Between 1969 and 1972, twelve men walked on the moon. This is humanity’s greatest accomplishment – that we managed to send astronauts more than 200,000 miles through the unknowns and the dangers of space, that these astronauts set foot on another world, and that they returned home safely. We did this to explore. Yes, there was an element of Cold War competition, but in the end, these missions were about science, about discovery, and about challenging the limits of possibility.

John Young has died, leaving only five surviving astronauts who walked on the moon. Their ages: 87, 85, 85, 82, and 82. Young was 87. Eight others who flew to the moon without landing are still alive; the youngest is 81. I hope that these men have many years left, but realistically, the day will soon come when no living person has set foot on the moon or even left low Earth orbit. Despite all of the advances we have made, we have not yet surpassed this accomplishment from nearly fifty years ago. NASA’s priorities have certainly changed, and they still do lots of wonderful, important work. And perhaps sending an astronaut back to the moon would serve little purpose. But I cannot avoid the sadness I feel knowing that some of our greatest heroes will soon be gone and that they will leave us without successors.

Gemini 3. Gemini 10. Apollo 10. Apollo 16. STS-1. STS-9. What an amazing career.

The image on the left shows Young in 1965, a few weeks before the Gemini 3 mission. The image on the right shows Young (seated, second from right) in 1983, about six months before the STS-9 mission. Young ultimately worked for NASA for 42 years. My words cannot do justice to his great career. Instead, let me share with you some quotes I find particularly meaningful in light of his death.

My favorite description of John Young comes from Andrew Chaikin’s A Man on the Moon:

Inside Young was an unwavering determination, an overriding sense of responsibility – to the space program, to the country, to his crew – and an almost childlike sense of wonder at the universe.

But more than this, I think, Young felt a responsibility – a commitment – to truth and to knowledge. Chaikin writes:

More than most astronauts, Mattingly thought, John Young seemed mindful of the risks of his profession. Around the Astronaut Office, his memos were well known, sounding the alarm about some engineering problem he’d uncovered. He wouldn’t rest until he knew every detail about the particular system or technique that worried him. And when he had learned all he could, then it was time to go fly – with his eyes wide open. That was the only way to handle this business; that was what made him so good. Maybe Young worried so much because he saw so clearly. But when it came down to the real question – Will you fly it? – John’s answer would always be yes.

John Young, intrepid explorer. Perhaps in looking at heroic figures from the past we see in them what we want to see. Maybe we look for the best of ourselves in them. In John Young, I see a man who lived for the thrill of discovery, a man for whom being bold was a way of life, a man who acknowledged challenges but saw past them, a man with a vision of limitless possibility. Consider the scene Chaikin describes as Young exits the Space Shuttle Columbia after its maiden flight:

Later, after the ground crews had arrived, Young emerged and bounded down the stairway to inspect his ship, punching the air with his fist like a relief pitcher who had just won the World Series. That day, Young told a crowd of well-wishers, “We’re really not too far, the human race isn’t, from going to the stars.”

Nearly thirty-seven years after that flight – and nearly forty-six years since Young walked on the moon – the stars still seem not too far off. NASA, SpaceX, Blue Origin, and others continue to push boundaries and extend our reach into the stars. But no amount of technological advancement can replace the boldness and the vision of men like John Young. We lost a great man on January 5, 2018. Ad astra, John Young. Ad astra.

# Discrete Math Project 1.1 – An Introduction to Logic

Section 1.1 covers the basics of logic: Statements, open sentences, truth tables, negation, conjunction, and disjunction. The authors use “Aunt Buosone” in several examples. I’m excited to see what other characters I meet. The math isn’t new to me yet, but the name “Buosone” sure is.

I have never taught logic to this extent. When I taught geometry, we spent quite a bit of time discussing logic and especially reasoning, but we never actually worked with truth tables. In early August each year, I would convince myself that spending time with truth tables and doing a deep dive into logic would be a great learning experience for my students and would pay off in the long run. But each year, I backed off and decided to only spend time on “geometry” logic and not on “discrete math” logic. The payoff for truth tables didn’t seem great enough, and we did a ton of great thinking and reasoning without them. I wonder if many geometry teachers include truth tables in their curriculum. If not, do they show up anywhere in the curriculum? I learned about truth tables in PDM – Precalculus and Discrete Mathematics – using the Chicago series (UCSMP). Given the emphasis on coding, I’m curious where discrete math fits into the curriculum.

Anyway, I like making truth tables. Well, I don’t mind it. I guess it gets tedious after a while, especially once you understand them. I did find the word problems interesting. Here’s an example:

That’s just a good problem, the sort I might have given my geometry students back in the day. Although the problem does not specifically call for a truth table, I made one anyway:

While this made the questions extremely easy to answer, I wonder what value the truth table really has here. I thought about the questions and answered them before making the truth table and used it to confirm my answers. The truth table certainly adds clarity, but in doing so, does it actually remove some of the critical thinking necessary to answer the questions? Or does it seem that way only because I have a fairly strong background in logic?

Curriculum Connections

This chapter included a number of logic puzzles. You know the type – Alice and Bob go to the same school, Bob and Carol both major in history, … Which person studies physics and is under six feet tall? I’ve never done much with this sort of puzzle, but I know students tend to like them. Definitely a good resource to have around to allow for easy differentiation after an activity.

Questions to Ponder

Does constructing truth tables help students develop logical thinking and reasoning skills? Does the process become rote? Is it just another procedure to learn?

How does the knowledge gained through learning about elementary logic help students to learn about mathematical proof? What do we gain by studying logic at this level?

What are the consequences of differences between mathematical language and the vernacular? How can we help students learn how and when to use and, or, and not correctly within a mathematical context? Does the idea of an inclusive or pose a significant barrier to student understanding, and if so, how do we overcome it?

# The Discrete Math Project

I recently bought Discrete Mathematics for Teachers by Ed Wheeler and Jim Brawner. I have a bit of a habit of buying math books. Learning about math – even if it’s unrelated to the curriculum I teach –  makes me a better teacher. While I’ve taken a bunch of education courses over the last few years, it’s been quite some time since I took a math course. I’ve decided to replicate that experience by working through this textbook. I’m excited!

Discrete math is not an unfamiliar subject to me. I learned a fair amount of discrete math in high school, and as part of my math degree, I took a course in discrete math. I’ve specifically taught sequences, combinatorics, and probability, and I’ve also touched on some set theory and logic. I hope to learn some new things and to refresh some knowledge tucked deep away in the furthest recesses of my brain.

My plan is to work through roughly one section each week. I’ll probably do it on Friday or Saturday night. That’s just how my social life works. There are 39 sections, so in theory, this project could take me the better part of the year. As I go along, I’ll certainly explore certain topics in more depth, and I welcome suggestions for avenues that I might pursue. For each section, I hope to put together some sort of blog post – an interesting problem, curriculum connections, etc.

The following quote from the Preface stood out to me:

The goal is to develop teachers who not only know the mathematics they are teaching, but also understand the larger mathematical context in which the mathematics they teach has life.

My content knowledge has enriched my teaching in so many ways. I look forward to continuing to develop it in the coming weeks. I invite you to join me on this journey.

# Reflections on Choice Words

I recently finished reading Choice Words: How Our Language Affects Children’s Learning by Peter Johnston. Though the book is ostensibly about reading and literacy education, I found it spoke to much larger issues in education. Indeed, I consider it to be one of the best education books I’ve ever read. This post contains various thoughts about and comments on Choice Words.

Our communication reveals our beliefs about ourselves, our students, and teaching. Johnston writes:

The way we interact with children and arrange for them to interact shows them what kinds of people we think they are and gives them opportunities to practice being those kinds of people.

If we place students in the roles of thinkers, problem solvers, and mathematicians, then they can construct and refine each of those identities. If we value our students’ questions, ideas, and suggestions, then they can develop their curiosity, insight, and creativity. A former colleague once asserted that “if we want students to become responsible, then we need to give them responsibilities and see what happens.” Johnston advocates “creating an intellectual space into which [students’] minds can expand.” This classroom would necessarily encourage discussion, cherish student contributions, foster mutual respect, and cultivate independence and responsibility. Why? Because these conditions define the environment needed to nurture young minds.

But we can also see that it is not simply the names and labels we invoke that affect children, or for that matter the love with which we embrace them, but the ways we unwittingly use language to position them and provide them with the means to name and maim themselves.

Johnston argues that everything we do sends a message to our students. Avoiding a classroom discussion, for example, may suggest that we don’t believe our students can make thoughtful contributions, or that we don’t trust students to engage themselves in meaningful conversation, or that only the teacher has valuable knowledge. When we cling to right-wrong, good-bad, and other dichotomies – especially about what students do and say – we indoctrinate our students to this way of thinking. We shut down the intellectual space that developing minds need.

We cannot persistently ask questions of children without becoming one-who-asks-questions and placing children in the position of the one-who-answers-questions.

When we work on Three-Act Tasks or ask students to Notice and Wonder, we allow them to pose their own questions and to take on the role of mathematician. Language, Johnston says, “creates realities and invites identities.” A teacher who acts as the sole arbiter of right and wrong may preclude students from developing their own evaluative capacities. A teacher who refers to a classroom task as “work” or something students “have to do” may unintentionally set students up to dislike that task, especially in relation to activities they find “fun” or otherwise meaningful. The intended message need not, and often will not, match the received message.

Teachers’ conversations with children help the children build the bridges between action and consequence that develop their sense of agency. They show children how, by acting strategically, they accomplish things, and at the same time, that they are the kind of person who accomplishes things.

Agency involves the power to achieve, the means to bring about desirable results. To me, agency lies at the heart of what we do as teachers. We have all taught the persistent student, the tenacious student, the driven student, the student who relentlessly pursues success. These students have a strong sense of agency, that belief in their own competence and in their capacity for accomplishment. We have also all taught students whose sense of agency remains underdeveloped. It’s not enough to tell students to be tough or to display grit or to believe in themselves or to just give it a try. A student whose narrative involves doubt and failure needs our help in developing agency.

But when a child tries something and does not succeed, we need to turn that event toward a narrative and identity that will be useful for the future. If children are not making errors, they are not putting themselves in learning situations.

Success and failure play important roles in the learning process and in helping students develop their sense of agency. These successes and failures must belong to the student, though. Teachers may support, of course, but students must play the central roles in their own narratives. A passive student becomes a student without agency, a student who relies on others to do the thinking and to solve the problems.

Children with strong belief in their own agency work harder, focus their attention better, are more interested in their studies, and are less likely to give up when they encounter difficulties than children with a weaker sense of agency.

Language matters. How we interact with students matters. The ways learning occurs in our classrooms matter. Most of all, children matter.

I recently watched this video produced by YouCubed and Jo Boaler that talks about giftedness. Essentially, the video argues that labeling students as gifted presents equity issues and does a disservice to students by giving them a fixed idea of what they can learn and do as well as how they should behave.

Giftedness is real. One definition of “gifted” is “a high level of intelligence [indicative of] advanced, highly integrated, and accelerated development of functions within the brain” (Clark, 2013). The Elementary and Secondary Education Act defines gifted students as those who “give evidence of high achievement capability … and who need services or activities not ordinarily provided by the school in order to fully develop those capabilities.” Just as some individuals have extraordinary artistic or athletic talents, some students have significant intellectual gifts. Acknowledging this fact does not force us to believe that some students cannot learn math. Nor does it force us to set limits on what we think students can learn and do.

The problem, I think, is that YouCubed has conflated the concept of giftedness with how this concept has been applied in schools. Even if many teachers and schools wrongly label and limit kids, that doesn’t mean giftedness is not a useful concept. It simply means that teachers need to do better with how we use the idea of giftedness.

This argument refers to ineffective and inappropriate uses of giftedness to suggest that gifted education is inherently inequitable. But we can provide services to gifted students without limiting other children’s potential. It’s bad teaching to suggest that gifted students should always know the answer or should not ask questions. Similarly, it’s bad teaching to suggest that non-gifted students cannot learn high levels of math or to place false limitations on what students can do. But these are problems with teacher behavior. These are not problems with the idea of gifted education.

Indeed, our developing knowledge of neuroplasticity and the idea that brains experience significant growth and change actually support labeling students as gifted. Why? Because acknowledging the incredible potential that some students have forces us to consider ways to help them realize that potential.

Is it inequitable to provide services such as enriched classes to gifted students? No. Equity means allowing every student the opportunity to achieve his or her potential. Equity does not mean offering the exact same opportunities to every student. Our obligation as educators is to create an environment that helps every student to learn and grow as much as possible. We can do so while accepting that some students learn faster or slower, that some students require more support or greater challenges.

Is everyone gifted? No. But that doesn’t mean we should place artificial limits on what students can learn and do. It’s okay to acknowledge the great intellectual capacity and potential that gifted students have. We can do this without saying that gifted students are better or deserve more. We cannot afford to avoid labeling gifted students, however, because doing so will make it harder to meet the needs of exceptional learners.

Note: I wrote a draft of this post after initially viewing the YouCubed video last month. I’ve fleshed out some of my commentary, but it remains mostly the same as I left it late in the evening on November 9th.

References

Clark, B. (2013). Growing up gifted: Developing the potential of children at home and at school. Boston: Pearson.

Elementary and Secondary Education Act, 20 U.S.C. § 7801 (1965).

# What Do My Students Really Know? Part 1

My two Accelerated Math 1 classes have been working with sequences. They have completed some investigation problems to develop vocabulary and to grapple with the characteristics of different types of sequences. We have built definitions of explicit and recursive formulas as well as of arithmetic and geometric sequences, and as a class, we came up with explicit formulas for both of these types of sequences. Their work has impressed me, especially given that we’ve done all of this using function notation, which is totally new to everyone. Students don’t even seem particularly troubled by some of the more challenging problems (such as figuring out missing terms when given two specific terms or determining the term number for a far-off term) that have frustrated some of my past classes. I know that they need more practice, but overall, I’m pleased with their progress.

For practice, we completed a matching activity in which students had a card with either the formula for a sequence or the sequence itself. They had to find the student with the matching card. I designed the sequences to be fairly similar so that students really had to think about which sequence went with which formula. It wasn’t enough, for example, to notice that the first term of the sequence matched up with the formula because multiple sequences had the same first term.  My students handled this activity pretty easily.

I wanted to try something similar with Desmos, so I made this Polygraph activity. It features eight numerical sequences and eight explicit formulas. I expected students to ask questions like: “Is your sequence arithmetic?”; “Is the common difference positive?”; and “Is the first term negative?” Although I certainly saw questions like those, I noticed that many students used imprecise language and pursued a line of questioning that lacked a logical progression meant to eliminate possibilities. Their work surprised me and has made me wonder what they have really learned. Here is some of their work along with my commentary.

The questions about negative numbers make sense on the surface, but all decreasing arithmetic sequences will have negative numbers eventually. Is it important that this student was referring to the first three terms without saying so? Had I been the picker, I may well have answered “yes” to that question even if my sequence only displayed one negative number. Substantial mathematical background allows me to focus on this sort of nuance. How do I help students refine their language to remove the ambiguity that I see? Or am I seeing an issue where one really does not exist?

Does the formula for a sequence count as a sequence? Should that first question be worded as “a numerical sequence” to avoid confusion? Would $n-1$ count as having a negative? Does using “over” instead of “greater than” present a problem?

Although I’m happy to see “arithmetic” used correctly, the word “decimals” worries me. I’m not sure that “decimals” has a firm definition in my students’ minds. These students seem to be referring to numbers containing a decimal point, which makes sense, but I wonder what they think of $3.2\times 10^{3}$. Does that count as a decimal? Perhaps a better question would ask about non-integers.

Sure, these questions get the job done with minimal ambiguity, but I’m not a huge fan of the final two questions. Without a need for vocabulary like “common ratio,” I suppose it makes sense to ask these questions. Maybe I need to force students into choosing between two geometric sequences with the same first term. But would students then simply ask about “the number in parentheses” or something similar?

Ah! Here’s a misconception! Seeing multiplication in the equation led the picker to call it a geometric sequence. This is why I love Desmos activities – concrete evidence of students’ understandings and misunderstandings that I can immediately work to address.

Another opportunity for improvement on my part! The two sequences remaining after the first three questions differed in two regards – their first terms and their common differences. A possible revision might be to change the highlighted formula to $f\left ( n \right )=12+6\left ( n-1 \right )$. This would even make a question about subtraction useless, so it should elicit a question about the common difference.

Lots going on here. Plenty of great vocabulary here (e.g. “explicit formula” and “common difference”) as well as some great questions about odd and even numbers. I’m particularly fond of the question “do you add to get to the next term.” Sure, I’d love to see “arithmetic sequence” show up in that student’s vocabulary, but it’s hard to fault a student for a description that just works.

Just a fascinating line of questioning here. Only one of the three visible sequences has exactly two negative numbers, but the students here only considered the visible terms. Why did the guesser asked about negative numbers anyway? Wouldn’t asking if the sequence increases (or decreases) be simpler? And to follow one interesting choice with an even more interesting question…wow! Why does this student bother including the word “arithmetic” in this question? Both of the sequences are arithmetic. And why ask if it’s adding $7$ instead of simply asking if it’s increasing? This sort of specificity doesn’t really add anything other than the risk that one of the students miscalculates and leads the other astray. I wish I could have been inside this student’s head!

So much to say about these… One student was extremely curious about the starting number. What does that say about that student’s understanding of sequences? Several students asked about function notation rather than asking about formulas or equations. How about the student who asked about positive numbers, even numbers, double-digit numbers, AND the number $4$? Yes, these questions got the job done, but what do they really tell me about my students’ understanding of sequences?

As you can see, my students used all sorts of vocabulary. They used lots of different math terms, but they didn’t use as much of the sequence vocabulary as I expected. On the one hand, I’m proud of their ability to use what they know and what they’re comfortable with to find success. But I wonder what this all says about our work with sequences. This Polygraph activity certainly differs significantly from the work we’ve been doing. Students need to do more than just perform calculations and interpret numbers. Polygraph forces students to wrestle with sequences in a unique manner and to draw distinctions between sequences represented numerically or as equations. Perhaps in the face of such a challenge my students retreated to the most natural language. Perhaps their understanding of sequences isn’t what I thought it was. Perhaps the numerical sequences and explicit formulas I chose did not lend themselves to the sort of questioning and language that I had hoped for.

So what does this all mean for my teaching and for my students’ learning? I actually feel like I’ve done a better job with sequences than I ever have before. So much more of our work this year has involved thinking through and solving problems. But maybe I haven’t gone far enough in including rich, meaningful tasks. Or maybe this was just a necessary step on the path toward mastery. As we continue working with sequences and eventually use our sequence knowledge to develop linear and exponential functions, I look forward to further probing my students’ understandings and challenging them to engage more deeply with the mathematics.

# Day 20 in Room 224 (9/12/17)

I have not done a great job with #teach180 recently. Or with blogging in general. To make up for it, here are more interesting comments and observations from my students. Enjoy!