, but that number seemed to be an indication that something was going well. Then I looked at the numbers. Of those reassessments, eighty-eight of them were in the last week of the semester. Roughly 1/3 of the total were taken by the top five most frequent assessors.

I wasn’t getting to many of the students. Despite my efforts to constantly ask students to do so, talk about opportunities to reassess in class, and target students that needed extra help with suggestions to come see me after school, some students didn’t know that they could reassess. Some students didn’t make the time. In fact, most didn’t. I was also tolerating a fairly heavy end of semester reassessment rush that seemed higher than what I’d like to see happen.

The other concept that also bugged me was that in this system, I wasn’t placing value on those behaviors that would lead to higher standards scores. No, I wasn’t about to go back to a system of checking homework, and giving extra credit – the grades reported through standards were a pretty good measure of what students knew. On an end of semester survey, a majority of students confirmed this fact. Students also understood that reassessment without preparation wouldn’t result in increased scores. I had students that said, however, that doing homework, getting help after school, teaching concepts to other students – those things that result in improved understanding – weren’t accounted for in the grade. If I valued those as part of the process, why was I not assigning any value to them in the grade?

I spent a lot of time thinking about this over the winter break. In my system, there was a fundamental gap between the behaviors I wanted students to develop, and the way I was measuring the learning happening in my classroom. Students understand the system this way:

There is a visible wall between what students know I value and what they see measured in their grade. This is the philosophical basis of standards based grading – the grade is a measure of learning content, not necessarily of completion of activities. It is fundamentally different from other grading systems. We don’t see it this way as teachers, however.

We see it like this:

I have the conversation with students all the time about the difference between these two conceptions. They do understand. It just doesn’t fit with the rest of what they have been told in school, where learning sometimes takes second fiddle to being a student.

This is a communication design problem. After a lot of thinking and designing, I came up with a new computer based system using the Meteor JS framework that I’ve been using since January.

Some parts of it have helped bridge this gap (and other problems) in some ways that have been quite productive – I’ll go through them piece by piece.

Problem 1 – Google Forms are not reliable in China.

I previously used a Google form to collect reassessment sign-up data. Students would frequently (and with good reason) say that they couldn’t log on to sign up because of this issue. While there are still internet issues with the new site, this is less of a problem. They can log in with their own email address, their own secure password, and I can control the input/output of information much more reliably since it’s a database, not a spreadsheet.

Problem 2 – I don’t want students signing up for reassessments the same day.

I really want students planning ahead. The Google form didn’t have the option of limiting which days they could sign up for based on the time of sign up, at least I didn’t see how to do this easily. It’s true that I could simply see which students signed up from the Google form results spreadsheet, but that involved scrolling, searching, and mentally filtering results. That was more work than I wanted to be doing.

With some Javascript, I make it impossible to do this. Students log in, click on ‘Reassessment Sign-up’ and see this form:

Problem 3 – Students forget which standard they signed up to reassess on.

I received an email every time students signed up for a reassessment through the Google form. I never figured out a slick way to do this through forms – I’m sure it’s possible. With this system, students can click on ‘My Reassessments’ and see what they’ve signed up to do:

If they miss the bus and have to reschedule for later the same day, they can log in and edit the reassessment themselves. Because of Meteor’s realtime system, those changes instantly show on my browser when they happen.

Problem 4 – Students sign up to reassess without any preparation.

This was a big one I wanted to solve. On my previous form, there was just a check box that said “I promise to prepare for this reassessment” along with a box where they describe what they did to prepare. I wasn’t directly able to check this, and usually didn’t have time to do so beyond noting that it was more substantial than ‘I studied’.

When students log in and sign up for a reassessment now, they see a little badge called ‘Reassessment Credits’. Each reassessment costs them one credit. What happens when they try to sign up for a reassessment without any credits left?

They can’t do so. No way around it, at least without learning some Javascript to hack my system.

This is where I find I am doing the work to educate students in what I am looking for. I value the work that they do in preparing for reassessments. I show that value through giving students credits, a token for whatever it is they choose to do to improve their understanding. These credits then directly convert into opportunities to show what they have learned through reassessment. I have the flexibility to give credits for different things that students do – tutoring other students, sitting in the classroom and doing homework with me, or simply showing me the homework they did after the previous class. This value is separate from the grade. Students have also found that through doing the work to earn a credit, I have directly increased their standards grade because I’ve seen evidence of higher understanding there.

Problem 5 – Students don’t know that their peers are reassessing.

I’m not ashamed to say that I also wanted to call upon the fact that students (and all of us) are influenced to some extent by our social community. I added one more part to the page that some students have reported has pushed them to sign up and complete reassessments.

When I first added this feature, a student said she was disappointed when she saw her proportion of the total decreasing, so she felt compelled to figure out what she might reassess over based on her progress.

Problem 6 – Staying organized with reassessments from four different classes is difficult

This system is computerized, so it’s easy to sort which reassessments I have on a given day.

Rather than skimming a spreadsheet and figuring out which quizzes to prepare, I have an easy list that allows me to prioritize and plan which quizzes I need to have available for which part of the day.

I’ve taken a lot of the obstacles out of the way that have made it stressful for me to stay organized in reassessments, and for students to opt not to do so. I am not trying to make a point about quantity vs. quality here, but I’m really pleased with how the system has worked for students.

So far this semester, I’ve had 221 reassessments through the system at this point in the semester, and we’re only in the first quarter. Thirty seven out of 55 students have assessed multiple times. Eleven students thus far have not reassessed at all, and I’ve seen that information easily through some database queries, which has prompted some conversations with those students about why they haven’t. It’s far easier than it used to be to get this information out.

Students also have great ideas for making it better. They know themselves, and know what would help them stay on top of their needs. I enjoy the challenge of learning to build this for them, and they seem to appreciate the effort.

I am not sharing the URL quite yet since there is student user information in there, and haven’t built a nice sandbox login yet to see the features. That will come when I have the time, whenever that is.

What do you mean by ‘play with the equation’?

My professional obsessions lately have focused on using technology to turn traditional processes of learning into more inquiry based ones. A really big part of this is purely in the presentation.

Here’s the traditional sequence:

Calculate the force between the Earth (mass 5.97 x 10^²⁴ kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^⁶meters.
If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
Find at what distance from Earth’s center the gravity force would be equal to 0.5 Newtons.

My observation in this type of sequence is that the weaker algebra students (or students that use the presence of mathematics as a way out of things) will turn off the moment you say calculate. The strong students will say ‘this is easy’, throw these in a calculator, and write down answers without units.

A modified presentation would involve showing students a spreadsheet that looks like this:

Calculate the force between the Earth (mass 5.97 x 10^²⁴ kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^⁶meters.
If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
Find at what distance from Earth’s center the gravity force would be equal to 0.5 Newtons.

I would argue that this is, for many students, just as much of a black box as the first. In this form, however, students are compelled to tinker and experiment. Look for patterns. Figure out what to change and what to keep the same. In the last question, students will likely guess and check, which may get tedious if the question changes. This tedium might motivate another approach. A mathematically strong student might double click on the force cell and look up the function, or look up Newton’s Law of Gravitation on Wikipedia and try to recreate the spreadsheet on his or her own. A weaker student might be able to play with the numbers and observe how doubling the mass doubles the force, and feel like he or she has a way to answer these questions, albeit inefficiently. Both students have a path to wrestle with the question that forms the basis of physics: how do we model what we observe in our universe?

This approach makes obvious what it means to play with a mathematical object such as an equation. Playing with an equation is something that I’ve admittedly said to students before in a purely algebraic context. I know that I mean to rearrange the equation and solve for the variable that a given question is asking for. Students don’t typically think this way in math or science, or any equation that we give them. If they do, it’s because we’ve artificially trained them to think that this is what experimentation in math looks like. I think that this is primarily because the user interface of math, which has been paper and pencil for thousands of years, doesn’t lend itself to this sort of experimentation easily. Sure, the computer is a different interface, and has its own input language that is sometimes quite different from mathematical language itself, but I think students are better at managing this gap than we might give them credit for.

Moving in circles, broom ball, and Newton’s cannonball

In physics today, we began our work in circular motion. I started by asking the class three questions:

When do you feel ‘heaviest’ on an elevator? When do you feel ‘lightest’?
When do you feel ‘heaviest’ on an airplane? When do you feel ‘lightest?’
When do you feel ‘heaviest’ on a swing? Lightest?

We discussed and shared ideas for a bit. I tried my hardest not to nudge anyone toward thinking they were right or wrong, as this was merely a test for intuition and experience. We then played a few rounds of circular ‘book’-ball, a variation of the standard modeling curriculum activity of broom ball from the modeling curriculum in which students use a textbook to push a ball in a circular path on a table. The students could not touch the ball with anything other than a single book at one time. A couple of students quickly established themselves as the masters:

I then had students draw the ball in three configurations as well as the force and velocity vectors for the ball at those locations. Students figured the right configuration much more quickly than in previous years:
I think some of our work emphasizing the perpendicular nature of the normal force on surfaces in previous units may have helped on this one.

We then took a look at some vertical circles and analyzed them using what we knew from the last unit on accelerated motion together with our new intuition about circular motion.

We finished the class playing with my most recent web-app, Newton’s Cannonball. We haven’t discussed orbits at all, but I wanted them to get an intuitive sense of the concept of how a projectile could theoretically go into orbit. This was the latest generation of my parabolas to orbits exploration concept.

Just shut up and work with us, Weinberg

I have an issue with talking too much in class. I think many of us do.

I’ve already done some focused work identifying what my students need me to show them for a given topic, and it’s a lot less than I initially think. After a conversation with some smart educators, I decided to commit this week to not do whole class instruction unless it was absolutely necessary.

Sometimes I confuse necessity with convenience. The problem is that it’s always convenient to do whole class instruction. You look out and see eyes staring at you, and it seems at the moment to be maximally efficient to communicate to the entire group at once. The quality of that attention is never what it seems.

In my biggest class, I’ve been continuing to put direct instruction into videos. As I’ve written previously, these are videos (three minutes or so) that have the information distilled down to small chunks. In doing this, I get around to every student and make sure they are somehow engaging with that video through writing down important information, trying the problem being demonstrated, or completing the challenge I usually put at the end. It’s impossible for me to be instructing at the front of the class (or anywhere for that matter) and be aware of what every student is doing. With the video at every student’s seat, I can be there. I can ask them questions one-on-one to see what they understand. I can make notes of the students that are struggling. I can assess every student at some point while I walk around, leave alone those that are doing just fine without my dictating their attention, and focus on those that need more guidance.

This increased time away from blabbing at the front of the room means more assessment time. The class starts with a quick quiz (1-2 questions) that I can get back to students during the period. I can give every student some bit of feedback, and it ensures that I have a conversation with every single student during the class. That is awesome. It means I can ask higher level questions of the stronger students and push them forward. It means I can see what students are writing down within seconds of doing so.

Though I occasionally think to myself that the reason this works is because my students are well behaved and will stay on task when I am not directly focused on them, I don’t think this is why it has been successful. I’m in the middle of my students (rather than in one location) the whole time. I can see what they are all doing. If they do get off task, they know that I know if because chances are I’ll be there in a minute or so. The class is noticeably less structured, and I don’t feel as productive as I think I would if I was marching through a lesson plan. This is more a reflection of how I now have a more realistic awareness of how my students are doing with the material, rather than in ten minute chunks of independent work between lecture.

The students benefit most from interacting with each other. They do occasionally need help from me one-on-one, but the nature of that help varies greatly between students. I can give that help when I’m not spending so much time talking. The inverse is more powerful there – I can’t give that help if I’m talking too much.

I decided to give students a quick exit survey on whether they liked the new format, whether they wanted to go back, or whether they wanted something different from both classroom structures. Here’s what they said:

I’ve gotten this sort of strong message before, but I unfortunately go back to the old ways, for the old reasons. It’s easier to talk. It’s easier to do a developmental lesson. It’s easier to ask a question and conclude from a one or two student non-random sample that the class gets it. It just isn’t necessarily what works best for students. I need to keep that in mind.

Programming and Making Use of Structure in Math

A tweet from James Tanton caught my eye last night:

Quickie: What is the sum of all products in a 10-by-10 multiplication table? What is sum of all products axbxc in 10-by-10-by-10 table?

— James Tanton (@jamestanton) October 13, 2013

Frequent readers likely know about my obsession with playing around the borders of computational thinking and mathematical reasoning. This question from James has some richness that I think brings out the strengths of considering both approaches quite nicely. For one of the few times I can remember since starting my teaching career, I went to a computational solution before analyzing it analytically.

A computational approach is pretty simple. In Python:

sum = 0 for i in range(1,11): for j in range(1,11): sum += i*j print(sum)

…and in Javascript:
sum = 0 for(i=1;i

The basic idea is the same in both languages. We iterate over each number in the first row and column of the multiplication table and add them up. From a first look, one could call this a brute force way to a solution, and therefore not elegant from a mathematical standpoint.

Taking this approach does, however, reveal some of the underlying mathematical structure that is needed to resolve this using other techniques. The sequence below is exactly how I analyzed the problem once I had written the program to solve it:

For a single row of the table, we are adding together the elements of that row. Instead of adding the individual elements together one by one, we could instead think about finding the sum of the elements of a single row, and then add together all of the rows. For example: $1 + 2 + 3 + ... + 10 = 55$. This is a simple arithmetic series.
Each row is the same as the row before it, aside from each element being multiplied by the first element in the row. Every row's sum therefore is being multiplied by the numbers in the first column of the table. $1(1+2+3...+10)+2(1+2+3...+10)+3(1+2+3+...+10)+...+10(1+2+3+...+10)$.
Taking this one step further, this is equivalent to the sum of that first row multiplying the sum of the first column: $(1 + 2 + 3 + ... + 10)(1 + 2 + 3 + ... + 10)$. In other words, the answer to our problem is really the square of the sum of that first row (or column), or 55*55.

I bring up this problem because I think it suggests a useful connection between a practical method of solving a problem, and what we often expect in the world of classroom mathematics. This is clearly a great application of concepts behind a traditional presentation of arithmetic series, and a teacher might give this as part of such a unit to see if students are able to see the structure of the arithmetic series formulas within it.

My question is what a teacher does if he or she presents this problem and the students don't make that connection. Is the next step a whole class discussion about how to proceed? Is it a leading question asking how arithmetic series applies here? This, by the way, zaps the whole point of the activity if the goal was to see if students see that underlying structure based on what they already know. Once this happens, it becomes yet another 'example' presented to the class.

I wonder what happens if a computer/spreadsheet solution is consistently recognized throughout the class as a viable tool to investigate problems like this. A computer solution is really nothing more than an abstraction of the process of adding the numbers together one by one. If a student did actually do this by hand, we'd groan and ask if they thought there was a better way, and the response inevitably is 'yes, but I don't know a better way'. In the way I found myself thinking about this problem last night, I started from the computational method, discovered the structure from those computations, and then found a path toward a more elegant solution using algebraic techniques.

In other words, I made use of the structure of my program to identify an analytical approach. Contrast this with a more traditional approach where we start with an abstract definition of an arithmetic series (by hand), do practice problems (by hand) and once we understand how it works, use computational shortcuts.

The consistent power that I see in approaching and developing ideas with students from a computational standpoint first is not that it often makes it easier to find an answer, though that can be a good thing when the goal is to find an answer. Computational methods can make it easy to change things around and generalize a problem - what Polya termed generalization. It's easy to change the Javascript program to this and ask what multiplication table it models:

sum = 0 for(i=5;i

Computation makes the process of finding a more elegant way seems much more natural - in the best situations, it builds intellectual need for an easier way. It is arbitrary to say that a student should be able to do a problem without a calculator. Computational tools demand that we find a more compelling reason to solve problems by hand if computers are able to do them rapidly once they are set up to solve them through programming. It is a realistic motivation to show that an easier way speeds up finding a solution to a problem by a factor of 10. It means less waiting for a web page to load or an image to post.

The language of mathematics is difficult enough to throw in the additional complications of computer language syntax. I fully acknowledge that this is a hurdle. I also think, however, that this syntax is more closely related to the concepts that we are trying to teach our students (3*x is three times x) than we sometimes think. The power of computer programming to be a bridge between the hand calculations that our students do and the abstractions of the mathematical content we teach is too great to ignore.

Computational Thinking and Algebraic Expressions

I am still reviewing algebra concepts in my Math 9 course with students. The whole unit is all about algebraic operations, but my students have seen this material at least once, in some cases two years running.

Not long ago, I made the assertion that the most harmful part of introducing students to the world of real-world algebra looks like this:

Let x = the number of ________

Why is this so harmful?

For practiced mathematicians, math teachers, and students that have endured school math for long enough, there are a couple steps that actually happen internally before this step of defining variables. Establishing a context for the numbers and the operations that link them together are the most important part of producing a correct mathematical model for a given situation. A level of intuition and experience is necessary if one is going to successfully skip straight to this step, and most students don’t have this intuition or experience.

We have to start with the concrete because most people (including our students) start their thinking in concrete terms. This is the reason I have raved previously about the CME Project and the effectiveness of using their guess-check-generalize concept in introducing word problems to students. It forms an effective bridge between the numbers that students understand and the abstract concept of a variable. It encourages experimentation and analysis of whether a given answer matches the constraints of a problem.

This method, however, screams for computers to take care of the arithmetic so that students can focus on manipulating the variables involved. Almost all of the Common Core Standards for Mathematical Practice point toward this being an important focus for our work with students. I haven’t had a great point in my curriculum since I really started getting into computational thinking to try out my ideas from the beginning, but today gave me a chance to do just that.

Here’s how I introduced students to what I wanted them to do:

I then had them open up this spreadsheet and actually complete the missing elements of the spreadsheet on their own. Some students had learned to do similar tasks in a technology class, but some had not.
02 – SPR – Translating Algebraic Expressions

The students that needed to have conversations about tricky concepts like three less than a number had them with me and with other students when they came up. Students that didn’t quickly moved through the first set. I’d go and throw some different numbers for ‘a number’ and see that they were all changing as expected. Then we moved to a more abstract task:

It’s great to see that you know how to use different operations on the number in that cell. Now let’s generalize. Pick a variable you like – x, or N, or W – it doesn’t matter. What would each of these cells become then? Write those results together with the words in your notebook and show me when you’re done.

The ease with which students moved to this next task was much greater than it has ever been for me in past lessons. We also had some really great conversations about x*2 compared with 2x, and the fact that both are correct from an arithmetic standpoint, but one is more ‘traditional’ than the other.

Once students got to this point, I pushed them toward a slightly higher level task that still began concrete. This is the second sheet from the spreadsheet above:

Here we had multiple variables going at once, but this was not a stretch for most students. The key that I needed to emphasize here for some students was that the red text was all calculated. I wanted to put information in the black boxes with black text only, and have the spreadsheet calculate the red values. This required students to identify what the relationship between the variables needed to be to obtain the answer they knew in their head had to be true. This is CCSS MP2, almost verbatim.

This is all solidifying into a coherent framework of using spreadsheet and programming tools to reinforce algebra instruction from the start. There’s still plenty to figure out, but this is a start. I’ll share what I come up with along the way.

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