You Don’t Know Your Impact Until You Do.

There comes a time, often at the end of the semester, when you look around your classroom once the students have left, and let out a big sigh.

Am I doing the right things?

Am I helping students grow in ways that are best for them?

Then you get an email from a former student that says things like this:

I got selected to be a part of a research group in the department of PHYSICS! Can you believe it? The one subject I did not like at all is the first research opportunity for me!

All these great opportunities wouldn’t have happened to me if you didn’t have patience to make me understand physics. I now understand why you wanted me to figure out how to approach a problem all by myself instead of telling me what to do step by step.

I never realized how important it is to be able to do more than calculations until recently because I have been helping out a friend with her chemistry homework. However, I feel like that is all I do – help her finish her homework instead of helping her understand how to analyze a problem before jumping to equations.

I don’t want her to jump to equations because, at the end of the day, chemistry is a science, not math. We use math to help us, but a calculated answer means nothing by itself. It is the ability to analyze and interpret numbers than differentiates us from computers. Going to back to my friend and her chemistry homework, I noticed a lot of things that she says that reminded me of myself and physics.

For example, she would say “I don’t get it, it seems so easy, but I just don’t know which equation to use.” Then when I try to guide her to figure out which equations to use, she just interrupts me with “Just tell me which equation to use, and I can do the math.”

Doesn’t that sound like me in physics class? It frustrates me how she takes such a mathematical approach to a scientific problem. I mean it’s great that she can do math, but so can the computer.

I am telling you about my experience because I want to first let you know how much I appreciate your patience with me, and second, I want to apologize for that things I said about physics. It must not have been very pleasant to hear someone talk about something you are obviously interested in in such an aggressive tone.

I am sorry for complaining about physics the way I did last year, and if you students in the future complain about a subject feel free to relate my experience with physics to them. Also, I am very happy that you made me struggle with physics last year because now when I don’t see how to solve a problem immediately I know how to use the tools available to me to experiment to find the right answer.

Moreover, do continue to do explorations with your students because they are so helpful when it comes to critical thinking….

…I know you always take the opinions of your students seriously, and I know that you have stepped away from doing explorations because our class had such a negative attitude towards them; however, knowing how to use a different program can help student develop their problem solving skills, which makes them a more competitive student.

If you know me at all, you know that this hits many of the questions I have about my own teaching. One perspective is certainly not every perspective. I’m certainly not going to stop questioning. That said, this message made me grin with pride. It means a lot to hear that something you do in the classroom enables students to make opportunities for themselves.

With the student’s permission, I was eager to share the email as a way to help others remember why we do this job. You might never know the impact you have as a teacher until you do.

Keep this in mind as you approach the last teaching days of the year, everyone.

Why SBG is blowing my mind right now.

I am buzzing right now about my decision to move to Standards Based Grading for this year. The first unit of Calculus was spent doing a quick review of linear functions and characteristics of other functions, and then explored the ideas of limits, instantaneous rate of change, and the area under curves – some of the big ideas in Calculus. One of my standards reads “I can find the limit of a function in indeterminate form at a point using graphical or numerical methods.”

A student had been marked proficient on BlueHarvest on four out of the five, but the limit one held her back. After some conversations in class and a couple assessments on the idea, she still hadn’t really shown that she understood the process of figuring out a limit this way. She had shown that she understood that the function was undefined on the quiz, but wasn’t sure how to go about finding the value.

We have since moved on in class to evaluating limits algebraically using limit rules, and something must have clicked. This is what she sent me this morning:
[wpvideo 5FSp5JDn]

Getting things like this that have a clear explanation of ideas (on top of production value) is amazing – it’s the students choosing a way to demonstrate that they understand something! I love it – I have given students opportunities to show me that they understand things in the past through quiz retakes and one-on-one interviews about concepts, but it never quite took off until this year when their grade is actually assessed through standards, not Quiz 1, Exam 1.

I also asked a student about their proficiency on this standard:

I can determine the perimeter and area of complex figures made up of rectangles/ triangles/ circles/ and sections of circles.

I received this:
…followed by an explanation of how to find the area of the figure. Where did she get this problem? She made it up.

I am in the process right now of grading unit exams that students took earlier in the week, and found that the philosophy of these exams under SBG has changed substantially. I no longer have to worry about putting on a problem that is difficult and penalizing students for not making progress on it – as long as the problem assesses the standards in some way, any other work or insight I get into their understanding in what they try is a bonus. I don’t have to worry about partial credit – I can give students feedback in words and comments, not points.

One last anecdote – a student had pretty much shown me she was proficient on all of the Algebra 2 standards, and we had a pretty extensive conversation through BlueHarvest discussing the details and her demonstrating her algebraic skills. I was waiting until the exam to mark her proficient since I wanted to see how student performance on the exam was different from performance beforehand. I called time on the exam, and she started tearing up.

I told her this exam wasn’t worth the tears – she wanted to do well, and was worried that she hadn’t shown what she was capable of doing. I told her this was just another opportunity to show me that she was proficient – a longer opportunity than others – but another one nonetheless. If she messed up a concept on the test from stress, she could demonstrate it again later. She calmed down and left with a smile on her face.

Oh, and I should add that her test is looking fantastic.

I still have students that are struggling. I still have students that haven’t gone above and beyond to demonstrate proficiency, and that I have to bug in order to figure out what they know. The fact that SBG has allowed some students to really shine and use their talents, relaxed others in the face of assessment anxiety, and has kept other things constant, convinces me that this is a really good thing, well worth the investment of time. I know I’m just preaching to the SBG crowd as I say this, but it feels good to see the payback coming so quickly after the beginning of the year.

My tutor’s name is Geogebra CAS.

When I first started teaching, I learned that the best thing to have students do after factoring a trinomial was to have the students check by multiplying out the binomials. At the time, it naively made total sense – students don’t even need me to be there to practice! They can do this on their own while sitting on the subway or waiting for the bus – whatever dead time they have. The students that need to practice factoring can do as much of this as they need until they can factor with some degree of automaticity.

Some (not all) students took my advice. Of those that did, I often saw stuff like this:

x² – 4 = (x – 2)(x – 2)
= x² – 2x + x(2) + 4 = x² – 4

This was a worse situation than how we started – not only were they factoring incorrectly, but their inability to multiply binomials was giving them the false idea that they were doing a good job of factoring! This frustrated me to no end – even if I did give students time during class to practice and develop these skills, what could I tell them to do to improve outside of class? One colleague considered stopping giving homework because he saw it repeatedly reinforcing student errors. I didn’t go that far, but I did start grading homework to try to find mistakes.

The missing piece for these students is the lack of useful and correct feedback. Most of them learned the procedures, but made arithmetic or careless errors such as leaving out terms when simplifying. Without any correct data to make decisions on, these students were just going through a procedure and generating incorrect results, and using the incorrect results to validate an incorrect procedure. If they had a way to generate correct feedback, this experience would stop being worthless and instead become a useful method for developing student skills!

This is where CAS systems come in – Wolfram Alpha is nice, but Geogebra CAS is even better because of speed. I worked with a student that needed practice both in simplifying polynomial expressions and factoring polynomials completely. This is what I had him do while he sat with me:

  • Make up a pair of binomials of the form (x – 5)(4x – 5), multiply them, and then find the quadratic and linear coefficients. When you are ready, use the Simplify[] command to check your answer.
  • Make up a product of polynomials of the form 4x^2(x+5)(2x-5) . Multiply it out all the way on paper, and then check your result using the Simplify[] command.

After this step, we talked about how he could do this on his own and check his work. While we were sitting there, he made mistakes, but was able to catch them himself. He was the source of the problems, and was able to check and see if his final answers were correct. We then moved on to factoring practice:

  • Write out 15 products of binomials (3x-1)(x+5). For some of them, add a monomial factor. Include a couple sum and difference polynomials as well. Multiply any three of them out manually and check using Simplify[].
  • Use Geogebra to multiply any ten of the the rest of them and write down the resulting polynomials on a separate sheet of paper.
  • Eat dinner, watch TV, or something that has nothing to do with factoring.
  • Return to the paper and factor the ten polynomials you wrote down completely. Use Factor[] to check and make sure your final answers match what Geogebra produces – if there are differences, check to see if you have actually factored completely or not. Make a note of any repeat mistakes.

There is a whole lot of extra busy work involved in this process, but part of that is because it’s easy to factor a polynomial that you just generated moments before if you still remember the factors. For some students, this won’t matter, but it helps ensure that the exercises generates are actually useful. This student was on fire during class today, even though we were looking at a different topic entirely. I should have asked him directly whether this is the case, but perhaps the boost of confidence going through this process gave him is part of the reason. I also really like that this method allows the student to simultaneously work on multiplying polynomials and factoring them. My method beforehand would have been to stick to multiplying, then factoring, and then mix them up – there’s no reason to do this.

Computer algebra has been around for a while. The reason I think it’s now to the point where it can be transformational is that it’s easily accessible, easy to use, and almost instant. This idea of using technology (and particularly Geogebra) to help students develop their pencil and paper skills is one that really excites me. I’m excited to see if it works with the students that came in a bit behind but are willing to put in the time to catch up. I don’t want my class time to be spent learning algorithms – that defeats my strong belief that we should focus on teaching mathematical thinking, modeling, and problem formulation instead of algorithms. That said, students do need to be able to develop their skills, and this offers a personalized way to help them do so on their own.