I have been doing some thinking recently about kindness. In recent years, I am glad to say that I have always opted for being kind, during a conflict when others are scathing towards me. This has happened on several occasions when colleagues or student parents have come at me with misinformation, and because there is no hard evidence in my favor, they choose to interpret things in the worst possible light. Generally, I find that when an opinionated person reaches a conclusion, it is pretty much not helpful at that point of the conflict to try and convince them that they could be wrong. So, instead I have embraced a kindness policy. No matter how personal they try to make their attacks, I keep my end purely professional and courteous. I try to emphasize that we are all on the same side, wanting the best for the kid, and I almost always offer to do something extra to help the situation. (Anyhow I am always doing extra for everyone, so it is not at a cost to me to offer. But, it does make it seem like I am making concessions, which depending on whom you talk to and what their conflict style is, could either be seen as generous or weak.)

The result of this policy is generally good. Last year, one of my colleagues eventually recognized his own fault in the conflict, and although he never outright apologized for his inappropriate and condescending emails insinuating that I was dumb, I knew he came around to being grateful for the way I had handled the conflict, because he started being extra nice to me specifically. He would pull chairs, pour water for me, and even came to chat to me about his newborn for a good thirty minutes. (It was weird because we didn't have a personal connection like that. But because I had already actively let go of the disrespect, it was ok for me to enjoy the civility that had been a product of my choice to be kind.) I have enjoyed similar outcomes with very aggressive parents. One time, one of my students plagiarized part of his work, and during our investigation, we didn't have hard evidence so we let him eventually off the hook after grilling him on content he couldn't really explain. His mom tried to force me to apologize to her son, and was super aggressive and unreasonable. Her son was so embarrassed by her behavior, that he ditched class the next day and told a trusted adult that he couldn't face me after witnessing that conflict. Later, at the next parent meeting, despite me feeling anxious leading up to it, the parent came in ready to apologize. It is crazy, but often kindness begets kindness, and to me it is the only way to approach difficult people.

This year, there are a few students who have, for reasons I cannot fathom, adopted a sarcastic attitude in class that is toxic. People who know me know that I don't have it in me to be sarcastic, and that I am a straight-shooter. I have tried to reach them with kindness, and I don't think it is working. I think the root problem is that the material has really stepped up in difficulty, and as have my expectations for them in terms of homework, writing, working with assigned people, and retaking quizzes. The few kids who are having a hard time either with the topics or with the (actually very modest) workload are reacting not in a self-reflective way, but are shutting themselves off emotionally and intellectually. As I am going over again the calculator skills in class that will help them self-monitor accuracy on the requiz, for example, those few kids are busy snickering in the audience and one kid said, "Oh I just love math, especially this year." Their attitudes are preventing them from making real growth. As I see the other kids ramp up in their abstract reasoning and written accuracy and effort, I see this group potentially falling further behind, if they continue to close themselves off to the possibility of themselves needing to improve. I also see it splitting the class into two groups, those who think class is going fine and are in a way defending me, and those who don't want to contribute to a productive process. (Sarcasm seldom leads to productive discussion or outcome. It is a classic choice of easy cleverness over the harder choice of kindness.)

So, I am going to embrace this situation with as much kindness and firmness as I can balance. I would like to have a frank discussion with my class about this toxic attitude, not only to address it with those kids but also so that their peers can see me model what it is like to embrace a mean-spirited conflict with authenticity and kindness. But, I am going to be cautious about this and to invite another adult into the room (someone I trust and that those kids fear a little), in order to help keep the discussion productive and to not let those kids turn it into an opportunity to be nasty. I don't know if this is going to work. In fact, here is the hardest thing about choosing to be kind: you never do know how people will react to kindness, and it is possible that they will walk away thinking that they've won and not seeing what you are trying to achieve in the long term. But, it feels like the right approach for me, in line with who I am. Wish me luck!

## Saturday, December 14, 2013

## Saturday, November 23, 2013

### Thinking Flexibly About Exponential Form

I had a great discussion with my Calculus class about how the exponential sequence 2^x is

1, 2, 4, 8, 16, ... but that there is nothing special about the base 2. There has to be a number "a", such that (3^a)^x = 2^x, because there has to be an exponent "a" such that 3^a = 2. Together they found that number, and we said that g(x) = 3^(0.631x) is therefore an approximation of f(x) = 2^x. Similarly, there must be an exponent k such that (e^k)^x = 2^x. So, the kids took a minute or so to find that k = ln(2), so h(x) = e^(0.693x) must also be an approximation of f(x) = 2^x. We discussed how h is easily differentiable now that we are Chain Rule pros.

I then set the kids loose on an activity that asks them to think flexibly about the exponential form. The reason why I like this activity is because the kids distinguish between an exact form that is "natural" for the situation versus the usual e^(kx) approximation. (The kids know that they can substitute k with an exact expression without losing precision, however.) For example, for #1 in the worksheet, where the kids are looking at bacteria that double every 6 minutes, both the kids and I think that it's natural to think of the sequence as {10, 20, 40, 80, ...} and to observe the general form y = 10(2)^x. To fix the problem that we want it to double only ONCE by x = 6 minutes, not doubling 6 times, we divide our exponent by 6 to get y = 10(2)^(x/6). For me, this is the natural way of writing the equation and testing it initially to make sure that it fits the bill. If they then want to differentiate it, they then turn it into base e, where e^k = 2^(1/6), so that y = 10(e)^(kx) replaces y = 10(2)^(x/6). The kids figure out that k = (1/6)*ln(2) or around k = 0.116. So, y = 10(e)^(0.116x) is an approximation of our exact function, and it has the benefit of being easily differentiable.

In thinking about exponential form as being fluid, the kids can consider equivalent compound-interest scenarios. I gave them a couple of scenarios to play with and to explain, in order to get at that idea. I am pretty happy with the level of understanding they have with this concept, seeing that it's the second time this year we've seen exponential compounding. After that, they didn't seem to have much trouble working through our practice quiz on the exponential topic. Overall, I am pretty happy with the way our differentiation technique unit has gone. We're moving a little bit slower than I had hoped, but their understanding of the connections between concepts has been really great!!! Both they and I are still excited to walk into this class everyday, and that's a good feeling. I anticipate that by January, we'll be wrapped up with all the differentiation techniques (including related rates problems, which I've been sprinkling into the mix periodically), and the kids will be ready to start thinking backwards and/or to do a differentiation project.

Stay tuned!

1, 2, 4, 8, 16, ... but that there is nothing special about the base 2. There has to be a number "a", such that (3^a)^x = 2^x, because there has to be an exponent "a" such that 3^a = 2. Together they found that number, and we said that g(x) = 3^(0.631x) is therefore an approximation of f(x) = 2^x. Similarly, there must be an exponent k such that (e^k)^x = 2^x. So, the kids took a minute or so to find that k = ln(2), so h(x) = e^(0.693x) must also be an approximation of f(x) = 2^x. We discussed how h is easily differentiable now that we are Chain Rule pros.

I then set the kids loose on an activity that asks them to think flexibly about the exponential form. The reason why I like this activity is because the kids distinguish between an exact form that is "natural" for the situation versus the usual e^(kx) approximation. (The kids know that they can substitute k with an exact expression without losing precision, however.) For example, for #1 in the worksheet, where the kids are looking at bacteria that double every 6 minutes, both the kids and I think that it's natural to think of the sequence as {10, 20, 40, 80, ...} and to observe the general form y = 10(2)^x. To fix the problem that we want it to double only ONCE by x = 6 minutes, not doubling 6 times, we divide our exponent by 6 to get y = 10(2)^(x/6). For me, this is the natural way of writing the equation and testing it initially to make sure that it fits the bill. If they then want to differentiate it, they then turn it into base e, where e^k = 2^(1/6), so that y = 10(e)^(kx) replaces y = 10(2)^(x/6). The kids figure out that k = (1/6)*ln(2) or around k = 0.116. So, y = 10(e)^(0.116x) is an approximation of our exact function, and it has the benefit of being easily differentiable.

In thinking about exponential form as being fluid, the kids can consider equivalent compound-interest scenarios. I gave them a couple of scenarios to play with and to explain, in order to get at that idea. I am pretty happy with the level of understanding they have with this concept, seeing that it's the second time this year we've seen exponential compounding. After that, they didn't seem to have much trouble working through our practice quiz on the exponential topic. Overall, I am pretty happy with the way our differentiation technique unit has gone. We're moving a little bit slower than I had hoped, but their understanding of the connections between concepts has been really great!!! Both they and I are still excited to walk into this class everyday, and that's a good feeling. I anticipate that by January, we'll be wrapped up with all the differentiation techniques (including related rates problems, which I've been sprinkling into the mix periodically), and the kids will be ready to start thinking backwards and/or to do a differentiation project.

Stay tuned!

## Friday, November 15, 2013

### Friday Fun Day

For my calculus class, Fridays we meet in a different classroom. I like the change of scenery and it always makes me think about doing some extra fun, irregular exercise things on Fridays with the kids.

They've been working a lot on Chain Rule, but ironically only with polynomials embedded in polynomials, polynomials embedded in sine and cosine, and vice versa. They've also done an itty bitty related rates. We haven't seen the product or the quotient rule, and just the Chain Rule has kept us really busy.

Anyhow, on this Friday Fun Day, I thought of introducing the idea that reciprocal trig functions can be differentiated using Chain Rule. I told them that in reality, they won't often use those formulas, but after my hint at the Chain Rule, they were able to successfully differentiate f(x) = csc(x) and g(x) = sec(x).

Then, with only 10 minutes left in the class, I asked them to pair up and to assign one person as partner A, and the other as partner B. I asked each person to pull out their graphing calculator and to graph y=e^x. Then, they together pick an x value. Partner A needs to find f(x) and Partner B needs to find f'(x) using the calculator, and to compare answers to see what they notice.

In about 60 seconds, one kid said, "This is a trick!!"

haha. But, I wasn't prepared for their question that followed, as to WHY e^x is its own derivative. After class I did a bit of research and thought I'd post it here in case you Calculus teachers out there are wondering the same. The standard proof involves knowing how to differentiate ln(x), but as we haven't gotten there yet, I think this is a better explanation:

e is a value that comes from continuous compounding formula, namely the part (1 + 1/n)^n, limit taken as n approaches infinity.

e^x is therefore the limit as n approaches infinity of (1 + 1/n)^(nx)

By binomial expansion (which unfortunately my kids have never seen), this looks like:

1^(nx) + (nx choose 1) 1^(nx - 1)(1/n) + (nx choose 2) 1^(nx - 2)(1/n)^2 + (nx choose 3) 1^(nx - 3)(1/n)^3 + (nx choose 4) 1^(nx - 4)(1/n)^4 + ....

= 1 + (nx)(1/n) + [(nx)(nx - 1)/2] (1/n)^2 + [(nx)(nx - 1)(nx - 2)/(2*3)] (1/n)^3 + [(nx)(nx - 1)(nx - 2)(nx - 3)]/(2*3*4)(1/n)^4 + ....

As n approaches infinity, this becomes

e^x = 1 + x + x^2/2 + x^3/(2*3) + x^4/(2*3*4) + ...

So if you differentiate each term with respect to x, you get:

d(e^x)/dx = 0 + 1 + x + x^2/2 + x^3/(2*3) + .... which is the same as the original sequence.

BAM. It's a trick. ;)

PS. I am mad that I tried to tell my husband on Friday about my e^x discovery, and he said off-handedly (before I gave him any mathematical details), "Oh yeah, I remember expanding e^x and then each term in the expansion has a derivative term that maps back to the original expansion." DAMN. He's 32 and doesn't remember most things most times; why does he still remember this?!?! That's just not right.

PPS. Yes, we have this sort of conversations. It's very geeky.

They've been working a lot on Chain Rule, but ironically only with polynomials embedded in polynomials, polynomials embedded in sine and cosine, and vice versa. They've also done an itty bitty related rates. We haven't seen the product or the quotient rule, and just the Chain Rule has kept us really busy.

Anyhow, on this Friday Fun Day, I thought of introducing the idea that reciprocal trig functions can be differentiated using Chain Rule. I told them that in reality, they won't often use those formulas, but after my hint at the Chain Rule, they were able to successfully differentiate f(x) = csc(x) and g(x) = sec(x).

Then, with only 10 minutes left in the class, I asked them to pair up and to assign one person as partner A, and the other as partner B. I asked each person to pull out their graphing calculator and to graph y=e^x. Then, they together pick an x value. Partner A needs to find f(x) and Partner B needs to find f'(x) using the calculator, and to compare answers to see what they notice.

In about 60 seconds, one kid said, "This is a trick!!"

haha. But, I wasn't prepared for their question that followed, as to WHY e^x is its own derivative. After class I did a bit of research and thought I'd post it here in case you Calculus teachers out there are wondering the same. The standard proof involves knowing how to differentiate ln(x), but as we haven't gotten there yet, I think this is a better explanation:

e is a value that comes from continuous compounding formula, namely the part (1 + 1/n)^n, limit taken as n approaches infinity.

e^x is therefore the limit as n approaches infinity of (1 + 1/n)^(nx)

By binomial expansion (which unfortunately my kids have never seen), this looks like:

1^(nx) + (nx choose 1) 1^(nx - 1)(1/n) + (nx choose 2) 1^(nx - 2)(1/n)^2 + (nx choose 3) 1^(nx - 3)(1/n)^3 + (nx choose 4) 1^(nx - 4)(1/n)^4 + ....

= 1 + (nx)(1/n) + [(nx)(nx - 1)/2] (1/n)^2 + [(nx)(nx - 1)(nx - 2)/(2*3)] (1/n)^3 + [(nx)(nx - 1)(nx - 2)(nx - 3)]/(2*3*4)(1/n)^4 + ....

As n approaches infinity, this becomes

e^x = 1 + x + x^2/2 + x^3/(2*3) + x^4/(2*3*4) + ...

So if you differentiate each term with respect to x, you get:

d(e^x)/dx = 0 + 1 + x + x^2/2 + x^3/(2*3) + .... which is the same as the original sequence.

BAM. It's a trick. ;)

PS. I am mad that I tried to tell my husband on Friday about my e^x discovery, and he said off-handedly (before I gave him any mathematical details), "Oh yeah, I remember expanding e^x and then each term in the expansion has a derivative term that maps back to the original expansion." DAMN. He's 32 and doesn't remember most things most times; why does he still remember this?!?! That's just not right.

PPS. Yes, we have this sort of conversations. It's very geeky.

## Thursday, November 14, 2013

### Organizing Information for Related Rates

I started doing an itty-bitty bit of related rates with my kids today, and it was totally fun! I only did very basic problems with them so far, because they've been asking about where we would encounter chain rule. My goal this month/early next month is to finish off all the basic differentiation rules and skills, then to come back to related rates later to do a thorough study, so I gave them only the first few problems (Level 1) from Bowman today, just enough to wet their appetite.

And I gave them this grid to help them organize their thinking. I tried doing all the problems (up through Sam's problem on the two flies on the inflatable earth, which I consider Level 4), and they all fit into this grid pretty well. I hope that helps!

I LOVE RELATED RATES!!!!!

By the way, some of my kids solved for rate of change in radius like this:

dr/dt = dr/dA * dA/dt

others did:

dA/dt = dA/dr * dr/dt and divided both sides by dA/dr.

I like this! It's a good opportunity for us to come together as a class tomorrow to discuss equivalence of equations involving derivatives, so that they can learn to think more flexibly about the Chain Rule already.

I am also instituting very structured group work this term, which has been very helpful in approaching new tasks, at least in this class (and also in my Alg 2 classes). The kids haven't really commented on it, but I find that their discussions are more productive now that we are into our second grouping and they are more used to the idea of working with people and having assigned leadership roles. Excited about the promise of the rest of the year!!!

And I gave them this grid to help them organize their thinking. I tried doing all the problems (up through Sam's problem on the two flies on the inflatable earth, which I consider Level 4), and they all fit into this grid pretty well. I hope that helps!

I LOVE RELATED RATES!!!!!

By the way, some of my kids solved for rate of change in radius like this:

dr/dt = dr/dA * dA/dt

others did:

dA/dt = dA/dr * dr/dt and divided both sides by dA/dr.

I like this! It's a good opportunity for us to come together as a class tomorrow to discuss equivalence of equations involving derivatives, so that they can learn to think more flexibly about the Chain Rule already.

I am also instituting very structured group work this term, which has been very helpful in approaching new tasks, at least in this class (and also in my Alg 2 classes). The kids haven't really commented on it, but I find that their discussions are more productive now that we are into our second grouping and they are more used to the idea of working with people and having assigned leadership roles. Excited about the promise of the rest of the year!!!

### Graphically Analyzing Inequalities and Equations Flexibly

I've been doing inequalities and equations in the coordinate plane with my Algebra 2 kids, and I just

We started by looking at the meaning of <, >, and = inside the coordinate plane. We established quickly that when two things are

Then, we discussed that < means "lower than" and > means "higher than."

For any given equation or inequality, we would graph it as Y1 and Y2.

For example, 2x + 3 = 3x - 8, means that Y

For inequalities, the kids learn to first analyze them using graphs before reviewing how to solve them by hand. For example, 2x + 3 < 3x - 8, means that Y

It looks something like this:

and from that, the kids can conclude that the solutions for x are all to the right of x = 11, so they write down x > 11.

On the quiz for inequalities, my kids had

What I love about this graphical emphasis is how extendable this is to different topics. Today in one class I asked the kids to solve -20 < -4x + 5 < 11, and half the class decided to solve it by hand and the other half decided to solve it by just looking at the graph. They determined that only the middle part of the graph satisfies the inequality on both sides, and from that they determined -1.5 < x < 6.25.

And the graphing is extendable to absolute-value equations and inequalities, which loop in the idea of graphical symmetry (BEFORE the kids ever see quadratic graphs!).*

If, for example, my kids are asked to solve 2|x - 8| = 10, they can at this point sketch by hand the absolute-value graph centered at x=8, know that its slope is 2, and sketch another horizontal line at y=10. They can first assume the quantity (x - 8) is positive, so they can solve for x = 13 as one of the intersection points. Then, via symmetry around x = 8, they can then quickly figure out that x = 3 must be the other intersection point.

Although I plan to teach absolute-values a different way still, in order to reinforce the meaning of the various symbols, it has been delightful to preview the idea of functional symmetry and turning point before we ever see quadratics. The kids are doing so well on the abstract concepts so far! Totally kicking butt! YEAH.

*Also, the shading also is a good way of visualizing how to split the coordinate into different sections, delimited by x values, so their familiarity with this will really help out when we need to graph or analyze piecewise functions later this year.

*love*it. I see extensions from this topic to lots of others.We started by looking at the meaning of <, >, and = inside the coordinate plane. We established quickly that when two things are

*equal*in the coordinate plane, they must overlap. For example, (6, 2)= (6, 2) because they overlap. Also, y = x + 3 is*equal*to y = x + 3 because they overlap.Then, we discussed that < means "lower than" and > means "higher than."

For any given equation or inequality, we would graph it as Y1 and Y2.

For example, 2x + 3 = 3x - 8, means that Y

_{1}=2x + 3 has to*overlap*with Y_{2}=3x - 8. The kids sketch the graph, label the intersection, and answer in terms of x: x = 11. Easy breezy.For inequalities, the kids learn to first analyze them using graphs before reviewing how to solve them by hand. For example, 2x + 3 < 3x - 8, means that Y

_{1}=2x + 3 has to*be lower than*Y_{2}=3x - 8. The kids sketch the graph, label the intersection, and draw a vertical dashed line at x = 11 to separate the plane into two sections. They put their hands to cover one of the halves of the plane and ask, "Is Y_{1}actually lower than Y_{2}here?" If so, they shade it.It looks something like this:

On the quiz for inequalities, my kids had

*100%*accuracy on all the equations and inequality questions because they were asked to show their work two ways, one by hand and one by the graph, for each equation or inequality. They needed to get the same answer using both methods, in order to gain full reflection points (as well as accuracy and process points). Most of the kids were also able to explain why two parallel lines would cause an equation to have no solution, and inequalities to have either infinite solutions or no solutions, depending on the direction of the inequality symbol.What I love about this graphical emphasis is how extendable this is to different topics. Today in one class I asked the kids to solve -20 < -4x + 5 < 11, and half the class decided to solve it by hand and the other half decided to solve it by just looking at the graph. They determined that only the middle part of the graph satisfies the inequality on both sides, and from that they determined -1.5 < x < 6.25.

And the graphing is extendable to absolute-value equations and inequalities, which loop in the idea of graphical symmetry (BEFORE the kids ever see quadratic graphs!).*

If, for example, my kids are asked to solve 2|x - 8| = 10, they can at this point sketch by hand the absolute-value graph centered at x=8, know that its slope is 2, and sketch another horizontal line at y=10. They can first assume the quantity (x - 8) is positive, so they can solve for x = 13 as one of the intersection points. Then, via symmetry around x = 8, they can then quickly figure out that x = 3 must be the other intersection point.

And they can then extend their graphical analysis to solving 2|x - 8| > 10 to get x > 13 or x < 3 without memorizing procedures.

Although I plan to teach absolute-values a different way still, in order to reinforce the meaning of the various symbols, it has been delightful to preview the idea of functional symmetry and turning point before we ever see quadratics. The kids are doing so well on the abstract concepts so far! Totally kicking butt! YEAH.

*Also, the shading also is a good way of visualizing how to split the coordinate into different sections, delimited by x values, so their familiarity with this will really help out when we need to graph or analyze piecewise functions later this year.

## Tuesday, November 12, 2013

### Product Rule via Geometry

I was researching some justification for the product rule, and I am amazed that I never knew this before. It's so obvious (and much easier) if you approach it from a geometric perspective, thinking about how the length, width, and area of the rectangle change. Here are my visuals in thinking through this process, and also how I am planning to scaffold it for my kids. I named the product formula A = ... because I think that is most intuitive to think of that product as the Area that changes as a result of each length or width function changing.

Do you introduce product rule this way? Do you have recommendations for me?

Do you introduce product rule this way? Do you have recommendations for me?

## Tuesday, November 5, 2013

### Blowing Kids' Minds, One Combined Function at a Time

In my Calculus class, recently a discussion came up about when a derivative trig function would have a midline that is not 0. We quickly discussed the algebraic result of this question, but I thought afterwards that this was just too juicy of a discussion to let slide by.

So, on our new practice quiz, I decided to throw in one such bonus problem. (Go to the end.) They were AMAZED by the resulting graph when they worked on this in class today. One algebra-whiz kid was like, "This is NOT allowed! You cannot mix trig function with other functions!!" So hilarious!! I LOVE MY KIDS. I was pleased with their sense of wonder and surprise, but even more pleased that they were still able to look at and compare features between the derivative and original function graphs, even though they thought what they were looking at was super weird and not intuitive.

At the end of class, I off-handedly asked a couple of the fastest-working kids today, "So what's the average value of the derivative function?" They said, "...-1?" And then I asked, "How does that show up inside the original graph?" Those kids' eyes got so big and they said, "It has an average derivative of -1! OMG, you can see it!!!"

Cool graphical connections!!! We'll have to revisit this bonus problem as a class tomorrow, to make sure that everyone can appreciate the juiciness of this connection before we move on to other algebra goodness.

So, on our new practice quiz, I decided to throw in one such bonus problem. (Go to the end.) They were AMAZED by the resulting graph when they worked on this in class today. One algebra-whiz kid was like, "This is NOT allowed! You cannot mix trig function with other functions!!" So hilarious!! I LOVE MY KIDS. I was pleased with their sense of wonder and surprise, but even more pleased that they were still able to look at and compare features between the derivative and original function graphs, even though they thought what they were looking at was super weird and not intuitive.

At the end of class, I off-handedly asked a couple of the fastest-working kids today, "So what's the average value of the derivative function?" They said, "...-1?" And then I asked, "How does that show up inside the original graph?" Those kids' eyes got so big and they said, "It has an average derivative of -1! OMG, you can see it!!!"

Cool graphical connections!!! We'll have to revisit this bonus problem as a class tomorrow, to make sure that everyone can appreciate the juiciness of this connection before we move on to other algebra goodness.

### Exploring Sine and Cosine Derivatives

My onwards saga with differential calculus presented via explorations! (See previous post here.)

My kids recently did an exploration of sine and cosine derivatives. It was a little long, but I think well worth it. The idea is this:

* First, we spent a class period reviewing how to find a sine or cosine equation given a graph. This way I made sure that they'd have the pre-requisite exploration algebra skills.

* On the worksheet, they were given f(x) = sin(x) and asked to sketch its derivative using their knowledge of how to sketch derivative graphs. They're quite confident with this sketch, and then they decided that f'(x) = cos(x) by looking at its shape. I circulated and made sure that each kid used the dy/dx feature of their calculator to verify that the amplitude of f' is still 1, by making sure that the steepest part of f has a derivative value of 1.

* They then are given g(x) = cos(x), and again they started by sketching its derivative graph in order to determine that g'(x) = -sin(x) by inspection.

* They then made predictions for when the amplitude of the original function is not 1. In each case, they sketched the curve and then verified the max derivative value, in order to verify the amplitude of their resulting derivative graph.

* They repeated this for simple cases when the period changes. Based on their sketch of the derivative graph, they determined that the resulting derivative function would share the same period as the original functions, so they think k(x) = sin(2x) would have k'(x) = cos(2x). But, when they then checked the max derivative value along k(x), they realized that they were not correct. --WHY?? One of the kids immediately figured it out. I overheard him saying that because "sin(2x) is squished horizontally, that middle part of maximum steepness now becomes twice as steep as before, so obviously the amplitude will change." They checked this again using L(x) = cos(2x), whose max derivative (or steepness) is again 2, not 1, due to the horizontal compression. Brilliant!!

* They then tested their hypothesis out with M(x) = 3sin(2x), whose max steepness is now even steeper than 2.

* As a class, when we came together to discuss these exploratory problems together, I pulled up GeoGebra to guide our discussion. I entered something like M(x) = 3sin(2x) and had kids hypothesize just how steep that graph gets at x=0, since not everyone had quite reached the same conclusions before. If they said that the max derivative is 1, I entered y = 1x to show them that's unfortunately less steep than our curve is at x = 0. If they said 2, I entered y = 2x to show it's still less steep than our curve. Someone then guessed that the slope of the tangent at that point is 6, and we graphed it to see that y = 6x is indeed as steep as the curve gets. We then typed in M'(x) into the GeoGebra input bar, which automatically generates the formula AND the graph for the derivative. We discussed why the amplitude changes to 6, but why the period stays the same as that of M(x).

BAM! My kids are awesome!! They're able to now differentiate basic wave equations without knowing the first thing about chain rule. (Not yet, that's next, after we reinforce some other parts of our abstract analysis.) And they can explain the outcome of these sine and cosine derivative graphs! Yeah!!

My kids recently did an exploration of sine and cosine derivatives. It was a little long, but I think well worth it. The idea is this:

* First, we spent a class period reviewing how to find a sine or cosine equation given a graph. This way I made sure that they'd have the pre-requisite exploration algebra skills.

* On the worksheet, they were given f(x) = sin(x) and asked to sketch its derivative using their knowledge of how to sketch derivative graphs. They're quite confident with this sketch, and then they decided that f'(x) = cos(x) by looking at its shape. I circulated and made sure that each kid used the dy/dx feature of their calculator to verify that the amplitude of f' is still 1, by making sure that the steepest part of f has a derivative value of 1.

* They then are given g(x) = cos(x), and again they started by sketching its derivative graph in order to determine that g'(x) = -sin(x) by inspection.

* They then made predictions for when the amplitude of the original function is not 1. In each case, they sketched the curve and then verified the max derivative value, in order to verify the amplitude of their resulting derivative graph.

* They repeated this for simple cases when the period changes. Based on their sketch of the derivative graph, they determined that the resulting derivative function would share the same period as the original functions, so they think k(x) = sin(2x) would have k'(x) = cos(2x). But, when they then checked the max derivative value along k(x), they realized that they were not correct. --WHY?? One of the kids immediately figured it out. I overheard him saying that because "sin(2x) is squished horizontally, that middle part of maximum steepness now becomes twice as steep as before, so obviously the amplitude will change." They checked this again using L(x) = cos(2x), whose max derivative (or steepness) is again 2, not 1, due to the horizontal compression. Brilliant!!

* They then tested their hypothesis out with M(x) = 3sin(2x), whose max steepness is now even steeper than 2.

* As a class, when we came together to discuss these exploratory problems together, I pulled up GeoGebra to guide our discussion. I entered something like M(x) = 3sin(2x) and had kids hypothesize just how steep that graph gets at x=0, since not everyone had quite reached the same conclusions before. If they said that the max derivative is 1, I entered y = 1x to show them that's unfortunately less steep than our curve is at x = 0. If they said 2, I entered y = 2x to show it's still less steep than our curve. Someone then guessed that the slope of the tangent at that point is 6, and we graphed it to see that y = 6x is indeed as steep as the curve gets. We then typed in M'(x) into the GeoGebra input bar, which automatically generates the formula AND the graph for the derivative. We discussed why the amplitude changes to 6, but why the period stays the same as that of M(x).

BAM! My kids are awesome!! They're able to now differentiate basic wave equations without knowing the first thing about chain rule. (Not yet, that's next, after we reinforce some other parts of our abstract analysis.) And they can explain the outcome of these sine and cosine derivative graphs! Yeah!!

## Friday, November 1, 2013

### Mid-October Math Reflections

I had a very productive first marking term with my students. Even though there was some push back about writing in the math class (new for them), being assigned homework (new for some of them), and having a fairly strict grading system (or so they said), my students gave forth excellent efforts and have really come out on top of most of my expectations.

On the very last day of the marking term, I decided that instead of teaching something new right before their mini-vacation, I would ask the kids to write a self-reflection of their learning thus far. This assignment certainly didn't count for a grade, but I did explain that this was an opportunity for me to see who they were outside of my class, beyond the in-class efforts and their points on quizzes and projects. I gave them some specific prompts and specific timing guidelines per prompt, in order to encourage them to not rush through the task. They ended up giving me so many wonderfully rich details! It was so wonderful to read, that afterwards I went back to re-write some of my already finished, already polished narrative evals in order to acknowledge some of the new things I learned about them.

I integrated into this self-reflection three elements that I just loved:

1. I asked the kids to give me a short (5-minute) math autobiography. I had assigned math autobiographies in previous years, at the start of the school years, but because those students didn't yet know me and I didn't know them at the start of the year, it was not terribly useful for me to know a background story not associated with a name/face. This time, because I already knew them a little bit, their autobiographies gave me SO much insight into where they come from and why they behave the way they do in my class. I asked the kids to write about their general math history, as well as their favorite math teacher and what that person did to make math a good experience for them. Some kids wrote down that they've never had a great experience with math (ever!). Others wrote down that they had been educated previously in another country, and my class was the first time in years that they had learned anything new, so it was hard for them to adjust to having to work hard at something. One kid wrote down that he used to love math, then he hated math because of his relationship with one teacher, and now he likes it again because of me. He said that he's going to work harder the rest of the year in order to fully benefit from the course. Some other students wrote that they've preferred learning where the teacher shows an example and does a mini-lesson (which isn't how I run my class).

2. I asked the kids to wrote down as many things as possible from this marking term that they are proud of. I gave them specific examples, and I said that even if they didn't find the material to be easy, they should have many, many things from the class that they are proud of! Some of them wrote that they felt really proud of their persistence on taking re-quizzes without giving up. Others were proud of their good conceptual understanding as evidenced by their good scores. Some others said they've put in a lot of effort into the writing assignments and really tried their best on every assignment, even if I didn't collect them. Some others said they regularly helped their friends in class and over the phone at home. Some of them told me that they made lots of flashcards and studied them regularly, in order to help them focus on the important concepts. Some of them did the practice quizzes multiple times in order to make sure they understood what was important. Some were proud that they kept seeing me outside of class for help, showing great responsibility in their own learning. My kids were awesome!!

3. I asked the kids what was not working for them in my class, and for them to provide me with a proposal for improvement. This could be something that they're doing or something that I'm doing. From this, they gave me a variety of good ideas, such as writing their names on the board to line up for help (so that it's "fair"), rotating groups while sticking with a partner, so there is some consistency and someone who is still working at the same pace as you, etc. etc.

Fabulous! I was so impressed and overwhelmed by the frankness of their reflections. It was both humbling and inspiring to know that they already respond on an emotional and an intellectual level to their experience in my classroom, within just two short months. As much as learning is a dialogue, I felt that this was a really useful and timely feedback for me in my own teaching of these particular students.

Following that, my department had a ROCKIN' retreat today. We sorted out the sequence of topics from Algebra 1 to Precalc, placed them into all the courses, developed an emergency differentiation plan for our current Precalc classes, and came up with an amazing idea of students choosing math electives for the last couple of months of the school year in order to help us to better differentiate and to group them temporarily by both ability and interest. We came up with the mantra that we want our kids to be COURAGEOUS in math, more than anything else. (More than confidence, more than any skills or concepts.) I work with the best colleagues!!!!

On the very last day of the marking term, I decided that instead of teaching something new right before their mini-vacation, I would ask the kids to write a self-reflection of their learning thus far. This assignment certainly didn't count for a grade, but I did explain that this was an opportunity for me to see who they were outside of my class, beyond the in-class efforts and their points on quizzes and projects. I gave them some specific prompts and specific timing guidelines per prompt, in order to encourage them to not rush through the task. They ended up giving me so many wonderfully rich details! It was so wonderful to read, that afterwards I went back to re-write some of my already finished, already polished narrative evals in order to acknowledge some of the new things I learned about them.

I integrated into this self-reflection three elements that I just loved:

1. I asked the kids to give me a short (5-minute) math autobiography. I had assigned math autobiographies in previous years, at the start of the school years, but because those students didn't yet know me and I didn't know them at the start of the year, it was not terribly useful for me to know a background story not associated with a name/face. This time, because I already knew them a little bit, their autobiographies gave me SO much insight into where they come from and why they behave the way they do in my class. I asked the kids to write about their general math history, as well as their favorite math teacher and what that person did to make math a good experience for them. Some kids wrote down that they've never had a great experience with math (ever!). Others wrote down that they had been educated previously in another country, and my class was the first time in years that they had learned anything new, so it was hard for them to adjust to having to work hard at something. One kid wrote down that he used to love math, then he hated math because of his relationship with one teacher, and now he likes it again because of me. He said that he's going to work harder the rest of the year in order to fully benefit from the course. Some other students wrote that they've preferred learning where the teacher shows an example and does a mini-lesson (which isn't how I run my class).

2. I asked the kids to wrote down as many things as possible from this marking term that they are proud of. I gave them specific examples, and I said that even if they didn't find the material to be easy, they should have many, many things from the class that they are proud of! Some of them wrote that they felt really proud of their persistence on taking re-quizzes without giving up. Others were proud of their good conceptual understanding as evidenced by their good scores. Some others said they've put in a lot of effort into the writing assignments and really tried their best on every assignment, even if I didn't collect them. Some others said they regularly helped their friends in class and over the phone at home. Some of them told me that they made lots of flashcards and studied them regularly, in order to help them focus on the important concepts. Some of them did the practice quizzes multiple times in order to make sure they understood what was important. Some were proud that they kept seeing me outside of class for help, showing great responsibility in their own learning. My kids were awesome!!

3. I asked the kids what was not working for them in my class, and for them to provide me with a proposal for improvement. This could be something that they're doing or something that I'm doing. From this, they gave me a variety of good ideas, such as writing their names on the board to line up for help (so that it's "fair"), rotating groups while sticking with a partner, so there is some consistency and someone who is still working at the same pace as you, etc. etc.

Fabulous! I was so impressed and overwhelmed by the frankness of their reflections. It was both humbling and inspiring to know that they already respond on an emotional and an intellectual level to their experience in my classroom, within just two short months. As much as learning is a dialogue, I felt that this was a really useful and timely feedback for me in my own teaching of these particular students.

Following that, my department had a ROCKIN' retreat today. We sorted out the sequence of topics from Algebra 1 to Precalc, placed them into all the courses, developed an emergency differentiation plan for our current Precalc classes, and came up with an amazing idea of students choosing math electives for the last couple of months of the school year in order to help us to better differentiate and to group them temporarily by both ability and interest. We came up with the mantra that we want our kids to be COURAGEOUS in math, more than anything else. (More than confidence, more than any skills or concepts.) I work with the best colleagues!!!!

## Thursday, October 10, 2013

### A Spectrum of Relationships with Math

Our school had our annual "Curriculum Night" tonight. The parents followed their kids' schedules to come meet teachers, each for 10 minutes at a time for "class." 10 minutes isn't really enough to do an activity, but it was enough time to convey what I think is the most important about my approach to the class. It was my first time doing something like this, and what I chose to do was...

I started with asking the parents to stand along a spectrum on the floor, to indicate whether they LOVE and use math all the time at their jobs, or whether they feel nervous when they encounter math. Somewhere in between would be if they can try to help sometimes with their kid's math homework, but they need to first look at the textbook for a little while.

After this short little exercise, I asked the parents to sit, and I said that the reason I wanted to start with this is that all of these parents are fairly successful adults, and yet they all have different levels of affinity to math. So, as a teacher I try to keep in mind that it is only natural that kids in my class are also part of this spectrum, spanning from those kids who love abstraction and are ready to think 10 examples ahead, to those kids who need a lot of assurance everytime we progress into something unfamiliar. So, I have to plan lessons that can address all parts of the spectrum.

I spoke then a little bit about my techniques for differentiation. I said that many kids who are not so mathematically comfortable, are more comfortable with words. So, often times just asking them to write about math can help them to break it down into smaller parts, to help them understand and process each part. I also said that some kids are really comfortable with technology. So, allowing them to experiment first on a graphing calculator and to pull out numbers or observe patterns on the calculator can help them ease the transition into abstract concepts. Then some more kids are intuitive about the world, and they learn best when you anchor the theories to something very concrete that they already understand. This is why we do projects. In each class, I gave an example of a project that either we did recently (ie. bungee jumping regression project in Algebra 2), or one that is coming up soon (ie. video motion analysis via Logger Pro in Precalc), or one that will come about a little bit later in the year (ie. 3-d ceramic vase modeling in Calculus). All of these projects are ways for me to reach those kids who learn concretely, and they help to make the abstract topics more accessible to the kids. On the other end of the spectrum, for the kids who are always aching to move ahead, I always try to give them a little nudge towards what is to come, to help them anticipate the development in their mind before the entire class discusses and develops that concept. That helps this type of learner to stay challenged, because they tend to enjoy figuring things out on their own and then teaching their peers.

I then showed the parents a short sample of a piece of writing recently produced by one of my Precalc kids as his final draft to the triangular numbers and stellar numbers project. I walked them through my reasoning for writing in math -- discussing how even researchers in academia need the ability to write/communicate clearly, on the level of people much less specialized than them, in order to get funding and to get published for their discoveries. I also mentioned the importance of emphasis on testing formulas, and drew a parallel to the scientific process. The parents were just amazed when looking at the level of work and the clarity that the kid was writing with! Several parents came up to me afterwards to express their gratitude that their kids have to write so extensively in my class. I was honestly so floored by their warm enthusiasm!!

It was one of the best experiences I've ever had in meeting so many parents at once. I think the approach of starting parents off in a simple move-around activity (standing along a spectrum) helped to really engage them and helped them to consider not just their own child in my class, but also the other children who have diverse needs in the same class. We're a learning community, and I hope that I was able to convey that in my 10 minutes with each group of parents!

Although it has been a rough teaching week, tonight was truly a highlight for me to get such positive feedback from my students' parents!

I started with asking the parents to stand along a spectrum on the floor, to indicate whether they LOVE and use math all the time at their jobs, or whether they feel nervous when they encounter math. Somewhere in between would be if they can try to help sometimes with their kid's math homework, but they need to first look at the textbook for a little while.

After this short little exercise, I asked the parents to sit, and I said that the reason I wanted to start with this is that all of these parents are fairly successful adults, and yet they all have different levels of affinity to math. So, as a teacher I try to keep in mind that it is only natural that kids in my class are also part of this spectrum, spanning from those kids who love abstraction and are ready to think 10 examples ahead, to those kids who need a lot of assurance everytime we progress into something unfamiliar. So, I have to plan lessons that can address all parts of the spectrum.

I spoke then a little bit about my techniques for differentiation. I said that many kids who are not so mathematically comfortable, are more comfortable with words. So, often times just asking them to write about math can help them to break it down into smaller parts, to help them understand and process each part. I also said that some kids are really comfortable with technology. So, allowing them to experiment first on a graphing calculator and to pull out numbers or observe patterns on the calculator can help them ease the transition into abstract concepts. Then some more kids are intuitive about the world, and they learn best when you anchor the theories to something very concrete that they already understand. This is why we do projects. In each class, I gave an example of a project that either we did recently (ie. bungee jumping regression project in Algebra 2), or one that is coming up soon (ie. video motion analysis via Logger Pro in Precalc), or one that will come about a little bit later in the year (ie. 3-d ceramic vase modeling in Calculus). All of these projects are ways for me to reach those kids who learn concretely, and they help to make the abstract topics more accessible to the kids. On the other end of the spectrum, for the kids who are always aching to move ahead, I always try to give them a little nudge towards what is to come, to help them anticipate the development in their mind before the entire class discusses and develops that concept. That helps this type of learner to stay challenged, because they tend to enjoy figuring things out on their own and then teaching their peers.

I then showed the parents a short sample of a piece of writing recently produced by one of my Precalc kids as his final draft to the triangular numbers and stellar numbers project. I walked them through my reasoning for writing in math -- discussing how even researchers in academia need the ability to write/communicate clearly, on the level of people much less specialized than them, in order to get funding and to get published for their discoveries. I also mentioned the importance of emphasis on testing formulas, and drew a parallel to the scientific process. The parents were just amazed when looking at the level of work and the clarity that the kid was writing with! Several parents came up to me afterwards to express their gratitude that their kids have to write so extensively in my class. I was honestly so floored by their warm enthusiasm!!

It was one of the best experiences I've ever had in meeting so many parents at once. I think the approach of starting parents off in a simple move-around activity (standing along a spectrum) helped to really engage them and helped them to consider not just their own child in my class, but also the other children who have diverse needs in the same class. We're a learning community, and I hope that I was able to convey that in my 10 minutes with each group of parents!

Although it has been a rough teaching week, tonight was truly a highlight for me to get such positive feedback from my students' parents!

## Tuesday, October 8, 2013

### Backwards Intro to Differential Calculus

Towards the end of the summer I was brainstorming this idea of teaching Calculus backwards, starting with applications and graphing calculators, then manual Calculus skills, then finally tying those manual Calculus skills to various limits. It is now a little more than a month in, and I have to say that although I cannot compare this approach to a traditional curriculum because I've never taught Calculus the traditional way, I

After a unit on analyzing numerical and graphical derivatives, my kids have a very strong foundation of how to read/interpret graphs visually and how to sketch related derivative graphs carefully. Some of them even have a surprising ability to draw original functions given the shapes of the derivative graphs, even though we really didn't practice that as a class (not yet... I am saving it as part of our intuitive intro to integral Calculus).

And the best part of spending the first part of the year on application is that kids are now

And, because they've already done some mixed applications with f(x) and f'(x) analyses, I can immediately set them up tomorrow to answering a question such as, "When will the instantaneous rate reach 2 along this graph?" and the skills that we have already worked on for

I am so excited! Also, in context, we are reviewing various algebra skills incrementally. On their next quiz/practice quiz, they'll be expected to show work to find standard-form equations of polynomials by hand, and then to differentiate them in order to answer follow-up questions. My plan is to loop in all of the manual algebra modeling skills alongside the new derivative and integral skills, so that the kids would always know that they need to keep up with all of the transformational concepts from Algebra 2 and Precalc, as part of the norm for our class.

So, Calculus is pretty much my favorite class to teach right now. So incredibly fun!!!! I cannot wait until we dig into more of these derivative skills, in order for us to talk about some juicy applications that require both algebraic and geometric analysis. YES!!!!!

*love*the way that I am doing it!!!After a unit on analyzing numerical and graphical derivatives, my kids have a very strong foundation of how to read/interpret graphs visually and how to sketch related derivative graphs carefully. Some of them even have a surprising ability to draw original functions given the shapes of the derivative graphs, even though we really didn't practice that as a class (not yet... I am saving it as part of our intuitive intro to integral Calculus).

And the best part of spending the first part of the year on application is that kids are now

*itching*for algebra, because all of the visual introduction has really primed them for more specificity. They were so, SO excited today when I set them loose on an exploration regarding the power rule of differentiation. After just observing three examples that they generated via explorations on their calculator, they were fully able to generalize the pattern in partner pairs, and by the time they got to the back side of the worksheet, they thought it was just SO COOL that they could now differentiate by hand, without a calculator. It was so AWESOME to see. I've never seen kids so excited to differentiate by hand before! And, when they weren't sure whether they were doing something by hand correctly (ie. when they needed to differentiate a term already with a negative exponent on x), I would encourage them to put in an x value into their resulting f' formula and to check it against the dy/dx value that the calculator returns. Their tech skills from Unit 1 are now becoming a powerful tool for self-monitoring their process.And, because they've already done some mixed applications with f(x) and f'(x) analyses, I can immediately set them up tomorrow to answering a question such as, "When will the instantaneous rate reach 2 along this graph?" and the skills that we have already worked on for

*choosing*the correct tool between f and f' will come in handy then, to smoothly transition them into the new sets of manual skills (I hope!).I am so excited! Also, in context, we are reviewing various algebra skills incrementally. On their next quiz/practice quiz, they'll be expected to show work to find standard-form equations of polynomials by hand, and then to differentiate them in order to answer follow-up questions. My plan is to loop in all of the manual algebra modeling skills alongside the new derivative and integral skills, so that the kids would always know that they need to keep up with all of the transformational concepts from Algebra 2 and Precalc, as part of the norm for our class.

So, Calculus is pretty much my favorite class to teach right now. So incredibly fun!!!! I cannot wait until we dig into more of these derivative skills, in order for us to talk about some juicy applications that require both algebraic and geometric analysis. YES!!!!!

## Sunday, October 6, 2013

### Three Core Values in My Class

This blog post is my contribution to Mission #1 of Exploring the MathTwitterBlogosphere.

I already write a lot about what physically happens in my classroom, so today I am going to write about three values of my class that impact my students' experience and give it a stamp of uniqueness. They are what I consider core elements of my classroom culture.

The first:

I think that a lot of teachers out there hold the same value, so I wouldn't say that it's uniquely mine when taken alone, but it has practical implications in my interaction with the kids.

The most important implication of this value is that in my class, deadlines are not strictly held. If I tell kids that they should bring me an assignment on Monday, then on Monday I would discuss it a bit, answer last questions, and then say, "Okay. Please hand it in to me if you've already done your best on this assignment and feel that you have understood everything. If you need to still take another day or two in order to bring me your best work, then please do that instead of giving it to me now. I don't want you to turn in something that is less than your best efforts, because it's not useful to either you or me."

The same holds for quizzes and tests. Last Friday, a kid was struggling with a quiz I gave in class. After class, I gave him a full period of extended time, but as I was looking over his shoulders, I could see that he was still making the same mistake over and over again with signs, which was causing more frustration further down the line when he realized that his answers wouldn't check against the equations. Eventually, I stopped him and just said, "Alright, look, I don't want you to keep spinning in place and to feel more frustrated. Why don't you give this to me, and we'll plan for a requiz next week after we look over your errors together?" A part of me really struggled with that recommendation, because I think persistance is such an excellent trait, and allowing a kid to persist in face of difficulty is very valuable. But, a bigger part of me believes that each kid has to learn at their own rate, and forcing this quiz upon this kid then was going to do more harm than good.

Another manifestation of this value is that in class, I always try to touch upon a higher-order application of what we are learning, in order to keep the interest of those kids who just yearn for a little more depth beyond what everyone else can grasp immediately. Inevitably, those higher-order applications will come back as bonus questions on the test, not because I love to give bonus points but because I want a way to assess which students are accessing that knowledge based on our brief discussion in class. The real, core assessment is still based on the lowest common denominator, the core skills that we have thoroughly developed and practiced as a class, but on each assessment there should be room for the upper-end kids to stretch their understanding. It is one way that I make room for kids to show me that they are acquiring knowledge at a different rate.

The second:

This is a value that a lot of math teachers hold, but one that I really invest a lot of time to teach and to develop in my classes. Depending on the topic and what makes the most sense, I either expect them to check their results via the calculator or to check their results by hand.

At the beginning of the year, it can take me more class time to teach this and to ask kids to self-verify their work, than the time that it takes me to teach new algebra skills. This is why I don't think that other teachers really teach it; a lot of people value this skill, but when it comes down to it, they don't necessarily value it enough to put in the time to force every kid to develop this skill. But as time goes on, the time spent learning this skill shrinks rapidly, and kids get very used to doing this as part of learning and assessing any new topic. In fact, my current students have said to me, "You really force us to check every problem on a quiz!" ...And yes, I do! The time they spend completing a quiz should be roughly as follows: only 2/3 to 3/4 of the time spent on completing the quiz, and the rest of the time going through and gaining full confidence in all their answers. Ideally, they should be able to tell me after every quiz, "I

I try to get away from being the source of verification, because I want my students to one day exceed me in their knowledge and understanding of math, and if I don't teach them how to verify their own answers, their knowledge will always be upper-bounded by what I know.

The third:

Again, I think all teachers believe that learning strategies are important and incorporate them into our daily lessons. But, we don't all teach them explicitly or discuss their usefulness. In fact, some of the best content-teachers can still overlook the importance of explicitly teaching and discussing strategies for review in the classroom. Teenagers need modeling for learning strategies as much as they need modeling for how to do a math procedure. I find that once my students and I practice a certain strategy in the class, they often come back to me to request more support in learning to learn. And, in the long run, it helps them greatly in building their confidence with math.

The three most popular strategies with my students are: creating their own concept flash cards; doing practice quizzes; and doing white-board procedural practice (while they take photos of the problems). These are tried-and-true methods that the kids find to be the most useful in terms of self-diagnosing their gaps, ironing out consistent procedural errors, and increasing mental focus.

As I think about content delivery in my classroom, I tend to think that it is inseparable from the learning culture that I set up around the content. These three elements are not unique by themselves, but together they do help to form a strong culture of high expectations when it comes to self-reliance, self-monitoring, and self-knowledge. The learners cannot be viewed as helpless and passive, but in order for them to become successful and self-reliant, we need to cultivate the tools that would help them reach those expectations.

I already write a lot about what physically happens in my classroom, so today I am going to write about three values of my class that impact my students' experience and give it a stamp of uniqueness. They are what I consider core elements of my classroom culture.

The first:

*Each kid learns at a different rate*,*but what's non-negotiable is the quality of their efforts**and the fact that each student needs to be challenged*.I think that a lot of teachers out there hold the same value, so I wouldn't say that it's uniquely mine when taken alone, but it has practical implications in my interaction with the kids.

The most important implication of this value is that in my class, deadlines are not strictly held. If I tell kids that they should bring me an assignment on Monday, then on Monday I would discuss it a bit, answer last questions, and then say, "Okay. Please hand it in to me if you've already done your best on this assignment and feel that you have understood everything. If you need to still take another day or two in order to bring me your best work, then please do that instead of giving it to me now. I don't want you to turn in something that is less than your best efforts, because it's not useful to either you or me."

The same holds for quizzes and tests. Last Friday, a kid was struggling with a quiz I gave in class. After class, I gave him a full period of extended time, but as I was looking over his shoulders, I could see that he was still making the same mistake over and over again with signs, which was causing more frustration further down the line when he realized that his answers wouldn't check against the equations. Eventually, I stopped him and just said, "Alright, look, I don't want you to keep spinning in place and to feel more frustrated. Why don't you give this to me, and we'll plan for a requiz next week after we look over your errors together?" A part of me really struggled with that recommendation, because I think persistance is such an excellent trait, and allowing a kid to persist in face of difficulty is very valuable. But, a bigger part of me believes that each kid has to learn at their own rate, and forcing this quiz upon this kid then was going to do more harm than good.

Another manifestation of this value is that in class, I always try to touch upon a higher-order application of what we are learning, in order to keep the interest of those kids who just yearn for a little more depth beyond what everyone else can grasp immediately. Inevitably, those higher-order applications will come back as bonus questions on the test, not because I love to give bonus points but because I want a way to assess which students are accessing that knowledge based on our brief discussion in class. The real, core assessment is still based on the lowest common denominator, the core skills that we have thoroughly developed and practiced as a class, but on each assessment there should be room for the upper-end kids to stretch their understanding. It is one way that I make room for kids to show me that they are acquiring knowledge at a different rate.

The second:

*Kids should have ways of verifying and monitoring their own correctness, beyond asking me.*This is a value that a lot of math teachers hold, but one that I really invest a lot of time to teach and to develop in my classes. Depending on the topic and what makes the most sense, I either expect them to check their results via the calculator or to check their results by hand.

At the beginning of the year, it can take me more class time to teach this and to ask kids to self-verify their work, than the time that it takes me to teach new algebra skills. This is why I don't think that other teachers really teach it; a lot of people value this skill, but when it comes down to it, they don't necessarily value it enough to put in the time to force every kid to develop this skill. But as time goes on, the time spent learning this skill shrinks rapidly, and kids get very used to doing this as part of learning and assessing any new topic. In fact, my current students have said to me, "You really force us to check every problem on a quiz!" ...And yes, I do! The time they spend completing a quiz should be roughly as follows: only 2/3 to 3/4 of the time spent on completing the quiz, and the rest of the time going through and gaining full confidence in all their answers. Ideally, they should be able to tell me after every quiz, "I

*know*I got 100%."I try to get away from being the source of verification, because I want my students to one day exceed me in their knowledge and understanding of math, and if I don't teach them how to verify their own answers, their knowledge will always be upper-bounded by what I know.

The third:

*Learning to learn immediately supports learning of the content, so time should be spent in class to explicit teach, discuss, and practice various learning strategies.*Again, I think all teachers believe that learning strategies are important and incorporate them into our daily lessons. But, we don't all teach them explicitly or discuss their usefulness. In fact, some of the best content-teachers can still overlook the importance of explicitly teaching and discussing strategies for review in the classroom. Teenagers need modeling for learning strategies as much as they need modeling for how to do a math procedure. I find that once my students and I practice a certain strategy in the class, they often come back to me to request more support in learning to learn. And, in the long run, it helps them greatly in building their confidence with math.

The three most popular strategies with my students are: creating their own concept flash cards; doing practice quizzes; and doing white-board procedural practice (while they take photos of the problems). These are tried-and-true methods that the kids find to be the most useful in terms of self-diagnosing their gaps, ironing out consistent procedural errors, and increasing mental focus.

As I think about content delivery in my classroom, I tend to think that it is inseparable from the learning culture that I set up around the content. These three elements are not unique by themselves, but together they do help to form a strong culture of high expectations when it comes to self-reliance, self-monitoring, and self-knowledge. The learners cannot be viewed as helpless and passive, but in order for them to become successful and self-reliant, we need to cultivate the tools that would help them reach those expectations.

## Saturday, September 28, 2013

### Color-Worthy Handouts

My German teacher had a simple but very effective trick for organizing papers: make copies on colored paper, whenever it is a handout that you want the students to be able to quickly retrieve and to reference over time. I have started to do this this year. Regular worksheets are always on white paper, but quizzes and important reference handouts are always on colored paper. This way, after a unit is over, I can basically tell the kids to leave all the white handouts filed away at home and to just keep their colored handouts handy for future classes.

Here are some recent handouts that I think are useful enough to be on colored paper:

Here are some recent handouts that I think are useful enough to be on colored paper:

- Addressing common calculator entry errors, because seeing my kids enter them wrong repeatedly is making me want to tear my hair out. It's an indication that they are not getting enough exposure to math technology (on the computer or on their TIs) over the course of their education, because the way to enter these algebraic expressions is pretty standard across all platforms.
- Good writing transitions to help out when kids are communicating their problem-solving process via lengthy writing. I am super excited to share an example of student work,
*verrrry soon!*

### Using Logger Pro in Quadratic Modeling!

One of the wonderful things of teaching in different schools is that you get to learn from different teachers. My current school has a site-wide license for Logger Pro, which (I know, unfortunately) is a proprietary program that allows you to import and analyze videos. It pulls the scaling information based on your definition of what 1 meter looks like in the video, and it uses the timestamps built into the video to retrieve timing info. From that, this program is able to pull both position information over time, and estimated velocity information over time. (The velocity bit is not that precise, however.)

I was playing around with this piece of software this morning because our Precalculus course team wishes to incorporate it into our Quadratics unit. I imported a video from David Cox, which can be found at http://vimeo.com/16506894,

and I got this screenshot in Logger Pro. The red is the horizontal position of the yellow ball over time, for the frames that I chose. The blue is the vertical position of the yellow ball over time, for the frames that I chose.

I was playing around with this piece of software this morning because our Precalculus course team wishes to incorporate it into our Quadratics unit. I imported a video from David Cox, which can be found at http://vimeo.com/16506894,

and I got this screenshot in Logger Pro. The red is the horizontal position of the yellow ball over time, for the frames that I chose. The blue is the vertical position of the yellow ball over time, for the frames that I chose.

I love this! I can see letting my kids do the same, picking out points from a video that includes both dimensions of movement, and then discussing why height is always quadratic and the horizontal distance is not. And then, they will do quadratic modeling both by hand (by setting up a system of equations) and on the calculator (via regression) in order to find the curve that fits this graph. LOVE IT!

PS. If you are lucky enough to work at a school that would agree to get a site-wide license, the really nice thing is that you get to install it at home completely legally, which is great for both you and the students. So, keep that in mind when you are talking to your admin!

### Week 3 Teaching - Setbacks and Triumphs

We are in the thick of it now, the part of the semester when I see how kids handle setbacks and challenges. This is one of the ways I really get to know a kid, because I truly believe that how you handle setbacks defines your character. I tell the kids that they can keep reviewing and re-quizzing, or re-submitting drafts of a writing assignment, until

For me personally, I've always viewed myself as a second-try kind of gal. If I weren't, I would have given up the first time a teaching program told me that I wasn't their ideal candidate, and I would never have ended up doing what I do, and loving it. Too many adults give up on their goals too easily, and let other people decide for them what they can and cannot achieve. I don't want that to happen to my students.

Anyhow, enough philosophizing. More about teaching. My Calculus kids are entering a very interesting phase of the course. We had our first quiz, which was quite tricky and conceptual, even though it did not involve many numbers. To my delight, about two-thirds of the class did quite well on this quiz, and the top three or so scores were all girls!!! I cannot help myself but feel gleeful about that, especially because 1. the girl who did the best on the quiz (missed actually

Anyhow, Precalc is also going swimmingly. I heard feedback from one 11th-grade advisor that her advisee loves my class, and thinks that all the math we do thus far is very clear and very understandable. The students are in the process of finishing up their first big lab writeup, which was very exciting because when I took them down to the computer lab, I got to show them how to 1. enter and format equations properly into MS Word or Google Docs, and 2. how to connect their TI-84s to the computer via TI-ScreenConnect to

My Algebra 2 classes are moving pretty slowly through their regression project, because I have discovered that they have some holes in their Algebra 1 knowledge and am taking some daily class time to discuss homework problems before assigning new review assignments. In class, we are doing white-boarding practice about once a week, because it is a great time for me to make sure that everyone is doing some algebra practice together and getting instantaneous feedback/help as needed. Following our fairly difficult first quiz, which I had written about last week, lots of kids came to see me to do re-quizzes, which I loved. Although they didn't all get 100% on their re-quizzes, it started a very productive dialogue with kids about how they are studying, what study tips I can recommend, and why things always seem easier when you do a re-quiz. One international student in particular had an 180-degree shift of attitude towards me after the re-quiz. I think she's the kind of student who thrives particularly on positive feedback, so the fact that she had failed the first time but got 100% on the second try, really boosted her confidence and her feelings towards math (and I guess, me). So, although it had been a challenging/"discouraging" week last week, I think it was a necessary reality check for many kids and now they are much more focused and strategic in their learning. My 10th-graders, for example, took a lot of photos yesterday during our white-boarding practice, because the one student who had done that last time and who had practiced with those problems later at home, had done very well on the quiz and had offered that up as a study strategy to her peers. So, we are a growing community of learners, moving in the right direction, slowly but surely!

For both Algebra 2 classes, on Monday we will go and test out the kids' predictions for the bungee drop. They will build the cords, name their rubber chickens, take a photo with their chickens, and then we will go out to the balcony. It'll be a very fun day, but the hard work that is yet to come is to write the lab reports coherently. I am a little nervous about getting their rough drafts on Tuesday, and what those will look like, especially for my international kids....

But, I cannot complain. I love this time of the year!

*they*decide that their score is good enough to stop. No one is going to disallow them to keep working to get better, because I think that training kids to keep tackling something long after the class has "moved on" is how we can teach them to develop a persevering character.For me personally, I've always viewed myself as a second-try kind of gal. If I weren't, I would have given up the first time a teaching program told me that I wasn't their ideal candidate, and I would never have ended up doing what I do, and loving it. Too many adults give up on their goals too easily, and let other people decide for them what they can and cannot achieve. I don't want that to happen to my students.

Anyhow, enough philosophizing. More about teaching. My Calculus kids are entering a very interesting phase of the course. We had our first quiz, which was quite tricky and conceptual, even though it did not involve many numbers. To my delight, about two-thirds of the class did quite well on this quiz, and the top three or so scores were all girls!!! I cannot help myself but feel gleeful about that, especially because 1. the girl who did the best on the quiz (missed actually

*none*of the problems, including the bonus ones which we had only briefly seen during class) had previously said to me that she always felt a little behind in other people's math classes, and 2. our school has been having some conversations surrounding issues of diversity (mainly ethnic and socioeconomic), which has been making me wonder a bit about the role of "male privilege" in the math classroom, similar to the issues of "white privilege." Anyhow, the fraction of kids who didn't do so well on their quizzes are seeing me on Tuesday for a re-quiz, so as of now it is still too early for me to say whether they just had a bad day, or they are still learning how to study effectively, or they really need some serious intervention with the topics that we have covered. In this class, unlike the other classes, I have not been collecting/grading homework past discussing the answers as a group, since thus far we have been building up introductory concepts and there are not a lot of nitty-gritty skills checkpoints for me to look at and respond to. I think that contributed to the lack of individual feedback before the first quiz, which I will remedy in the coming unit by being more hands-on with grading their homework assignments, once we step into the realm of manual differentiation techniques. Overall, I have been very happy with the way that my kids have built their conceptual understanding around derivatives and instantaneous and average rates. We take every second Friday to review algebra skills from the past, and then I assign a review homework assignment for the weekend following. So far, we have reviewed: 1. factorization techniques, 2. how to solve for a parameter within an equation, and 3. when to use their calculators to solve complex equations; and already I can see their independence growing inside the classroom from these mini skills reviews. We just wrapped up a great worksheet (if I may say so myself), because every problem in this worksheet is anchored in something very real. Problem 1 was about investment, and I had talked to the kids about how my husband does real-estate investment and how he uses this type of math to calculate mortgage rates and monthly expenses on a rental property, in order to compare those fees with his rental income to make sure that he will clear a profit every month from his investment. (Related to this I talked to the kids about why they should invest, and why investment does*not*mean that they cannot be contributing productively to the society.) Problem 2 was about carbon-dating, but the kids needed to read the initial C14 levels out of a graph of atmospheric carbon levels to use in their carbon decay model. This is very realistic, and we got to talk a little bit about the science behind your body equalizing with the atmospheric carbon levels while you are breathing/alive, as well as about how cow-farming is causing historic carbon levels to rise (connecting this to what they see in the graph of historical atmospheric levels). Problem 3 on the worksheet is an ad that I actually found on the web for Gap Inc's credit card offers, so although it is another review problem for interest rates, it is steeped in real world context. I am trying to make my Calculus class as inter-curicular as possible on a day-to-day basis, so that kids can see the reason/motivation behind studying what we study. Interestingly enough, a couple of the faster-moving kids have already started on our end-of-unit economic mini-project (Part 1 and Part 2), and the first question they asked me before even starting the math was, "Why would anybody care about marginal costs?"*--Aren't my kids fabulous?*I*want*them to ask me questions like this, so that Calculus can come alive for them.Anyhow, Precalc is also going swimmingly. I heard feedback from one 11th-grade advisor that her advisee loves my class, and thinks that all the math we do thus far is very clear and very understandable. The students are in the process of finishing up their first big lab writeup, which was very exciting because when I took them down to the computer lab, I got to show them how to 1. enter and format equations properly into MS Word or Google Docs, and 2. how to connect their TI-84s to the computer via TI-ScreenConnect to

*prove*that they are doing the tests of their formulas. Some of the kids, in fact, didn't even know how to construct tables or write subscripts, so there was lots of tech education there, besides helping them out with the mathematical language. The kids thought that typing up their revisions to the rough drafts was going to be easy, but it did in fact take them two full (45-minute) class periods, and many still had to go home to take some time this weekend to re-read through it to make sure that they have hit every part of the project rubric. Anyhow, I prepared a graphical organizer template so that sometime next week, we can discuss how this idea of approaching and analyzing math sequences is going to be the big idea through the entire first Quint. (We have 5 Quints a year, as opposed to 4 Quarters.) Also, something quite cool that I tried recently was to put kids into groups and let them do mixed analysis of linear and quadratic sequences, and instead of me telling them whether they were correct, they got to check using the web interface of visualpatterns.org! The kids were super into it, and I think seeing the two types side by side really helped them to clarify mentally the different strategies for each type. Alongside the writeups, the kids have just about finished reviewing all the algebra skills for lines, so I will give another quiz next week before moving on to reviewing quadratic and transformational skills.My Algebra 2 classes are moving pretty slowly through their regression project, because I have discovered that they have some holes in their Algebra 1 knowledge and am taking some daily class time to discuss homework problems before assigning new review assignments. In class, we are doing white-boarding practice about once a week, because it is a great time for me to make sure that everyone is doing some algebra practice together and getting instantaneous feedback/help as needed. Following our fairly difficult first quiz, which I had written about last week, lots of kids came to see me to do re-quizzes, which I loved. Although they didn't all get 100% on their re-quizzes, it started a very productive dialogue with kids about how they are studying, what study tips I can recommend, and why things always seem easier when you do a re-quiz. One international student in particular had an 180-degree shift of attitude towards me after the re-quiz. I think she's the kind of student who thrives particularly on positive feedback, so the fact that she had failed the first time but got 100% on the second try, really boosted her confidence and her feelings towards math (and I guess, me). So, although it had been a challenging/"discouraging" week last week, I think it was a necessary reality check for many kids and now they are much more focused and strategic in their learning. My 10th-graders, for example, took a lot of photos yesterday during our white-boarding practice, because the one student who had done that last time and who had practiced with those problems later at home, had done very well on the quiz and had offered that up as a study strategy to her peers. So, we are a growing community of learners, moving in the right direction, slowly but surely!

For both Algebra 2 classes, on Monday we will go and test out the kids' predictions for the bungee drop. They will build the cords, name their rubber chickens, take a photo with their chickens, and then we will go out to the balcony. It'll be a very fun day, but the hard work that is yet to come is to write the lab reports coherently. I am a little nervous about getting their rough drafts on Tuesday, and what those will look like, especially for my international kids....

But, I cannot complain. I love this time of the year!

Labels:
algebra2,
calculus,
grade11,
grade9,
math stuff,
precalculus

## Saturday, September 21, 2013

### Week 2 Teaching - the Gentle Push Back

The second full week of school has been a very meaty one. The kids seemed very eager to learn after the first few unstructured socializing/cohort retreat days. And I am starting to see the various personalities starting to emerge, which is both wonderful and more challenging because now it is real teaching and real learning.

In my Algebra 2 classes, we had our first quiz, which challenged all the students in different ways. My vision for the start of Algebra 2 had been to lay down a solid tech foundation alongside review or re-teaching of linearity skills, so that kids realize that a corner stone of Algebra 2 has to be using technology flexibly in order to self-monitor accuracy. I told the kids that I don't want them to ask me, "Is this right?" but I would be very happy to hear them ask, "How can I check this using the calculator?" So, on the first quiz they needed to demonstrate this skill throughout the algebra problems, in order to earn full points on reflection. (I gave separate points for: Communication, Approach, Accuracy, and Reflection.) In the end, my two groups were challenged differently; the Grade 10 native-speaker class had holes in their algebra skills and couldn't complete all problems, and the Grade 9 largely non-native speaker class generally showed stronger algebra skills (even with less in-class whiteboarding practice), but struggled with using the calculator flexibly. But, some of the kids shared their effective studying strategies in class, and some of the other kids are planning to see me on Monday for a requiz, so I feel quite hopeful that this first quiz is just the start of a learning dialogue.

By the way, my relationship with the international kids is developing in an interesting way. Because I speak Chinese, I am able to help the kids in my class without watering down the level of tasks I am asking them to complete. But, at the same time I can carry some weight when I see them being off-task and I offer to call their parents in China to have the dialogue directly about their efforts, in Chinese. What an interesting situation for me and them. Interestingly, they are better-behaved for me, and they try hard to speak English in my class, except when they need to help each other translate something. I am curious how they are going to do on their first big writing assignment, which we will be doing next week...

In my Precalculus class, kids are wrapping up the rough drafts of their first big project on special (triangular and stellar) numbers. I had to be very explicit in helping them to format their writeup. (I had to say at one point, "Take out a sheet of paper. At the top, write, 'In this task, I was asked to...' Now, complete that thought in your own words. I give you 30 seconds to do that. ...Now, write down, 'In order to accomplish that, first we had to...' and go ahead and complete that thought, make sure you insert a diagram here." But, after about 5 minutes of modeling, I think they all got the idea and were able to continue the rest at home, because all the drafts that they brought back to me the next day looked pretty coherent. So, it has been a tough project for them, for sure, but I still think it had tremendous learning value. We also had a quiz, and the kids are doing fine with function identification, interpretation of f(3)=7, and writing both recursive and explicit equations for arithmetic sequences. Some of the more clever ones were able to write formulas for quadratic sequences already, based on their learning from the Special Numbers project. So, I am pretty happy so far. As we wrap up our project (meaning, as I read over their drafts), the kids are doing mixed lines review and checking all answers via their calculator. They use either the Table or Trace to check all equations that they write from given info, and they use [2nd][Math] to verify equivalent expressions after simplifying. So far, so good, because kids in this class seem to be quite independent.

In my Calculus class, I had one student come forward to say that he really enjoys the exploratory nature of our class, and two others who came to ask me to do more examples followed by practice. I thought over this carefully and decided that although I think it is awesome that kids are being advocates for their own learning, and I really wanted to acknowledge that and to encourage that, the issue is really much more complex than their individual learning styles. I ended up describing to the class two contrasting learning models, direct instruction and inquiry-based learning. I said that in most math classes they have had, they probably experienced the former (intro, example, guided example, individual practice, closure, and eval), and that that is fine. It is comfortable, you know what to expect when you come to class. But, that way only reaches the top half of the class, the half that is fortunate enough to maintain focused attention and to comprehend at the speed of material presentation. Then I showed them a diagram of inquiry-based learning, which is a cycle of asking questions, investigation, creation of model or new knowledge, discussion, reflection, and back to asking questions. I explained to them that what we do in class is NOT true inquiry, because true inquiry would be like me saying, "Go. Find out how much universal health care is going to cost our country, both in the short run and in the long run." The problem would be entirely open-ended, complex, and vast, and we would learn all the necessary math skills as we move along. I explained to them that what we do typically in our class is a smaller version of this; within the individual topics of Calculus, I try to think about ways to structure our class so that they can create their own understanding. I ended the class with showing a little clip from Sir Ken Robinson's tedX talk (the animated one), and saying that teaching creativity is hard, and that our traditional schools have been doing a good job killing creativity. I told the kids that, yes, I think even in math there is room for creativity in the classroom, and unfortunately we don't get that by me doing an example and then handing out 25 problems that look the same. So, although I will try to find a balance between direct instruction and exploratory learning, I want the kids to keep an open mind and to appreciate opportunities for creativity in any discipline.

After this talk, I heard from another student in this class who said that she really enjoys math this year, and that our chat helps her to understand and appreciate my philosophy even more. So, one point for being authentic with kids and treating them as intellectual equals.

Good second week. Our school is awesome, by the way. I love that kids clean the school three times a week, and I adore my colleagues!!!!!

In my Algebra 2 classes, we had our first quiz, which challenged all the students in different ways. My vision for the start of Algebra 2 had been to lay down a solid tech foundation alongside review or re-teaching of linearity skills, so that kids realize that a corner stone of Algebra 2 has to be using technology flexibly in order to self-monitor accuracy. I told the kids that I don't want them to ask me, "Is this right?" but I would be very happy to hear them ask, "How can I check this using the calculator?" So, on the first quiz they needed to demonstrate this skill throughout the algebra problems, in order to earn full points on reflection. (I gave separate points for: Communication, Approach, Accuracy, and Reflection.) In the end, my two groups were challenged differently; the Grade 10 native-speaker class had holes in their algebra skills and couldn't complete all problems, and the Grade 9 largely non-native speaker class generally showed stronger algebra skills (even with less in-class whiteboarding practice), but struggled with using the calculator flexibly. But, some of the kids shared their effective studying strategies in class, and some of the other kids are planning to see me on Monday for a requiz, so I feel quite hopeful that this first quiz is just the start of a learning dialogue.

By the way, my relationship with the international kids is developing in an interesting way. Because I speak Chinese, I am able to help the kids in my class without watering down the level of tasks I am asking them to complete. But, at the same time I can carry some weight when I see them being off-task and I offer to call their parents in China to have the dialogue directly about their efforts, in Chinese. What an interesting situation for me and them. Interestingly, they are better-behaved for me, and they try hard to speak English in my class, except when they need to help each other translate something. I am curious how they are going to do on their first big writing assignment, which we will be doing next week...

In my Precalculus class, kids are wrapping up the rough drafts of their first big project on special (triangular and stellar) numbers. I had to be very explicit in helping them to format their writeup. (I had to say at one point, "Take out a sheet of paper. At the top, write, 'In this task, I was asked to...' Now, complete that thought in your own words. I give you 30 seconds to do that. ...Now, write down, 'In order to accomplish that, first we had to...' and go ahead and complete that thought, make sure you insert a diagram here." But, after about 5 minutes of modeling, I think they all got the idea and were able to continue the rest at home, because all the drafts that they brought back to me the next day looked pretty coherent. So, it has been a tough project for them, for sure, but I still think it had tremendous learning value. We also had a quiz, and the kids are doing fine with function identification, interpretation of f(3)=7, and writing both recursive and explicit equations for arithmetic sequences. Some of the more clever ones were able to write formulas for quadratic sequences already, based on their learning from the Special Numbers project. So, I am pretty happy so far. As we wrap up our project (meaning, as I read over their drafts), the kids are doing mixed lines review and checking all answers via their calculator. They use either the Table or Trace to check all equations that they write from given info, and they use [2nd][Math] to verify equivalent expressions after simplifying. So far, so good, because kids in this class seem to be quite independent.

In my Calculus class, I had one student come forward to say that he really enjoys the exploratory nature of our class, and two others who came to ask me to do more examples followed by practice. I thought over this carefully and decided that although I think it is awesome that kids are being advocates for their own learning, and I really wanted to acknowledge that and to encourage that, the issue is really much more complex than their individual learning styles. I ended up describing to the class two contrasting learning models, direct instruction and inquiry-based learning. I said that in most math classes they have had, they probably experienced the former (intro, example, guided example, individual practice, closure, and eval), and that that is fine. It is comfortable, you know what to expect when you come to class. But, that way only reaches the top half of the class, the half that is fortunate enough to maintain focused attention and to comprehend at the speed of material presentation. Then I showed them a diagram of inquiry-based learning, which is a cycle of asking questions, investigation, creation of model or new knowledge, discussion, reflection, and back to asking questions. I explained to them that what we do in class is NOT true inquiry, because true inquiry would be like me saying, "Go. Find out how much universal health care is going to cost our country, both in the short run and in the long run." The problem would be entirely open-ended, complex, and vast, and we would learn all the necessary math skills as we move along. I explained to them that what we do typically in our class is a smaller version of this; within the individual topics of Calculus, I try to think about ways to structure our class so that they can create their own understanding. I ended the class with showing a little clip from Sir Ken Robinson's tedX talk (the animated one), and saying that teaching creativity is hard, and that our traditional schools have been doing a good job killing creativity. I told the kids that, yes, I think even in math there is room for creativity in the classroom, and unfortunately we don't get that by me doing an example and then handing out 25 problems that look the same. So, although I will try to find a balance between direct instruction and exploratory learning, I want the kids to keep an open mind and to appreciate opportunities for creativity in any discipline.

After this talk, I heard from another student in this class who said that she really enjoys math this year, and that our chat helps her to understand and appreciate my philosophy even more. So, one point for being authentic with kids and treating them as intellectual equals.

Good second week. Our school is awesome, by the way. I love that kids clean the school three times a week, and I adore my colleagues!!!!!

Labels:
algebra2,
calculus,
grade11,
grade9,
math stuff,
precalculus

## Thursday, September 12, 2013

### Here's a Graph that Makes a Statement

Although I do realize that it's complicated, and that there are a lot of factors that go into this data, it's worthy of discussion in a class that cares about real-world statistical implications.

(from http://www.forbes.com/sites/danmunro/2012/12/30/2012-the-year-in-healthcare-charts/ ).

And, here's a follow up, to make the discussion even better:

(from http://ucatlas.ucsc.edu/spend.php )

## Wednesday, September 11, 2013

### Week 1 Teaching

My year at school has begun, and as of today, we have had a full 5 days' worth of classes, even though lots of kids were missing class here and there for special retreat-type of activities. I feel quite settled, and I am starting to learn most of the kids' names despite having a terrible memory.

I am thrilled about how Precalculus is going! It's almost too good to be true -- the kids can build their own patterns from colored blocks; they can write recursive equations to represent those patterns; they can explain the limitations of knowing only the recursive formulas; they can write explicit equations that turn out to be linear; they can sum arithmetic elements and explain why that sum must be quadratic, both from a graphical standpoint and an algebraic standpoint. And they're ready to break down quadratic patterns into linear elements, in order to find the formula for the n-th quadratic element! ...Rock ON, 11th-graders!! They will start their first project very soon, either tomorrow (if they understand all preceding concepts) or next week. It is a project that I adopted from the IB, watered down quite a bit simply because I don't want the kids to be scared off right away.

My vision of tying linearity and quadratics neatly together for the kids with the concept of sequences is working out well so far; we'll see how they fare on the first quiz! :)

In Calculus, I discovered on Day 1 that the kids were strangers to the powers of the graphing calculator. I spent the next day walking them through basic features, and since then they've been

My Algebra 2 classes are two diverse groups. One is full of international students, some of who had just stepped off the plane to arrive in America with very little English comprehension. This has made for a very interesting challenge, trying to teach those kids in the same class as the native speakers. I think it is great -- it helps to teach all kids the virtues of patience and kindness, because some of the non-native speakers are much stronger in math than some of their native-speaking peers, so in my mind at least, it's a bit of give-and-take for everyone. The other Algebra 2 class is full of bubbly tenth-graders, and most of them are my advisees. I simply love that class! They don't work as fast as the other group through the material, but they're very responsible, eager to learn, and

How are your school years going??

I am thrilled about how Precalculus is going! It's almost too good to be true -- the kids can build their own patterns from colored blocks; they can write recursive equations to represent those patterns; they can explain the limitations of knowing only the recursive formulas; they can write explicit equations that turn out to be linear; they can sum arithmetic elements and explain why that sum must be quadratic, both from a graphical standpoint and an algebraic standpoint. And they're ready to break down quadratic patterns into linear elements, in order to find the formula for the n-th quadratic element! ...Rock ON, 11th-graders!! They will start their first project very soon, either tomorrow (if they understand all preceding concepts) or next week. It is a project that I adopted from the IB, watered down quite a bit simply because I don't want the kids to be scared off right away.

My vision of tying linearity and quadratics neatly together for the kids with the concept of sequences is working out well so far; we'll see how they fare on the first quiz! :)

In Calculus, I discovered on Day 1 that the kids were strangers to the powers of the graphing calculator. I spent the next day walking them through basic features, and since then they've been

*flying*through the conceptual material which I've presented through explorations. Unfortunately, this class has had the most amount of absences, since the Seniors and various special leadership folks had to miss class starting on Tuesday of this week. That has caused the pacing to be a bit chaotic. The kids who have been here everyday now have a solid grasp of the derivative: what it is, how to read it from the calculator, and how it relates to average rate. They can visualize derivative graphs when looking at an abstract original graph of f, and in general I think their calculator skills are slowly maturing. Yay!My Algebra 2 classes are two diverse groups. One is full of international students, some of who had just stepped off the plane to arrive in America with very little English comprehension. This has made for a very interesting challenge, trying to teach those kids in the same class as the native speakers. I think it is great -- it helps to teach all kids the virtues of patience and kindness, because some of the non-native speakers are much stronger in math than some of their native-speaking peers, so in my mind at least, it's a bit of give-and-take for everyone. The other Algebra 2 class is full of bubbly tenth-graders, and most of them are my advisees. I simply love that class! They don't work as fast as the other group through the material, but they're very responsible, eager to learn, and

*nice*. Kids from my Grade 10 class keep saying that that seems like the fastest-passing period, and they seem to genuinely enjoy their time in class even though we're still on the nitty gritties of basic algebra. We're working our way through some knowledge about how to use the graphing calculator flexibly to check our work alongside reviewing skills from Algebra 1 and wrapping our mind around visual linear patterns. The first big project will be coming up soon though (next week!), and it's the rubber chicken bungee jumping project (I think at most schools, they use barbies), which will involve a significant amount of writing and*really*show kids what I expect this year from their analyses. Whee! I'm excited!How are your school years going??

## Tuesday, September 3, 2013

### 2013 - 2014... ready to rock it!

I'm so-so-SO excited this year about teaching. I'm going to try to keep my collection of lessons running online at http://bit.ly/mimisMath2013-2014. My goal is to produced better-documented units this year -- ones that can be more easily shared with colleagues and tweeps, that have clearly defined goals and rationale.

Happy September!!

Happy September!!

## Wednesday, August 28, 2013

### Grading by Criteria

This is an excerpt from the grading section of my course syllabi this year. My new school is very good about letting each teacher choose an appropriate system for their own class. Two of my math colleagues saw this and thought it was interesting, so I thought that maybe it'd be interesting for some of you to consider as well. For me, this is similar to the grading system used in the IB Middle Years Programme. It also just generally makes more sense to me to be grading by criteria instead of by the type of assignments, especially because as a department we are trying to foster some of those important qualities in our kids.

Even though I still have a lot of things to prepare before the year starts, I am very excited! I think it's going to be a great year.

**Grading:**Your grades are a direct feedback to you. They will speak to your areas of excellence, versus areas where I believe that you can still grow and develop your skills. They should, therefore, reflect the values that we hold in our course. To this end, your class grade will break down into grading criteria as follows:

Communication in Math 30% All collected written explanations (classwork and projects)

Mathematical Approach 40% All collected “algebra work” (quizzes, tests, and projects)

Accuracy and Precision 20% All collected “algebra work” (quizzes, tests, and projects)

Reflection on Results 10% On-going observation (class participation and written projects)

Grading CriterionGrading Criterion

__Assessed using…__Communication in Math 30% All collected written explanations (classwork and projects)

Mathematical Approach 40% All collected “algebra work” (quizzes, tests, and projects)

Accuracy and Precision 20% All collected “algebra work” (quizzes, tests, and projects)

Reflection on Results 10% On-going observation (class participation and written projects)

This might be different from what you are used to. Do not worry! You can always request a re-test or a re-quiz if there is something that you wish you had done better on. Also, grading rubrics will be provided in advance of project due dates, in order to clarify how these criteria will be applied specifically to the project.

This might be different from what you are used to. Do not worry! You can always request a re-test or a re-quiz if there is something that you wish you had done better on. Also, grading rubrics will be provided in advance of project due dates, in order to clarify how these criteria will be applied specifically to the project.

Even though I still have a lot of things to prepare before the year starts, I am very excited! I think it's going to be a great year.

Subscribe to:
Posts (Atom)